The Internet Encyclopedia of Philosophy

 Time

Time has been studied by philosophers and scientists for 2,500 years, and it is much better understood today than long ago. Nevertheless, many questions remain to be resolved. Philosophers discuss whether time travel is possible; whether time had a beginning; whether time is an objective feature of reality or only a product of subjective experience; and whether it is sensible to speak of time flowing. Philosophers also investigate and debate about whether the future and past are real or, instead, only the present is real; whether time exists when nothing is changing; what gives time its arrow one way and not the other way; and whether time is infinitely divisible or instead composed of discrete moments. This article explores both what is known about time and what is controversial and unresolved. The article is structured so that it discusses answers to the following questions about time:

Supplement: frequently-asked questions:

 

What should a philosophical theory of time do?

Can we begin with a short definition of time? Yes, but there are two considerations that must be faced. First, the definition will not be able to to define time in terms of more primitive, yet familiar, notions. Second, succinct definitions of time are rarely helpful unless they are backed up with a more elaborate and systematic treatment of time. The brief definitions that stand alone are either trivial (Time is what keeps everything from happening all at once) or too imprecise (Time is the dimension of causality) or circular (Time is the collection of instants) or simply cryptic (Time is the flow of events past the stationary I). When philosophers ask, "What is time?", they normally are asking for a philosophical theory designed to answer many of the philosophical questions about time such as whether the past-present-future distinction is objective and how we should understand the flow of time. A succinct definition of time will be adequate only insofar as it is backed up by this more elaborate theory.

Consider what a more systematic theory of time should do. It should reveal, among other things, the relationship between time and the mind. It is easy to confuse time itself and the perception of time. Does time exist for beings that have no minds?

A theory of time should reveal what physical science presupposes and implies about time. Does it imply the possibility of time travel, for instance? What does it assume about the relationship between time and spacetime? Physicists say that, locally, time is made of a linear continuum of instants, with each instant lasting for zero seconds. Being a continuum implies that between any two instants there is another instant. No time measurement is so fine grained that it could detect whether this is true for instants that are extremely close together in time. If so, then on what grounds do scientists know that time is a continuum?

A philosophical theory of time should describe the relationship between instants and events. Does the instant that we label as "11:01 AM" for a certain date exist independently of the events that occur then? In other words, can time exist if no event is happening? This question raises the thorny metaphysical issue of absolute vs. relational theories of time.

A theory of time should address the question of time's apparent direction. If the projectionist in the movie theater shows a film of cream being added into black coffee but runs the film backwards, we in the audience can immediately tell that events couldn't have occurred this way. We recognize the arrow of time because we know about the one-directional processes in nature--that brown coffee never unmixes into black coffee and cream, for example. This arrow becomes less and less apparent to the viewer as the film subject gets smaller and smaller and the time interval gets shorter and shorter. Philosophers disagree about the explanation of this arrow. The arrow appears to be very basic for understanding nature, yet it is odd that asymmetries in time don't appear in most of the basic dynamical laws of physics. Philosophers also wonder what life would be like in some far off corner of the universe if the arrow of time were reversed there. Would our counterparts walk backwards up steps while remembering the future?

Another philosophical problem about time concerns the questions, "What is the present moment and why does it move into the past?" Present events seem to flow by, receding ever farther into the past. Many philosophers are suspicious of this notion of the flow of time. They doubt whether it is a property of time as opposed to being some feature of human perception. There are also suspicions about the present, the feature that is referred to by the indexical word 'now.' If the present is real, then why isn't there a term for it in the laws of science?

For a last example of a philosophical problem regarding time, some philosophers argue that the future is not real, but the present is. These philosophers have a problem with the apparent implication that, if the future were real, then it would be fixed now, and we would not have the freedom to affect that future. Other philosophers disagree.

A full theory of time should address this constellation of philosophical issues and paradoxes about time.

 

How is time related to mind?

Kant believed our sense of time is a necessary condition of our experience. Mach believed our sense of time is a simple sensation. Most researchers today would reject both these positions and claim that our sense of time is indirectly related to our ability to sense all sorts of changes; our sense of time is an intellectual construction that helps us account for our experience.

Does this time, which is sensed, exist objectively? It can be very difficult to distinguish a genuine objective aspect of reality from an appearance of reality or from the particular perspective from which we regard that reality. This difficulty arises when we ask: If consciousness were absent, would time be absent, too? Consciousness probably requires a brain-based sense of time, but does time require consciousness?

This metaphysical question has been much discussed. Aristotle raised the question: "Whether, if soul (mind) did not exist, time would exist or not, is a question that may fairly be asked; for if there cannot be some one to count there cannot be anything that can be counted..." [223a] He doesn't answer his own question because, he says rather profoundly, it depends on whether time is the conscious numbering of movement or instead is just the capability of movement's being numbered were consciousness to exist. Aristotle's distinction foreshadows the modern distinction between psychological time and physical time.

Physical time is public time. Psychological time is private time. We are referring to psychological time when we say that time passes slowly for someone who is waiting anxiously for the water to boil on the stove. We are referring to physical time when we speak of the time that a clock measures. When a physicist defines speed to be the rate of change of position with respect to time, the term 'time' refers to physical time. Psychological time is best understood as being consciousness of physical time. Psychological time stops when consciousness does, but physical time does not. Physical time is more basic for helping us understand our shared experiences in the world. It is more useful than psychological time for doing science.

St. Augustine said time is nothing in reality but exists only in the mind's apprehension of that reality. Henry of Ghent and Giles of Rome both said time exists in reality as a mind-independent continuum, but is distinguished into earlier and later parts only by the mind. In the 11th century, the Persian philosopher Avicenna doubted the existence of physical time, arguing that time exists only in the mind due to memory and expectation. In the 13th century, Duns Scotus recognized both physical and psychological time.

At the end of the 18th century, Kant suggested a subtle relationship between time and mind--that our mind structures our perceptions so that we know a priori that time is like a mathematical line. Time is, on this theory, a form of conscious experience.

The controversy in metaphysics between idealism and realism is that, for the idealist, nothing exists independently of the mind. If this controversy is settled in favor of idealism, then time, too, would have that subjective feature--physical time as well as psychological time.

The philosophical issue of the flow of time concerns whether this flow is an objective feature of reality or, instead, is entirely a feature of human perception.

Finally, the question about the relationship between time and the observer's frame of reference is not a question about the relationship about time and mind, the question of whether time requires consciousness.

 

Definitions of time

A wide variety of short answers have been given to the question "What is time?" Some of these are backed up by more elaborate theories of time, and some are not. Plato, for example, said time is the circular motion of the heavens. Aristotle said it's not motion but the measure of motion. Kant, taking a very different approach to time, said it is a form that the mind projects upon the external things-in-themselves. A more modern definition says time is the dimension of causality.

Aristotle provides an early, careful answer to the question "What is time?" when he says time is the "number of movement in respect of the before and after, and is continuous.... In respect of size there is no minimum; for every line is divided ad infinitum. Hence it is so with time." [Physics, 220a] Occasionally Aristotle speaks as if time were motion, but in these passages, he also asserts that time, though linked to motion, is neither the circular motion of the heavens (Plato's view) nor any other motion. He believes time is something by which we measure motion.

In the 17th century, the English physicist Isaac Barrow rejected Aristotle's linkage between time and change, or between instants and events, by saying that time is something which exists independently of motion and which existed even before God's creation. Barrow's student, Isaac Newton, agreed. Newton argued very specifically that time and space are an infinitely large container for all events, and the container exists with or without the events. Space and time are not material substances, but are like substances, he added.

Gottfried Leibniz objected. He argued that time is not an entity existing independently of events. Leibniz insisted that Aristotle and Newton had overemphasized the relationship between time and duration, and underemphasized the fact that time ultimately involves order as well. Time is an ordering of changes, the overall ordering of all non-simultaneous events. This is why time needs events, so to speak. Leibniz added that this order is also a "something" as Newton had been insisting, but it is an ideal entity. Triangles, numbers, and relations are also ideal entities.

In the 18th century, Immanuel Kant said time and space are forms that the mind projects upon the external things-in-themselves. He spoke of our mind structuring our perceptions so that space always has a Euclidean geometry, and time has the structure of the infinite mathematical line. Kant's idea that time is a form of apprehending phenomena is probably best taken as suggesting that we have no direct perception of time but only the ability to experience things and events in time. Some historians distinguish perceptual space from physical space and say that Kant was right about perceptual space. It's difficult, though, to get a clear concept of perceptual space. If physical space and perceptual space are the same thing, then Kant is claiming we know a priori that physical space is Euclidean. With the discovery of non-Euclidean geometries in the 1820s, and with increased doubt about the reliability of Kant's method of transcendental proof, the view that truths about space and time are apriori truths began to lose favor.

In 1924, Hans Reichenbach defined time order in terms of possible cause. Event A happens before event B if A could have caused B but B couldn't have caused A. This was the first causal theory of time. Its usefulness depends on a clarification of the notorious notions of causality and possibility without giving a circular explanation that presupposes an understanding of time order. Reichenbach's idea was that causal order can be explained in terms of the 'fork asymmetry'. That is, outgoing processes from a common center tend to be correlated with one another, but incoming processes to a common center are uncorrelated. [Do you remember tossing a rock into a still pond? Imagine what the initial conditions at the edge of a pond must be like to produce correlated, incoming, concentric water waves that would expel the rock and leave the water surface smooth.] The usefulness of the causal theory also depends on a refutation of David Hume's view that causation is simply a matter of constant conjunction, a temporally symmetric notion. For Hume, there is nothing metaphysically deep about causes preceding their effects; it's just a matter of convention that we use the terms 'cause' and 'effect' to distinguish the earlier and later members of a pair of events which are related by constant conjunction.

One way to answer the question "What is physical time?" is to declare that it is whatever the time variable t is denoting in the best-confirmed and most fundamental theories of current science. Many philosophers complain that this answer is incomplete because, although a philosophical theory of time should be informed by what science requires of time, the philosophical theory should progress beyond current physical theory.

Is time like a cirle or like a line? The concept of linear time first appeared in the writings of the Hebrews and the Zoroastrian Iranians. The Roman writer Seneca advocated linear time, but most of the ancient Greeks and Romans believed time to be ultimately cyclic. Aristotle gave psychological reasons for why the circle is an appropriate model. His point was that we cannot conceive of a first time; for any first time we could conceive of a time before that. Neither Aristotle nor Plato envisioned their cosmic cyclicity as requiring any detailed endless repetition such as the multiple births of Socrates, though some Stoic philosophers did adopt this drastic position. Rejecting circularity, Islamic and Christian theologians adopted the Jewish notion that time is linear with the universe being created at a definite moment in the past. Augustine explicitly objected to Aristotle's belief that time is circular, insisting that human experience is a one-way journey from Genesis to Judgment, regardless of any recurring patterns or cycles in nature. In the Medieval period, Thomas Aquinas agreed. It was not until 1602 that the concept of linear time was clearly and explicitly formulated--by the English philosopher Francis Bacon. In 1687, Newton captured some of this viewpoint when he represented time mathematically by using a line rather than a circle, although he ultimately believed time to be cyclic. The concept of linear time was promoted by Barrow, Leibniz, Locke and Kant. In 19th century Europe, the idea of linear time became dominant in both science and philosophy, and it remains so today.

 

What science requires of time

Quantum field theory and Einstein's general theory of relativity are the most fundamental theories of physics. According to these theories, spacetime is a collection of points called "spacetime locations" where physical events occur. Spacetime is four-dimensional and a continuum, with physical time being a distinguished, one-dimensional sub-space of this continuum.

In 1908, the mathematician Hermann Minkowski had an original idea in metaphysics regarding space and time. He was the first person to realize that spacetime is more fundamental than time or than space. As he put it, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." The metaphysical assumption behind Minkowski's remark is that what is 'independently real' is what does not vary from one reference frame to another. It's their "union," what we now call "spacetime," that doesn't vary. It follows that the division of events into the past ones, the present ones, and the future ones is also not 'independently real'.

Newton would have disagreed. He declared that every observer can in principle determine time intervals that depend in no way on the observer's frame of reference. If the time interval between two lightning flashes is 100 seconds on someone's clock, then the interval also is 100 seconds on your clock, even if you are flying by at an incredible speed. Albert Einstein rejected this piece of common sense in his 1905 special theory of relativity when he declared that the time interval (and the distance) between two events depends on the observer's reference frame. As Einstein expressed it, "Every reference-body has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event." Each reference frame (or reference-body) divides spacetime differently into its time part and its space part. This feature of our universe is what Einstein calls the "relativity of simultaneity." It is a relativity for distant events only, not for events happening at the same place.

