Contents

## Why Special Relativity Is Interesting

Special Relativity (SR) is one of the main cornerstones of modern physics. If SR is wrong, then everything based on it will crumble too, like a house of cards. (For example, quantum field theories and the standard model of particle physics are based on SR.) If you can falsify SR, you get 10 Nobel Prizes, not just 1, and you get a gold star from your grade 3 teacher.

SR lets you understand the rules for time machines. Time machines into the past are strongly forbidden. Time machines going into the future are permitted, and use high speeds (or strong gravity fields).

SR has something very unusual. SR has a Rule Zero: a rule for making all the other rules. This Rule Zero is interesting because it's a set of handcuffs: it's very tightly constraining. Being tightly constrained is a good thing for a scientific theory, because it's easier to falsify. The handcuffs come from ideas about symmetry groups and invariance. With SR, fundamental physics starts to be guided by these important ideas.

SR is mathematically simple. If you know what a square root is, then you can understand the core of SR. For bonus points, if you're not afraid of a cosine, then you can understand what SR says about optics (photons and light rays).

SR is a reboot of physics. It goes back to the beginning, and fixes an important mistake. When modern physics began in the 1600s, they had certain ideas about space and time. We now know that some of those ideas were mistaken. (Those mistakes were completely natural, since they had no data for objects moving at really high speeds.) Albert Einstein was the first person to figure this out. He pointed out the mistake, which has to do with time. In short, the mistake was assuming that time is absolute. Instead of absolute time, you have an absolute speed limit, and an absolute space-time interval (see below).

When you learn SR, you get to understand how time dilation works. The discovery of time dilation is one of the most important discoveries in all of science. Some have compared its discovery with the discovery that the Earth is spherical, for instance.

When you know SR well, you can spot a common mistake - the mistake of forgetting about time dilation. (This mistake can even be made by trained physicists who should know better. In some cases, they know the theory, but they haven't really internalized it.) Take this statement as an example: 'The Pleiades are 400 light-years away. So, it will take at least 400 years travelling at near light speed to get there.' This statement is appropriate only in the world of Isaac Newton. In the world of SR, this statement, in unqualified form, is wrong. Why? Because in the world of SR, the time between two events in general depends on who's doing the talking. For a traveller going to the Pleiades at near light speed, the time that they care about is the time on their own wristwatch. And measured by their own wristwatch, that time is going to be less than 400 years. In principle, there's no minimum travel-time, as measured by the traveller. The closer you get to light speed, the smaller the travel-time becomes. That's time dilation.

## Summary of Important Results

Here's a quick summary of some important results in SR. More details are found below.

Speeds:

• there's a speed limit c for all signals (signals transfer packets of energy - matter, photons)
• v = c when the signal has no mass (photons)
• v < c when the signal has mass (matter)
• there's no speed limit for things that aren't signals (geometrical ideas, not matter or photons)
Time:
• a moving clock runs slowly compared to two stationary clocks
• high speed one-way trips are wormholes
• high speed return-trips are time machines into the future
• time machines can go to the future, but not to the past
• there's no minimal wristwatch-time for any trip between any two places in the Universe
Geometry:
• moving objects are flattened in the direction of their motion, with respect to stationary identical objects
Pairs of events:
• for any two events, the quantity c2(Δt)2 - (Δd)2 is invariant
• the time interval between the events is not invariant (absolute)
• sometimes the time-order of the pair is invariant, sometimes it's not
• a pair that's simultaneous in one grid, may not be simultaneous in another
Optics:
• high-speed travel changes the measured color, brightness, and direction of stars, moving them towards the direction of travel
• high-speed jets of matter have beaming effects that change their measured brightness
• high-speed jets of matter moving towards you can superficially appear to move faster than c, but this is only a geometrical effect
Other:
• space-time is an idea, not something you can point to
• there's a rule zero, a rule for making all the other rules
• principles are most effective (and beautiful) when they put strong limitations on possible theories

## The Idea of Space-Time is Pretty Simple

A ball rolls across the floor in a straight line from A to B. It starts by moving quickly, then it slows down and eventually stops. You can draw a picture of the ball's motion in different ways:

1. Draw a straight line from A to B.

This isn't very good, because there's no information about the speed of the ball. So this picture is incomplete.

2. Draw a straight line from A to B, but also add tick marks to the the line, indicating its position once per second.

This is better, because it has information about both the position and the speed. You could say that the first try has only information about where, and the second try has information about both where and when.

