In Defense of a Three-Dimensional Spacetime

Part Two: Special relativity

At last it came to me that time was suspect! ‐ Albert Einstein (1879-1955)

Ten years after The Time Machine, Einstein's first paper on relativity was published, changing our understanding of the relationship of time and space, particularly at high speeds. Its essence is the declaration that all observers will measure light to have the same speed; the consequences of that fact are many, including that nothing can go faster than light and that observers may differ in their measurements of time, in a way similar to the way Galileo demonstrated that observers in various states of motion may differ in their measurements of distance. As it dealt with only the special case of uniform, unaccelerated motion, this part of Einsteinís work is known as "special relativity." Relativity directly contradicts Wells' concept of time, but has often been presented in ways which may seem to the lay person to argue in favor of something like Wells spacetime. Indeed, when surveying modern physics literature, a better-educated lay person may wonder to what extent Wells spacetime has kept an undue hold even on the minds of educators. Even the experts appear to slip into this mindset from time to time.

Einstein's concept of the relationship of space and time has come to be represented geometrically by Minkowski spacetime, named after the mathematician who proposed in 1907 that Einsteinís relativity be understood in the context of a four-dimensional space. Einstein was at first unimpressed and "dismissed Minkowskiís formulation as excessively pedantic, joking that he scarcely recognized his own theory once the mathematicians got hold of it." (Ferris, p. 197) Nevertheless, Hermann Minkowski's insight became key to Einstein's effort to generalize relativity to include gravity and accelerated motion. Building on the foundation of curved geometries established by Riemann and others, Einstein endeavored "to assign the fourth dimension to time and make the whole, complicated affair come out right." (Ferris, pp. 199-200) Einsteinís "general relativity" was completed in November 1915 and published the following spring.

For those who understand it, special relativity destroys the notion of a universal time which is agreed upon by all observers in all locations. The whole of creation does not tick forward in unison in the way a movie progresses frame by frame through time, and thus it cannot be rewound and replayed. Due to Einstein's work, we now realize that events that are too widely separated to have any causal relationship cannot be meaningfully said to have any universal order in time. Different observers moving at different speeds and in different directions may measure such events to occur simultaneously or one after another, and in different orders in time; and this is perfectly harmless because neither event is capable of affecting the other. Though measurements of space and time may differ between observers, we see that one quantity remains constant no matter who is doing the measuring; this quantity is a combination of distance and time measurements between two events and is called the spacetime interval. In its simplest form, it looks like this (in all that follows, let us assume that units of time and space are chosen to make the speed of light equal to one, thus eliminating the need for the factor "c"):

Equation 1. s2 = d2 - t2

This formula defines the interval s between any two events in terms of the spatial distance d measured between where the two events occur, and the elapsed time t between when they are measured to happen. It is a simple yet deeply meaningful equation and any student who finds it in a physics text will likely react with great enthusiasm after having been thoroughly intimidated by the much more complex Lorentz transformation formulas (one of which is presented here without further explanation for the mere sake of cruelty):

t = t'(1-v2/c2)-1/2

In science texts, the distance d of the spacetime interval is often broken down into its components in three dimensions like this:

Equation 2. s2 = x2 + y2 + z2 ‐ t2

An attentive geometry student may notice that this formula looks somewhat similar to the Pythagorean theorem, and indeed some texts will point out the similarity. The Pythagorean theorem is one of the rules of Euclidean geometry and it relates the lengths of the sides of a right triangle this way:

Equation 3. a2 + b2 = c2

200px-Pythagorean.svg

Illustration: The Pythagorean theorem states that the sum of the areas of squares having sides a and b will be equal to the area of the square with side c, if and only if a and b form a right angle. Among other things, the theorem is a carpenterís best friend, also known as the "3-4-5" rule. To check whether an angle is truly square, a carpenter can mark off three feet on one side and four on the other, then measure whether the diagonal is five feet long. 3*3 + 4*4 = 5*5.

