All events in the universe take place in space-time. Space-time is actually a mathematical model. It fuses three spatial dimensions with one time dimension into a geometric whole. How can we conceptualize this? There is a common trap in physics, which is to confuse the theory or mathematical model with the actual thing, something physical that can be observed and measured. A wealth of experimental evidence, listed in the 4-minute video below, supports the theory of special relativity, which describes the space-time model:
This is good evidence that the space-time model used in special relativity accurately accounts for real observable and testable phenomena.
What is a Dimension?
Mathematically speaking, a dimension is the smallest number of coordinates you need to specify the location of a point. A line, for example, has a dimension of 1 because you only need one coordinate to specify a point on it. A surface or plane has a dimension of two and the inside of a sphere has three dimensions. To describe the location of a point inside a sphere, for example, you need three coordinates - we usually call them x, y and z coordinates. Building on this we can say that a point within a sphere (or any three-dimensional space) can potentially move in any combination of three possible directions - in the x, y and/or z direction. Borrowing from statistics, we can say that point possesses three degrees of freedom.
It's easy to visualize the three dimensions of space we live in. We've got up/down, latitude and longitude. An everyday example of a location in three-dimensional space might be "on the third floor in the northwest corner of the Black Building." We can pinpoint exactly where to go if we are given these directions. In our everyday world, time, however, feels different. Locations in space can be fixed but we experience time as fluid. It flows as events constantly recede into our past and reach into our future. Now when we revisit our directions, we notice that we need a "when." What if these are instructions for a dentist appointment, for example? The dentist wants me at a specific location at a specific time, such as in his chair at 2 pm on Wednesday. I've got all four coordinates you need. At 1 pm or 3 pm, someone else will be occupying that chair, those identical spatial coordinates. But only I will be in that chair at this specific time coordinate of 2 pm Wednesday. Therefore, we can see that to describe any event we'll need both spatial coordinates and a time coordinate.
But how is time related to space? I can explore this by describing my walking path to the dentist's chair. I will need to describe both space and time. I enter the east door of Black Building at 1:45, approach the elevators at 1:50, wait there for one for a minute for the door to open and then travel up to the third floor during the next minute before I walk down a corridor to the dentist office at 1:55. I plunk myself down in the dentist chair at 2 pm and remain there for 50 minutes. I am describing a series of events that flow through space AND time.
During these events spanning between 1:45 and 2:50 pm, I am moving through space and time, in other words, I am moving through four changing coordinates, or dimensions. However, I notice that I may be moving through one or two spatial dimensions simultaneously but in every case as I move through space I must also move through time. Time seems to have some rules that space doesn't. I can never "stop" in a time dimension. I can never move into a negative time dimension. Time, then, appears to have fewer degrees of freedom than any spatial coordinate. I, or anyone, can only move in one direction along this timeline and at a rate I can't control. Is time a dimension? If it is, it does not appear to be a fourth dimension in the same way as the other three spatial dimensions.
We can map out my progress between 1:45 and 2:50 pm on Wednesday using a framework that describes changes in all four coordinates of time (1) and space (3). Yet, as we just noticed we encounter some problems when we think about time as a fourth dimension equivalent to a spatial dimension. We can describe this impasse mathematically. As we will get into later, it was Hermann Minkowski, not Einstein, who formulated the dimensions of space-time, and these four dimensions are NOT equivalent to four-dimensional (Euclidean) space. Space-time is not, for example, this albeit mesmerizing four-dimensional rotating cube:
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| JasonHise;Wikipedia |
Time Was Once an External Stage
Isaac Newton: We probably all started our exploration of physics with this great man, a key figure in the scientific revolution in mid-seventeenth century Europe. In his time, a four-dimensional framework for any physical event would seem unnecessary and probably ridiculous. To describe his laws of motion, he needed only three spatial dimensions, the ones we experience every day, and he assumed it all happened while an absolute time progressed at a specific rate independent of everything else going on. Even physical space was likewise treated as outside all events. Every object either had an absolute state of motion or an absolute state of rest relative to the absolute space it found itself in. Time and space were treated much like an external stage on which all phenomena in the universe take place. The assumption made sense and it worked for a long time. It is how we experience space and time every day.