3. Draw a curve in space-time. To do that, just represent the time as another coordinate in the diagram, at a right angle to the x-axis.

In this case, the 1-dimensional line is replaced by a 2-dimensional plane. The tick marks that were used to indicate the time are replaced by a time coordinate. The time axis is placed vertically here, simply to follow a common convention. This is a 2-dimensional space-time. The points in this plane are events, consisting of a where (x) and a when (t). The path of the ball is a continuous sequence of connected events in space-time, which form its history, or world-line.

When the ball is motionless, the history is vertical. When the ball is moving, the history bends away from the vertical. The faster the ball is moving, the more its history bends away from the vertical.

In order to make these diagrams, you need convert the unit of time (seconds) into a unit of distance (meters). The conversion looks like this:

`seconds * (meters / second) => meters`
The conversion factor is a fixed quantity having the unit of speed (meters per second). The speed you pick is up to you. But, as explained below, most of the time the speed is chosen to be Nature's speed limit c (see below).

The basic idea of space-time is to convert times to lengths, and simply pretend that this time/length is an extra dimension, in addition to the space dimensions. It's important to understand that space-time, with its 'extra' dimension of time, is a geometrical idea, a mental abstraction. It's not something you can physically point to.

The idea of representing the history of an object as a curve in space-time is not just a mathematical curiosity. 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 (by convention), 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:
• 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.

## Rebooting Physics

SR is a reboot of physics from the beginning. Physics begins with examining motion, and the examination of motion begins like this:
• using grids to measure motion
• the rule of inertia
• the inertial grid family: the group of grids in which the rule of inertia holds
So let's examine these ideas in sequence.

## Grids Measure Motion

The motion of an object is always relative to something else. A grid (or a frame of reference) is used to measure positions and times. A grid is most easily pictured as a cubic lattice. The lattice is constructed out of identical meter sticks (or whatever you prefer). In the simplest case, you can picture the lattice as being in empty interstellar space, far away from large objects (which create gravity fields). The objects making up the grid are simply placed into position, and aren't rigidly attached to each other. Here are some pictures to illustrate (these are diagrams of space, not space-time):

The idea of course is to use numbers to state the position of an event in an object's history. Let's take the 3D case (3 space dimensions):

• traditionally, the 3 numbers are labeled x, y, and z.
• each of these 3 numbers corresponds to a distance measured along one of the axes (x, y, or z).
• there is an origin point, where the axes all meet, and where x, y, and z are all 0.
• these axes are at right angles to each other.
• the sign of the numbers indicates direction, for example the +x direction, or the -x direction.
So far, this scheme only measures positions. Next we'll look at measuring time with a grid.

## Grids Measure Time With Many Sensors, Not One

To measure time, you might think that only a single master clock would be needed for a grid. But that's a bit more work, because you need to figure out how to measure the time of events that occur at some distance from the clock. It's simpler to use multiple synchronized clocks, that are spread out over the entire grid. The clocks are all identical. (You can picture these clocks as being placed at the intersection points of the meter sticks.) When you measure the position of an object, you can measure the time as well, just by using the clock at the same location. When you simultaneously measure both the position and the time, then you measure the coordinates of an event, a where and a when. This is simpler than using a single master clock, since everything remains completely local.

Now we have a system for measuring the positions of things in space, combined with many stationary, synchronized clocks (or sensors) spread out over that system to measure the time as well. Calling them sensors instead of clocks is useful, since they have to detect and record local events.

Now picture making measurements with such a grid. Imagine being in interstellar space, building a grid, and then throwing objects through the grid. As the object passes by each sensor, the sensor detects the object, and records the event: at time t, the object was at some sensor location (x,y,z). The sum total of all these events (t,x,y,z) is a measurement of the history of the object. You could imagine a master computer that gathers together all these events, and reconstructs the history of the object.

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 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.

To summarize, when you think of a grid, you should have this sort of picture in your head:

• a cubic grid of sensors spread out over an area or volume (either 2D or 3D)
• the sensors are separated by meter sticks
• the sensors and the meter sticks 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
• the grid can have any orientation, location, or speed, but it's not allowed to be spinning or accelerating (more on acceleration below)
• the measurements can be sent to a central computer where you can analyze them
Some authors use an alternative scheme modeled on a radar station instead of a grid. In that scheme, light signals are emitted from a base station, bounce off a target event, and then return back to the base station. Such schemes actually represent a more practical method for measuring co-ordinates than the grids described above.