In English, this means that the diagonal side of the right triangle times itself is equal to the sum of the other two sides times themselves. In general, the generic variables x and y are commonly used to represent two dimensions of space, and the distance between them may be represented by the letter s:

Equation 4. s2 = x2 + y2

The relevance of this theorem to our current discussion is that it can be used to find the distance between any two points on a flat surface. And there is beauty in the fact that no matter which direction the measurer is facing ‐ whether the side-to-side x of one personís measurement is the front-to-back y of someone elseís ‐ all measurers will agree on the distance s. In this way, the theorem provides a consistent metric for all orientations in three-dimensional space in the same sense that the spacetime interval above provides a consistent, agreed-upon value in Minkowski spacetime. All spaces need a reliable measuring-stick, or metric, though one spaceís metric may not be valid in anotherís.

If we look at this consistency in reverse, starting with a fixed distance from a single point on a plane and seeking possible locations for a second point in every possible direction, these locations together form a circle. The mathematical formula for any circle centered on the point (x,y) = (0,0) is of the same form as Equation 4 above. The value s is the radius of the circle, and the circle itself represents all possible combinations of x and y values which result in that radius.

Unit_circle3

A circle of radius 1 which satisfies the equation 12 = x2 + y2

The Pythagorean theorem can be extended to three dimensions to show how, for instance, the height a, width b and depth c of a box are added to give a single diagonal length d between opposite corners:

Equation 5. d2 = a2 + b2 + c2

Pythagoras_3D

Or, using more standard variables, we have:

Equation 6. s2 = x2 + y2 + z2

This presentation of the Pythagorean theorem looks curiously like the formula for the spacetime interval in Equation 2 above. If we just add a term "-t2" to the Pythagorean theorem for time, we have the spacetime interval. There is even a mathematical operation that can change the minus sign for the t2 term in the spacetime interval to a plus sign, resulting in all terms being added rather than subtracted, bringing all terms into compliance with the Pythagorean formula. We may replace t with t times the square root of -1, also known as the imaginary number i, to obtain this:

Equation 7. s2 = x2 + y2 + z2 + (i*t)2

Seeing all four of these right-hand-side terms in four-dimensional symmetry, it is tempting to suppose that Wells spacetime is therefore supported by relativity; that time is a fourth physical dimension at right angles to space. I fell into this trap for a time, and struggled to find my way out. The textbook for the relativity course I enrolled in was unhelpful in this regard. When discussing gravity and general relativity, the authors wrote that "the presence of a large body of mass causes spacetime to warp in the region near it so that spacetime becomes non-Euclidean." (Resnick and Halliday, p. 296) From this, one might infer that, by contrast, an empty region of spacetime would be Euclidean and four-dimensional.

Einstein's own description of relativity for lay audiences would likewise have done little to liberate me from my Wellsian preconceptions. In Relativity - the Special and General Theory, Einstein does make note of the difference between Pythagorean addition of dimensions on one hand and the form of the spacetime interval on the other. As he explains it, that pesky little minus sign (or the number i which makes the minus sign go away) is a matter of giving "due prominence" to the relationship between Minkowski spacetime and Euclidean space, a relationship which he calls "pronounced." In this form, time "enters into natural laws in the same form as the space coordinates;" and thus "We can regard Minkowskiís Ďworldí in a formal manner as a four-dimensional Euclidean space (with imaginary time coordinate)." He writes of the Lorentz transformations ‐ formulas that enable us to translate space and time coordinates between observers in relative motion ‐ as corresponding to a "rotation of the coordinate system in the four-dimensional world." (Hawking 2007, pp. 72,223-224) Supporting that idea, one text on relativity uses an imaginary replacement for time to show how the Lorentz transformations may be derived from rotations in a Cartesian (Euclidean) system. (Dalarsson and Dalarsson pp. 116-119)

This same perspective is reinforced by Stephen Hawking, whose popular-science book A Brief History of Time sold over 10 million copies. In it, Hawking points out that though the orbits of the planets are elliptical in three dimensions, according to general relativity these and other freely-falling bodies "always follow straight lines in four-dimensional spacetime." (p. 30) Introducing the idea of "imaginary time," he writes that for some quantum mechanical applications, "one must measure time using imaginary numbers, rather than real ones. This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean[.]" (p. 134)

We are thus given the impression that physical reality is to be considered a four-dimensional Euclidean space, and the differences between what two observers may measure are merely differences in perspective due to their vantage point in this four-dimensional reality. In the sources quoted above, the distinction between "time" and "imaginary time" seems to be little more than a formality. But the distinction is an important one; the minus sign in the spacetime interval ‐ or the "imaginary" in "imaginary time" ‐ marks the difference between the "Wells spacetime" of science fiction and the "Minkowski spacetime" of relativity. Hawking states that one would presumably be free to travel both forward and backward in "imaginary time," though he says that the same ambiguity does not exist with regard to real time (Hawking 1990, pp. 143-144).