The idea was basically unquestioned until the mid-nineteenth century, shortly after James Clerk Maxwell and others, started to tinker with electricity, magnetism and light. Maxwell discovered that all three were a) related to each other and b) traveled as disturbances through (Newtonian three-dimensional) space.
Maxwell worked out that "light and magnetism are affectations of the same substance and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws." The "field" in this statement is the epicentre of what would become one of the most hotly contested debates in the history of science. What exactly was this field? His work would ultimately lead to the connection between space and time. To start with, he and others knew only two well-established facts: 1) light seemed to have a constant velocity through air and 2) light slowed down when it traveled through other transparent media such as water.
Aether Was the Medium of Space
Scientists at this time knew that electromagnetic disturbances act like waves so they figured they must travel through some kind of medium, which they called aether. Experimental results on aether, however, were confounding. For example, the speed of light through air was constant in any direction. Wind or air density had no effect. Even more confusing, the speed of an electromagnetic disturbance seemed to be independent from speed of the source of that disturbance. In other words, light seemed to disregard Newtonian physics. How? What about this aether substance could accommodate such findings? It proved frustratingly impossible to define electromagnetic waves mechanically. They did not act like other mechanical waves such as sound waves or water waves.
Like many other scientists at the time, Einstein, pondered how aether worked. Like physicists Paul Dirac, Louis de Broglie, Maxwell and others, he thought that aether was some kind of medium with physical properties filling empty space. There must be something there that carries the electromagnetic disturbance, and perhaps the results could be explained by some kind of elastic force through which the waves are propelled, analogous to water waves or sound waves. Even in 1920, after he developed special relativity, he stated that there must be something that allows for the "existence of standards of space and time (measuring-rods and clocks)" to allow for space-time intervals in the physical sense.
Aether Theories Run Into Trouble
Various aether experiments designed to find out how it worked mechanically, and there were many, yielded either contradictory or nonsensical results. An interesting and well-known example of this problem was the Fizeau experiment, conducted in 1851. At that time, a number of scientists were comparing the speed of light (an electromagnetic disturbance) in air versus water. They could observe that a beam of light slowed down in water. They wondered what process slowed the light-bearing aether down in the denser medium.
If you are wondering how light slows down when it has one invariant velocity, you ask a fantastic, often overlooked, question. A good concise answer can found here.
This phenomenon, called refraction, has been observed since ancient times and was described mathematically in the 1600's by Snell's Law. A number of researchers revisited refraction as possible evidence that aether could be partially dragged by moving matter such as water. The idea being tested was that aether moving against the direction of water flow might be slowed down. Vice versa, aether dragged along with the water flow might boost the flow rate of the aether and therefore the speed of light. Fizeau designed an experiment that compared the speed of light through water moving in the same direction as the light beam with the light's speed moving against the direction of the water flow. If their theory was correct, light would move faster along the same direction of water flow and slower when it's against the flow. They didn't know how or if aether interacted with matter but this experiment was designed as a first step to the answers. They made the assumption that because light could penetrate all transparent media, such as water, those media were permeable to the aether. If the medium is moving, does it carry the aether along with it? Is the aether partially dragged or is not affected at all?
Hippolyte Fizaeu got puzzling, but not entirely unexpected, results from his experiment. He showed that the speed of light in same-flow water was boosted but it was less than the sum of the speed of light in air plus the speed of the water flow, as Newton's laws would have predicted. The aether appeared to be dragged along, but only partially. It turned out that Augustin-Jean Fresnel had already established a dragging coefficient in the late 1700's, based on several earlier aether experiments that appeared to support the idea of partial aether dragging. All of these experimental results seemed to suggest that the aether might be denser inside mediums such as water than it was in air or in a vacuum and that light traveled more slowly through denser media. Fresnel's dragging coefficient was proportional to the refractive index of the medium.
Scientists at the time knew that the reduction in the speed of a beam of light depends on the index of refraction, which in turn depends on the light's wavelength. The refractive index decreases with increasing wavelength, so, for example, blue light bends more than red light when a light beam passes from air into water. This is why white light disperses into a rainbow when it passes through a prism. This presented a snag. The experimental data pointed to a seemingly complex scenario where aether is partially dragged by matter and the aether must flow (simultaneously!) at different rates for different colours of light, as a white light beam, containing all the colours, was used in their experiments. Were there a seemingly infinite number of aethers, one for each wavelength of light? The use of polarized light in this experiment presented the same problem. Light polarized in opposite directions both exhibited the same partial aether drag, suggesting that the aether was carrying two opposite directions of motion at the same time. These partial-dragging results were confirmed by numerous other experiments as well. Aether theory was offering complication rather than simplification.