Science assigns numbers to times because, in any reference frame, the happens-before order-relation on events is faithfully reflected in the less-than order relation on the time numbers (dates) that we assign to events. In the fundamental theories, the values of the time variable t are real numbers, with each number designating an instant of time. Time is a linear continuum of instants, similar to the mathematician's line segment. Therefore, physical time is one-dimensional rather than two-dimensional, and continuous rather than discrete. One can't be sure from this that time is linear rather than circular because a segment of a circle is also a linear continuum. If it were circular, then Homer might write his Iliad and Odyssey epics in the future, a possibility that appealed to the ancient Stoic philosophers. The logic of the term 'time' doesn't rule out a nonlinear structure, but there is no reason to believe it occurs.

Regarding the instants, time's being a linear continuum implies there is a nondenumerable infinity of them. It also implies they are so densely packed that between any two there is a third, and yet no instant has a next instant. There is little doubt that the actual temporal structure of events can be embedded in the real numbers, but how about the converse? That is, to what extent is it known that the real numbers can be adequately embedded into the structure of the instants? The problem is that, although time is not quantized in quantum theory, for times shorter than about 10 to the minus 43 seconds, the so-called Planck time, science has no experimental grounds for the claim that between any two events there is a third. Instead, the justification is that the assumption of continuity is convenient and useful, and that there are no better theories available.

Because of quantum mechanical considerations, physicists agree that the general theory of relativity must fail for durations shorter than the Planck time, but they don't know just how it fails. That is, there is no agreement among physicists as to whether the continuum feature of time will be adopted in the future theory of quantum gravity that will be created to take account of both gravitational and quantum phenomena.

In 1922, the Russian physicist Alexander Friedmann predicted from general relativity that the universe should be expanding. In 1929, the American astronomer Edwin Hubble made careful observations of clusters of galaxies and confirmed that the universe actually is undergoing a universal expansion. Each galaxy cluster is moving away from most every other. So, at any earlier moment the universe was more compact. Projecting to earlier and earlier times, and assuming that gravitation is the main force at work here, the astronomers now conclude that 13.7 billion years ago (plus or minus 1%) the universe was in a state of infinite density and zero size. Because all substances cool when they expand, physicists believe the universe itself must have been cooling down over the last 13.7 billion years. Therefore, the universe started out very hot and very small. This beginning process is called the "big bang." As far as we know, the entire universe was created in the big bang, and time itself came into existence 'at that time'.

In the literature in both physics and philosophy, descriptions of the big bang often assume that a first event is also a first instant of time and that spacetime did not exist outside the big bang. This intimate linking of a first event with a first time is a philosophical move, not something demanded by the science. It is not even clear that it's correct to call the big bang an event. The big bang event is a singularity without space coordinates, but events normally must have space coordinates. One response to this problem is to alter the definition of 'event' to allow the big bang to be an event. Another response, from James Hartle and Stephen Hawking, is to consider the past cosmic time-interval to be open or unbounded at t=0 rather than closed or bounded by t=0. Looking back to the big bang is then like following the positive real numbers back to ever smaller numbers without ever reaching a smallest positive one. If Hartle and Hawking are correct that time is actually like this, then the universe had no beginning event, but it has a finite amount of past time, and the term 'the big bang' refers not to any single event. But in order to simplify the discussion ahead, this article will speak of 'the' big bang event as if it were a single event.

There are serious difficulties in defending the big bang theory's implications about the universe's beginning. They are based on the assumption that the universal expansion of clusters of galaxies can be projected all the way back. Yet physicists agree that the projection must fail in the Planck era, that is, for all times less than 10 to the minus 43 seconds after 'the' big bang. Therefore, current science cannot speak with confidence about the nature of time in the Planck era, nor whether time existed before that era. If a theory of quantum gravity does get confirmed, it should provide information about the Planck era, and it may even allow physicists to answer the question, "What caused the big bang?" However, at present, the best answer is probably "Nothing; it just happened." The philosophically radical, but theologically popular, answer, "God caused the big bang, but He, himself, does not exist in time" is cryptic because it is not based on a well-justified and detailed theory of who God is, how He caused the big bang, and how He can exist but not be in time. It is also difficult to understand St. Augustine's remark that "time itself was made by God." On the other hand, for a person of faith, belief in God as creator is usually stronger than belief in any scientific hypothesis or in any epistemological desire for a scientific justification of the remark about God or in the importance of satisfying any philosopher's demand for clarification.

The big bang theory is accepted by the vast majority of astronomers, but it is not as firmly accepted as is the theory of relativity. Relativity theory challenges a great many of our intuitive beliefs about time. The theory is inconsistent with the common belief that the order in which two events occur is independent of the observer's point of view. For events occurring at the same place, relativity theory implies the order is absolute (independent of the frame), but for distant events occurring close enough in time to be in each other's absolute elsewhere, event A can occur before event B in one reference frame, but after B in another frame, and simultaneous with B in yet another frame.

Science impacts our understanding of time in many other fundamental ways. Relativity theory implies there is time dilation between one frame and another. For example, the faster a clock moves, the slower it runs, relative to stationary clocks. Time dilation shows itself when a speeding twin returns to find that his (or her) earth-bound twin has aged more rapidly. This surprising dilation result has caused some philosophers to question the consistency of relativity theory, arguing that, if motion is relative, then from the perspective of the speeding twin, he should be the one who aged more rapidly. This argument is called the twins paradox. Experts now are agreed that the mistake is within the argument for the paradox, not within relativity theory. The argument fails to notice the radically different relationships that each twin has to the rest of the universe as a whole.

There are two kinds of time dilation. Special relativity's time dilation involves speed; general relativity's involves acceleration and gravitational fields. Two ideally synchronized clocks need not stay in synchrony if they undergo different accelerations or different gravitational forces. This effect would be especially apparent if one of the two clocks were to fall into a black hole. A black hole can form when a star exhausts its nuclear fuel and contracts so compactly that the gravitational force prevents anything from escaping the hole, even light itself. The envelope of no return surrounding the black hole is its event horizon. As a clock falls toward a black hole, time slows on approach to the event horizon, and it completely stops at the horizon (not just at the center of the hole)--relative to time on a clock that remains safely back on earth.

The supplement to this article continues with the topic of what science requires of time, and it provides background information about other topics discussed in this article.

 

Time travel

Let's begin by discussing activities that have been taken to be time travel but aren't under serious consideration by philosophers today. (a) Remembering an earlier event in your life may be a kind of mental time travel, although there's a difference between experiencing an event and remembering it. Because it's at best travel only in psychological time rather than physical time, philosophers haven't shown much interest in this sort of time travel. (b) If you get on a plane on the earth's surface and travel west, you will cross a time zone and instantly go back an hour. All you've really done though is changed your reference frame, so this is a trivial form of time travel. (c) If your body were quick-frozen in the year 2,000 and thawed in 2,088, then you would have traveled forward 88 years in clock time but only a few seconds of your biological time. This is a case of biological time travel, not a case of physical time travel and so has not been of interest to philosophers. (d) A change in the direction of the arrow of time isn't considered to be a case of time travel.

Travel to the future

Time travel can be to the past or the future. Because travel to the future is easier and more practical, it will be discussed first. There are two straightforward ways to travel into another person's future. In the twins paradox, a person speeding away from his twin who remains on earth will, upon reunion, have entered the earth-twin's future. Second, according to general relativity, if the twin goes to a stronger gravitational field by leaving the dinner table and descending to the cellar for a bottle of wine and then returns, he will have entered the future of his twin who stayed at the dinner table. When someone enters a relatively stronger gravitational field, their time slows down relative to the time of those who don't enter the stronger field.

Regarding the first kind of time travel to the future, if you have a fast enough spaceship, you can travel to the year 4,500 A.D. and see the future of earth. You can affect that future, not just see it. This is a direct consequence of the time dilation described in the theory of relativity. You can travel to someone else's future, not your own. You're always in your own present. Unfortunately, once you go to 4,500 A.D. (as judged in a frame of reference in which the earth is considered stationary), you are stuck in the earth's future. You can not reverse course in your spaceship and return to the 21st century on earth. You must live with the consequence that all your friends have died centuries ago. Visits to the future are permanent, not temporary.

On this trip to 4,500 A.D., how much time would elapse on your own clock? The answer depends on how fast your spaceship goes, what accelerations occur, and whether gravitational forces are acting. The faster your spaceship goes, the less time it will take--actually take, not just appear to take. As you approach infinitesimally close to the speed of light, the trip to 4,500 A.D. will take essentially no time at all. That's from your own perspective though; observers who remained stationary on earth and judged your flight from that perspective will have observed your speedy travel for thousands of years.

In science fiction movies, which almost always depict nonrelativistic time travel, time travelers suddenly appear from out of the past, and other travelers suddenly disappear from now and pop into the future. These phenomena have never been observed, despite the parapsychological literature. If they were reliably observed, then we would consider the hypothesis that spacetime has an extra dimension allowing time travel. The discontinuous worldline in ordinary 4-d spacetime could actually be a continuous trajectory in 5-d spacetime. One would wonder, though, how anyone could ever verify (check) that the time traveler took one trajectory in the higher dimension rather than another.

Travel to the past

One of the major metaphysical assumptions made in the analysis of time travel to the past is that the world is never logically contradictory. This is the heart of the Grandfather Paradox. According to this paradox, you step into a time machine, go back and kill your grandfather before he's met your grandmother, so you prevent your own birth. Therefore, you both exist and don't exist right now. This result violates the law of noncontradiction, so we may conclude that we erred in assuming the possibility of this sort of time travel. If time travel is going to exist, it can't permit any change in what is known to have happened--presuming that logic is more fundamental than metaphysics. More generally, John Earman proposed that we grant law-status to consistency constraints on all spacetimes with closed timelike curves.

How about influencing history instead of changing it? The time traveler helps make history what it was. For example, Joe Stalin, the dictator of Russia, was 21 years old in 1900. Let's suppose time machines are invented in 2030. In that year, Sam decides to assume the identity of Stalin. He knows Russian history, speaks fluent Russian, is 21 years old, and looks like Joe Stalin did at 21. Sam enters the newly invented time machine, goes back to 1900, secretly murders Stalin, then starts calling himself "Stalin". Sam never reveals his past [as Sam], and he eventually becomes the dictator of Russia.

This possibility requires altering our normal assumptions about personal identity. Because Stalin really died in 1953, Sam must die in 1953, many years before he is born. To accept that the time travel occurred, we'd have to revise our current notion of personal identity as well as our notion of what can be remembered, assuming that Sam-Stalin remembers life before stepping into his time machine.

Sam's world line will be composed of discontinuous segments. The world lines of more continuous time travelers might be a loop, a closed timelike curve. Either possibility implies backward causation. Some philosophers believe backward causation can be ruled out by the definition of 'cause,' just as they can rule out Monday ever immediately following Friday. Many other philosophers disagree on the grounds that backward causation is improbable or nonexistent, but not impossible. 

Another implication of Sam's time travel is his apparent violation of the law of conservation of matter by popping into existence in 1900. Must we also revise that law? The modern version of the law of conservation of matter-energy is that the conservation is statistical; matter is conserved on average. The shorter the time span and the smaller the mass involved then the more likely that there can be violations in conservation.

There are other significant implications involved with this sort of participatory time traveling--traveling back in time to participate in what actually happened. The future is oddly constrained by the time traveling. After Stalin's death, the world's events must allow Sam at age 21 to enter the time machine. Nothing can happen to prevent Sam getting to the machine. All his enemies somehow must botch their attempts to kill him. Attempted sabotage of the time machine must also fail. Scientists viewing these attempts will be surprised that they are continually yet inexplicably frustrated by unfavorable circumstances. Looking back from the year 2030 it will appear as if the world conspired to ensure that a predestined event occurred. It has been argued that because we've never seen the world conspire with massive coincidences, this sort of time travel never occurs even if it is logically and conceptually possible.

An additional argument against time travel of the kind that influences past events but doesn't change them is that by now we should have seen all sorts of time traveler tourists from the distant future. Nobody has ever seen one, despite some unreliable witnesses described in supermarket tabloids. Therefore, time travel most probably never occurs even if it could. The principal counter is that there might be very good reasons why our time hasn't yet been visited. The travelers might be uninterested in us. It might be very expensive to go to our time. They might be here but be invisibly cloaked so as not to interfere with us. Therefore, it is jumping to conclusions to be so pessimistic about the probability of time travel.

Admittedly, though, no one has any practical and realistic plans for how to build a time machine. The best plans use such phrases as "First, take a worm hole and...." Kurt Godel was the first person other than Einstein to have realized that the general theory of relativity does permit a physical object to travel at less than the speed of light and yet arrive at its own past. This travel requires the warping of spacetime itself, tipping the light cone of the traveling object. The time line dips back into the past and could form a closed curve in spacetime. Since Godel's initial work in 1949, mathematicians and theoretical physicists have described other time machines, or at least universes containing backward time travel, that are consistent with Einstein's equations of general relativity. Stephen Hawking believes all these time machines are ruled out by the laws of general relativity. General relativity theory is so complex that it isn't always clear, even to the experts, what is and isn't allowed by the theory.  Other physicists accept that Einstein's equations do allow time travel, but they rule out these solutions as being physically impossible or improbable for other reasons, such as those mentioned above.