Wells spacetime appears in most ‐ if not all ‐ books discussing relativity. John Archibald Wheeler uses it to show the congruence of the paths of a fast and a slow ball on separate trajectories ‐ one mostly straight and another arcing ‐ between two points. (Wheeler, p. 28-29) Any graph showing time as perpendicular to space is ipso facto a diagram of Wells spacetime. This practice was begun in the seventeenth century by Galileo and Descartes.

From the beginning, Minkowski spacetime has been pervasively conflated with Wells spacetime, in some cases leading to the viewpoint that the future is unchangeable. For example, the physicist and mathematician Hermann Weyl, a contemporary of Einsteinís, echoed the assertion of Wellsí Time Traveller that "There is no difference between Time and any of the three dimensions of space except that our consciousness moves along it," According to Weyl,

The objective world simply is; it does not happen. Only to the gaze of my consciousness, crawling upward along the world line of my body, does a section of the world come to life as a fleeting image in space which continuously changes in time. (Smolin 2013, p. 61)

What would this mean regarding free will? Are we active participants in the unfolding of time, or do we incrementally experience a future that is determined by the present (and thus by the past)? In that it allows us to present time as a dimension which we can examine from an outside perspective, and thus gives us a view of a fixed eternity, Smolin calls Galileoís and Decartesí innovation "the scene of the crime." (Smolin 2006, p. 257)

Light cone diagrams are ubiquitous in the literature on relativity, typically showing a horizontal line to represent space and a perpendicular line for time. The intersection of these two lines can represent any given event A in spacetime, and a cone coming to a point at this intersection ‐ and extending upwards from it ‐ represents the set of events that can be reached by a pulse of light originating at A. The interior of the cone represents the "future" of A: the events that can affected by any event at A or that can be reached by a particle present at A. Light cones are extraordinary teaching tools, but as they show time as perpendicular to space, they are a representation of Wells spacetime rather than Minkowski spacetime.

200px-Light_cone.svgA light cone diagram in which point A is "immediate" in both time and space, point B is "future," and point C may be called "impresent."

Misleading as it may be to show time as a dimension perpendicular to space, what makes the light cone diagram so instructive is that it shows that (since nothing can go faster than light) every event falls into one of five categories or regions, which I name as follows. Relative to any event, there is what I call the immediate, the "here and now" where the space and time axes cross in the light cone diagram. There are the past and future interiors of the light cones that contain all possible paths of any mass to and from the "here and now." I choose to call the surfaces of the light cones the present, as they are what is visible from that event or, conversely, capable of seeing that event. The areas outside the light cones I call "impresent," unreachable from ‐ and having no possible effect on ‐ the immediate "here and now."

There is nothing we can do at this moment or at any time in the future that can have any effect on any event outside our future light cone. We certainly cannot affect the past, but neither can we affect any event that is so far away that even a light beam sent by us right now would arrive at that place after the event had already passed. Events outside our future light cone are as beyond our influence as if they had already happened, although we cannot know about them yet unless they are in our past light cone. We may come to know of them and be affected by them later, but we cannot know of the "impresent" now. If the sun were to suddenly flicker and be extinguished, we would not know of it for the several minutes it would take for the light waves to reach us. As far as we were concerned, it would not have happened. Nothing outside our past light cone can have any effect on us. We cannot see it or even know about it. Although we may predict the future or have miraculous foresight, we cannot literally see or know what is still future any more than we can change what is already past. Impresent events are too far away for light from them to have reached us yet. Events outside our past light cone are as beyond our knowledge as if they had not happened yet, although we cannot do anything to change them unless they are in our future light cone.