Electromagnetism Re-imagined
It wasn't until around 1892 that Hendrik Lorentz approached these baffling experimental results from a new angle and a solution began to take shape. He assumed, first of all, that the aether was completely stationary. He was then able to derive Fresnel's coefficient by using Maxwell's equations and an undragged aether. He looked at the problem as one of light speed transitioning between two reference frames, one where the system is at rest in the aether and the other where the system is in motion in the aether. By rest, he meant absolute rest in the absolute space of Newton. By doing this he introduced a clear distinction between matter (this time in the form of electrons) and aether. This meant a departure from any mechanical theory of aether.
Referring the work of Maxwell and others, he described the aether as "states" in an electromagnetic field. By doing so, he introduced an abstract aether replacing the previous and problematic mechanistic model. He also planted the seed for special relativity: A moving observer with respect to the aether will observe the same electromagnetic phenomena as an observer at rest in the stationary aether.
Lorentz's interpretation was that partial dragging was something that happened to the electromagnetic wave itself and not something that happened to the aether, which was stationary. This proved to be a critical first step in the evolution toward our modern theory of space-time. It was, however, a first step. It transformed mechanical aether into an abstract electromagnetic aether, but it still held onto the presence of some kind of aether and it held onto Newton's concept of absolute space.
The idea that space must contain something to support the propagation of electromagnetism (and gravity as well) was extremely difficult to let go. By 1901, as Lorentz was developing the theory underlying his famous Lorentz transformation, which would provide a bedrock for Einstein's theory of special relativity, Henri Poincaré wrote (in a state of philosophical angst?) that there must be no absolute space nor absolute time. Even so, Poincaré would not let go of aether: "If light takes several years to reach us from a a distant star, it is no longer on the star, nor is it on the Earth. It must be somewhere, and supported, so to speak, by some material agency."
Aether Evolves into Space-time and Fields
Quoting from Einstein once again in 1920, after he published his theory of relativity, "To deny the aether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view." Again, one can detect the angst.
Today, we can argue that Poincaré and Einstein were correct about space, but only in a sense. We can argue that the field itself has replaced the aether. Electromagnetic radiation, such as visible light, which was the focus of Fizeau's and other aether experiments, is carried as waves through an electromagnetic field. These waves operate by quantum rules rather than mechanical rules. We now have a concept of a physical universe that is described by the mathematics of space-time geometry, and within that geometry we describe various fields carried by force-carrying particles called bosons. Photons are the force-carrying particles of the electromagnetic field. Starting with the development of the theory of electromagnetism in the late 1800's and continuing through the development of quantum mechanics later in the early 1900's, physicists gradually abandoned the idea of a background medium altogether. Aether isn't required to explain how special relativity works.
How "real" is a field? Is it physical or strictly a mathematical construct? The concept of the field arose as a fundamental physical quantity that independently exists. For example, physicists envisioned the electromagnetic field extending indefinitely throughout space in all directions as a physical field interacting with matter. Maxwell's electromagnetic field equations were developed using classical field theory which obeyed Newton's laws. Later on they were refined further by incorporating special relativity and quantum mechanics. We now know that an electromagnetic field is carried by force-carrying photons. Subatomic particles, such as photons, are treated as excited states in the quantum field that obey the laws of quantum mechanics.
As treated in quantum field theory, a field is strictly mathematical and doesn't physically exist. The field still extends all over space and we can make an observation of the field by taking a measurement of it at a particular moment and location. In this sense, the field interacts with matter and we can argue that the field is physically real. We experience evidence of it all the time, such as when we see a burst of visible light photons when we turn on a light. We feel a static charge or watch iron filings arrange themselves according to the lines of force exerted by a magnet. In the case of magnetism, for example, we indirectly detect the virtual photons that carry the magnetic force. We understand these photons as mathematical wave functions.
