Using Pictures To Describe Motion

Oops. You're in the kitchen, and you've just dropped an orange. It rolls across the floor, then comes to a stop. Here's a simple question for you: how would you draw a picture of how the orange moves?

First try: you put some masking tape on the floor, from the place where you dropped it to the place where it stops rolling. Well..., that's a start, but it's not complete. The masking tape shows the start and end of the orange's movement, it's true, but there's something missing. There's no information about the speed. The orange rolls quickly at first, then it slows down, and eventually it stops. How could you add information about the speed of the orange?

Second try: you drop the orange again, and it rolls along the masking tape, just as before. But this time, you have a marker in your hand, and at regular intervals of time, you put a tick mark beside the orange, as it's rolling. You could hum a little tune like "Row, row, row your boat..." as you're doing so, and at each beat, you make the next tick mark. This is to make sure the time intervals are evenly spaced. At first, the spaces between the ticks are larger, then the spaces between them get smaller, and then finally at the end there are no spaces between the tick marks at all.

The masking tape now has information about both the position and the speed of the orange. Much better. You could say that the first try has only information about where, and the second try has information about both where and when. The combination of a where and a when is called an event.

But there's a third variation that you can also try. You can use the third dimension for the time. Instead of making tick marks, you can represent the time as a length, going straight up from the floor, vertically.

How would that work? Well, let's keep the same masking tape with the same tick marks. We'll keep the start of the tape attached to the floor, but we're going to lift up the rest of the tape from the floor, in a very specific way. Move along the masking tape to the first tick mark after the start, and lift the tape up off the floor say, 10 centimeters (cm), and hold it there somehow. Now go the next tick mark, and lift the tape up off the floor, but this time 20 cm. Keep doing this, adding 10 cm each time, till you get to the end. Canvas not supported. Please upgrade your browser.

Now what do you see? The masking tape is curved upwards, sort of like a letter "J". Notice the angle the tape makes with the floor. At first it's a small angle, and then at the end it's going straight up. This is interesting, since now the speed of the orange is indicated both by the tick marks, and by how steeply the tape is climbing. As a matter of fact, we can now get rid of the tick marks altogether, since the speed information is now indicated by how vertical the tape is.

This is the basic idea of space-time - representing times by lengths, and simply pretending that this time/length is an extra dimension, in addition to the space dimensions. That's it. It's a simple idea. Space-time uses geometry to represent both where and when information. In space-time, the entirety of an object's motion is represented by a curve or a line, called its history or its world-line.

The idea of representing the history of an object as a curve in space-time was originally just a mathematical curiosity. But things changed when it was discovered that space-time actually has a lot of interesting and important geometrical structure. Nature has many important rules that can be expressed elegantly only using the idea of space-time. Indeed, it's not an exaggeration to say that the geometry of space-time is at the very core of physics.

Example Histories

When an object is moving along a straight line, you can draw its history in a space-time diagram having two axes at 90° to each other, one for the time (t), and one for the distance (x). These are the simplest kinds of space-time diagrams, and sometimes they're all you need. Some examples are shown below: Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

Sharp corners in the history correspond to abrupt changes in the motion. Gradual changes in speed are shown with curving lines.

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

These simple animations also show how simple motions appear as histories in space-time.

Grids: Using Numbers To Describe Motion

You've dropped far, far too many oranges on your old kitchen floor. Now it's time to renovate it. Let's cover the kitchen floor with square tiles. These tiles form a grid. Let's pick one tile as the origin, a place from which you'll make your measurements. It doesn't matter which tile you pick; let's pick an origin near the middle of the kitchen floor.

Without any other tools, you can now make a rough measurement of the position of an object on the floor (like the orange) with respect to your origin. You can just count the number of tiles between the origin and the orange. But you have to count the tiles in a certain way. You're not allowed to count diagonally, so don't even think about it. You need to count twice, once in the left-right direction, and once in the up-down direction, as in "To get from the origin to the orange, move 3 tiles to the right, and then 4 tiles up." To distinguish between left and right (or up and down), you just let the number of tiles be either positive and negative - for example, positive to the right (or up), and negative to the left (or down).

It's traditional to label the directions using letters x, y (and z if you need to measure in a third spatial dimension). Which direction is x and which direction is y doesn't matter. The measurements in the various directions are called rectangular coordinates. Using that language, the orange might have (x, y) coordinates of (3, 4) with respect to our origin. In 2 dimensions you need 2 numbers to measure a position, in 3 dimensions you need 3 numbers to measure a position.

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

To get a better measurement, you can use a meter stick instead of just counting tiles, but the basic idea is the same. You measure one distance in the x-direction, and another distance in the y-direction, and both measurements can be either positive or negative, as before. So much for the measuring the position.

Grids Use Many Sensors To Measure Time, Not One

How exactly should time be measured in such a grid system? To measure time, you might think that only a single master clock would be needed. That's not a good idea, since at high speeds there are strange effects related to time, and you have to be really careful about how you make these measurements. A better procedure is to use multiple, synchronized clocks, that are spread out over the entire grid. When you measure the position of an object, you can measure the time as well, just by using the clock at the same location. This is actually much simpler than using a single master clock, since everything remains completely local. If you used a single clock, you would need to solve the problem of sending the correct time from one place to another. Using many clocks lets you avoid all those shenanigans. When you have simultaneously measured both the position and the time, then you have measured the coordinates of an event, a where and a when.

Let's call this a sensor grid, or just grid for short: a system for measuring the positions of things in space, combined with many stationary sensors spread out over that system to measure the time as well. (The official name for it is frame of reference.) Calling them sensors instead of clocks is useful, since they have to detect and record local events.

There are ways of making measurements that don't use grids. For example, in astronomy, there is only a single sensor (the telescope), not many, so there's no grid at all. The reason grids are emphasized here is that it's by far the simplest way to describe the basic rules of how measurements of space and time work. The key point is this: grids are simplest because each sensor is responsible for recording only the local events that happen in its own little neighbourhood. Any other way of looking at measurements is more complicated.

When you think of a grid, you should have this sort of picture in your head:

A Camera Is Not A Grid

Let's use the term camera to represent a single sensor. With a camera, you're usually concerned with incoming light signals from far away objects. This corresponds with actual observations made by astronomers (and by your own eyes too).