You may have noticed dualities among the above statements concerning light cones. These dualities arise from the dual light cones of the "present:" any point in space corresponds to a point on both light cones. The time at which that point touches the lower light cone is an event which the observer at A is presently able to see. The time at which that point touches the upper light cone is the event which the observer at A is presently able to illuminate by directing a light pulse toward it. This observer is able to affect this "present" point now, but can only witness the results of this interaction later, when the event of this interaction is echoed back, by flowing down in the diagram as time passes and crossing the lower cone. This duality is given mathematical expression by the distance formula in the Pythagorean theorem (equation 6, above): we solve for the distance s to any point in space by finding the square root of x2 + y2 + z2. That square root has both positive and negative solutions, which we may take as representing the two ways in which that point is now present to us: the event flowing into the past which we are only now able to see, and the event flowing from the future which we now have our last chance to illuminate, be seen from, or affect in any way.

Note that my past is not the same as your past and neither is my future (or more accurately, the set of all my possible futures) the same as your future. My past at this time is not what will be my past five minutes from now, and the same is true of my future. Past and future are relative to a particular place and time: an event. Nor is my "now" your "now," even if our clocks remain in perfect synchronization. Our unique position in space and time determines our present in the sense of our perspective on the past and our ability to affect the future. We each experience time differently. This non-universality of past, present, and future is the essence of special relativity; the four-dimensional construct that we use to visualize it is not.

"Dimension" is perhaps too familiar a term for newcomers to relativity, carrying with it connotations that are not appropriate when referring to time. Time as a "coordinate" is arguably less loaded with such connotations. In the highly popular book Hyperspace, Michio Kaku writes of time as a dimension, emphasizing its distinction as a temporal dimension, but placing it in a context that can be truly unfortunate for the uncritical reader: discussions of time travel in a universe which he speculates may have as many as nine physical dimensions. He makes an unfortunate segue from a discussion of four-dimensional spacetime to one of three-dimensional space possibly being multiply connected due to a fourth dimension. (pp. 94-95) Kakuís central theme appears to be that "more is better" in terms of dimensions being used to explain the physical world; my thesis here is that Kakuís point of view ‐ shared by many others ‐ may be preventing us from seeing all that is possible with the visible, familiar three dimensions. It is too seldom that Minkowski spacetime is presented with strong cautions that it is "just a man-made scheme for understanding the world through mathematics" (Mazur, p. 185) rather than a Godís-eye view of eternity; "a mathematical device" that is not meant to "imply that space and time are now to be regarded as basically similar physical quantities." (Lawden, pp. 8,11)

As I recall, my motivation to abandon the idea of Wells spacetime came not from the reading of textbooks but from an article I found on the internet called "The Ontology and Cosmology of Non-Euclidean Geometry." One sentence in particular made an impression on me: "Just because the math works doesn't mean that we understand what is happening in nature." In other words, just because you can draw a graph of something doesnít mean that the resulting physical shape exists in any other sense. Smolin has recently written a similar caution:

The pragmatist will argue that thereís nothing wrong with checking hypotheses about laws of motion by converting motion into numbers in tables and looking for patterns in those tables. But the pragmatist will insist that the mathematical representation of motion as a curve does not imply that the motion is in any way identical to the representation. The very fact that the motion takes place in time whereas its mathematical representation is timeless [in the sense of being viewed from outside time] means they arenít the same thing. (Smolin 2013, p. 24)

Consider the following statement: "Since time is one-dimensional (the history of Rome, for instance, can be ordered on a single line) and space has three dimensions, their combination is a four-dimensional realm." (Schutz, p. 214) This brings us to the heart of the matter. Mathematically, events must be treated as having four coordinates; to identify them fully, one must specify a time coordinate as well as three coordinates of space. But the physical reality in which those events occur is three-dimensional; time is measured by observing events that unfold in one or more of the three dimensions of space, such as the back-and-forth reflection of a light beam between two parallel mirrors that form the "light clock" spoken of in many texts on relativity (see below for greater detail).