The Speed Of Light Forces a Shift In Thinking
Einstein, wondering about space and time and perplexed by the nature of aether, turned his focus to the experimental findings. He knew that Lorentz was beginning to approach the aether problem in terms of changing frames of reference. He focused on one blaring observation. The speed of light appeared is a universally unchanging value. In itself this was a truly mind-blowing observation in a then largely Newtonian universe.
Imagine the headlight of a train approaching at the speed of light. The photons in that light beam would also be travelling at the speed of light, and never exceeding it. Why don't these two velocities add together, as they would if a man threw a ball forward from a train travelling forward at everyday speed? If photons followed the same rules of Newtonian dynamics as balls do, the two speeds would add up. What is it that slows the light down and keeps it in check? What is that process if it is not the work of some kind of elastic aether? Einstein knew that something in the description of this thought-experiment must give. One thing he could conclude with some certainty was that he did not yet have the entire picture of space as the medium through which light travels. By following a tactic similar to Lorentz by allowing the question of medium to take a back seat, he could reframe the problem. If the speed of light never changes, then time or space, or both, must. Put mathematically, space and/or time must transform.
Time Can Vary
The concept of transformation itself isn't new in physics. Galilean transformations operate in Newtonian physics. They tell us that any event that takes place in one frame of reference will operate under the same physical laws if it takes place in a different frame of reference. For example, barring all other sight cues, a car traveling at 50 km/h passing a car traveling at 30 km/h in the same direction will appear to the passengers of the 30 km/h car to be traveling at 20 km/h. It's the basic addition/subtraction of velocity vectors, operating under the same rules as the ball being thrown from a train example above. These kinds of transformations presume that the passage of time is the same for observers in different frames of reference. They presume that time is absolute in other words. These Newtonian rules are still useful and that's why we learn them. They work perfectly until we are dealing with velocities approaching light speed (or near gravitational fields). Lorentz and others tried to understand how the speed of light breaks these well-established common-sense rules. In any reference frame the speed of light is always the same. It does not obey the Newtonian laws that underlie a Galilean transformation.
A Galilean transformation holds up for events that happen at everyday velocities, but as an object approaches the speed of light in one reference frame compared to a stationary reference frame (we can call this frame a stationary observer), both space and time, for that observer, transform. Space and time depend on the reference frame. Any object approaching the speed of light experiences time dilation (time stretching or slowing down) and length contraction as observed relative to a stationary observer. To that observer, the object contracts in the direction it is traveling* and a clock attached to that object slows down.
*An object travelling near light speed will actually appear rotated even though its measured length will be contracted. The object is moving so fast that light from the along the object reaches the observer at slightly different times. A receding object will appear contracted and an approaching object will appear elongated, while a passing object will appear skewed or twisted. This optical effect is called Terrell rotation).
This means that observers moving at different speeds relative to one another can observe different distances, different elapsed times and even, as a result of these transformations, experience different orderings of events. These transformations are not illusions. At the expense of getting ahead of myself, consider an example of a proton (a particle of matter) in the Large hadron Collider. It is accelerated to almost light speed and as it does so it experiences a Lorentz factor of about 10,000. The Lorentz factor is the factor by which time and length change for an object that is moving. To put this in perspective, if you could shrink down and ride on top of this proton from Earth to Alpha Centauri, your trip would you take only a couple of days. Alpha Centauri is four light-years away, which means it takes (traveling at light speed!) four years for its photons to make that same length of trip. An observer on Earth would record that your trip to Alpha Centauri took a little over four years.
Putting his thought-experiment observations into a formal framework, Albert Einstein published his game-changing theory of special relativity in 1905. It incorporated Lorentz transformations in space and time. A few years later, Hermann Minkowski formulated a geometric interpretation of the Lorentz transformations, and this is now the mathematical structure, called Minkowski space-time, on which the theory of special relativity rests.
Minkowski space-time mathematically combines three-dimensional Euclidean space with time to create a four-dimensional structure called a manifold. A manifold is a strictly mathematical concept that is nicely explained here.
