Special Relativity Part I - Speed Limits and Geometry

Using Pictures To Describe Motion

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

• A) an object that's not moving at all (stationary) is just a vertical line
• B) an object moving at a speed that doesn't change is a line that makes an angle with the vertical; the faster it's moving, the greater the angle with the vertical
• C) stationary, then moving, then stationary again
• D) a round-trip, and going slower on the way back

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

• E) moving in the +x direction with an increasing speed
• F) moving in the -x direction with an increasing speed
• G) playing catch: an object bouncing back and forth between two places is a zig-zag
• H) two objects move towards each other, collide, then bounce back to where they came from

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

Grids: Using Numbers To Describe Motion

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.

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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 grid of sensors spread out over an area or volume (either 2D or 3D)
• the grid can have any orientation, location, or speed, but it's not allowed to be spinning or accelerating (more on acceleration below)
• the sensors don't move with respect to each other
• sensors record local events only, and not distant ones
• the sensors have a clock, and record the time of each event
• the sensor clocks are synchronized with each other

A Camera Is Not A Grid

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

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:

• the speed increases
• the speed decreases
• the direction of motion changes

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.

Inertial Grid Family

• move in space or time (displacement): change the place selected for the origin (x y z)=(0 0 0), or the moment in time for which t=0
• rotate: change the directions of the spatial xyz axes, to point to some other fixed direction
• mirror reflection: change the positive versus negative directions for one or more spatial axes, or even for the time
• boost: let the grid move in some fixed direction, at a constant speed (this is the only change that involves the motion of one grid with respect to another)

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:

• a grid which is spinning or tumbling in any way
• a grid which whose speed is changing, either speeding up or slowing down (a spacecraft firing its engines, for example)
• a grid which whose direction of motion is changing (going in a circle, for example)

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 Under	Then This Item Is Conserved
Displacement in time	Energy
Displacement in space	Momentum
Rotation in space	Angular Momentum

Invariants: What Family Members Agree On

Questions:

• what can all members of the inertial grid family agree on?
• how do we move from the raw data of measurements, specific to each sensor grid, to the rules of physics?
• in summary, how do we go from the parochial to the universal?

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:

• the maximum speed limit for all signals (discussed below)
• the space-time interval between any two events (discussed below)
• action integrals, which are related to forces and fields (discussed below)
• the electric charge and rest-mass of any particle
• the volume of a region in space-time
• E² - H², the difference in the squared sizes of electric and magnetic fields
• R, the scalar curvature of space-time
• ...and many others...

Rule #1: The 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

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

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:

• speed - the speed only, without the direction of motion
• velocity - includes both the speed and the direction of motion, taken together as a unit

• if β is a speed: 0<=β<1
• if β is a velocity: -1<β<1

Geometric Points Have No Speed Limit

• lighthouse: a moving spot from a spinning laser, as projected on a distant object like the Moon
• scissors: the moving intersection point of two scissor blades, as the scissors close
• beach: the place where a wave is breaking, as it moves laterally along the shoreline
• oscilloscope: the moving spot on the screen of a cathode-ray oscilloscope (very similar to the example of the laser on the Moon)

ct Is Better Than t

(meters/second) x seconds = meters

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.

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Invariant #2 - The Space-Time Interval

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

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

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.

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

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.

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

• the events inside the light cone (that have s² > 0, with respect to A)
• the events outside the light cone (that have s² < 0, with respect to A)

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

• event A can be affected by signals from events in its past, and only its past.
• event A can send a signal to events in its future, and only its future.
• none of the events in the elsewhere of event A can either send a signal to A, or receive one from it; that is, events in the elsewhere of event A cannot interact with event A in any way.

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

c2 * Δt2 - Δd2 = constant

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

• for positive (time-like) constant intervals, the hyperbolas 'open up' along the time axis.
• for negative (space-like) constant intervals, the hyperbolas 'open up' along the space axis.

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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:

• they show multiple events near event A in a single grid (which is what we have done so far)
• or, they show a single event B near event A, but as measured in multiple grids

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

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:

• event A: the start of the skip, when the rope hits the floor
• event B: the end of the skip, the next time the rope hits the floor

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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

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.