However, in the radar base scheme:

• measurements are never local
• small calculations are needed to deduce the space and time co-ordinates from the radar data
• as a mental model, they aren't as simple as the grid scheme

## 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.

It's important to keep the ideas of a grid and a single camera very clear in your head. They are very different things. You need to know which of these two ideas is being used in a given context.

Many references use the word observer. That term is completely avoided here, because of its ambiguity. The word observer, in its typical usage, connotes a single thing - a single camera/sensor. But in SR, in most cases, it's really referring to a grid.

## Inertial Grid What's the simplest kind of motion in the Universe? To see 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 idealized 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 (an inertial frame of reference). 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:

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

In principle, there are an infinite number of possible inertial grids. 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

It's very important than inertial grids form a family (or what mathematicians call a group). If you have one inertial grid, then you can make another one by applying one or more of the four allowable grid transformations:
• change the starting point (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 directions.
• mirror reflection: swap the directions of the positive and negative directions for 1 or 3 axes. (The case of swapping 2 axes is actually the same as a rotation, so it's left out here.) You can do this for the time as well, by having the clocks go backwards into the future.
• boost: let the grid move in some fixed direction, at a constant speed (this is the only change that involves one grid moving with respect to another).
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:

• 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). Here are the most important ones:

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.

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 specific to the universal?

One way of moving from the specific 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 volume of a region in space-time (discussed below)
• the electric charge and rest-mass of any particle
• E2 - H2, the difference in the squared sizes of electric and magnetic fields at a given place and time
• ...and many others...
A better name for theory of relativity might be the theory of invariance, the theory of space-time, or something similar. The name theory of relativity is mediocre: it's ambiguous, leads to confusion, 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 (if you'll pardon the pun).

## What Does Relative Mean?

Something is relative when it's not invariant. Items that change according to which grid is being used are said to be relative to that grid.

One of the main points of SR is that it changes the list of items that are invariant versus relative. Here's a list of some items of note (the list is incomplete):

ItemBefore SR (Newton)SR (Einstein)Comment
Simultaneous eventsinvariantrelativeNewton's time was absolute
Time between eventsinvariantrelativeNewton's time was absolute
Rate of a moving clockinvariantrelativeNewton's time was absolute
Length of a moving stickinvariantrelativeSR has the pancake effect
Spatial geometry of an objectinvariantrelativeSR has the pancake effect
Distance between eventsrelativerelativeWas always relative, even before SR
Frequency of lightrelativerelativeDoppler effect was known before SR
Direction of lightrelativerelativeAberration of light was known before SR

## Rule Zero: The Rule About All The Other Rules

Invariants lead directly to an important 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 a sense, relativity is the search for invariants - the search for what we can all agree on. This is a kind of 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 falsify.

[An aside for experts] The rule-about-rules is often stated like this: The rules of physics have the same form in all inertial grids. But what exactly does it mean to say that a rule has the same form? What exactly is the same? The 'same-ness' of differential equations can be precisely expressed only with 4-vectors and covariance. But that language isn't appropriate for beginners. It's too complicated.

It's also unnecessary. The fundamental covariant differential equations can be derived from a corresponding action integral. Action integrals are scalar invariants - quantities that are the same for all inertial grids. Hence, the principle of relativity can (and should) be stated in a simple form which speaks only of invariants - concrete numbers that can be derived from measurements - as opposed to more abstract ideas about the form of covariant equations.

## 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? A physical object (matter, radiation) that moves a packet of energy 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 one way to synchronize all of the clocks in a grid. Choose a reference clock. The reference clock sends out a flash of light, say at time 0. When the flash of light reaches another clock, the flash is detected and it triggers the clock to start it running.

The trick is to preset the start-time of each clock to a certain value. That value is calculated using the known distance Δd of the clock from the reference clock, and the formula cΔt = Δd.

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 importantly, how could they measure the same value for the speed limit?

So it appears that the these ideas are a complete failure, 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.

## 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 Collider0.9999999895816072
Primary cosmic rays - most energetic0.99999999999999999999995598229

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
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:
• if β is a speed: 0 ≤ β ≤ 1
• if β is a velocity: -1 ≤ β ≤ 1

## Geometric Points Have No Speed Limit

The speed limit rule applies to signals - to physical objects carrying energy 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:
• 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)
What's moving in these cases is not a single physical 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.