I eventually began to see, as Lincoln Barnett put it: "All measurements of time are really measurements of space." (Cole, pp. 134-35) In what seemed a great revelation to me, I realized that time was ‐ in a very important sense ‐ a radial measurement from the observer, with "here" corresponding to "now," and "there" corresponding to "then." I can touch and be touched, affect and be affected now by things that are here. I cannot affect nor be affected by things that are there until later. They cannot affect me nor can I affect them now, for we are beyond one anotherís immediate reach. I see my own body as it is now, or at least very close to now; I see the stars as they were long ago. This is true in a very quantitative way: when I look at the night sky, what I see now are events that are one light year away and one year old, four light years away and four years old, eight light years away and eight years old, and so on. The events I see now that are sixteen years old originated in a sphere that has a radius of sixteen light years. Even on a small scale, light from events that occur ten feet away will not reach me for ten nanoseconds, though this lag is much too small for human beings to notice. Thus I say that physically, time can be considered a radial measurement within the three dimensions of space.

Some may object to this idea that time is contained in space rather than being apart from it. In an imaginary dialog with his pet, one physicist writes: "I said that time and space were different aspects of the same thing, not that they were identical. Theyíre obviously different, because you can only move forward through time, not backwards." (Orzel, pp. 108-109) There is a flaw in this argument, in that it can be made analogous to imagining oneself at the north pole and saying that the dimension of "north-south" is different from the two dimensions of the Earthís surface because one can only travel in the "south" direction from there. North and south are radial from the observer at the north pole, just as time can be seen as radial from the observer in space. It is neither "this way" nor "that way"; it is merely "outward" or "inward." A universal system that is unbiased toward any observer must ignore this radial aspect and thus is not surprisingly four-dimensional. Though useful, it does not correspond to nor contain the perspective of any single observer.

As Joseph Mazur writes, "If all motion were to cease in the universe for an interval of time, what could we possibly mean by that interval? If motion is not taking place, then the time span of the interval is not either . . . every time interval must represent the motion of something in the universe." (Mazur, p. 37) And by inference, every interval of time must represent the distance which that "something" transits.

The idea that spacetime is three-dimensional in this physical sense need not be considered contradictory to the fact that physical events meaningfully occupy a mathematical, four-dimensional vector space, which is a term familiar to those who have studied linear algebra. The four dimensions of spacetime can be considered analogous to the four dimensions of a demographic analysis that includes age, income, weight, and resting heart rate. Each person would occupy a single point in this "space," measured along four numerical coordinates that are not necessarily related nor fully independent of one another, just as time is not fully independent of space.

The spacetime interval is presented as a consistent "measuring stick" for four-dimensional spacetime, but what exactly is it measuring? The curious nature of the spacetime interval ‐ and the Minkowski spacetime that it measures ‐ is most evident in the fact that the interval between events widely separated in both space and time may be zero. Think about that for a moment. In what kind of space might the distance between two completely different sets of coordinates be zero?

A spacetime interval of zero or of a small value in no way implies proximity. In fact, the only event pairs that may have zero interval between them are events that cannot be observed as being simultaneous or local. If they are simultaneous but separate, they will have an interval corresponding to their separation in space. If they are local (in the same place) but separate, they will have an interval corresponding to their separation in time. If, however, their separation is such that a beam of light from one event can reach the place and time of the other, then these two events have an interval of zero between them; and as with all these intervals, its value will be measured the same for all observers.

Though Einstein writes of "neighboring events" and "adjacent points of the four-dimensional spacetime continuum," (Hawking 2007, pp. 171,198-99) it is difficult to say what meaning this neighboring or adjacency could have. These events neighbor one another no more than two people neighbor one another whose birthdates, number of children, and shoe sizes happen to (closely) coincide. Rather than being an expression of physical proximity, Minkowski space must be considered a four-dimensional "causality structure," with a zero interval between two events meaning that the two events can be causally connected only at the speed of light. Non-zero interval values pertain to causal relationships as well. Using the convention shown in Equations 1 and 2 above to calculate the spacetime interval, values of s2 greater than zero indicate that the events can have no causal relationship; a value less than zero indicates that the events may be causally connected at speeds slower than the speed of light.

I believe that it is a mistake to compare the spacetime interval to the Pythagorean theorem, even if care is taken to point out that it is in some way "modified" (e.g. Wolfson and Pasachoff, p.1035) Any similarity between the two is wholly superficial, as should be evident by now in our discussion; if you, the reader, are not yet convinced, I hope the following comparison between the circular geometry of Euclidean space and the hyperbolic geometry of Minkowski space will suffice.