Probing the possibility of a contradiction in backwards time travel, John Earman has described a rocket ship that carries a very special time machine. The time machine is capable of firing a probe into the past. Suppose the ship is programmed to fire the probe on a certain date unless a safety switch is on. Suppose the safety switch is programmed to be turned on if and only if the 'return' of the probe is detected by a sensing device on the ship. Does the probe get launched? The way out of Earman's paradox seems to require us to accept that (a) the universe conspires to keep people from building the probe or the safety switch or the sensing device, or (b) time travel probes must go so far back in time that they never make it back to the time when they were launched, or (c) past time travel is impossible. 

Feynman diagrams in particle physics were described by Feynman himself as illustrating how a particle's moving forward in time is actually its antiparticle moving backward in time. However, physicists don't take Feynman's suggestion literally. As a leading particle theorist, Chris Quigg of Fermi National Accelerator Laboratory, explained, "It's not that antiparticles in my laboratory are actually moving backward in time. What's really meant by that is that if I think of a particle moving from one place to another forward in time, the physical process is the same as it would be if we imagine running the film backward and also changing the particle into an antiparticle."

In addition to time travel that changes the past and time travel that participates in the past, consider a third kind, time travel that reaches the past of a different universe. This idea appeals to an unusual interpretation of quantum mechanics, the parallel universes interpretation. According to this interpretation, everything that can happen does happen in some universe or other. There's a universe in which the Nazis won World War II and Stalin was assassinated. There's another universe in which the Nazis won World War II and Stalin escaped all assassination attempts. On this theory of time travel, for you to travel back in time and have lunch with President Abraham Lincoln is for you to stop existing in the present universe as you enter the time machine and for you to appear earlier in time in a parallel universe, one in which you in fact did have lunch with Abraham Lincoln. The theory implies that we must change our current view of what makes a person the same person through time [say, bodily identity and continuity of consciousness through time in a single universe] and accept some kind of trans-universe identity.

 

Is the relational theory of time preferable to the absolute theory?

When you set your alarm clock for 7:00, does the time of 7:00 cause your alarm to go off? No, although it wouldn't go off if it weren't 7:00, under the circumstances. Such is the nature of causality. It is generally agreed that time causes nothing. Another question, underlying the point about your alarm clock, is whether 7:00 exists despite what happens. Absolute and relational theories of spacetime offer opposing answers to the question.

Absolute theories say time exists independently of the spacetime relations exhibited by physical events. Relational theories say it does not. Some absolute theories describe spacetime as being like a container for events. The container exists with or without events in it.

The term 'absolute' in this context does not mean independent of the observer, but independent of the events. The absolute theories imply that spacetime could exist even if there were no physical objects and events in the universe, but relational theories imply that spacetime is nothing but objects, their events, and the spatiotemporal relationships among them, though as we shall discuss in a moment, much depends on whether spacetime also involves possible events in addition to actual events. Everyone agrees time cannot be measured without there being changes, but the present issue is whether it can exist without changes.

Aristotle took a position regarding the relationship between time and change when he remarked that, "neither does time exist without change [218b]." However, the battle lines were most clearly drawn in the 17th century when Leibniz explicitly said there is no time without actual change and Newton protested that time exists regardless of whether anything changes. They offered several arguments for their positions.

Leibniz's principal argument: There is no difference between the presence and absence of absolute space. If there were, then God could create the world at any absolute place He wanted or at any time that He wanted. He could have made everything be five miles east of where it is and have had all the universe's events occur five minutes later. But that's absurd; there's no way to detect such a difference. So, by the Principle of the Identity of Indiscernibles, the two different universes would actually be one. Thus a contradiction is revealed with the Newtonian view of absolute space.

Newton's two-part response: (1) The bucket thought-experiment shows that acceleration relative to absolute space is detectable [by looking for the presence centrifugal forces]; thus absolute space is real. (2) Besides, there don't have to be discernible differences for humans; God might have his own reasons for creating the world at a given place and time even though mere mortals cannot comprehend His reasons. Leibniz is correct, says Newton, to accept the Principle of Sufficient Reason and the Principle of the Identity of Indiscernibles, but Leibniz is wrong to use these to argue against absolute space.

Huygens, Berkeley, and Mach entered the arena on the side of Leibniz. In the 20th century, Reichenbach and the early Einstein declared the special theory of relativity to be a victory for the relational theory. However, some philosophers say Reichenbach and the early Einstein may have been overstating the amount of metaphysics that can be extracted from the physics. Newton's own absolute theory of space used the notion of a space-filling material aether at rest in absolute space with distances and times being independent of reference frames, and this is admittedly inconsistent with special relativity, but other absolute theories are consistent with current science.

Absolute theories were dominant in the 18th and 19th centuries, and the relational theories were dominant in most of the 20th century, but at the end of the century, absolute theories have gained some ground and there is no convergence of opinion on this prominent issue.

Absolute theories are called 'substantival' or 'substantial' if they require spacetime to be a substance. These are the kind of absolute theories discussed here. Absolutists disagree among themselves about what it means to be a substance. It does not mean that spacetime is a kind of stuff out of which physical events are composed. Absolutists have described spacetime as "an antecedent arena for events" and "ontologically prior to events" and "an irreducible object of predication" and "the substrata for properties" and "the domain of the intended models of the basic physical theories." The container metaphor may work for special relativity, but general relativity requires that the curvature of spacetime be affected by the distribution of matter, so today it is no longer plausible for an absolutist to assert that the 'container' is independent of what it contains.

What is implied by saying time is a relationship among events? For example, if events occur in a room a second before and after 11:01 AM, but not at that instant, must the relationist say there never was a time of 11:01 AM in the room? One relationist response is to say 11:01 exists because somewhere something is happening then. There can be no 'empty' time, the relationist says. Will this relationist strategy for time work also for space? Can there be no empty space? No merely possible places? That is a bigger philosophical problem. For elegant science, we need to speak of an electron's taking a different path from the one it actually took. Is there a coherent role for these paths of nonexistent events in the relationist's 'relationships among events'?

If the relational theory were to consider spacetime points to be permanent possibilities of the location of events, then the relationist theory would collapse into substantivalism, and there would no longer be a difference between the two theories, John Earman has argued. To the absolutist, a spacetime point is also just a place where something could happen. Lawrence Sklar says that if relationists are going to talk about locations between material objects where no objects exist, then they "had better allow talk about possible objects and their possible spatial relations" because "versions of relationism that eschew such notions are pretty implausible...." The same point applies to possible events.

Hartry Field argues for the absolute theory by pointing out that modern physics requires gravitational and electromagnetic fields that cover spacetime. They are states of spacetime. These fields cannot be states of some Newtonian aether, but there must be something to have the field properties. What else except substantive spacetime points?

 

The flow of time

"It is as if we were floating on a river, carried by the current past the manifold of events which is spread out timelessly on the bank," said Plato. Other writers describe the flow of time as the "moving now" that cleaves the past from the future. "The passage of time...is the very essence of the concept," said Gerald Whitrow, and philosophers are eager to expose the real story behind the metaphor. It is universally agreed that time doesn't pass by at a rate of one second per second.

There have been three major theories of time's flow. The first, and most popular among physicists, is that the flow is an illusion, the product of a faulty metaphor. The second is that it is subjective, but deeply ingrained due to the nature of our minds or brains. The third is that it is objective, a feature of the mind-independent reality that has so far been missed by today's scientific laws.

Some philosophers have argued that the passage of time is a feature of the world to be explained by noting how events change. An event such as the death of Queen Anne can change from having the property of being future, to having the property of being present, to having the property of being past (to one of her contemporaries). Agreeing that events can change their properties in this manner, J. M. E. McTaggart argued that the concept of time itself is absurd because it is contradictory for Queen Anne's death to be both present and past. Many other philosophers believe events do not change any of their properties. An event's 'changing' from being future to being present to being past is not a real change in its properties, but only in its relations to the observer. So, it is concluded by these philosophers that the notion of time's flow is a myth. [It is not concluded that the notion of time is a myth.]

Ludwig Wittgenstein approached the question of why the flowing conception of time is such a compelling myth by asking us to be more attentive to the proper use of our words--and quit considering time to be a queer medium:

In our failure to understand the use of a word, we take it as the expression of a queer process. (As we think of time as a queer medium, of the mind as a queer kind of being.)

Most physicists do not believe time flows from future into past. Instead they accept the idea that events merely exist in spacetime. This idea is called the ‘block universe' idea. The term was coined by William James. Advocates of the block universe commonly argue that the notion of time's flow is simply a mistake or else that it is a subjective feature of psychological time to be explained, say, by a person's having more memories and more information at later times. They argue that the only sense that can be made of the metaphor "Time flows" is that time exists.

Other physicists and philosophers, however, do not consider time's flow to be a myth and have not been satisfied with these analyses. 

 

The arrow of time

What gives time its direction or 'arrow'? Actually, time is directional in two senses. In one sense, which is not the sense meant by the phrase "the arrow of time", time is directed from the future to the past. This is the sense in which any future event is temporally after any past event. Because this is implied by the very definition of the terms 'future' and 'past,' to say "Time is directed from future to past" is to express a merely conventional truth of little interest to the philosophical community.

However, time is directed in a second sense, one that isn't merely a matter of the definition of the relevant terms. This is time's arrow. Time's arrow is the universe's events being ordered in time. It is what distinguishes events ordered by the happens-before relation from those ordered by its converse, the happens-after relation. It is still an open question in philosophy and science as to what it is about events that gives them an arrow. This question is not the same as asking what it is about events that allows us to create a clock.

This arrow is evident in the fact that footprints in the sand are traces of the past, never traces of the future. This arrow is also evident in certain processes, such as the process of mixing cool cream into hot coffee. You soon get lukewarm, brown coffee, but you never notice the reverse--lukewarm coffee unmixing into a cool part and a hot part . Such is the way this irreversible thermodynamic process goes. The arrow of an irreversible physical process is the way it normally goes, the way it normally unfolds through time--if not through all time, then through time in the present epoch of the universe's history. The amalgamation of the universe's irreversible processes [and of the temporally asymmetric features such as the footprints in the sand being traces of past events, not future events] produces the cosmic arrow of time, the master arrow. Usually this arrow is what is meant when one speaks simply of "time's arrow."

The goals of a theory of time's arrow are to understand why this arrow exists, what it would be like for the arrow to reverse direction, and what the relationships are among the various more specific arrows of time--the various temporal asymmetri processes such as entropy increases [the thermodynamic arrow], causes preceding their effects [the causal arrow], the universe's spatial expansion, our knowing the past more easily than the future, and so forth.

One problemmatic sub-issue is whether time is absolute or relational, because a solution to this sub-issue can affect, say, what it is for time to reverse and what it is for time to have an arrow. There seems to be little point in assigning an arrow to absolute time itself because its direction is unaffected by the direction of any physical process. If the absolute theory is correct, then the so-called cosmic arrow of time is not the arrow of time, but rather the arrow of the universe's processes. On the relational theory, this distinction collapses. Therefore, to avoid taking a stand on the relational-absolute issue, it's best to consider the issue of the arrow of time as concerning the asymmetry, not of time itself, but of events and processes in time.

If physical processes in time do have an arrow, and if the processes obey scientific laws, and if these laws are to be accounted for by the basic laws of physics (the laws governing the microscopic constituents of matter), then you might think that an inspection of these basic laws would readily reveal time's arrow. It won't. Except for the fact that it takes more than a trillion times longer for a kaon to decay into pions than for a kaon to be produced by motion reversal [which is essentially time reversal] from the pions, all the basic laws are time symmetric. This means that if a certain process is allowed by the equations, then that process reversed in time is also allowed. In other words, the basic laws of science are insensitive to the distinction between past and future.

To illustrate, let's suppose you could have a movie of a basic physical process such as two atoms bouncing off each other. You can't have such a movie because the phenomenon is too small, but let's forget that fine point for a moment. If you had such a movie, you could run it forwards or backwards and both showings would illustrate a possible process according to the laws of science. Time's arrow isn't revealed in this microscopic process. The reason why this result is so interesting to scientists and philosophers is that, if you had a movie of the mixing of hot, black coffee and cool cream, then you could tell which way is the right way to show the movie. The arrow of time that was absent in the microscopic movie would be evident in the macroscopic movie. This difference between microscopic movies and macroscopic movies is odd because ordinary processes are supposed to be composed of more basic processes. Why does the arrow of time appear in one movie but not the other?

Ludwig Boltzmann had an answer. He was the first to attempt to show how an irreversible macroscopic phenomenon may arise from reversible microscopic laws. He showed that macroscopic thermodynamic processes such as heat in a gas are irreversible because the probability of their actually reversing is insignificant. There are more lukewarm microstates of the set of its constituent molecules than there are microstates with separated hot and cold regions, so the system evolves in the 'direction' of what is most probable. To express the point somewhat more precisely, let A be the set of microstates of an isolated container in which one part of the container contains hot gas and a separate part contains cold gas. Let B be the lukewarm microstates. Assume all the microstates are equally probable apriori. The number of B states is dramatically larger than the number of A states, so the probability that one of the A states will soon lead to one of the B states is almost one whereas the probability that a B state will soon lead to an A state is almost zero. That is why the process of heat in a gas is irreversible.