The Space-time Interval
If we take the simple concept of distance, we can get a feel for how Minkoswki space-time works. In Newtonian physics, the distance between two points is invariant. It will be the same regardless of reference frame. In special relativity, however, that distance will depend on whether the observer is moving or not (length contraction). In four-dimensional space-time, a new invariant "yardstick" called the space-time interval replaces distance. Whereas in Newtonian physics, time and space (distance) are invariant, in Minkowski space-time, time dilates as distance contracts. These measurements are dependent on the frame of reference. The space-time interval of an event, which combines space and time, is the same in any frame of reference. A space-time interval extends from one place and time to another place and time. We can even build space-time by taking successive snapshots of space over time and adding them all together. Measurements of space and time can vary between observers but the space-time interval, obtained by measuring the distance and time between two events, will always be the same in every frame of reference. It doesn't matter how fast or in what direction an object is traveling with respect to the observer. The space-time interval displays Lorentz invariance.
To visualize how time and distance relate to one another geometrically in a space-time interval, as well as how the speed of light is constant in every frame of reference, try this 7-minute video:
How To Get From Length to a Space-time Interval
How did we get to this new invariant idea of "length" in space-time? When Minkowski developed the space-time manifold, he imagined that the space-time interval could be related to Pythagoras' theorem, but in four dimensions rather than the two we are all probably familiar with when we draw a right triangle on a sheet of paper, shown below right.
With a little mathematical finesse, we end up with three dimensions of space and one dimension of time: s2 = w2 + x2 + y2 + (ict)2. We've changed our variable z to s, to show that we are now measuring distance as a space-time interval. We've also added a new variable, i, to our time axis. The term i is an imaginary unit, also known as √ (-1). The old-fashioned picturesque descriptor "imaginary" doesn't mean an imaginary number is made up. It just helps us find solutions to mathematical problems. In our case, imaginary time is real time that undergoes a mathematical transformation called a Wick rotation. A Wick rotation is a way to convert a problem in Euclidean four-dimensional space into a problem in Minkowskian four-dimensional space-time. It trades one spatial dimension for a time dimension and allows the dimension to undergo a Lorentz transformation, which mathematically is a rotation of coordinates.
Since the ict term is squared we can multiply (ct) by -1. We end up with s2 = w2 + x2 + y2 - (ct)2. Again, we can think of this (ct)2 variable as "distance" along the time axis.
The introduction of an imaginary unit hints to us that even though we've put our equation into the form of a Pythagorean equation, the time dimension in it doesn't "act" like the other spatial dimensions. It does not have a simple Euclidean geometrical relationship with space.
Physicists now describe space-time in terms of a newer mathematical construct called a metric tensor. General relativity also describes space-time, but in that case, the space-time needs to curve under gravity. We can think of the metric tensor is a device that makes corrections to Pythagoras' theorem to enable the right triangle we used as our starting point to map onto curved space-time. It also does away with the imaginary unit (i) we discussed earlier by describing events in real time instead. The negative sign, however, is preserved in the metric tensor but it now describes how distance changes with time when space-time curves. The original Minkowski equation describes an incorrect but simplified flat space-time.
By creating a space-time interval, we can understand both the invariance of the speed of light as well as time dilation and length contraction.
Speed of Light Invariance
Imagine a photon traveling at the speed of light, c. All observers will observe that same velocity no matter what their velocity might be relative to it. The distance traveled by the photon (let's say it's traveling in the x direction so we'll call it distance x) for t seconds can be written as:
x = vt where x is the distance traveled, v is velocity and t is time
Let's begin to transform this simple equation for distance into one for a space-time interval. First we'll incorporate it into the Pythagorean theorem in three dimensions, like we did earlier. I'll start using s for the distance even though I'm not quite correct yet because we haven't incorporated time.
s2 = x2 + y2 + w2
There is no motion in any direction except the x direction so the w and y axes are zero. We need to describe this relationship in terms of space-time so we add the time dimension [-(ct)2]. We can now properly describe the distance (x) in terms of a space-time interval (s):
s2 = x2 + 02 + 02 - (ct)2
We can swap out x by incorporating our earlier equation x = vt.
s2 = (vt)2 - (ct)2. Our object is traveling at the speed of light so we know v = c.
s2 = (ct)2 - (ct)2. We get s2 = 0 so s = 0.