It's important to understand that a camera is not a grid. A grid is always a collection of a large number of sensors, not just one. With a grid, you can make calculations using numbers from multiple sensors (as seen below). With a camera, you can't do that, since there's only one sensor. These perspectives are different, and must be kept in mind.

The question of what a camera (or your own eye) actually sees is an interesting one. It's also a practical one, since astronomers use such sensors in their instruments, to view the behaviour of objects having large speeds (jets, active galactic nuclei, and so on). We'll return to this later, in Part II.

Inertial Grid

Physics begins by finding ways to describe how things move. What's the simplest kind of motion in the Universe? To find the simplest motion of all, you need to go to interstellar (or intergalactic) space, and look at the motion of an object which is far, far away from everything else (except for your grid, of course). Such an object is called a freely moving object - it doesn't interact with anything; no forces, no friction, nothing. How does it move? It drifts perfectly and serenely, in a straight line, at constant speed. By definition, this straight line motion is only true in an inertial grid (this is Newton's first rule of motion):

In an inertial grid, the speed and direction of motion of a freely moving object doesn't change.

Important: in English, the word acceleration means to speed up. But in physics, the word acceleration is more broad, and means any kind of a change in how an object is moving:

So, you can say that in an inertial grid, a freely moving object doesn't accelerate. You can test a grid to see if it's inertial by observing a freely moving object; if it accelerates at any time, then the grid isn't inertial.

There are many different possible inertial grids. (In principle, there's an infinite number of them.) Each inertial grid is parochial ("confined to a narrow area"), in the sense that the raw measurements of event coordinates made in each grid are specific to that grid, and only that grid. The event coordinates don't match between grids.

There are very few things as well-named as space. Space is remarkably well-named because there's just so much of it. Picture this, for example: you have a spacecraft whose sole navigational instrument is a big red button that says "Push Me". When you push this button, the spacecraft transports you immediately to a random place in the Universe. You look out the window. What do you see?

Nothing. You usually wind up in intergalactic space, and you usually won't be able to see any galaxies without a telescope. So you usually see absolutely nothing. You travel all that distance, and all you see is a big fat zero; not a single star, only an empty, desolate, lonesome blackness. On average, you will have to push the big red button about a million times before you finally wind up inside a galaxy, where you'll finally be surrounded by shiny, happy stars.

Inertial Grid Family

Inertial grids form a family. If you have one inertial grid, then you can make another one by applying one or more of the four allowable grid transformations: If you have an inertial grid to start with, then you can gleefully mix and match the above transformations in any way you want, and you will always end up with another inertial grid; you will always stay within the inertial grid family. (You can explore these transformations using this animation.)

But if you go outside the above kinds of transformations, then you are outside the family. You'll have a non-inertial grid, in which the motion of a freely moving object isn't uniform. Here are some examples of non-inertial grids:

One of the deepest ideas in physics relates these sorts of transformations with things called conservation rules. (It's called Noether's Theorem, after Emmy Noether.) A conservation rule is a rule that states that some physical quantity (a number) doesn't change over time. There's a deep connection between conservation rules and ideas of symmetry (transformations which leave things unchanged). For example:

If Symmetric UnderThen This Item Is Conserved
Displacement in timeEnergy
Displacement in spaceMomentum
Rotation in spaceAngular Momentum

Invariants: What Family Members Agree On

So far, all we have is a big pile of data about events. Even worse, each member of the inertial grid family has their own version of the data, with different values for the where and when coordinates of each event.


One way of moving from the parochial to the universal is to look for special combinations of quantities, calculated (derived) from the raw data, that happen to have the same value for all members of the inertial grid family. If you find such a combination, then you'll likely be able to make a new rule of physics with it. This leads to an important definition:

If a quantity is the same for all members of the inertial grid family, then it's said to be an invariant quantity.

(For brevity, this site will always use the word invariant in the above sense, as applying to all members of the inertial grid family.) Here are some examples of the many invariant quantities that have been found:

A better name for theory of relativity would be the theory of invariance, or something similar. The phrase theory of relativity is a mediocre term: it's ambiguous, leads to misunderstandings, and doesn't emphasize the most important idea. According to wikipedia, this dissactisfaction with the word relativity was even held by the theory's inventor, Albert Einstein. The term should really be abandoned, but it's difficult to change the inertia of long-standing tradition.

Rule #1: The Rule About All The Other Rules

This leads directly to the first rule. It's a strange one, since it's really a rule about all the other rules:

Rules must be expressible using only invariants.

In other words, relativity is the search for invariants - the search for what we can all agree on. This is a mission statement which starts a quest to find out what these invariants are, and what are their consequences. The most beautiful thing about this rule is that it's so deeply constraining. Constraints don't hurt a scientific theory, they actually help it, by making it easier to test.

Invariant #1: Speed Limit

The first and mightiest of all invariants is this:

There's an invariant universal speed limit for all signals.

What's a signal? Any physical object (matter, radiation) that moves from one place to another. So, the above rule says that no signal is instantaneous, and that the speed of any signal, anywhere in the Universe, at any time in the history of the Universe, can only be between 0 and a fixed maximum speed, as measured in any inertial grid. The speed limit is a universal constant: it's the same at all times, and in all places. Canvas not supported. Please upgrade your browser.

Is this rule consistent with the rule-about-rules? Yes, because it speaks only about an invariant (the speed limit). In fact, the speed limit rule is simply an assertion that a certain invariant exists.

The value of the speed limit is 299,792,458 meters per second, usually denoted by the letter c. Most people refer to this speed as the speed of light. But that's not the best way to think about it. For one thing, it's the speed of any massless particle, not just the speed of photons. The most important thing about this speed is that it's a speed limit for all signals.

The existence of this speed limit means that the night sky is a time machine. When you look up at night and gaze at the stars, the starlight you see has different ages, from a minimum of 4.4 years (Alpha Centauri) to a maximum of 2.5 million years (the Andromeda Galaxy). The largest telescopes can detect light that is older than any dinosaur fossil, and even older than the Earth itself. The oldest light is the cosmic microwave background (detected by radio telescopes). Its photons were emitted about 380,000 years after the Big Bang, which makes them about 13.8 billion years old. In this sense, astronomers are also archaeologists, looking backwards in time, using a fossil record made of light. The light has travelled undisturbed for even billions of years, until, at last, in a final act before ceasing to be, it offers up its quantum of energy, perfectly preserved for aeons, into the waiting bucket of a telescope, pointed by a curious human towards the night sky.