If we return to Equation 1, not introducing any imaginary numbers or breaking the distance down into three-dimensional components, the difference between the Pythagorean theorem and the spacetime interval is more stark. In both cases, we have a left-hand term that represents an invariant quantity of some sort. To compare the two metrics correctly, it is important that we keep the invariant quantity on the left-hand side of each; it would not do, for instance, to make the terms of the spacetime equation change sides algebraically until we end up with the same number of terms on each side and all terms being added so:

Equation 8. d2 = s2 + t2

This is mathematically correct, but it is not useful for our current purpose of comparing the invariant metrics for these two different spaces, Euclidean space and Minkowski spacetime. The spacetime interval s, not the distance d, remains invariant between observers in different inertial reference frames that measure various values for time and distance. The length of a line segment remains invariant, though different right triangles may be constructed using this line segment as the diagonal (or "hypotenuse"), and though those triangles may have various measurements of lengths x and y, the other two sides of the triangle. The difference between these two invariant metrics is that in Euclidean space, the two components on the right-hand side are added; in Minkowski spacetime, the two are subtracted. With Equation 4, we saw that the Euclidean (Pythagorean) relationship describes a circle. What we have not yet discussed is that the spacetime relationship of Equation 1 describes a hyperbola, quite a different thing altogether. A hyperbola (below) looks how we might imagine a circle turned inside out.

Of the spacetime interval, Chad Orzel writes, "[space and time] distances add in a hyperbolic manner ... The different rules for adding distances seem kind of strange compared to the ordinary Pythagorean theorem, but the end result has a certain mathematical elegance that is very pleasing." (Orzel, p. 161)

550px-Drini-conjugatehyperbolas_svg

The above plot (in blue lines) of the equation 1= x2 ‐ y2can represent a set of measurements between two events having a spacetime interval of 1, with distance on the x axis and time on the y axis (12 = d2 ‐ t2). The (x,y) coordinates (1,0) satisfy this equation and represent this pair of events being measured as having 1 unit of separation in space and no separation in time; in other words, the events are simultaneous in the reference frame in which they are measured as being one unit distant. Note that in all other frames of reference (represented by all other points on this blue hyperbolic curve) the events are measured as being both more distant in space and more separated in time. It is not the case that space "rotates into" time as a matter of four-dimensional perspective, with space decreasing and time increasing in the way that length may rotate into depth in Euclidean space; though Kaku seems to be saying otherwise when he writes, "By a simple rotation, we can interchange the any of the three spatial dimensions. Now if time is the fourth dimension, then it is possible to make Ďrotationsí that convert space into time and vice versa." (Kaku, p. 85) There is one context in which Kakuís analogy seems more appropriate and I will discuss it below.

In Hiding in the Mirror, Lawrence M. Krauss is diligent in cautioning the reader not to equate Minkowski spacetime with Euclidean space. Perhaps it is for this reason that I wince all the more at his statement that "one manís space interval can be another manís time interval," as this implies the same sort of rotation that Kaku seems to describe above. I would rather it be said that "one personís pure space measurement between two events is anotherís measurement of greater space and additional time." Also, in what might be seen as a contradiction of the main presentation he gives regarding Minkowski spacetime, Krauss states much later in his book that "there is nothing about the four-dimensional spacetime of Minkowski Ö that is remotely non-Euclidean." In this context he argues that such a space is flat, and in this he is of course correct; but this may tend to undermine his earlier clarification that "space and time are tied together in a way quite unlike the way up and sideways are tied together." As he points out, "people most often hear what they want to hear, and consequently they often tend to interpret the new results of science in terms that justify their previous expectations." (Krauss, pp. 46-47,88-89) Effective teaching will keep these expectations in mind and be careful to make the necessary distinctions.

In the spacetime interval, time measurements must be subtracted from space measurements in order to prevent counting the same aspect of separation twice. This is because in one sense, time is space; if more of one is measured between two events, then more of the other is measured as well. This also supports my earlier suggestion that time may best be seen as a radial measurement within the three dimensions of space rather than a fourth-dimensional extension to it. In such a paradigm, the spacetime formula mixes Cartesian and polar coordinates in the same three-dimensional space, taking three measurements in one system and one (radial) in the other, and it would of course be inappropriate to add all four since they are not independent. Hence the subtraction.