This means that the space-time interval for any object traveling at light speed is zero. It is invariant. It doesn't matter what reference point you measure the object from. You could be accelerating in the w or y direction as you measure its velocity. It will always be light speed and its space-time interval will always be zero. Put another way, only an object traveling at light speed will have a zero space-time interval. All observers will observe the same (zero) space-time interval for that object, which means they all observe it to have a velocity of c.
How does a photon experience the universe? We can get a feel for this surprisingly complex situation by comparing the world lines of three objects, all traveling at different constant velocities in the same direction, shown below in a simple space-time graph.
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| Jheise;Wikipedia |
In this graph, t is time and x is distance along one space coordinate. We could draw a more complex space-time graph by incorporating all three space coordinates with one time coordinate, representing Minkowski space-time.
An object at rest would be a vertical line originating at the same origin point as the coloured lines and where x = 0. Its world line is space-like. A space-like world line could likewise describe the length of a physical object such as a ruler, as the distance between two space-like events. Each of the three coloured lines represents the world line of an object traveling at a specific constant velocity (hence all the straight lines). Their world lines, and the world lines of any objects traveling less than the speed of light, are time-like curves in space-time. Even though only straight lines are drawn here, any world line is considered to be a special type of curve in space-time.
This graph represents all times (future and past) and all possible distances along x in space-time. It is a simple representation because all three objects are traveling along in the same direction, along the x-axis. They all originate at the origin of time and distance on the graph. At that origin point, they share the same space-time interval. A physical example might be a single particle decaying into three particles, each having a different velocity.
A stationary object moves in time but not in distance. A slow object moves further in time than it does in distance. A faster object moves further in distance than it does in time. A very fast object moves in distance but very little in time. A photon has the fastest possible velocity. It moves in distance but not in time. It follows a light-like curve, which would be represented as a horizontal line moving along the x-coordinate, where t = 0. A light-like curve is a straight line in this simple graph where two spatial dimensions are not shown. Often the convention is to draw this horizontal line at a fixed 45-degree angle. By doing this we can draw the light-like curve in three spatial dimensions as an easier-to-visualize three-dimensional cone, directed upward into the future and downward into the past, shown below.
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| MissMJ;Wikipedia |
Time Dilation
Imagine setting up an array of synchronized clocks over a very large table in space, a table on the scale of thousands of kilometres across with no gravitational field are nearby. From one edge of the vast table you take a photo of all the clocks. You find that the clock closest to you is running a little faster than those furthest away. After a little thinking, you realize you need to take into account the transition time for the light from each clock to reach you. The speed of light is constant so this is a fairly straightforward synchronization calculation. You go back and adjust all of your clocks. Now they all read the same time when you take your photo of them. A friend flies past your clock arrangement at 0.95% light speed and takes his own photo of the array at exactly the same time you take your next photo. Comparing photos, you notice that his clocks are all a bit behind yours. Your frame of reference is at rest compared to your clock table so you don't experience time dilation. However, you and your clock table are in motion compared to his frame of reference. He records time dilation. His present moment was not the same as your present moment. The two events you experienced were desynchronized.
If an additional initially synchronized clock were glued to the outside hull of your friend's ship beforehand and you repeated the experiment, you might guess that you will see it as running faster than your clocks. Instead, you read it as running slower as he flies past you. For you, he is the moving frame of reference. For him, once again you are the moving frame of reference. You once again see in your photos that his clocks are slower. And yet, for him, your clocks were slower. This counter-intuitive effect is known as the twin paradox.
What is different in the two frames of reference has nothing to do with the mechanisms of the clocks. A moving mechanism doesn't get heavier or something such as that. The key difference is that the moving clock is traversing a longer distance between events (ticks). The events do not have to be hands moving on a clock face. The clocks could be mechanical, quartz digital, atomic or even hourglasses.
To get a feel for this it might be easier to imagine that our clocks are made of pulses of light bouncing between two mirrors. One trip from mirror to mirror is equivalent to one tick of our earlier clock. The speed of light is invariant, so the ticking mechanism of this clock will be perfectly constant. The clock moving with respect to the stationary clock will tick slower. The moving clock will, in its frame of reference, experience the stationary clock as the one moving and it will tick slower than the former one. This brings home the fact embedded in special relativity that there is no absolute motion and there is no absolute rest. The only absolute is the speed of light. We can visualize time dilation (and the twin paradox) in the set-up in the gif below.