Here's where the fun begins. You can combine the speed-limit rule with the rule-about-rules, and that's when weird things start to happen. If the speed limit rule is valid, then all inertial grids need to measure the same value for the speed limit. But remember that one of the allowable transformations for the inertial grid family talks about a boost, where one inertial grid is moving with respect to another. If two inertial grids are moving with respect to each other, then how can they possibly agree on the speed of anything? Most important, how could they measure the same value for the speed limit?

So it appears that the these ideas are a complete flop, since they seem to contradict each other. But they actually don't contradict each other. The reasoning above is wrong because it assumes that speeds always behave as they do at low speeds. They don't. When speeds approach the speed limit, the behavior of measurements of space and time gets weird. This is the crux of the matter: if you insist that the speed-limit rule and the rule-about-rules are both valid, then you are forced into a serious re-examination of the basic facts regarding measurements of distance and time.

The idea of a family of inertial grids, with the same set of four allowable transformations, is old. The same concept was used at the dawn of science, by Galileo and Newton. But Galileo and Newton didn't have accurate measurements, as we do, and that led them to a critical mistake: they chose the wrong things as their fundamental invariants. They chose time-intervals and space-intervals. But these are not invariant! The fundamental invariant isn't times or distances, it's the speed limit c. This is the single most important fact about relativity.

Examples of Fast Objects

DescriptionSpeed β = v/c
Jets in microquasars ~0.95 (max)
Secondary cosmic rays - muons at sea-level ~0.99 (mean)
Electrons from radioactive decays ~0.99 (max)
Protons in the Large Hadron Collider0.999 999 991
Primary cosmic rays - most energetic 0.999 999 999 999 999 999 999 995

The existence of a speed limit in Nature means we can use it as a natural unit for expressing any other speed as a simple fraction or percentage of the speed limit. In relativity, speeds are usually expressed in this way, using the conventional Greek letter β (beta), as in β = v/c. When the speed is a significant fraction of c, the motion is said to be relativistic. If β is nearly 1, then the motion is called ultra-relativistic.

For most people, the speed limit seems incredibly fast. But in most astronomical contexts, the speed limit is actually very slow, because the distances are so vast. How much time does it take a photon to go from the Sun to the Earth? A little over 8 minutes. Even the Sun itself is so large that a photon needs 4.64s just to go a distance equal to its diameter. News may travel fast on Earth, but it sure doesn't in space. See this animation for a simple demonstration.

In English, the words speed and velocity mean the same thing, but in physics, they don't:

The symbol β can be used in both ways. When it appears in a formula, β is usually treated as a velocity. When β is a velocity, you just use the sign of the value to indicate the direction of motion (usually along one of the spatial xyz axes). It's important to understand that the range of values taken by β depends on whether it's treated as a speed or a velocity:

Geometric Points Have No Speed Limit

The speed limit rule applies to signals - to physical objects moving through space, like a spacecraft, a banana, or a photon. These are physical objects. A geometric point isn't a physical object, and doesn't obey the speed limit rule. Here are some examples of such geometric points: What's moving in these cases is not a single object. Over time, the moving point is made up of different objects. For the laser lighthouse, the photons emitted by the laser all travel at the speed limit. What's moving on the Moon is a collection of different events, represented by different photons hitting the moon at a different places, one after the other. Each photon obeys the speed limit, but the spot on the Moon can travel at any speed, with no limit.

ct Is Better Than t

There's another way of representing the time on the vertical axis in a space-time diagram. Instead of using just the time t, you can multiply t by the speed limit c, and use the result ct. This is usually the preferred way since it has two pleasant side-effects. First, the product ct has the unit of length, not time. As you can see, the seconds cancels out:
(meters/second) x seconds = meters
Thus, the time axis will have the same unit as the 3 space axes (meters, for example).

Second, anything moving at the speed limit will be represented as a line at an angle of 45° with respect to the vertical. Since the speed limit is so important, it's nice to have it appear in such a simple way. (It's also easy to draw.)

If no object can travel faster than the speed limit c, then in diagrams using ct instead of t, the tangent of the history of any moving object can never be at an angle greater than 45° with respect to the vertical.

We know how to draw the history for an object having β=0 (not moving at all) and β=1 (moving at the speed limit). But what about values of β between 0 and 1? How do you draw those? Well, that's pretty easy as well. You can find its angle with respect to the vertical using a simple geometrical construction, using your space-time diagram and a ruler. The example below is for β=0.3. As you can see, the trick is to take your ruler and divide the line BC into equal parts. It's important to understand that the tick marks on BC are evenly spaced, but the corresponding angles with respect to the vertical are not.


Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

Invariant #2 - The Space-Time Interval

There's a second invariant which can be constructed out of any pair of events A and B whatsoever, using any inertial grid's measurements for the time interval and space interval (the distance) between them, combined in a special way. This number is simply called the space-time interval (or just interval for short). The square of the interval (and thus the interval itself) is calculated like this:

Canvas not supported. Please upgrade your browser.

s2the square of space-time interval between the 2 events
cthe speed limit
Δt the time interval between the 2 events
Δd the distance between the 2 events

Each pair of events has a value for s2. The value may be positive, negative, or 0. To state it explicitly, the rule about the interval is this:

The value of s2 for any two events is invariant.

Neither the time-interval nor the space-interval is invariant by themselves. But, when you combine them in the specific way shown here, then you get something which is invariant. It's important to realize how revolutionary this idea is: it says that both time intervals and distances are parochial, not universal. Measurements of time intervals and distances differ from one grid to another.

"The discovery that time is relative brought about a profound revolution in our picture of nature. It represents one of the greatest victories of the human mind over the distorted notions acquired over the ages. It can be compared only with the revolution in human ideas brought about by the discovery of the fact that the Earth is a sphere."
- What Is Relativity?, L. D. Landau

The most important thing to know about the space-time interval is this:

The formula for s2 defines the geometry of space-time.

General relativity (also called geometrodynamics) is a theory of gravity. Like special relativity, it also centers on the idea of the interval as defining the geometry of space-time. The rules of geometrodynamics (rather complicated) describe how the presence of mass or energy changes the formula for the interval, and thus how mass or energy warps the normal geometry of space-time seen in special relativity.

