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8. Does the Theory of Relativity Imply Time Is Partly Space?

In 1908, 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.

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 at, say, the event of Mohammed’s birth, you may point the x-axis east or toward Mecca or away from the center of Earth, but you may not point it forward in time–you may do that only with the t-axis, the time axis.

9. Is Time the Fourth Dimension?

Yes and no; it depends on what you are talking about. Time is the fourth dimension of 4-d 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, although in 1895 H. G. Wells treated time as a fourth dimension in his novel The time Machine. Spacetime is represented mathematically by Minkowski as a space of events, not as a space of ordinary geographical places.

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 points on a plane could be specified with just one number.  If so, then the line and plane must have the same dimensions according to the Riemann definition. To avoid this problem and to keep the plane being a 2-d object, the notion of dimensionality of a space has been given a new, but rather complex, definition.

10. 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.

11. 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.

Isn’t time relative to the observer’s coordinate system in the sense that in some reference frames there could be fifty-nine seconds in a minute? No, due to scientific convention, it is absolutely certain that there are sixty seconds in any minute in any reference frame. How long an event lasts is relative to the reference frame used to measure the time elapsed, but in any reference frame there are exactly sixty seconds in a minute because this is true by definition. Similarly, you do not need to worry that in some reference frame there might be two gallons in a quart.

In a deeper sense, time is relative, not just to the coordinate system, but to the reference frame itself. 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. Einstein illustrated his idea for two observers, one on a moving train in the middle of the train, and a second observer standing on the embankment next to the train tracks. If the observer sitting in the middle of the rapidly moving train receives signals simultaneously from lightning flashes at the front and back of the train, then in his reference frame the two lightning strikes were simultaneous. But the strikes were not simultaneous in a frame fixed to an observer on the ground. This outside observer will say that the flash from the back had farther to travel because the observer on the train was moving away from the flash. If one flash had farther to travel, then it must have left before the other one, assuming that both flashes moved at the same speed. Therefore, the lightning struck the back of the train before the lightning struck the front of the train in the reference frame fixed to the tracks.

Let’s assume that a number of observers are moving with various constant speeds in various directions. Consider the inertial frame of reference in which each observer is at rest in his or her own frame. Which of these observers will agree on their time measurements? Only 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 occurring at time zero (the origin of the time axis). If 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.

In some reference frame, 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, that is, only if one event is in the absolute elsewhere of the other. Despite the relativity of time to a reference frame, any two observers in any two reference frames should agree about which of two causally connectible events happened first.

12. What Are the Relativity and Conventionality of Simultaneity?

Leibniz suggested that simultaneous events are those that in principle could not have causally affected each other. The suggestion is unacceptable in relativity theory which implies that causal influence moves no faster than the speed of light and that simultaneity is relative to reference frame. If the universe obeys relativistic physics, then events that occur simultaneously with respect to one reference frame will not occur simultaneously in another reference frame that is moving with respect to the first frame. This is called the “relativity of simultaneity.”  It applies only to pairs of events in each other’s absolute elsewhere because, if the two aren’t in each other’s absolute elsewhere, then there is a reference-frame-free answer to which of the two occurred first.

…  Because Lorentz’s timeline is a straight line we can tell that he is moving at a constant speed.  The two flashes of light arrive at Einstein’s location simultaneously, creating spacetime event B.  However, Lorentz sees flash 2 before flash 1.  That is, the event A of Lorentz seeing flash 2 occurs before event C of Lorentz seeing flash 1.  So, Einstein will readily say the flashes are simultaneous, but Lorentz will have to do some computing to figure out that the flashes are simultaneous in the frame because they won’t “look” simultaneous.  However, if we’d chosen a different reference frame from the one above, one in which Lorentz is not moving but Einstein is, then Lorentz would be correct to say flash 2 occurs before flash 1 in that new frame.  So, whether the flashes are or are not simultaneous depends on which reference frame is used in making the judgment.  It’s all relative.

This relativity of simultaneity is philosophically less controversial than 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, especially if they are very far apart. One 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. Instead of using the midway method, we could take the distant clock and send a signal home to our master clock, one already synchronized with our standard clock; the master clock immediately sends a signal back to the distant clock with the information about what time it was when the signal arrived. We at the distant clock notice that the total travel time is t and that the master clock’s signal says its time is, say, noon, so we immediately set our clock to be noon plus half of t.

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 Grünbaum 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, Michael Friedman, and Graham Nerlich object 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. They 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.

13. 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 absolutely past events are the events in or on the backward light cone of your present event, your here-and-now. The backward light cone of event Q is the imaginary cone-shaped surface of spacetime points formed by the paths of all light rays reaching Q from the past. An event’s being in another event’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” and these past events will be in your present as judged by some other reference frames.  The absolute elsewhere is the region of spacetime containing events that are not causally connectible to your here-and-now. Your absolute elsewhere is the region of spacetime that is neither in nor on either your forward or backward light cones.  No event here now, can affect any event in your absolute elsewhere; and no event in your absolute elsewhere can affect you here and now.  A spacetime point’s absolute future is all the future events outside the point’s absolute elsewhere.

A single point’s absolute elsewhere, absolute future, and absolute past partition all of spacetime beyond the point into three disjoint regions.  If point A is in point B’s absolute elsewhere, the two events are 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” or “causally connectible.”

14. What is Time Dilation?

According to special relativity, two properly functioning clocks next to each will stay synchronized.  Even if they were to be far away from each other, they’d stay synchronized.  But if one clock moves away from the other, the moving clock will tick slower than the stationary clock, as measured in the inertial reference frame of the stationary clock.  This slowing due to motion is called “time dilation.”  If you move at 99% of the speed of light, then your time slows by a factor of 7 relative to stationary clocks. In addition, you are 7 times thinner than when you are stationary, and you are 7 times heavier. If you move at 99.9%, then you slow by a factor of 22.

Time dilation is about clocks in different frames disagreeing with each other.  Suppose your twin’s spaceship travels to and from a star one light year away. It takes light from your Earth-based flashlight two years to go there and back. But if the spaceship is fast, your twin can make the trip in less than two years, according to his own clock.  Does he travel the distance in less time than it takes light to travel that distance?  No, according to yourclock he takes more than two years, and so is slower than light.

We sometimes speak of time dilation by saying time itself is “slower,” but time isn’t going slower in any absolute sense, only relative to some other frame of reference. Does time have a rate?  Well, time in a reference frame has no rate in that frame, but time in a reference frame can have a rate as measured in a different frame, such as in a frame is moving relative to the first frame.

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. Time dilation isn’t affected by direction of motion.  The Doppler effect is affected by direction of motion, which we detect in the difference between a blue shift and a red shift.

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, in Newton’s physics the speed of light is 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.

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 Positioning System. The atomic clocks on the satellites must be programmed to compensate for the relativistic dilation effects of both gravity and motion.