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| Cleoris;Wikipedia |
Length Contraction
We can do another thought experiment to explore how length contraction occurs. Imagine two light clocks like the ones described above in which light bounces between two mirrors. We can put them close together or far apart and we can orient them in any way we want with respect to each other. If they are both at rest with respect to us, as observers, they will run at the same rate. Any direction or location in space (barring any and all gravitational influences) is physically the same and physical laws work the same anywhere in the universe. If we set them perpendicular to one another and then set them both in motion at 99 % light speed in the direction of one of the clocks, we get some interesting results.
We, as observers, remain at rest with respect to the clocks. This means that as the clocks fly by us, the light is bouncing parallel to the motion in one clock and the light is bouncing perpendicular to the motion in the other clock. We will find that they will both run slower, as we expect, but we also find that they are still both running at the same rate. This observation is not what we expected.
The clock that is perpendicular to the motion should slow down, as we figured out above. We can imagine that those light pulses must travel a longer distance because they are making a stretched out zigzag path of motion between the mirrors. The stretching is where the extra distance comes from. But what happens to the clock that is parallel to the motion? Aren't those light pulses going to take a much longer time to reach the front-facing mirror that's going just 1 % slower than them?
We can put together a more concrete example to show what's going on here. Let's say the mirrors in the clocks are 300,000 km apart, the distance light travels in one second. At rest with respect to us, the light will take one second to travel from one mirror to the other. Now we set them in motion like we did above, with one set of mirrors oriented perpendicular to the motion and one parallel to it. If the clocks are now moving at 99 % light speed and the light is moving perpendicular to the direction of motion in one of them, we can calculate that the light will now take about 10 times longer, or 10 s, to make one trip between mirrors. The clock is now going 10 times slower relative to us.
What about the parallel clock? At rest, the light takes one second to go from one mirror to the other. If the light pulses in the clock are moving in the same direction, at 99 % light speed, the light has to chase a rapidly receding mirror in one direction. We can figure out that it will take about 100 s to reach the front mirror when it's moving away at 99 % light speed. It will take just a tiny fraction of a second to make its return trip to the other mirror because the back mirror is approaching at almost light speed. We discover that the light takes ten times longer to bounce mirror to mirror in the parallel clock (100s rather than 10 s). Yet we measured them and they are both running at the same rate - 10 times slower than they did at rest with us. How is the parallel clock still keeping the same time as the perpendicular one? The only way it can is to physically shrink in the direction of the motion, shortening the bounce distance. In fact, it will shrink to 1/10th of its rest length at 99 % light speed. Length contraction and time dilation have a perfectly inverse relationship. Length and time compensate one another to preserve the invariance of the space-time interval we explored earlier.
If one of the clocks could travel at light speed (and it cannot because it has mass) the light pulses in it would not move at all. There would be no ticking forward in time. If we could somehow ride along with a photon of light, we would discover that it does not experience time. If we consider the effect of length contraction as well, we come to a startling conclusion. There is no distance at all between the two mirrors in the parallel clock. In that clock the mirrors themselves would have no depth. Time slows down and length contracts for objects travelling very fast. A photon's path of travel is shortened to zero. Proper distance, like proper time, does not exist for a photon. Realizing this gives new weight to the concept that light has a space-time interval of zero.
Some Parting Thoughts
To accept the well-established fact that we live inside time as part of four-dimensional space-time is a bit like accepting that we experience only a tiny sliver of visible colour within a far vaster array of electromagnetic radiation. We know that it exists but we don't directly experience it in our everyday lives. We intuitively understand the universe in three-dimensional space but it is almost impossible to conceptualize four-dimensional space-time.
The movie Interstellar plays with the fact that time is a dimension. In that movie, a future "us" has figured out how to manipulate time to make it act like a tangible physical dimension. We currently can't do that and I don't know if that ever could be possible, but the mathematical formulation of space-time, in particular the Wick and Lorentz rotations, appear to treat time and space as two facets of the same thing. We got to our understanding of time as a dimension of space-time by way of the speed of light. At the speed of light, both time and space reach their limits. Does space-time exist at the speed of light? Do space and time fully unfold to our perception only when we experience the universe at rest?
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