The law of physics describing heat processes is the second law of thermodynamics, an irreversible law that says a change occurring in an isolated, macroscopic system will most probably not lead it into a state of lower entropy. Entropy is approximately a measure of a system's disorder, so an entropy increase is a trend toward decay, running down, rusting, the conversion of useful energy into heat. Isolated systems change toward disorder because disorder is so probable, Boltzmann would say. So, entropy's relentless increase accounts for the irreversibility of thermodynamic processes, and this is the basis of time's arrow, according to Boltzmann.

A dynamic system is a system defined by the values of the positions and velocities of all the system's particles--such as the places and speeds of the atoms in a cup of coffee) Henri Poincare's recurrence theorem in statistical mechanics says every isolated dynamical system will eventually return to a state as close to the initial state as we might wish. Wait long enough, and the lukewarm brown coffee will separate into hot black coffee and cool white cream. In other words, strictly speaking, there are no irreversible processes. So, there is a contradiction between Poincare's theorem and Boltzmann's proof. The second law implies that entropy increases, but Poincare's theorem implies that entropy remains the same over the long haul. The resolution of the problem is to revise the second law: it's probable, for period of time short compared to the Poincare period, that the higher entropy state is the later entropy state.

If the thermodynamic arrow of time [for periods much shorter than Poincare's recurrence time] is to be explained by entropy increase, as Boltzmann hoped, then we want to know why entropy was so low in the past. You wouldn't expect it to be low in the past if you started from the present and applied the basic time symmetric laws of science. That is, hasn't Boltzmann really shown only that entropy change is associated with entropy increase, so that entropy change toward the past implies high entropy in the past? And Boltzmann has failed to show why entropy in our past was so low. These criticisms come from Boltzmann's colleague in Vienna, Franz Loschmidt. Boltzmann's response used the anthropic principle. He said we are the kind of creatures whose physiology is such that we are "bound to regard the future as being the direction in which entropy increases", so THAT explains why entropy increases toward the future, but not the past.

Loschmidt took a different view. He said the observed occurrence of only entropy-increasing processes in the present era of history must be a consequence of the particular initial conditions in our region of the universe and not a consequence of the laws governing molecular motions, nor a consequence of some anthropic principle. Boltzmann didn't realize that the temporal asymmetry he got out of a system if just the asymmetry he put in. Agreeing with Loschmidt, Einstein argued that the asymmetry we see when a wave expands from its source but never converges coherently to a point is just as statistically based as Boltzmann's entropy flow; and both asymmetries rest on what the initial conditions happen to be like.

If so, then entropy increase is not the deep reason behind time's arrow. Instead, the arrow of time depends essentially on the universe's having started with the initial configuration that it had. This analysis leads naturally to the request for an explanation of the initial configuration of our universe, an explanation we would hope to get from cosmologists.

Our original question was: Why does the arrow of time appear in the coffee movie but not in the atomic movie? There is Boltzmann's answer involving entropy and statistics and the anthropic principle. There is a second answer that appeals to order in the past and ultimately to the initial conditions of the universe; things just started out so as to make it that way. The Swiss physicist Walther Ritz and, more recently, the English physicist Roger Penrose, offer a third answer: we must not yet have found the true laws (or invented the best laws) underlying nature's behavior; we need to keep looking for basic, time asymmetrical laws in order to account for time's arrow. However, the more commonly accepted theory of temporal asymmetry today is that the asymmetry arises from asymmetric boundary conditions on mostly symmetric laws rather than from asymmetric laws.

Let's consider how the various arrows of time relate to each other. The direction of increasing entropy is the thermodynamic arrow. Additional arrows exist in the following processes:

a. It is easier to know the past than the future.

b. Waves spread out from, and never converge into, a point.

c. Quantum mechanical measurement collapses the wave function.

d. Kaon decay is different in a time reversed world.

e. We see black holes but never white holes.

f. Conscious actions affect the future but not the past.

g. Explanation uses the past information, not the future.

h. Causes precede their effects.

i. Higgs boson decay is different in a time reversed world.

For a process to be classified as an arrow of time, it must work differently or not at all if time were reversed. Many physicists suspect all these arrows are linked and that they must somehow involve conditions very early during the big bang.

Many physicists believe that someday the growth of physics will show that arrow i in our list is the most fundamental. The temporal asymmetry of the Higgs boson particle is already known to be the key reason why the universe contains what little matter it now does after all the primordial antimatter collided with primordial matter.

If the arrows of time are not all linked, then some may reverse while others do not. The question of whether the arrows are or are not linked is one to be settled by the physicists, not the philosophers.

There has been some controversy among physicists regarding whether the universe's expansion is an arrow of time. What is now clear is that if the universe's expansion does stop and the universe begins to collapse back upon itself, then the contraction will take place forward in time and the earlier expansion events (such as last century's World War I) won't occur in reverse order.

 

Is the future real?

Duns Scotus and A. N. Prior argue that only the present is real. This presentist viewpoint maintains that the past and the future are not real. There are two camps opposed to presentism, the camp that says the past and present are real, and the camp that says the past, present, and future are real. Aristotle and C. D. Broad believe the past and present are real. Reality 'grows' with the coming into being of determinate reality from an indeterminate potential reality. Radically opposed to either of the previous two treatments of the future, Hermann Weyl, J.J.C. Smart, and others argue that the objective world simply is. There is no objective ontological difference, they say, between the past, the present, and the future just as there is no ontological difference between here and there. This position is called the "block universe" position because it regards reality as a single block of spacetime with its time slices ordered by the temporally-before relation. It is mental perspectives only that divide the block into a past part, a present part, and a future part.

A presentist would complain that this block universe position misses the ontologically significant point that the present is so much more 'vivid' than any other time-slice of spacetime. The advocates of the block universe position argue that only the block universe can make sense of the relativity of simultaneity. Einstein's theory of relativity implies that an event in person A's present can be in person B's future, provided the two persons are in relative motion. Presentism and the growing-past theories must suppose that this event is both real and unreal because it's real for A but not real for B.

This philosophical battle between the camps has taken a linguistic turn in the 20th century by focussing upon the question, "Are predictions true or false at the time they are uttered?" Those who believe in the block universe (and thus in the determinate reality of the future) will answer "Yes" and their opponents will answer "No." The issue is whether contingent sentences uttered now about future events are true or false now rather than true or false only in the future at the time the predicted event is supposed to occur.

Suppose someone now says, "Tomorrow the admiral will start a sea battle." Suppose that tomorrow the admiral orders a sneak attack on the enemy ships. And suppose that this action starts a sea battle. Advocates of the block universe argue that, if so, then the sentence was true all along. Truth is eternal or fixed, they say, and 'is true' is a tenseless predicate. These philosophers point favorably to the ancient Greek philosopher Chrysippus who was convinced that a contingent sentence about the future is either true or it is false and not any value in between. Many others, following a suggestion from Aristotle, argue that the sentence is not true until it's known to be true, namely at the time at which the sea battle occurs. The sentence wasn't true before the battle occurred. In other words, predictions have no truth values at the time they are uttered. Predictions fall into the "truth value gap." This position that contingent sentences have no truth values is called the Aristotelian position because many researchers throughout history have taken Aristotle to be holding the position--although today it is not so clear that Aristotle himself held it.

The principal motive for adopting the Aristotelian position arises from the belief that if sentences about future human actions are now true, then humans are determined to perform those actions, and so humans have no free will. To defend free will, we must deny truth values to predictions.

The Aristotelian argument against predictions being true or false has been discussed as much as any in the history of philosophy, but it faces a series of challenges. If there really is no free will, or if free will is compatible with determinism, then the motivation to deny truth values to predictions is undermined.

A second challenge arises from moral discussions about the interests of people who are as yet unborn. Quine argues that if we have an obligation to conserve the environment for these people, then we are treating them as being as real as the people around us now. Only the block universe view can make sense of this treatment.

A third challenge, from Quine and others, claims the Aristotelian position wreaks havoc with the logical system we use to reason and argue with such predictions. For example, here is a deductively valid argument:

There will be a sea battle tomorrow.

If there will be a sea battle tomorrow, then we should wake up the admiral.

So, we should wake up the admiral.

Without the premises in this argument having truth values, that is, being true or false, we cannot properly assess the argument using the standard of deductive validity because this standard is about the relationships among truth values of the component statements. Unfortunately, the Aristotelian position says that some of these components are neither true nor false.

In light of these various challenges to the Aristotelian position, some philosophers conclude that Aristotle should retract his claim that predictions fail to be true or false at the time they are uttered and conclude that the block universe viewpoint is correct. However, philosophers are still very divided on the issue.

 

Temporal indexicals and essentially tensed facts

What is the significance of saying that an event occured in the past, the present, or the future? There are two major answers. One answer is that these distinctions represent objective features of reality that aren't captured by the popular 'block universe' approach. This answer takes tenses very seriously and is called the tensed theory of time. A second answer is that these distinctions are subjective features of the perspective from which we view the universe. On the latter view, called the tenseless theory of time, whether the assassination of President Kennedy occurred here depends on the speaker's perspective; similarly, whether the assassination occured now is equally subjective. The proponent of the tenseless view will say that talk about the past is really talk about our own relation to events. For example, on the tenseless theory, if it's now Jan. 1, 2008 for us, then to say now that some event is past is to say that it happens [tenselessly] before all the events of Jan. 1, 2008. The goal of the tenseless theory is to provide an adequate analysis of talk about the past, the present, and the future using only talk about happens before, happens after, and happens at the same time as.

This controversy is often presented as a dispute about whether tensed facts exist, with advocates of the tenseless theory objecting to tensed facts. The primary function of tensed facts is to make tensed sentences true. For the purposes of making the point, let's uncritically accept the Correspondence Theory of Truth, and apply it to the past tense sentence:

Custer died in Montana.

If we apply the Correspondence Theory directly to this, we would say that

The sentence "Custer died in Montana" is true because it corresponds to the tensed fact that Custer died in Montana.

Opponents of tensed facts argue that the Correspondence Theory should be applied only indirectly. It should be applied only to the result of analyzing tensed statements into equivalent statements that don't involve tenses, as follows:

There is a time t such that Custer dies in Montana at time t, and time t is before the time of the utterance of the sentence "Custer died in Montana."

In this analysis, the verb dies is tenseless in the same way that the verb "is" is tenseless in "7 + 5 is 12". The sentence isn't used to say "7 + 5 is presently 12." Ditto for the verb is in the above phrase "time t is before...." Applying the Correspondence Theory to this new sentence will produce:

The sentence "Custer died in Montana" is true because it corresponds to the tenseless fact that there is a time t such that Custer dies in Montana at time t, and time t is before the time of the utterance of "Custer died in Montana."

Advocates of the subjective theory of tenses argue that the truth conditions of any tensed sentence can be explained without tensed facts. Then Ockham's Razor is applied. If we can do without essentially tensed facts, then we should say essentially tensed facts do not exist. To summarize, tensed facts were presumed to be needed to account for tensed talk; but the analysis shows that ordinary tenseless facts are adequate.

This analysis of tenses is challenged. It can work only for utterances, but a sentence can be true even if never uttered by anyone. Even if that problem could be repaired, there are other challenges. For example, Roderick Chisholm and A. N. Prior claim that the 'is' in the sentence "It's now midnight" is essentially present tensed because there is no equivalent sentence using tenseless verbs. Trying to analyze it as, say, "There is a time n such that n = midnight" is to miss the essential reference to the present in the original sentence. The latter sentence is always true, but the original is not. They say that true sentences using the temporal indexical terms 'now,' 'before now,' and 'happened yesterday' are part of the facts of the world that science should account for, but science fails to do this because it doesn't recognize them as being real facts. Science so far restricts itself to eternal facts, such as in the Minkowski-like spacetime representation of events. These events are sets of spacetime points. For such events, the reference to time and place is explicit. A Minkowski spacetime diagram displays only what happens before what, but not which time is present time, or past, or future. What is missing from the diagram, say Chisholm and Prior, is some moving point on the time axis representing the observer's 'now' as time flows up the diagram.

In the same spirit, Michael Dummett argues that you can have a complete description of a set of objects in space even if you haven't said which objects are near and which are far, but you cannot have a complete description of those objects without specifying which events are present and which are not.

Russell, Quine, Grunbaum, and Horwich object to assigning special ontological status to the present. According to Quine, the analysts should in principle be able to eliminate the temporal indexical words because their removal is needed for fixed truth and falsity of our sentences [that is, fixed regardless of the time], and having fixed truth values is crucial for the logical system used to clarify science, the system of first-order predicate logic. "To formulate logical laws in such a way as not to depend thus upon the assumption of fixed truth and falsity would be decidedly awkward and complicated, and wholly unrewarding," says Quine.