Here's a summary of the logical progression so far:

  1. Inertial grid: In an inertial grid, the speed and direction of motion of an object that's not interacting with anything doesn't change.
  2. Invariant: If a quantity is measured to be the same in all inertial grids, then it's said to be an invariant quantity.
  3. Rule about all other rules: Physics rules must be expressible using only invariant quantities.
  4. Invariant #1: There's an invariant universal speed limit for all signals.
  5. Invariant #2: The space-time interval between any two events is invariant.
(The last rule is actually a consequence of the preceding two.) Everything else follows from these basic rules. All of the space-time diagrams and ideas presented below are essentially ways of helping you think correctly about the consequences of these rules.

Light-like, Time-like, and Space-like

You begin investigating the structure of space-time simply by looking at the numeric sign of s2. Since all inertial grids agree on the value of s2, then they must necessarily agree on its sign too. Take any event A and then calculate s2 between A and the surrounding events. You can see that there is a surface where s2 is zero, and this surface separates the regions where the value of s2 is alternately positive or negative.

The events having s2 = 0 with respect to A are said to have a light-like relation to A. In a two-dimensional space-time diagram, they form diagonal lines forming a big 'X' centered on A. If you add another space dimension, then these events will be in the shape of two cones with their pointy ends at A, with one opening upwards along the +ct axis, called the future light cone, and the other opening downwards along the -ct axis, called the past light cone. Both of these cones taken together are called the light cone of the event A. The history of any object travelling through event A at the speed limit will be somewhere on the light cone of A.

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

The light cone of event A divides the Universe neatly into two parts:

The s2 > 0 events are centered on the time-axis. They are said to have a time-like relation with A. They are separated into two unconnected regions, according to whether or not the other event happens before or after event A. The region inside the past light cone is called the absolute past of A, and the region inside the future light cone is called the absolute future of A.

The s2 < 0 events are centered on the spatial axes. They are said to have a space-like relation with A. In the two-dimensional space-time diagram, it looks as if there are two unconnected negative regions, but that's incorrect. When another space dimension is added (as in the middle diagram above), you can see that it's all connected together, on the outside of the light cone, and there's only one such negative region, not two. It's called the absolute elsewhere.

Note that events can't cross over from one region to another. That would violate the rules, since it would change the value of the s2. For example, if an event is in the past of event A according to one inertial grid, then it must be in the past of event A for all inertial grids.

Cause And Effect

In space-time diagrams having ct as the time axis, the history of a signal travelling at the speed limit c is drawn as a line, 45° with respect to the vertical. Histories drawn at an angle greater than 45° to the vertical represent impossible signals, because they represent motions faster than the speed limit. Canvas not supported. Please upgrade your browser. For an event A and its division of space-time into past, future, and elsewhere, this rule means:

If two events are related as cause and effect, then the cause must always precede the effect. In a space-time diagram, if event A is the cause of event B, then B must be able to receive a signal from A; in other words, B must be in the future of A.

For any history, you can construct a light cone at any event along the history. Again, because of the limiting nature of the speed limit, the entire history will always remain inside the light cone.

Surfaces With Constant Interval

As shown above, the sign s2 defines the basic structure of space-time - the light cone, the past, the future, and the elsewhere. But instead of looking at just the sign of s2, let's go a step further and look its value. What do you get if you insist that s2 is constant with respect to an event A? Mathematically, this is expressed by the formula:
c2 * Δt2 - Δd2 = constant
Geometrically, this equation defines a hyperbola in space-time near the event A. For different values of the constant, you get different hyperbolas. In a 2-dimensional space-time diagram, the hyperbola is a curve. If you add another space dimension, the hyperbola is a surface (called a hyperboloid of revolution).

There are two cases, according to the sign of the constant:

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

The most important point about hyperbolas is that they approach limits: they get closer and closer to straight lines, called asymptotes, but never actually touch them. Hyperbolas are said to be asymptotic. In our case, the light cone is an asymptote. In special relativity, this sort of asymptotic behavior is seen again and again. For example, a moving object can approach the speed limit, but it can never reach it (unless it has no mass, like a photon).

If you had to describe space-time using a single word, that word would be hyperbolic.

These surfaces appear repeatedly in special relativity. Instead of calling them "hyperbolas of constant interval", this site uses the more concise term interval shell (or simply shell for short).

These space-time diagrams, showing constant s2 with respect to an event, can actually be viewed or interpreted in two different ways:

The second interpretation is usually the more interesting, since it relates directly to one of the main rules: s2 between any two events is invariant across all inertial grids. So, you can use these sorts of diagrams to examine how the coordinates of an event B will change with respect to an event A, when measured in grids that are moving around with respect to each other. The event B can change its coordinates with respect to A, but it can only do so in accordance with the rule that the A-to-B square-interval must remain the same. In other words, the coordinates of B relative to A must remain on the same interval shell, as seen from any inertial grid. (There's a second rule: in the time-like case, event B is not allowed to go from the past of A to the future of A, or vice versa.)

The Faster The Object, The Slower The Physics

The strangest part of special relativity is the effect called time dilation. Time dilation is very small at low speeds, but becomes larger and larger as β approaches 1. The effect is this: take any pair of events having a time-like relation. Then, the time interval between these events can increase without limit, simply by choosing a different grid.

Pretend you're an astronaut in a spacecraft named Zoomer, moving through space at a speed of β=0.87 with respect to a second spacecraft, named Home Base. You need to stay in good physical condition, so you exercise every day. You always start by skipping rope. (In the weightlessness of space, you are held down by springs, or some other mechanism, as you do this.) When you skip rope, you count the number of skips you've taken, and your movements are very regular. In effect, you're acting like a human clock, repeating the same motions with a regular beat. Let's look at one full skip, and how it appears in space-time diagrams. There will be two events defining a skip:

Let's measure the coordinates of B with respect to A. In fact, we'll measure them in two sensor grids, one attached to the Zoomer spacecraft, and a second attached to the Home Base spacecraft. Here's what you'll see:

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

The space-time diagram on the right is a bit unusual, since it combines information from two different grids. This just makes it easier to compare their measurements. Note that the event B appears twice in the last diagram, once for each grid. This is intentional, since it's the exact same event, as measured in the two different grids. Let's review what these diagrams are saying.