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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:

• the motions of the exercising astronaut
• the astronaut's heart rate
• how long it takes a banana in the kitchen to turn brown
• the half-life of radioactive material on board the ship

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:

• the tempo (the beat) will be slower by some factor
• the frequencies of the sounds will all be lowered, also by the same factor

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:

• first, the phrase "they are both going slow at the same time" assumes what is not true - that the passage of time is universal.
• second, each case uses 3 sensor clocks, as mentioned above. This makes a total of 6 sensor clocks to make all the measurements, and they are all different sensors; none of them are the same. There's no contradiction since the measurements are all made by different clocks.

Warp Factor

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.

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:

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Measuring Intervals With Only A Single Sensor

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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)

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 hard way: make the usual where and when measurements as it passes through your grid. Then, for any pair of events, calculate the interval using the interval formula (see above).
• the lazy way: just attach a sensor to the moving object. As it passes through your grid, your grid sensors will talk to the moving sensor, and ask it for its time reading. Then, the interval between two events in the object's history is just the difference between two such time readings, multiplied by c.

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.

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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

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

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

Moving Children Age More Slowly

• the elapsed time for the parent is the greatest
• the elapsed time for the children varies, but is always less than the parent

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

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:

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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:

• graphically, uniform motion is just a straight-line history connecting A and B.
• numerically, uniform motion has the maximum reading on a wristwatch attached to the object, compared to the same reading for any alternative wiggle history between A and B. This is sometimes called the maximum ageing rule.

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

• classical mechanics
• special relativity
• electrodynamics
• geometrodynamics
• quantum mechanics

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

• begin with a start-event A and an end-event B
• pick a wiggle history connecting A and B
• calculate a number for a small part of the wiggle history; this number is called the lagrangian.
• sum up the lagrangian values along the wiggle history; this sum is called the action.
• single out one wiggle history from all the others, as having a minimum (or maximum) value of the action.

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:

• S: the total action
• a and b: the events at the start and end of the history
• m: the (rest) mass of the object
• c: the speed limit
• ds: the interval s for a tiny part of the object's history

Boost Transformation - Overview

The simplest case of a boost looks like this:

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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:

(ct,x,y,z)=(0,0,0,0)

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:

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Notice these things about the flow of events:

• events never cross the light cone
• events always stay within the same space-time region
• during a boost, events flow like toothpaste: some events are squeezed towards the origin, while others stretch away from it
• the flow is symmetric with respect to the light cone
• when β changes sign, the flow reverses

• put your left hand in front of you, with the palm facing you.
• point your fingers to the right. This indicates the direction of motion of grid 2 (with β>0).
• point your fingers up 45°, to imitate the 'X' of the light cone. (Coincidentally, this is the most comfortable position when holding your hand in front of your face.)
• imagine squeezing a tube of toothpaste with your hand in that position; this will indicate the direction of the flow of events.

Boost Transformation - Details

• transform a single event.
• transform multiple events that define a history. This is an important animation, and you should experiment with it.

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

• use a geometrical construction
• use a formula

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):

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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.

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

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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:

Directional Distance Compression

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:

• Δx=20 meters long (in the direction of motion)
• Δy=8 meters wide
• Δz=5 meters high

• Δx=20/Γ meters long (in the direction of motion)
• Δy=8 meters wide
• Δz=5 meters high

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

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

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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:

• 2-dimensional (ct, x): meters² (an area)
• 3-dimensional (ct,x,y): meters³ (an actual volume)
• 4-dimensional (ct,x,y,z): meters⁴ (there's no standard name for this)

High-Γ Travel

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

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?

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

• times are measured with clocks, not meter sticks
• the geometry of space-time is defined by the formula for s². In that formula, time and space are not treated the same - there's a difference in sign. The time is not treated as just another space coordinate.

Space-Time Geometry Constrains The Rules

• it transforms one set of 4 numbers (ct,x,y,z) to another set of 4 numbers (ct',x',y',z').
• the 4 numbers (ct,x,y,z) share the same units
• it uses only β and Γ, which are both dimensionless - they have no unit, and are pure numbers.
• mathematically, it guarantees the invariance of this quantity, built out of the 4 parts: c²t² - (x² + y² + z²). This is just the squared interval with respect to the origin event (0,0,0,0).

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

Boost Velocity

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

• the boosted speed never exceeds the speed limit (the large outer circle).
• when the object's speed is 1 in the unboosted grid, its speed remains 1 in the boosted grid.
• when the object's speed is 1, only the direction of the velocity vector changes. In this case, the transformation of velocity leads to the aberration of light.

End of Part I