 

The symbolic logic of time

In the 1950s, A. N. Prior created a new symbolic logic to describe our use of time words such as 'now', 'happens before', 'afterwards', 'next', 'always', and 'sometimes'. He was the first to appreciate the similarity in structure between time concepts and modal concepts such as 'it is possible that' and 'it is necessary that.' He applied a logic having infinitely many truth-values to create a 'tense logic' in which the relationships that propositions have to the past, present, and future help to determine their truth-value. In classical logic, there are only two truth-values, namely true and false. Dummett and Lemmon also made major, early contributions to tense logic. In one standard system of the logic of past time, the S4.3 system, the usual modal operator 'it is possible that' is re-interpreted to mean 'at some past time it was the case that.' Let the letter 'M' represent this operator, and add to the axioms of classical propositional logic the modal axiom M(p v q) iff Mp v Mq. The axiom says that for any two propositions p and q, at some past time it was the case that p or q if and only if either at some past time it was the case that p or at some past time it was the case that q. S4.3's key axiom is the equivalence

Mp & Mq iff M(p & q) v M(p & Mq) v M(q & Mp).

This axiom captures our ordinary conception of time as a linear succession of states of the world. Logicians disagree about what additional axioms and revisions are needed to make more of our beliefs about time be theorems of a symbolic logic of time.

The first person to give a clear presentation of the implications of treating declarative sentences as being neither true nor false was the Polish logician Jan Lukasiewicz in 1920. To carry out Aristotle's suggestions that future contingent sentences don't yet have truth values, he developed a three-valued symbolic logic, with all grammatical sentences having the truth-values of True, False, or else Indeterminate [T, F, or I]. Contingent sentences about the future, such as predictions, are assigned an I. Truth tables for the connectives of propositional logic are redefined to maintain logical consistency and to maximally preserve our intuitions about true and falsehood.

 

SUPPLEMENT

This supplement answers a series of questions designed to reveal more about what science requires of physical time, and to provide background information about other topics discussed in this article.

 What is a reference frame?

What is spacetime?

What is an event?

Does the theory of relativity imply time is partly space?

Is time the fourth dimension?

Is time infinite?

Is there more than one kind of physical time?

How is time relative to the observer?

What are the relativity and conventionality of simultaneity?

What is the difference between the past and the absolute past?

What is time dilation?

How does gravity affect time?

What happens to time near a black hole?

What is the solution to the twins paradox?

What is the solution to Zeno's paradoxes?

How do time coordinates get assigned to points of spacetime?

How do dates get assigned to actual events?

What is essential to being a clock?

What is our standard clock?

Why are some standard clocks better than others?

 

What is a reference frame?

A reference frame is a standard point of view or a perspective for making observations and judgments.

A reference frame is often specified by selecting a solid object that doesn't change its size and saying that the reference frame is fixed to the object. We might select a reference frame fixed to the Rock of Gibraltar. Another object is said to be at rest in the reference frame if it remains at a constant distance in a fixed direction from the reference body used to define the frame. For example, our house is at rest in a reference frame fixed to the Rock of Gibraltar. When we say the sun rose over our house this morning, we are implicitly choosing a reference frame fixed to our house. The sun is not at rest in this frame.

To define places in the reference frame in three dimensional space, the analyst needs to apply a coordinate system to the frame. The analyst needs to specify four distinct points on the reference body, or four objects mutually at rest somewhere in the frame. One point is the origin, and the other three can be used to define three independent axes, the familiar x, y and z directions, if a rectangular coordinate system were to be placed on the frame. Two point objects are at the same place if they have the same x-values, the same y-values and the same z-values. To keep track of events, you will also need a time axis, a t axis. Two point events are simultaneous if they occur at the same place and also at the same time.

Choosing a good reference frame can make a situation much easier to describe. If you are trying to describe the motion of a car down a straight highway, you would not want to choose a reference frame that is fixed to a spinning carousel. Instead, choose a reference frame fixed to the highway.

Special relativity is intended to apply only to special reference frames, inertial frames. The speed of light in a vacuum is the same when observed from any intertial frame of reference. The speed of light in a vacuum isn't affected by which inertial reference frame is used for the measurement, unlike measurements of speeds of cows and rockets.

How do you tell if you are in an inertial frame? You check that objects accelerate only when acted on by forces. That is, you check that Newton's first law of motion holds: any object's acceleration is zero if no net force acts on the object. If no unbalanced external forces are acting on a moving object, then the object moves in a straight line. It doesn't curve. And it travels equal distances in equal amounts of time. Any frame of reference moving at constant velocity relative to an inertial frame is also an inertial frame. A reference frame spinning relative to an inertial frame is never an inertial frame.

The presence of gravitation normally destroys any possibility of finding a frame that is a perfect inertial frame. So, in practice, when trying to use special relativity, inertial frames are distinguished not by being absolutely unaccelerated, but rather by being unaccelerated relative to some suitably defined average of all the matter in the universe. A reference frame in which star motion is ignored and the stars are assumed to be at rest is approximately an inertial reference frame and is often adequate for many purposes. This is the so-called inertial frame of the 'fixed stars.'

 

What is spacetime?

Spacetime is a certain 4-d space (or 4-d manifold, to use Riemann's term for space). It's the 4-d continuum we live in. Spacetime is the intended model of the general theory of relativity. This requires it to be a differentiable space in which certain geometrical objects obey the covariant field equations of general relativity, and in which physical objects obey the equations of motion of the theory.

The metaphysical question of whether spacetime is a substantial object or a relationship among events, or neither, is taken up above in the discussion of the relational theory of time. Regardless of how that question is answered, spacetime is more fundamental in science than either space or time alone.

Spacetime is a continuum in which we can define points and straight lines. However, these points and lines do not satisfy the principles of Euclidean geometry when matter is present. Einstein showed that the presence of matter affects geometry by warping space and time. Einstein's principal equation in his general theory of relativity implies that the curvature of spacetime is directly proportional to the density of mass in the spacetime. The equation can be interpreted as implying that curvature in the spacetime geometry is matter, or vice versa. In this sense, Einstein has geometricized reality.

The region of spacetime at the center of a black hole develops infinitely large curvature. The earth's curving of space-time is very slight but still significant enough that it must be accounted for when synchronizing two Global Positioning Satellites.

The spacetime interval between any two events is intrinsic to spacetime and does not vary with the reference frame, unlike a distance or a duration.

Although relativity theory assumes that spacetime is fundamental, there have been serious attempts over the last few decades to construct theories of physics in which spacetime is not fundamental but is a product of more basic entities. The primary aim of these new theories is to unify relativity with quantum theory. So far these theories have not stood up to any empirical observations or experiments that could show them to be superior to the presently accepted theories. So, for the present the concept of spacetime remains fundamental.

Regions of spacetime are frequently pictured with a Minkowski diagram using a rectangular coordinate system. The vertical time axis is the product of time and the speed of light so that world lines of light rays leaving the origin make a forty-five degree angle with each space axis. The Minkowski diagram applies to a particular observer who experiences the event that occurs at the point indicated by the diagram's origin. In a Minkowski diagram, an ideally small physical particle is not represented as occupying a point of spacetime but as occupying a line containing all the spacetime points at which it exists. The line is called the "world line" of the particle. If two world lines intersect, then the two particles have collided. A person's world line is composed of the world lines of the person's parts. Inertial motion corresponds to straight world lines, and accelerated motion corresponds to curved world lines.

 

What is an event?

An event might be defined simply as whatever is temporally before or after anything else. In ordinary discourse, an event is a happening during which some object changes its properties. For example, this morning's event of buttering the toast involved the toast's changing from unbuttered to buttered. In ordinary discourse, an event has more than an infinitesimal duration, but in the technical discourse of physics, all events are composed of point events, events with zero duration. Also, a point event is considered by physicists to be a spacetime point's having some property other than those it has just by being a location in spacetime. The point event is the having of some property at some point in space for an instant. Notice that no change is mentioned here.

We could try to define a point events in terms of objects and properties and times. We might speak of the point event of this spacetime point having the property of containing butter at a certain time this morning. However, point events and spacetime points actually are more basic in physics than are objects and their properties. Point events are what all objects and events are made of, and spacetime points are what have the properties.

These metaphysical assumptions of modern science are not part of common sense, the shared background beliefs of most people. They also are not acceptable metaphysical assumptions for many philosophers. In 1936, in order to avoid point events, Bertrand Russell and A. N. Whitehead developed a theory of time based on the assumption that all events in spacetime have a finite duration. However, they had to assume that any finite part of an event is an event, and this assumption is no closer to common sense than the physicist's assumption that all events are composed of point events.

 

Does the theory of relativity imply time is partly space?

In 1908, when Minkowski remarked that "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality," many people took this to mean that time is partly space, and vice versa. C. D. Broad countered that the discovery of spacetime did not break down the distinction between time and space but only their independence or isolation. He argued that their lack of independence does not imply a lack of reality. The Broad-Minkowski disagreement is still an issue in philosophy, but if Broad is correct, then time is time; it's not space at all. Nevertheless, there is a deep sense in which time and space are 'mixed up' or linked. This is evident from the Lorentz transformations of special relativity that connect the time t in one inertial frame with the time t' in another frame that is moving in the x direction at a constant speed v. In this equation, t' is dependent upon the space coordinate x and the speed. In this way, time is not independent of either space or speed. It follows that the time between two events could be zero in one frame but not zero in another. Each frame has its own way of splitting up spacetime into its space part and its time part. Nevertheless, the reason why time is not partly space is that, within a single frame, time is distinct from space. Time is not simply an arbitrary one-dimensional sub-space of spacetime; it is a distinguished sub-space. That is, time is a distinguished dimension of spacetime, not an arbitrary dimension. What being distinguished amounts to is that when you set up a rectangular coordinate system on spacetime with an origin, say, at the signing of the Declaration of Independence in Philadelphia, you can point the x-axis east or north or up or anywhere in between, but you are not allowed to point it forward in time--you can do that only with the t-axis, the time axis.

 

Is time the fourth dimension?

Yes and no; it depends on what you are talking about. Time is the fourth dimension of spacetime, but time is not the fourth dimension of space, the space of places.

Mathematicians have a broader notion of the term 'space' than the average person; and in their sense a space need not consist of places, that is, geographical locations. Not paying attention to the two meanings of the term 'space' is the source of all the confusion about whether time is the fourth dimension. The mathematical space used by mathematical physicists to represent physical spacetime is four dimensional and in that space, the space of places is a 3-d sub-space and time is another 1-d sub-space. Minkowski was the first person to construct such a mathematical space, and he was the first person to call time the fourth dimension, because it was the fourth dimension of his abstract space for spacetime. But spacetime is represented mathematically as a space of events, not a space of ordinary geographical places such as the place where the sun is located and the place where London is located.

In any coordinate system on spacetime, it takes at least four independent numbers to determine a spacetime location. In any coordinate system on the space of places, it takes at least three. That's why spacetime is four dimensional but the space of places is three dimensional. Actually this 19th century definition of dimensionality, which is due to Bernhard Riemann, is not quite adequate because mathematicians have subsequently discovered how to assign each point on the plane to a point on the line without any two points on the plane being assigned to the same point on the line. The idea comes from Georg Cantor. Because of this one-to-one correspondence, the line and the plane have the same number of points. If so, then the line and plane must have the same dimensions according to the previous definition. To avoid this problem, the notion of dimensionality of a space has been given a new, but rather complex, definition.

 

Is time infinite?

There are three ways to interpret this question: (a) Was there an infinite amount of time in the past? No, not if time began with the big bang and the relational theory of time is correct. Yes, if the absolute theory of time is correct. (b) Is time infinitely divisible? Yes, because general relativity and quantum mechanics require time to be a continuum. (c) Will there be an infinite amount of time in the future? This is difficult to judge. Yes, if the absolute theory of time is correct, because for that theory, time exists even when there are no events. But if the relational theory of time is correct, then the answer depends on whether events will keep occurring. The best estimate from the cosmologists these days is that the expansion of the universe will continue forever. There always will be the events of galaxies getting farther apart, and so future time will have an infinite duration.

 

Is there more than one kind of physical time?

Every reference frame has its own physical time, but the question is intended in another sense. At present, physicists measure time electromagnetically. They define a standard atomic clock using periodic electromagnetic processes in atoms, then use electromagnetic signals (light) to synchronize clocks that are far from the standard clock. In doing this, are physicists measuring 'electromagnetic time' but not other kinds of physical time? In the 1930s, the physicists Arthur Milne and Paul Dirac worried about this question. Independently, they suggested there may be very many time scales. For example, there could be the time of atomic processes and light, which is measured best by atomic clocks. There also could be the time of gravitation and large-scale physical processes, which is measured best by the rotation of a pulsar (pulsating star). The two physicists worried that the atomic clock and the astronomical clock might drift out of synchrony after being initially synchronized, yet there would be no reasonable explanation for why they don't stay in synchrony. Ditto for clocks based on the pendulum, on superconducting resonators, on the spread of electromagnetic radiation through space, and on other physical principles. Just imagine the difficulty for physicists if they had to work with electromagnetic time, gravitational time, nuclear time, neutrino time, and so forth. Current physics, however, has found no reason to assume there is more than one kind of time for physical processes. In 1967, physicists did reject the astronomical standard for the atomic standard because the deviation between known atomic and gravitation periodic processes could be explained better assuming that the atomic processes were the more regular of the two. Physicists had no reason to believe that a gravitational periodic process, that is just as regular initially as the atomic process and that is not affected by friction or impacts or other forces, would ever drift out of synchrony with the atomic process, yet this is the possibility that worried Milne and Dirac.