In the Zoomer grid, the events are separated by a time, but not by a distance, since they happen at the same place. A typical value for the time between A and B would be about 0.5 seconds, for skipping rope. The value of the squared interval, for this grid, depends only on the time interval, since the space interval (the distance) is 0:

s2 = c2 * (Δt)2 - (Δx)2
s2 = (300,000)2 * (0.5)2 - 0 km2
The interval shell corresponding to this value is shown in the third diagram above. It must necessarily connect the two B's on the diagram, because of the basic rule that says that the square-interval between two events is invariant for all inertial grids.

In the Home Base grid, the events are separated by both a time and a distance, since the Zoomer spacecraft is moving swiftly with respect to it. And, of course, the event B must be on the interval shell.

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

Next, if you compare just the time interval between A and B, along the vertical ct axis only, as measured in the two grids, you can see there's a large disagreement. If you actually measure it on the screen, you'll see that the Home Base time interval is twice the Zoomer time interval. If the Zoomer time interval for skipping the rope is 0.5s, then the Home Base time interval is 1.0s. (The value of β=0.87 was chosen specifically to get this doubling. For other values of β, the factor is different.) You can also see from the shape of the interval shell that the disagreement in the time intervals is very small when β is small, and gets increasingly large as β increases.

We can draw this stunning conclusion:

The faster the object, the slower the physics.

To restate what we've seen earlier, measurements of time intervals are parochial, not universal. That is, it's only s2 between two events which is invariant. The time-interval between events can take different values, according to the grid used to make the measurements.

Some people describe time dilation this way: moving clocks go slow. This is true, and it's not a bad way of remembering it. But it's important to understand that time dilation applies to everything related to the moving object, not just its clocks: it applies to time itself.

Does an astronaut on the Zoomer feel anything strange in their experience of time? No, the astronaut experiences no change at all in their experience. The fact that the Zoomer is moving with respect to the Home Base makes no difference at all to the astronauts on the Zoomer. It's only the measurement of time as seen from the Home Base that slows down. But, from the point of view of Home Base, those measurements are completely real. From the perspective of the Home Base, absolutely everything sloooooows doooooown on the Zoomer, including, for example:

An excellent mental picture of time dilation comes from musical recordings on magnetic tape. If the tape is recorded at one speed, and then played back at a slower speed, then the playback will sound different from the original in two ways:

For example, if the playback-speed is exactly half the recording-speed, a high C note becomes a low C note, played for twice as long. (The song Strawberry Fields Forever by the Beatles uses such effects, with different parts of the recording made at slightly different speeds.)

It's important to recall here something mentioned earlier: a grid is a grid of sensors having synchronized clocks, each of which records only the nearby events. That is, there are many sensors in a grid, not just one. So, for our astronaut skipping rope on the Zoomer spacecraft, the times recorded for the two events A and B in the Home Base grid are recorded by two different sensors. In fact, the various measurements shown above involve 3 different sensors - one in the Zoomer grid, and two in the Home Base grid.

What happens when you reverse the perspective, and take the Home Base spacecraft as moving at β=0.87 with respect to the Zoomer spacecraft instead? You see the same effect - the physics on the moving object goes slow. On the surface, this seems to be a contradiction: how can the physics be slowed down on both spacecraft at the same time?

It's actually not a contradiction, because there are two things wrong with that statement:

Warp Factor

The factor by which the time interval is increased is officially called the Lorentz factor, but here we'll simply call it the warp factor. It relates to the distortion of times and distances when high speeds are involved. The traditional symbol for the warp factor is the Greek letter gamma Γ. (The small gamma 'γ' renders poorly in many browsers, so this site will use a capital gamma 'Γ' instead.) Like β, you will see the warp factor Γ appearing repeatedly.

The warp factor Γ depends only on β. It increases slowly from 1, and then increases very rapidly as β approaches 1. It has no upper limit, and increases without bound.

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

The diagram on the left is interesting, since it combines three important things all in one picture: the speed β, the interval shell, and the warp factor Γ. Here's a larger version of the same diagram, suitable for printing. There's also an animated version. You can use that diagram to find the warp factor corresponding to each value of β, simply by drawing two lines: one for the speed β, to find its intersection with the hyperbola; then a second line, going horizontally over to the left, to find the corresponding value of Γ. Here's an example, using β=0.75, showing that Γ is 1.5:

Canvas not supported. Please upgrade your browser.

Measuring Intervals With Only A Single Sensor

In general, space-time intervals are measured with both sensor clocks and meter sticks. It's also possible to measure any time-like space-time interval with just a single sensor, but you have to choose the correct grid. We've already seen an example of this in our skipping rope experiment with the astronauts on the Zoomer and Home Base spacecraft. The consolidated space-time diagram for the two events A and B for the astronaut skipping rope on the Zoomer looked like this:

Canvas not supported. Please upgrade your browser.

In the Zoomer grid, the interval was measured just with a single sensor clock in the Zoomer spacecraft, because there was no distance between the two events A and B:

s2 = c2 * (Δt)2 - (Δx)2
s2 = c2 * (Δt)2 - 0
s = c * (Δt)
This is true in the Zoomer grid. But for any grid which is moving with respect to the Zoomer grid, it's no longer true.

In general: for any events A and B that have a time-like separation, you can always find a grid in which the the spatial distance between A and B is 0. This just corresponds to the 'lowest point' of the interval shell. In such a grid, the interval is measured just by a single sensor clock in that grid, located at the same position as A and B. A time interval measured by such a sensor clock is called proper time, but it's important to remember that the proper time is just the same old space-time interval, measured in a certain way.

An Object's Wristwatch Measures The Interval

The above can be interpreted in a slightly different way. Pretend there's a freely moving object moving through a sensor grid. How can you measure the interval between any two events on the object's history? Well, there are two ways: You can remember this like so: An object's wristwatch-time is proportional to the interval between events in the object's history.

Time Dilation Revisited

Let's return to the example of the two spacecraft, Zoomer and Home Base. Let's focus on the history of a single sensor, a sensor belonging to the Zoomer's grid, but let's record that history in both the Zoomer grid and the Home Base grid. This is almost the same space-time diagram as before, but now we have a full history of the sensor, not just two separate events.

Canvas not supported. Please upgrade your browser.

In the Zoomer grid, the sensor doesn't move, β=0. In the Home Base grid, the Zoomer-sensor is moving at β=0.87. The dots show three ticks of the Zoomer-sensor's clock. You can easily see the effect of time dilation. As the Zoomer-sensor passes through the Home Base grid, it shares its current time reading with the various Home Base sensors it meets along the way. (Remember that sensors can only record local events.) Note that the Home Base sensors record an event like this, for example: at a Home Base time of 6.0s, the passing Zoomer-sensor says that its own time was 3.0s.