Before we proceed further, I find it necessary to more thoroughly explain the meaning and significance of the "light clock." As the essence of relativity is the realization that all observers must measure the same value for the speed of light, the light clock is a most appropriate means of envisioning the measurement of time. Furthermore, it makes some of the consequences of relativity apparent in a very simple geometric way. As mentioned before, the light clock consists of two facing mirrors, at a fixed distance from one another, between which rays or pulses of light may be reflected and timed. Imagine such a pair of mirrors lying parallel to the ground, one low and one higher. If the mirrors are one foot apart, a beam of light will be able to cross the gap between them in roughly one nanosecond. We can imagine that each crossing is a "tick" of this clock. Imagine now that the mirrors are set in very fast motion along the ground. If a beam of light is sent across the gap to strike the same point on the opposite mirror as before, it must now travel not only that same vertical foot, but an additional distance along the ground. If the speed of light is to remain constant, it must travel that longer distance in a longer time, greater than one nanosecond. And indeed it does; we thus measure that the clock ticks more slowly. However, if someone is "riding along" with the clock, they will notice neither a change in the distance the light travels nor in the time it takes to get there. Any clock the rider may use will run at the same slower rate as the clock itself. The conclusion is that for the speed of light to be measured equal by all observers, moving clocks will run more slowly.

You may study this strange "thought experiment" with great skepticism, looking for an inconsistency in the logic until you say, "Aha! But what about a pair of mirrors whose beams travel and are reflected in the direction of the mirrorsí motion? The path is elongated on one direction, but shortened in the other so that one round trip has the same distance whether the mirrors are stationary or in motion. If a moving clock runs slower than stationary one, they will disagree on the speed of light!" But the resolution to this apparent dilemma is that moving bodies (and measuring-sticks) are contracted in the direction of motion, resulting in all measurements of the speed of light being equal. This is no trick of mathematics; high-precision clocks have verified this theory experimentally. Nor does the design of the clock have any particular bearing on the result; time truly does run more slowly in moving frames of reference.

200px-Time-dilation-001_svg 500px-Time-dilation-002_svg

Left: The light in a "light clock" travels the distance L in one "tick." Right: the clock is measured as being in motion, and during each tick, the light must travel the distance D, which is greater than L. If the speed of light is to remain constant, we must conclude that moving clocks run more slowly.

In fairness to Professors Krauss and Kaku (and indeed to my own former viewpoint), I must now confess that there is one sense in which length and time measurements change in opposite ways, giving one the impression of rotation in a four-dimensional context in which space is at right angles to time. We may picture for ourselves a train car at rest, having a length that we measure and containing a number of clocks along that length which we may see are all synchronized. Once we set that train car in motion, we (at rest) will measure the carís length to have diminished in its direction of motion in the same manner as we would see a light clock do so. We would also find that the clocks in the car are out of sync from back to front. While the carís length has diminished, the time difference between its clocks has grown. In this way, we might suppose there is an analogy to rotation. But Minkowski space is a domain of events, namely their locations and times according to the observer; the spacetime interval which defines its metric is not a relation between observed object length and clock rates. My failure to see this distinction is what hindered my understanding of relativity more than anything else. While we and the train measure one another to be contracted in one direction, the events we each measure to be simultaneous and of a certain spatial separation are measured by the other to have greater separation in both space and time. Likewise, the events which each of us measure to happen in the same place at different times are measured by the other to be at different relative distances and even more separated in time.

My final argument for the three dimensions of space encompassing the so-called "dimension" of time as well ‐ final as far as our discussion of special relativity is concerned, that is ‐ is that our "speed through time" increases with our acceleration through space. The so-called "twin paradox" of relativity concerns the astronaut who travels at high speed to a destination in far-away space and returns to find his twin brother to have aged much more than he. Relativity predicts it, and experiment has confirmed it on a much smaller scale. The astronaut accelerates into space and back, his clock slows, and thus it may be said that he travels "into the future" without aging proportionately. His acceleration through space has accelerated his speed "through time." "When your speed through space increases, your speed through time also increases." (Orzel, p. 161). There may be some who believe as I once did that speed through time is diminished by acceleration through space and that our speed through "spacetime" thus remains constant, but we should instead understand that there is a Pythagorean relationship only between an objectís relative speed through space and the relative speed of its clock.