 

How is time relative to the observer?

Physical time is not relative to any observer's state of mind. Wishing time will pass does not affect the rate at which the observed clock ticks. On the other hand, physical time is relative to the observer's reference system--in trivial ways and in a deep way discovered by Albert Einstein.

In a trivial way, time is relative to the chosen coordinate system on the reference frame, though not to the reference frame itself. For example, it depends on the units chosen as when the duration of some event is 34 seconds if seconds are defined to be a certain number of ticks of the standard clock, but is 24 seconds if seconds are defined to be a different number of ticks of that standard clock. Similarly, the difference between the Christian calendar and the Jewish calendar for the date of some event is due to a different unit and origin. Also trivially, time depends on the coordinate system when a change is made from Eastern Standard Time to Pacific Standard Time. These dependencies are taken into account by scientists but usually never mentioned. For example, if a pendulum's approximately one-second swing is measured in a physics laboratory during the autumn night when the society changes from Daylight Savings Time back to Standard Time, the scientists do not note that one unusual swing of the pendulum that evening took a negative fifty-nine minutes and fifty-nine seconds instead of the usual one second.

In a deeper sense, time is relative, not just to the coordinate system, but to the reference frame. That is Einstein's principal original idea about time.

Einstein's idea is that without reference to the frame, there is no fixed time interval between two events, no 'actual' duration between them. To illustrate Einstein's idea for special relativity, let's assume that a number of observers are at rest in their inertial frames of reference. Which of these observers will agree on their time measurements? Observers with zero relative speed will agree. Observers with different relative speeds will not, even if they agree on how to define the second and agree on some event occuring at time zero (the origin of the time axis). If the two observers are moving relative to each other, but each makes judgments from a reference frame fixed to themselves, then the assigned times to the event will disagree more, the faster their relative speed. All observers will be observing the same objective reality, the same event in the same spacetime, but their different frames of reference will require disagreement about how spacetime divides up into its space part and its time part.

This relativity of time to reference frame implies that there be no such thing as The Past in the sense of a past independent of reference frame. This is because a past event in one reference frame might not be past in another reference frame.

Relative to any observer, was Adolf Hitler born before George Washington? No, because the two events are causally connectible. That is, one event could in principle have affected the other since light would have had time to travel from one to the other. We can select a reference frame to reverse the usual earth-based order of two events only if they are not causally connectible. Despite the relativity of time to a reference frame, all observers should agree about what happens before what when it comes to describing causally connectible events.

 

What are the relativity and conventionality of simultaneity?

If the universe obeys Newtonian physics, two events will be judged as simultaneous whatever the frame of reference in which we make a judgment of simultaneity, but the velocity of an object will depend upon which frame of reference is used. If the universe obeys relativistic physics, then events that occur simultaneously with respect to one reference frame may not occur simultaneously in another reference frame that is moving with respect to the first frame. This is called the "relativity of simultaneity".

This philosophically uncontroversial feature of time in relativity theory is different from the philosophically controversial feature called the "conventionality of simultaneity". To appreciate the difference, consider what is involved in making a determination regarding simultaneity. Given two events that happen essentially at the same place, physicists assume they can tell by direct observation whether the events happened simultaneously. If we don't see one of them happening first, then we say they happened simultaneously, and we assign them the same time coordinate. The determination of simultaneity is more difficult if the two happen at separate places. One proper way to measure (operationally define) simultaneity at a distance is to say that two events are simultaneous in a reference frame if unobstructed light signals from the two events would reach us simultaneously when we are midway between the two places where they occur, as judged in that frame. This is the operational definition of simultaneity used by Einstein in his theory of relativity.

The 'midway' method described above of operationally defining simultaneity in one reference frame for two distant signals causally connected to us has a significant presumption: that the light beams travel at the same speed regardless of direction. Einstein, Reichenbach and Grunbaum have called this a reasonable 'convention' because any attempt to experimentally confirm it presupposes that we already know how to determine simultaneity at a distance. This is the conventionality, rather than relativity, of simultaneity. To pursue the point, suppose the two original events are in each other's absolute elsewhere; they couldn't have affected each other. Einstein noticed that there is no physical basis for judging the simultaneity or lack of simultaneity between these two events, and for that reason said we rely on a convention when we define distant simultaneity as we do. Hillary Putnam objects to calling it a convention--on the grounds that to make any other assumption about light's speed would unnecessarily complicate our description of nature, and we often make choices about how nature is on the basis of simplification of our description. Putnam would say there is less conventionality in the choice than Einstein supposed.

The 'midway' method isn't the only way to define simultaneity. Consider a second method, the 'mirror reflection' method. Select an earth-based frame of reference, and send a flash of light from earth to Mars where it hits a mirror and is reflected back to its source. The flash occurred at 12:00, let's say, and its reflection arrived back on earth 20 minutes later. The light traveled the same empty, undisturbed path coming and going. At what time did the light flash hit the mirror? The answer involves the so-called conventionality of simultaneity. All physicists agree one should say the reflection event occurred at 12:10. The controversial philosophical question is whether this is really a convention. Einstein pointed out that there would be no inconsistency in our saying that it hit the mirror at 12:17, provided we live with the awkward consequence that light was relatively slow getting to the mirror, but then traveled back to earth at a faster speed. If we picked the impact time to be 12:05, we'd have to live with the fact that light traveled slower coming back. There is a physical basis for not picking the impact time to be less than noon nor later than 12:20, because doing so would violate the physical principle that causes precede their effects. One requirement we place on the concept of simultaneity is that distant events which are simultaneous could not be in causal contact with each other. We can satisfy that requirement for any choice of impact time from 12:00 to 12:20.

 

What is the difference between the past and the absolute past?

The events in your absolute past are those that could have directly or indirectly affected you, the observer, now. These are the events in or on the backward light cone of your present event, your here-and-now. The backward light cone of event E is the imaginary cone-shaped surface of spacetime points formed by the paths of all light rays reaching E from the past. An event's being in a point's absolute past is a feature of spacetime itself because the event is in the point's past in all possible reference frames. The feature is frame-independent. For any event in your absolute past, every observer in the universe (who isn't making an error) will agree the event happened in your past. Not so for events that are in your past but not in your absolute past. Past events not in your absolute past will be in what Eddington called your "absolute elsewhere." This is the region of spacetime containing events that are not causally connectible to your here-and-now. For example, the nearest star to the sun, Proxima Centauri, is four light-years away. Any event happening there now is in our absolute elsewhere and couldn't affect us. An explosion of that star four years ago, or earlier, could affect us now, but no later event there could have any effect on us. Your absolute elsewhere contains all present events [the events simultaneous with your here-and-now event] and also all the future events that are not in your absolute future. The absolute elsewhere is the region of spacetime that is neither in nor on either your forward or backward light cones.

A single point's absolute elsewhere, absolute future, and absolute past partition all of spacetime. Event A is in event B's absolute elsewhere if it is far enough away in distance but close enough to B in time that an unobstructed signal could not have arrived at A from B. If A is in B's absolute elsewhere the two events are also said to be "spacelike related." If the two are in each other's forward or backward light cones they are said to be "timelike related."

 

What is time dilation?

According to special relativity, a properly functioning clock moving relative to you will tick slower than your clock, assuming that measurements are made in inertial reference frames. The moving clock will show a smaller number of seconds have passed if it is used to measure the duration of the same event that your clock is used to measure. We sometimes speak of time dilation by saying time itself is 'slower' or dilated, but time isn't going slower in any absolute sense, only relative to some other frame of reference. Time doesn't actually have a rate.

Time dilation is not an illusion of perception; and it's not a matter of the second having different definitions in different reference frames. Also, it's not a Doppler effect. For example, if a flashing green light is accelerating rapidly away from you, then each pulse of light is redder and more delayed than the one sent out before it. So, the time between pulses appears to be longer than it properly is. If the flashing light were, instead, moving toward you, the effect would be reversed. The light would be bluer and the pulses closer together. However, the red-shifts and blue shifts due to the Doppler effect are not examples of time dilation. Time dilation isn't affected by the direction of motion, only by speed.

Time dilation due to difference in constant speeds is described by Einstein's special theory of relativity. The general theory of relativity describes a second kind of time dilation, one due to different accelerations and different gravitational influences. For more on general relativistic dilation, see the discussion of gravity and black holes.

Newton's physics describes duration as an absolute property, implying it is not relative to the reference frame. However, he describes the speed of light as being relative to the frame. Einstein's special theory of relativity reverses both of these aspects of time. For inertial frames, it implies the speed of light is not relative to the frame, but duration is relative to the frame. In general relativity, however, the speed of light can vary within one reference frame if matter and energy are present.

To quantitatively illustrate time dilation due to motion, consider a properly functioning clock moving with a constant velocity v in an inertial frame. The time which elapses between two ticks of its second hand is not really the one second it has when it's at rest in the frame, but is the longer time of 1/square root(1-vý/cý) seconds. The moving clock takes longer to tick. Its second lasts longer, and so we observers at rest in the frame judge the clock's ticking to be 'dilated' or spread out and thus slowed down relative to our clock. The moving clock is still accurate, though. Time really is going slower in moving inertial frames than in stationary ones.

Time dilation due to motion is relative in the sense that if your spaceship moves past mine so fast that I measure your clock to be running at half speed, then you will measure my clock to be running at half speed also, provided both of us are in inertial frames. If one of us is affected by a gravitational field or undergoes acceleration, then that person isn't in an inertial frame and the results are different.

Both types of time dilation play a significant role in time-sensitive satellite navigation systems such as the Global Positionining System. The atomic clocks on the satellites must be programmed to compensate for the relativistic dilation effects of both gravity and motion.

 

How does gravity affect time?

Einstein's general theory of relativity (1915) is a generalization of his special theory of special relativity (1905). It is not restricted to inertial frames, and it encompasses a broader range of phenomena, namely gravity and accelerated motions. According to general relativity, gravitational differences affect time by dilating it. Observers in a less intense gravitational potential find that clocks in a more intense gravitational potential run slow relative to their own clocks. People live longer in basements than in attics, all other things being equal. Basement flashlights will be shifted toward the red end of the visible spectrum compared to the flashlights in attics. This effect is known as the gravitational red shift. Even the speed of light is slower in the presence of higher gravity.

 

What happens to time near a black hole?

A black hole is a volume of very high gravitational field or severe warp in the spacetime continuum. Astrophysicists believe black holes are commonly formed by the inward collapse of stars that have burned out. The center of a spherical black hole is infinitely dense. It is surrounded by an event horizon, a concentric sphere marking the point of no return. Anything getting that close could never escape the inward pull, even if it had an unlimited fuel supply and could travel at near the speed of light. Anything crossing the event horizon from the outside would quickly crash into the center of the black hole and be crushed to a point, according to relativity theory. Because even light itself could not escape from inside a black hole, John Wheeler chose the name 'black hole'.

In relativity theory, the proper time along a world line is the time that would be shown on a clock whose path in spacetime is that world line. The proper time is not the same as the coordinate time, namely the time that would be measured for the same events along the world line by an ideal clock at the origin of the coordinate system. In fact, proper time is independent of coordinate time, although the usual convention is to measure both times in the same units, namely seconds. As judged by a clock on earth, an astronaut flying into a distant black hole will take an infinite coordinate time to reach the event horizon of the black hole. But as judged by the astronaut just outside a black hole, it will take only a few microseconds of proper time to pass the event horizon and crash into the center of the black hole.

When you fall slowly into a black hole, you'll notice that people on earth are talking faster. Their lives are speeded up, and you'll compute that their clocks are ticking too fast, assuming that yours is ticking correctly. There is no symmetry here. Earth people watching you won't say the same thing about you. Instead, they will notice that your speech is developing a slow drawl, that your emitted light is getting redder and that your clock is slowing down relative to theirs. If you were to escape the pull towards the black hole and return home, you'd discover that you were younger than your earth-bound twin and that your initially synchronized clocks showed that yours had fallen behind. It is in this sense that you've experienced a time warp, a warp in the time component of spacetime. According to the general theory of relativity, gravitation is the warping of both space and time, not just of space. The warping of time is revealed by time dilation of one frame's durations relative to another's.

No local physical quantity is singular (infinite) at the event horizon. That is, if you were to freely fall through the event horizon, you wouldn't notice anything special there about your time or the speed of light. You'd notice something very odd about distant events, though. You'd notice distant processes speeding up. You'd never notice them achieving an infinite rate, though, because you'd crash into the black hole's center before the information could reach you.