It's important to have this exact picture in your head when thinking about time dilation. The clock moves through a sensor grid, and as it moves through the grid, the moving clock's time is compared with the time on the many grid sensors it meets along the way. To see the effect of time dilation, you need at least 3 clocks: the moving clock, and at least 2 grid sensors to compare it to.

This example uses a sensor clock, but the same effect is seen with any object. If a banana takes 7 days to turn brown on the Zoomer spacecraft, then, according to the the sensor readings in the Home Base grid, it will take 14 days (if β=0.87). That is, moving bananas take longer to turn brown.

Three Weeks To The Pleiades

As β gets closer to the speed limit of 1, the value of the warp factor Γ increases without limit. The warp factor keeps going up, and up, and up, the closer β gets to 1. The warp factor Γ gives the size of the time dilation effect. For the Zoomer spacecraft moving with respect to Home Base, there's no limit to the amount of time dilation seen by Home Base. For a traveller in Zoomer, the travel-time from one place to another, from their point of view, is just the time on their wristwatch. (The wristwatch is a sensor in Zoomer's grid.)

So, for the Zoomer-astronaut, the travel-time to any destination can approach zero time.

Here's an example, using a proton in the Large Hadron Collider. Pretend that one of these protons is sent on a proton vacation to the Pleiades, a cluster of stars about 385 light-years from Earth. If you were able to attach a wristwatch to this proton, what would it record as the travel-time of the proton to the Pleiades?

The answer is given by our friend, the warp factor Γ. Here are the numbers:

β = 0.999 999 991 
Γ = 7454
travel-time on the proton's wristwatch:
  = 385 years / 7454 
  = 18 days, 21 hours
This is a stunning fact. There is a universal speed limit c, but from the point of view of an actual traveller, this does not limit the travel-time seen by a traveller in any way. In principle, a traveller's travel-time from any place in the Universe to any other place in the Universe can be made as small as desired, by getting closer to the speed limit. Zounds!

As β approaches 1 and Γ climbs higher and higher, the geometry of space-time lets you travel quickly to the stars.

For an advanced species travelling to the stars, the main limitation is the maximum warp factor Γ. As we've seen, a larger warp factor means a proportionately shorter travel-time for space travellers. If one civilization contacts another, and if the goal is a trip between civilizations, then the single most important piece of information is the maximum value of Γ that each civilization is capable of engineering.

Notice that Γ is a pure number, and has no related unit. That is, Γ is dimensionless. Since it's a pure number, it could likely be communicated between civilizations, using radio waves and a binary encoding for numbers. For example, pairs of β and Γ values might be sent in sequence, ending at the maximum attainable value. Of course, the receiving civilization would need to figure out this little puzzle, along with the encoding.

Moving Children Age More Slowly

Imagine that a parent is at home, being annoyed by the children. "Children, I'm truly thankful for your existence, but at this moment, you are annoying me. Please go outside and play, while I sit here and have my coffee... Be back in an hour for supper." Everyone is carrying a device (like a wristwatch, or a phone) that measures the time to an extreeeeeeme accuracy. Before the children go outside, all of the watches are synchronized, and set to read the same time. The children run around for a while playing kick-the-can, and then go back inside the house. Everyone compares the time reading on their devices. What do they see? Well, they see two things: This is just the effect of time dilation, of course. Moving clocks go slow. Here's a strange fact to chew on: each child's watch reflects how much exercise the child had. The more running done by the child (either for longer times, or at greater speeds), the greater the warp factor Γ, and the greater the time difference between the parent's watch and the child's watch.

You can also look an it this way: when you take a trip, the distance travelled is shown by your odometer. Everyone knows that its reading depends on the route you took. If you took a longer route, the odometer shows a larger number. But a similar sort of thing can also be said of the elapsed time on your wristwatch. If two people leave from event A, separate, and then arrive later simultaneously at event B, then the elapsed time on their watches won't agree, and will depend on the details of how they made the trip. Again, time is parochial.

Rule For Uniform Motion

The above scenario with the playing children provides a rule to distinguish between accelerated and unaccelerated motion.

In an inertial grid, a freely moving object moves without any acceleration. It doesn't change speed, and it doesn't change the direction in which it's moving. In a space-time diagram, the history of a freely moving object is the simplest kind of history - a straight line, at some angle with respect to the vertical ct axis. For any such object, we know that we can pick a grid where the object is actually at rest; in that grid, the history is a vertical line.

So, the above scenario with the children going out to play looks like this in space-time, where, simply for convenience, we've chosen a grid in which the parent isn't moving:

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

Everyone's history joins event A to event B. Everyone starts out at one place-and-time (event A), and then, later, they join up again at the same place (event B). The history for the stationary parent is vertical, while the histories for the running children wiggle around the vertical, because they run around while playing outside. In other words, the children's histories show accelerated motion, while the parent's history does not.

The histories of the parent and the children form a set of histories that share a start-event A and an end-event B. Other than that, they can wiggle around a bit in between. Let's call this a set of wiggle histories. It's useful to include the parent's history as being just another wiggle history, where the wiggle happens to be zero.

There are two ways of thinking about the motion of a freely moving object, between two events A and B:

As we saw before, the reading on the wristwatch, multiplied by c, is just the space-time interval. So, you can also say that uniform motion has the maximum total interval in comparison with the nearby wiggle histories.

Action Principles

The above idea of wiggle histories, along with one history corresponding to a maximum value of some quantity, is an example of a action principle. There's deep-dish pizza, and then there's deep-dish physics. Deep-dish physics often uses an action principle. They provide powerful and elegant ways of looking at a problem. They can describe how objects move, or how fields change in space and time. Action principles appear in many parts of physics:

Skipping the mathematical details, here's a summary of how an action principle works:

In the case of the maximum ageing rule in the previous section, the numbers being added up are the wristwatch time intervals, along the wiggle histories. The rule is that the total wristwatch time is maximum for the history that has no wiggle at all.

The invariants we've seen so far are simple - the speed limit and the space-time interval. When you go searching for more invariants, guess what you find? You find action sums - or action integrals, as they are really called. That is, the remaining invariants are (mostly) action integrals. We aren't going to look at these action integrals in detail here. But to give you a taste, here's the simplest action integral of all, the one for the motion of a freely moving object:

Here are the details:

Notice that the product mcds is an invariant. This means that the sum S of many such items, represented by the long ∫ symbol, is also an invariant.

Boost Transformation - Overview

In the inertial grid family, there are 4 kinds of allowable transformations. Only one of these transformations involves a speed: the boost transformation, where one grid is moving with respect to another, at a fixed speed, and in a fixed direction.

The simplest case of a boost looks like this:

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

These two grids are almost the same. Their spatial axes (xyz) all point in the same directions, and they share the same event for the origin:

The only difference is that grid 2 (in red) moves along the x-axis, in either the positive or negative direction, with respect to grid 1 (in black). Here, β is a velocity - it has both a size (the speed) and a direction (along the + or - x-axis, as indicated by the sign of β). Given the event coordinates in grid 1, the task is to find the coordinates for the same event in grid 2, which is moving in this specific way with respect to grid 1.

As always, each event must keep the same interval with respect to the origin event. In each boosted grid, each event will 'flow' along its interval shell. Let's use one space-time diagram, to show how the coordinates of 4 events (one for each part of space-time near the origin) will appear in different grids. The general direction of the flow looks like this:

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

Notice these things about the flow of events:

You can remember the direction of flow by using your hands. Here's an example for β>0: For β<0, you do the same thing, but with your right hand instead of your left. That's the general picture of how event coordinates change from grid to grid, during a boost.

Boost Transformation - Details

Here are some tools for exploring boost transformations in more detail:

There are two ways of finding out the event coordinates in the boosted grid:

Both of these ways depend on just two numbers, our old friends β and Γ. Let's start with the geometrical construction. Here are two examples, for β=+0.5 (grid 2 moving to the right), and β=-0.5 (grid 2 moving to the left):

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

These space-time diagrams are a bit unusual, since they have two sets of axes. The blue set of axes (for grid 2) is at an angle with respect to the black set of axes (for grid 1). Don't attach too much importance to the distorted appearance of grid 2. The distortion isn't "real", it's just how grid 2 looks with respect to grid 1. From grid 2's point of view, grid 1 looks distorted, in a similar but reciprocal way (with β having the opposite sign).

The geometrical construction combines two things you've already seen before:

  1. Find the directions of the axes, using β. This is done using the same technique as before, with β marked off on a horizontal scale, from the ct axis to the light cone. The sign of β controls which way the angle goes - to the left or the right of the ct axis. The angle between the ct and ct' axes is the same as the angle between the x and x' axes.
  2. Find the scale of the axes, using Γ. Above, the scale of each axis is shown by regularly spaced dots. If you look closely, you'll see that the blue dots have a slightly larger spacing, controlled by the value of Γ. In our case β=0.5, and Γ=1.15. So, the spacing of the blue dots is 1.15 times the spacing of the black dots. As we've seen already, you can find the value of Γ by construction using this graph. (Or, you can just use the formula for Γ.)
  3. From the event whose coordinates you want to find in grid 2, draw lines parallel to the new axes, and read off the corresponding numbers on the ct' axis and the x' axis.
It's interesting that this geometrical construction exists. But because the axes are bent, it's not possible to get a good feel for how histories really appear in the second grid, with perpendicular axes. Instead of this geometric construction, you can also easily calculate the event coordinates in grid 2. This will give you a better picture of how histories appear in a boosted grid. As you would expect from the above, the formulas depend only on c, β, and Γ:

Remember that -1<β<1. That is, β is the velocity along the x axis, not just the speed. It can be either positive (grid 2 moving in the +x direction) or negative (grid 2 moving in the -x direction). These equations describe how an event changes coordinates during a boost from one inertial grid to another. Let's call these equations the boost transformation (also called the Lorentz transformation).

Time Order Can Change

We're usually concerned with time-like intervals, since only time-like intervals occur along the history of an object. Let's take a look at space-like intervals for a moment. Here's a reminder of what the interval shell looks like for a space-like interval. As usual, we have an origin event A, and an event B having a space-like separation from A. That is, event B is in the absolute elsewhere of A:

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

This shell shows the set of possible coordinates for event B, as seen from different grids that are moving with respect to each other. Notice that event B can come before A (appears lower than A in the diagram), or it can come after A (appears higher than A in the diagram). It can also come at the same time as A (simultaneous). In other words, the time order of events having a space-like interval is not invariant. It can change between different grids. Also, if two events with a space-like interval are simultaneous in one grid, then they will not be simultaneous in a boosted grid.

Earlier, we made some remarks above about cause and effect. Events that might be related by cause and effect must be in each other's light cone. They also have to keep the same order in time: the cause must always come before the effect. With space-like intervals, the opposite is the case. In summary:

Time order is invariantyesno
Cause-effect relation allowedyesno

Directional Distance Compression

A grid measures the coordinates of events, and it can easily measure the distance between two events. But how does it measure the distance between two objects, when they may be moving around? To measure the distance between two objects, you first need their histories. Then you pick a time, and draw a corresponding horizontal line ct=constant in space-time. The horizontal line will connect the two histories, and the distance between the two objects at that time is then just the Δx along that horizontal line.

Let's return to the Zoomer and Home Base spacecraft, and look at the spatial dimensions (the size) of the Zoomer spacecraft.

In the Zoomer grid, the dimensions of the Zoomer spacecraft have some fixed values. If we take the direction of motion of the Zoomer spacecraft as the x-direction, then we might say, for example, that it has these dimensions:

Does the Home Base grid measure the same dimensions in its grid, for the moving Zoomer spacecraft? Nope. It measures these dimensions: For y and z, the dimensions are the same. For the dimension in the direction of motion, the size is divided by Γ (with Γ calculated using the speed of Zoomer), and is smaller, since Γ is greater than or equal to 1. The same can be said of the volume of the Zoomer spacecraft as well: the volume is also divided by the same number Γ.

This effect is true for the distance between any two objects, not just the front and rear of a spacecraft. Let's call this effect directional distance compression, or distance compression for short. Both time dilation and distance compression have the same underlying cause: the geometry of space-time. You can think of both of these effects as being two aspects of the same thing.