 

What is the solution to the twins paradox?

This paradox, also called the clock paradox and the twin paradox, is an argument about time dilation that uses the theory of relativity to produce a contradiction. The argument considers two twins at rest with their clocks synchronized. One twin climbs into a spaceship and flies far away at a constant speed, then reverses course and flies back at the same speed. When they reunite, will the twins still be the same age? No. Relativity theory implies that the twin on the spaceship will return and be younger than the stay-at-home twin. However, it's all relative, isn't it? That is, we could have considered the spaceship to be stationary. Wouldn't relativity theory then imply that the twin we previously called the “stay-at-home twin” would race off then return and be the younger of the two? If so, we have a contradiction: when the twins reunite, each will be younger than the other.

Almost all philosophers and scientists agree that it is not a true paradox, but merely a complex puzzle that can be adequately solved within special relativity theory. The way out of the paradox is to notice that the argument has two halves. The first half describes the twin in the spaceship flying away, then turning around and flying back to the stay-at-home twin who remains fixed in an inertial frame. The second half describes the stay-at-home twin flying away from the spaceship and then returning to the spaceship, while the spaceship remains fixed in an inertial frame. The production of the paradox depends on the assumption that these two halves of the analysis are working with two equivalent descriptions of the same process. That assumption, in turn, is based on a heuristic principle that the description of the world is equally valid from the point of view of any observer. However, that heuristic principle fails in the twins situation; the two descriptions are not equivalent.

The easiest way to see that the two situations do not have equivalent descriptions is to notice that the twin in the spaceship is accelerated whereas the stay-at-home twin feels no forces.

If the first half of the analysis is correct in assuming that the stay-at-home twin remains in an inertial frame, then the second half is incorrect in assuming that the spaceship could remain in some inertial frame. There is no inertial frame for the spaceship.

A Minkowski diagram of the situation can help one see why there is no such frame. Of course one could fix the spaceship in a non-inertial frame. This would complicate the analysis, but the result would be the same, namely no contradiction.

Can’t we start by assuming that the spaceship is at rest in an inertial frame? Yes, one could start the argument toward a paradox by assuming that there is an inertial frame for the spaceship, and one could try to re-describe the twins paradox by beginning with what we've called the second half. If so, then the twin we called the “stay-at-home twin” would fly away and return younger than the spaceship twin. But if so, then the first half of the analysis would now fail because it needs to assume that the stay-at-home twin remained fixed in some inertial frame. There is no such frame. So, it can’t be shown that the spaceship twin is the younger.

The production of the paradox depends upon a careless use of the heuristic principle that the description of the world is equally valid from the point of view of any observer. This principle is misinterpreted in the twins paradox. What is always correct in relativity theory and what underlies the heuristic principle is the symmetry principle: the invariance of the laws under Lorentz transformations. These transformation equations give the relations between the coordinates of a single event [such as our spaceship’s flight away from the stay-at-home twin] as measured by observers in two different inertial reference frames in motion relative to one another. But there aren’t two inertial frames to use in the case of the twins paradox, so the symmetry principle is correct, but the heuristic principle is not applicable. The argument of the twins paradox applies the heuristic principle anyway, and thus draws an incorrect conclusion that there is a contradiction.

It was remarked above that the easiest way to see the dissimilarity in the two halves of the argument is to notice the forces on the spaceship. The asymmetry in the two halves is indeed due to the accelerations involved, but the age difference in the twins is not due to any accelerations. To appreciate why, it is helpful to use a version of the twins paradox created by Halsbury, as reported by Bondi in 1957. Halsbury created a version of the paradox using three clocks, none of which undergoes any acceleration. The three move inertially; that is, they are in free fall. Clock 1 is at rest in an inertial frame. Clock 2 moves rapidly past clock 1. When they meet, the clocks are synchronized. This point in time is analogous to the time at which one twin enters the spaceship. After clock 2 is far away from clock 1, it passes clock 3 which is moving at the same speed in the opposite direction back toward clock 1. When clocks 2 and 3 pass, the reading on clock 3 is synchronized with the reading on clock 2. When clock 3 eventually meets clock 1, will 1 and 3 show the same time? No, the reading on the stay-at-home clock 1 must be greater, due to the time dilation involved. Why is clock 3 retarded relative to the stay-at-home clock 1? It’s not the accelerations involved. It’s not any force on the clocks. Ultimately, this question we are asking isn’t a good question if it is presumed that there must be some “cause” for time dilation involving a force. Instead, it is simply because the clocks have different paths in spacetime and thereby have different proper times for the events they undergo.

 

What is the solution to Zeno's paradoxes?

In about 445 B.C., the Greek philosopher Zeno of Elea offered several arguments that led to conclusions contradicting what we all know from our physical experience. The paradoxes had a dramatic impact upon the later development of mathematics, science, and philosophy. His most familiar paradox, the paradox of Achilles and the Tortoise, involves the fast-running Achilles and the slow-crawling tortoise. The tortoise has a head start. If Achilles hopes to overtake the tortoise, he must at least run to where the tortoise is, but by the time he arrives there, the tortoise has crawled to a new place. So, Achilles must run to the new place; but of course the tortoise isn't there, having crawled on to yet another place, and so on forever. Therefore, Zeno argues, good reasoning shows that fast runners never can catch slow ones. So much the worse for our belief that motion really occurs, he says in defense of his mentor Parmenides who had argued that change is an illusion.

In his Progressive Dichotomy Paradox, Zeno argued that a runner will never reach the goal line because he first must have time to reach the halfway point to the goal, but after arriving there he will need time to get to the 3/4 point, then the 7/8 point, and so forth. If the distance to the goal is, say, 1 meter, then the runner must cover a distance of 1/2 + 1/4 + 1/8 + ... meters. Zeno believed this sum is infinite and concluded that the runner will never have the infinite time it takes to reach this infinitely distant goal. Because at any time there is always more time needed, motion can never be completed. Worse yet, argued Zeno in his Regressive Dichotomy Paradox, the runner can't even take a first step. Any first step may be divided into a first half and a second half. Before taking a full step, the runner must have time to take a 1/2 step, but before that a 1/4 step, and so forth. The runner will need an infinite amount of time just to take a first step, and so will never get going.

Zeno's Arrow Paradox takes a different approach to challenging the coherence of the concepts of time and motion. Consider one instant of an arrow's flight. For that entire instant the arrow occupies a region of space equal to its total length, so at that instant the arrow isn't moving, he reasoned. If at every instant the arrow isn't moving, then the arrow can't move.

Yet another paradox created by Zeno attacks the notion that there are shorter and shorter times. Consider a duration of one second. It can be divided into two non-overlapping parts. They, in turn, can be divided, and so on. At the end of this infinite division we reach the elements. Here there is a problem. If these elements have zero duration, then adding an infinity of zeros yields a zero sum, and the total duration is zero seconds, which is absurd. Alternatively, if that infinite division produced elements having a finite duration, then adding an infinite number of these together will produce an infinite duration, which is also absurd. So, a second lasts either for no time at all or else for an infinite amount of time.

These paradoxes by Zeno can be considered to challenge the notion that time and space are continuous. Some of his other paradoxes, not discussed here, challenge the presumption that time might be discrete or discontinuous. Today we consider that the moral of his paradoxes is not that motion is impossible, but rather that his reasoning reveals an underlying incoherence in ancient Greek thought, an incoherence that was not adequately resolved for 2,300 years.

The way out of Zeno's paradoxes requires revising the concepts of duration, distance, instantaneous speed, and sum of a series. The relevant revisions were made by Leibniz, Newton, Cauchy, Weierstrass, Dedekind, Cantor, and Lebesque over two centuries. The notion of infinite sums of numbers had to be revised so that an infinite series of numbers that decrease sufficiently rapidly can have a finite sum. Although 1/2 + 1/3 + 1/4 +... is infinite, the more rapidly decreasing sum 1/2 + 1/4 + 1/8 +... is 1. Second, mathematics decided they needed to allow infinitely sized sets to be of different sizes, such as the countable infinity of integers and the uncountable infinity of real numbers. A third key idea was to appreciate that durations and distances must be topologically like an interval of the linear continuum, a dense ordering of uncountably many points. Although individual points of the continuum have zero measure (that is, zero 'total length'), the modern notion of measure on the linear continuum does not allow the measure of a segment (continuous region) to be the sum of the measures of its individual points, as Zeno had assumed in his argument against plurality.

With these contemporary concepts, we can now make sense of Achilles covering an infinite number of distances in a finite time while running at a normal, finite speed. The new concepts restore the coherence of mathematics and science with our experience of space and time, and they are behind today's declaration that Zeno's arguments are based on naive and false assumptions.

Unfortunately, the solution has engendered new oddities within mathematics. Not only must we live with there being two infinite sets of different sizes and with space-filling curves, but the Banach-Tarski Theorem of 1924 implies that that we can divide a sphere E, into a finite number of disjoint parts and then rearrange them into two new spheres F and G such that E, F and G are congruent to each other. 2 from 1, in other words! This result does not reveal an actual inconsistency in mathematics, but it does shock our intuitions. Luckily, because the pieces F and G are not Lebesgue measurable, the pieces cannot be physically created by any physicist and so don't produce an obvious contradiction when applied to reality, unlike the case with Zeno's paradoxes.

 

How do time coordinates get assigned to points of spacetime?

A reference system is a reference frame plus either a coordinate system or an atlas of coordinate systems placed by the analyst upon the space to uniquely name the points. These names or coordinates are frame dependent in that a point can get new coordinates when the reference frame is changed. For 4-d spacetime, a coordinate system is a grid of smooth timelike and spacelike curves on the spacetime that assigns each point three space coordinate numbers and one time coordinate number. Inertial frames can have global coordinate systems, but if we are working with general relativity where we cannot assume inertial frames, then the best we can do is to assign a coordinate system to a small region of spacetime where the laws of special relativity hold to a good approximation. General relativity requires special relativity to hold locally, and thus for spacetime to be Euclidean locally. So spacetime allows coordinate systems locally. Consider two coordinate systems on adjacent regions. For adjacent regions we make sure that the 'edges' of the two coordinate systems match up in the sense that each point near the intersection of the two coordinate systems gets a unique set of four coordinates and that nearby points get nearby coordinate numbers. The result is an "atlas" on spacetime.

For small regions of spacetime, we create a coordinate system by choosing a style of grid, say rectangular coordinates, fixing a point as being the origin, selecting one timelike and three spacelike lines to be the axes, and defining a unit of distance for each dimension. We cannot use letters for coordinates. The alphabet's structure is too simple. Integers won't do, either; but real numbers are adequate to the task. The definition of 'coordinate system' requires us to assign our real numbers in such a way that numerical betweenness among the coordinate numbers reflects the betweenness relation among points. For example, if we assign numbers 17, pi, and 101.3 to instants, then every interval of time that contains the pi instant and the 101.3 instant had better contain the 17 instant. There is no way to select one point of spacetime and call it the origin of the coordinate system except by reference to actual events. In practice, we make the origin be the location of a special event, such as the birth of Jesus, or a selected tick of our atomic clock in Greenwich, England.

The choice of the unit presupposes we have defined what 'distance' means. The metric for a space specifies what is meant by distance in that space. The natural metric between any two points in a one-dimensional space, such as the time sub-space of our spacetime, is the numerical difference between the coordinates of the two points. Using this metric, the duration between the 11:00 instant and the 11:05 instant is five minutes. The metric for spacetime defines the spacetime interval between two spacetime locations, and it is more complicated than the metric for time alone. The spacetime interval between any two events is invariant or unchanged by a change to any other reference frame, although the spatial distances and durations do vary. More accurately, in the general theory, the infinitesimal spacetime interval between two neighboring points is invariant. The units of the spacetime interval are seconds squared.

Philosophers dispute the extent to which the choice of metric is conventional rather than forced by nature. Taking the conventional side, Adolf Grunbaum argues that time is metrically amorphous. It has no intrinsic metric in the sense of its structure determining the measure of durations. Instead, we analysts establish durations between instants by the way we assign coordinates to instants. If we were to say the instant at which Jesus was born and the instant at which Abraham Lincoln was assassinated occurred only 24 seconds apart, whereas the duration between Lincoln's assassination and his burial is 24 billion seconds, then we can't be mistaken. It's up to us to say what is correct when we first create our conventions about measuring duration. We can consistently assign any numerical time coordinates we wish, subject only to the condition that the assignment properly reflect the betweenness relations of the events that occur at those instants. That is, if event J (birth of Jesus) occurs before event L (Lincoln's assassination) and this in turn occurs before event B (burial of Lincoln), then the time assigned to J must be numerically less than the time assigned to L, and both must be less than the time assigned to B. t(J) < t(L) < t(B). A simple requirement. It is other requirements that lead us to reject the above convention about 24 seconds and 24 billion seconds as unhelpful. What requirements? We've found that, for doing science, certain processes are more 'regular' than others. Pendulum swings are more regular than repeated barks of a dog. Periodic appearances of the sun overhead are more regular than rainstorms. A good convention for what is regular will make it easier for scientists to explain what causes other events to be irregular. It is the search for regularity that leads us to adopt the conventions for numerical time coordinate assignments that we do.