Both time intervals Δt and space intervals (distances) Δx are parochial. They change values from one grid to another. The values of Δt and Δx in any one grid have no special status. For example, the assertion that the Pleiades has a specific distance of 385 light years is a parochial statement attached to a specific grid, a grid comoving with the Earth. In other grids, moving with high Γ with respect to the Earth, and in the direction of the Pleiades, it's a different story: the effect of directional distance compression compresses the distance to the Pleiades, and divides it by the value of Γ.

Invariant #3: Volumes in Space-Time

As seen above, the boost transformation makes events flow like toothpaste in space-time. Some pairs of events are stretched out, while others are squeezed together. Time dilation corresponds to the stretching (expansion by a factor of Γ), and distance compression corresponds to the squeezing (reduction by a factor of Γ). When you consider volumes of space-time, then these two effects actually cancel each other out.

The volume of a region of space-time is invariant.

Canvas not supported. Please upgrade your browser. Canvas not supported. Please upgrade your browser.

A volume of space-time is defined by a closed boundary of some sort. Roughly speaking, the boundary needs to "go in a circle" in order to get back to where it started. You can't make such closed boundaries using only histories, because histories can't go in loops. (Any loops in a history would imply speeds faster than the speed limit, which is against the rules.) But you can make a space-time volume using a mixture of histories and non-histories, as shown above. In the above diagram for Grid 1 on the left, the vertical sides are histories, while the horizontal sides are not. You can measure the areas of the two grey areas shown above: they are the same. (The area of a parallelogram is 'base times height'.)

A space-time volume has different units, depending on the number of dimensions you're using:

It's interesting to note that the "flow" of space-time volumes resembles the flow of an incompressible fluid.

High-Γ Travel

If a civilization can engineer spacecraft that can travel at very high speeds, where Γ takes a large value, then the effects of time dilation and distance compression will significantly affect the experience of travellers. As a threshold for high-Γ travel, you might take the value Γ=2, corresponding to β=0.87.

To the traveller, high-Γ travel is experienced as a combination of wormhole and time machine.

When you travel from the Earth to another star with high Γ, your travel time will be shrunken by time dilation. Alternatively, you could say that the distance to the star is reduced by distance compression, in the direction of travel. Either way, when Γ is sufficiently high, the result is that you arrive at the destination much more quickly than you would have without these effects. In the language of science-fiction, the trip feels as if you've passed through a wormhole - a weird short-cut through space. (As well, you will also see optical distortions in the directions of the stars when you look out the window of the spacecraft.)

After the travellers have reached the other star, then, in a grid attached to the Earth, the travellers are also in the deep future. From the Earth's perspective, it's taken a long time for the travellers to reach their destination. From the traveller's perspective, they will know intellectually that they are now in Earth's deep future, but they won't feel it in their bones: when they turn their radio telescopes back towards the Earth, they will pick up transmissions that were emitted long ago. To the traveller, nothing very strange will be seen in the apparent date of these transmissions. Of course, if they make the return trip back to the Earth, that impression will be corrected, and they will finally experience the time-machine aspects of high-speed travel.

The Here-And-Now Is Stretched Out

In ordinary geometry, each point in space is a certain non-zero distance from all other points. In the hyperbolic geometry of space-time, however, the same cannot be said of events. In space-time, each event A has its light cone, the set of events having light-like (zero) intervals with respect to A. The event A is, roughly speaking, "nearby" these other events, and those other events are stretched out across the Universe.

Of course, to take advantange of the "closeness" of the events on the future light cone, travellers need to have a high value of Γ. If you take seriously the idea that no one member of the inertial grid family has special status, then the fact that high speeds are needed is seen as a detail.

Is Time The Fourth Dimension?

Yes and no. As we saw in the beginning, the basic idea of space-time is a pretty simple one. You just translate times into distances, in order to draw a picture of an object's history. In this rather trivial sense, yes, time is treated as a fourth dimension. But there's nothing deep or strange about that.

Is time a kind of fourth spatial dimension? Definitely not. Here's why:

Because of this, many people refer to space-time as having a 3+1 structure, to emphasize the difference.

Space-Time Geometry Constrains The Rules

To create new rules that are compatible with special relativity, you need to express them using invariants. The boost transformation is the template for building all other invariants:

The basic facts about the boost transformation are: You can treat these equations as a little machine that translates one set of 4 numbers into another set of 4 numbers (with the above restriction about the units). Let's label the 4 numbers as (a,b,c,d), with 'a' corresponding to 'ct'. If these 4 numbers satisfy the boost transformation, then, mathematically, this quantity must also be an invariant:

a2 - (b2 + c2 + d2)

If you find a new 4-pack of numbers that behaves according to the rules of this little machine, then you will also have found a corresponding new invariant. These 4-packs are called 4-vectors.

The game played by physicists is to make guesses about new rules, with new 4-vectors, and their corresponding invariants. They then see if those guesses are in agreement with experiment. It's stunning that the geometry of space-time so tightly constrains the creation of new rules.

Electrodynamics Pops Out Of Special Relativity

As stated above, any rule that claims to be compatible with special relativity must be expressible using invariants. So, you can play at simply guessing new rules, by imagining equations that are compatible with the rule-about-rules. Of course, such guesses may or may not agree with experiment. These guesses can take many forms. One of the simplest such guesses is this (don't worry about the details):

Does this correspond to anything in Nature? Yes. This precisely describes motion of a particle in an electromagnetic field! In fact, all of the equations of electrodynamics (electricity and magnetism) are expressed most elegantly using the language of space-time and 4-vectors.
Pure mathematics starts with general principles or axioms, and then deduces consequences. In math, the only constraints are those of logic. Physics is almost always presented to the student in imitation of mathematics, with many separate results deduced from a few general principles (as is done here on this website).

It's interesting and important that physics imitates mathematics in this way. But, in the end, you must remember that in physics the final arbiter is not logic - it's experiment. Strictly speaking, there's no idea of "proof" in physics; at least, not in the formal sense of the word used by mathematicians.

Boost Velocity

Velocity is a vector, and has both direction and speed.

During a boost transformation, the xyz components of the velocity vector change from one grid to another. Of course, this transformation follows the rule that the speed of any object cannot exceed the speed limit. This always remains true, no matter how big the boost, and no matter how big the original speed in the unboosted grid.

This animation demonstrates how the velocity changes during a boost transformation. Note that:

End of Part I

This concludes Part I, which covers the fundamental ideas of special relativity. Please continue on to Part II, which explores light signals and optics.