In this discussion, there is no need to worry about the distinction between change in metric and change in coordinates. For a space that is topologically equivalent to the real line and for metrics that are consistent with that topology, each coordinate system determines a metric and each metric determines a coordinate system. More precisely, once you decide on a positive direction in the one-dimensional space and a zero-point for the coordinates, then the possible coordinate systems and the possible metrics are in one-to-one correspondence.

There are still other restrictions on the assignments of coordinate numbers. The restriction that we called the "conventionality of simultaneity" fixes what time-slices of spacetime can be counted as collections of simultaneous events. An even more complicated restriction is that coordinate assignments satisfy the demands of general relativity. The metric of spacetime is not global but varies from place to place due to the presence of matter and gravitation. Spacetime cannot be given its coordinate numbers without our knowing the distribution of matter and energy. However, for very small regions of spacetime, the general relativistic metric tensor reduces to the metric for special relativistic spacetime.

 

How do dates get assigned to actual events?

Our purpose in choosing a coordinate system or atlas to assign real numbers to all spacetime points is to express relationships among actual and possible events. The relationships we are interested in are order relationships (Did this event occur between those two?) and magnitude relationships (How long after A did B occur?). The date of a (point) event is the time coordinate number of the spacetime location where the event occurs. We expect all these assignments of dates to events to satisfy the requirement that event A happens before event B iff t(A) < t(B), where t(A) is the time coordinate of A. The assignments of dates to events also must satisfy the demands of our physical theories, and in this case we face serious problems involving inconsistency as when a geologist gives one date for the birth of earth and an astronomer gives a different date.

It is a big step from assigning numbers to points to assigning them to real events. Here are some of the questions that need answers. How do we determine whether a nearby event and a distant event occurred simultaneously? How do we operationally define the second so we can measure whether one event occurred exactly one second later than another event? How do we know whether the clock we have is accurate? Attention must also be paid to the dependency of dates due to shifting from Standard Time to Daylight Savings Time, to crossing the International Date Line, and to switching from the Julian to the Gregorian Calendar.

Let's design a coordinate system. Suppose we have already assigned a date of zero to the event that we choose to be at the origin of our coordinate system. To assign dates to other events, we first must define a standard clock and declare that the time intervals between any two consecutive ticks of that clock are the same. The second will be defined to be so many ticks of the standard clock. We then synchronize other clocks with the standard clock so the clocks show equal readings at the same time. The time at which a point event occurs is the number reading on the clock at rest there. If there is no clock there, the assignment process is more complicated.

We want to use clocks to assign a time even to distant events, not just to events in the immediate vicinity of the clock. To do this correctly requires some appreciation of Einstein's theory of relativity. A major difficulty is that two nearby synchronized clocks, namely clocks that have been calibrated and set to show the same time when they are next to each other, will not in general stay synchronized if one is transported somewhere else. If they undergo the same motions and gravitational influences, they will stay synchronized; otherwise, they won't. For more on how to assign dates to distant events, see the discussion of the relativity and conventionality of simultaneity.

As a practical matter, dates are assigned to events in a wide variety of ways. The date of the birth of the Sun is measured very differently from dates assigned to two successive crests of a light wave. For example, there are lasers whose successive crests of visible light waves pass by a given location every 10 to the minus 15 seconds. This short time isn't measured with a stopwatch. It is computed from measurements of the light's wavelength. We rely on electromagnetic theory for the equation connecting the periodic time of the wave to its wavelength and speed. Dates for other kinds of events also are often computed rather than directly measured with a clock.

 

What is essential to being a clock?

Clocks record numerical information about time. They measure the quantity of time, the duration. Every clock has two parts: a part that generates a sequence of regular ticks such as swings of a pendulum and a part that counts these ticks and converts them into a measurement in seconds and minutes and hours and years. In an atomic clock, the ticks are oscillations of laser light. In a light clock the ticks are elapsed distances covered by a light wave. A principal goal in clock-building is to make each tick last the same duration as any other tick. When this goal is achieved, the clock is said to tick uniformly, or regularly. If the counted periods or processes have the same duration as the standard clock, then the clock is said to be synchronized with the standard clock. A second principal goal is to count the ticks accurately, a serious problem when the ticks are occurring a trillion times a second.

To calibrate a clock, that is, to synchronize it with the standard clock, we want our clock to show that it is time t just when the standard clock shows that it is time t, for all t. Another goal in clock building is to ensure there is no difficulty in telling which clock tick is simultaneous with which event that occurs in the immediate vicinity of the clock, so that the clock is ready to report the time of an event in its immediate vicinity. An event external to the clock is assigned the same time number as the internal tick that is simultaneous with the external event.

Because we do have problems of determining simultaneity for distant events, when we calibrate a clock, we prefer to place it in the immediate vicinity of the standard clock. A clock isn't really measuring the time between two events in any reference frame other than one fixed to the clock. In other words, a clock measures the elapsed proper time between events that occur along its own world line.

Usually in the discussions in this article we have assumed that a clock is very small, that it can count any part of a second and that it can count high enough to be a calendar, although this is not the case with real clocks. Despite the practical problems, there is the problem of there being a physical limit to the shortest duration measurable by a given clock because a clock cannot measure time more accurately than the time it takes light to travel between the components of that clock, the components in the part that generates a sequence of regular ticks.

 

What is our standard clock? 

By current convention, the standard clock is the clock we agree to use for defining the standard second. The current standard second is defined to be the duration of 9,192,631,770 periods (cycles, oscillations, vibrations) of a certain kind of microwave radiation in the standard clock. More specifically, the second is defined to be the duration of 9,192,631,770 periods of the microwave radiation required to produce the maximum fluorescence of cesium 133 atoms (that is, their radiating a specific color of light) as the atoms make a transition between two specific hyperfine energy levels of the ground state of the atoms.

Atoms of cesium with a uniform energy are sent through a chamber that is being irradiated with these microwaves. The frequency of these microwaves is tuned until the maximum number of cesium atoms flip from one energy to the other, showing that the microwave radiation frequency is now precisely tuned to be 9,192,631,770 vibrations per second. Because this frequency for maximum fluorescence is so stable from one experiment to the next, the vibration number is accurate to so many significant digits. The National Institute of Standards and Technology's F-1 atomic fountain clock, which was adopted in late 1999 as the primary time standard of the United States, is so accurate that it drifts by less than one second every 20 million years.

The standard clock is used to fix the units of all lengths. The unit of length depends on the unit of time. The meter depends on the second. It does not follow from this, though, that time is more basic than space. All that follows is that time measurement is more basic than space measurement. And this has to do with convention and with the fact that current science is capable of measuring time more precisely than space.

The meter is defined in terms of the pre-defined second as being the distance light travels in exactly 0.000000003335640952 seconds or 1/299,792,458 seconds. That number is picked so that the new meter will be nearly the same distance as the old meter, which was the distance between two marks on a platinum bar that was kept in the Paris Observatory.

These standard definitions of the second and the meter amount to defining or fixing the speed of light in a vacuum in all inertial frames. The speed is exactly one meter per 0.000000003335640952 seconds or 299,792,458 meters per second, or approximately 186,282 miles per second or about a foot per nanosecond. There can no longer be any direct measurement to see if that is how fast light REALLY moves in an inertial frame; it is simply defined to be moving that fast. Any measurement that produced a different value for the speed of light would be presumed initially to have an error in, say, its measurements of lengths and durations, or in its assumptions about the influence of gravitation and acceleration, or in its assumption that the light was moving in a vacuum. This initial presumption comes from a deep reliance by scientists on Einstein's theory of relativity. However, if it were eventually decided by the community of scientists that the theory of relativity is incorrect and that the speed of light shouldn't have been fixed as it was, then the scientists would call for a new world convention to re-define the second. Some physicists believe that a better system of units would first define the speed of light, then define the second, and then make the meter be a computed consequence of these.

 

Why are some standard clocks better than others?

We choose as our standard clock our best clock, the one with the least drift, the one with the most regularity in its period. Other clocks ideally are calibrated by being synchronized to this clock. A practical goal in selecting a standard clock is to find a clock that is relatively insulated from environmanetal impact such as stray electric fields or the presence of dust. The principal theoretical goal is to find a periodic (cyclic) process that, if adopted as our standard, makes the resulting system of physical laws much simpler and more useful than if we were to have chosen some alternative periodic process such as the periodic dripping of water from our goat skin bag or even the revolution of the earth about the Sun.

The standard clock was once defined astronomically in terms of the revolution of the earth. The second was defined to be 1/86,400 of the mean solar day, the average rotational period of the earth with respect to the Sun. Now we've found a better standard clock, an atomic clock. All atomic clocks measure time in terms of the natural resonant frequencies of various atoms and molecules. The periodic behavior of a super-cooled cesium atomic clock is the best practical standard clock we have so far discovered.

Why is choosing it better than choosing an astronomical process such as the yearly motion of the earth around the Sun? The brief answer is that the sloshing of the tides, among other things, is affecting the rotation of the earth so that by sticking to the earth clock we have trouble accounting for simultaneous accelerations and retardations of the orbital motions of the other planets and the simultaneous accelerations and retardations of atomic motions such as those in cesium-133 atoms. Our atomic theory says that these atomic processes should behave uniformly as time goes on, so sticking to the earth clock forces awkward changes in our atomic theory. On the other hand, by switching to the cesium atomic standard, these alterations are unnecessary, and we can readily explain the non-uniform wobbling of the earth's yearly revolutions by reference to the tides on the earth, the gravitational pull of other planets, dust between planets, and collisions with comets. These influences affecting a solar clock do not affect the cycles of the cesium atom. One other advantage of the cesium clock is that it provides a standard that is reproducible anywhere in the universe where there is cesium, and the behavior of the cesium atom is relatively isolated from other processes such as a comet bombarding the earth.

However, in order to keep our atomic-based calendar in synchrony with the rotations and revolutions of the earth, say, to keep atomic-noons occurring on astronomical-noons and ultimately to keep Northern hemisphere winters from occurring in some future July, we systematically add leap years and leap seconds in the counting process. These changes don't affect the duration of a second, but they do affect the duration of a year because, with leap years, not all years last the same number of seconds.

We are lucky to live in a universe having a large number of different processes that bear consistent time relations or frequency of occurrence relations to each other. For example, the frequency of a fixed-length pendulum is a constant multiple of the half life of a specific radioactive uranium isotope; the relationship doesn't change as time goes by (at least not much and not for a long time). The existence of these sorts of relationships makes our system of physical laws much simpler than it otherwise would be, and it makes us more confident that there is something we are referring to with the time-variable in those laws.


Suggestions for Further Reading:

Bondi, Hermann. “The Space Traveler’s Youth,” Discovery, Dec. 1957, pp. 505-510.

This popular article includes a description of the Halsbury version of the twins paradox with all clocks in inertial frames.

Davies, Paul. About Time: Einstein's Unfinished Revolution, Simon & Schuster, 1995.

An easy to read survey of the impact of the theory of relativity on our understanding of time.

Gell-Mann, Murray. The Quark and the Jaguar, W. H. Freeman and Co., 1994.

A Nobel Prize winning physicist discusses time's arrow in chapter 15.

Hawking, Stephen. A Brief History of Time, Updated and Expanded Tenth Anniversary Edition, Bantam Books, 1996.

A leading theoretical physicist provides introductory chapters on space and time, black holes, the origin and fate of the universe, the arrow of time, and time travel.

Horwich, Paul. Asymmetries in Time, The MIT Press, 1987.

A monograph that relates the central problems of time to other problems in metaphysics, philosophy of science, philosophy of language and philosophy of action.

Price, Huw.Time's Arrow and Archimedes' Point, Oxford University Press, 1996.

This technical monograph adopts the block universe view and argues that physicists have failed to achieve the Archimedean standpoint that is required for a proper understanding of time's asymmetry.

Prior, A. N. Past, Present and Future, Oxford University Press, 1967.

A pioneering work in temporal logic, the symbolic logic of time.

Prior, A. N. "The Notion of the Present," Studium Generale, volume 23, 1970, pp. 245-8.

A brief defense of presentism, the view that the past and the future aren't real.

Sciama, Dennis. "Time 'Paradoxes' in Relativity," in The Nature of Time edited by Raymond Flood and Michael Lockwood, Basil Blackwell, 1986, pp. 6-21.

A good account of the twins paradox or clock paradox.

Van Fraassen, Bas C. An Introduction to the Philosophy of Time and Space, Columbia University Press, 1985.

An advanced undergraduate textbook by an important philosopher of science.

Whitrow. G. J. The Natural Philosophy of Time, Second Edition, Clarendon Press, 1980.

A broad survey of the topic of time and its role in physics, biology, and psychology. Pitched at a higher level than the Davies book.

 Author Information:

Bradley Dowden
Email: mailto:dowden@csus.edu?subject=Your Time Article
California State University Sacramento

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