βAngle
0.6031.0°
0.7035.0°
0.7536.9°
0.8038.7°
0.9042.0°
0.9543.5°
0.9944.7°
βAngle
0.0000.0°
0.1005.7°
0.2011.3°
0.2514.0°
0.3016.7°
0.4021.8°
0.5026.6°

## 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: SymbolMeans
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. When s2 is negative, the value of s is imaginary. (Warning: people often refer to both s and s2 as the interval, but that's sloppy language.) To state it explicitly, the rule is this:

The value of s2 for any two events is invariant.

The space-interval Δd has always been specific to a grid: when one grid is moving with respect to another, then the distance Δd between 2 events will, in general, vary between the 2 grids. That's not new. What is new is that the same applies to the time-interval Δt as well. That's new, because when modern physics began in the 1600s, the time-interval was assumed to be universal. But that was a fundamental mistake. Instead, it's the space-time interval s, constructed from the space-interval Δd and the time-interval Δt, that's really universal, and neither of its parts. That is, time is specific to a grid, not universal.

Roughly speaking, time and space can be converted into each other, but only as long as s2 stays the same.

"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

Another important thing to know about s2 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 energy changes the formula for s2, and thus how energy warps the normal (or flat) 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. Time is relative, not absolute.
(The last rule can actually be seen as 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.
Notation alert #1. s2 is really a habit brought over from the geometry of space. It may not really be appropriate to the geometry of space-time. The problem is the sign: in the geometry of space, the distance is always greater than or equal to 0. In space-time, s2 can be of either sign (or zero). If you take the square root of s2, you can end up with an imaginary number.

It's not actually necessary to define

c2(Δt)2 - (Δd)2

as being the square of something (for example, see J. Synge, Relativity: The Special Theory).

Notation alert #2. The expression for s2 is usually written without parentheses as:

s2 = c2Δt2 - Δd2

s2 = c2(Δt)2 - (Δd)2

## 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 cone opening upwards along the +ct axis, called the future light cone, and the other cone 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.

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 (or just the past) of A, and the region inside the future light cone is called the absolute future (or just the 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 (or just the elsewhere).

Note that events can't cross over from one region to another when you change grids. 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:
• 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 A; 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.

One can imagine the term light cone being replaced with something less specific to light:
• null cone (which is actually used by some authors)
• causality cone
• speed limit cone
• limit cone

## Surfaces With Constant s2

As shown above, the sign of 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(Δt)2 - (Δd)2 = 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:

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

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 having mass can approach the speed limit, but it can never reach it.

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 s2", this site uses the more concise term interval shell (or simply shell for short).

From The Special Theory of Relativity, by David Mermin:

So although space and time can, to some extent, be transformed into each other in a change of reference frame, there are certain limits on this transformation, in the sense that an interval inside the light cone cannot be transformed into one outside the light cone, or on the light cone.

So a certain subtle kind of difference between space and time is preserved (in the sense of a lack of complete interchangeability under transformation)....

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.)

## Measure Time Dilation

In principle, the measurement of time dilation in the lab is simple. (See this video from 1962 for a demonstration of measuring time dilation with muons.) In the following diagram, the clock X moves past two stationary clocks A and B.

Pay attention to 2 events: X coincides with A, and X coincides with B.

For both of these events, you take the readings on the two clocks. You will have 2 readings for X, and 1 reading each for A and B. Question: is the time interval between these 2 events the same, as measured by the 1 moving clock, and the 2 stationary clocks? The answer is no.

It's helpful to write it as a ratio, where the numerator is the difference in readings on the 2 stationary clocks (A and B), and the denominator is the difference in readings on the 1 moving clock (X).

(B - A) / ΔX

Experimentally, what you find is that the ratio is not 1.0, but greater than 1.0:

(B - A) / ΔX > 1.0

The strangest part of special relativity is this effect, called time dilation. At low speeds, the ratio is very close to 1.0, and the effect is hard to detect. But as the moving clock moves faster and faster, the ratio gets larger and larger. A moving clock runs slowly with respect to two stationary clocks.

## The Faster The Object, The Slower The Physics

Let's look at time dilation using space-time and intervals. 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:

• 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
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:

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.

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 specific to a grid, 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.

Here's a good way to describe time dilation: a moving clock runs slowly with respect to two stationary clocks.

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
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.

## Reverse The Perspective

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 slowly with respect to stationary clocks. 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:

• first, the phrase "they are both going slow at the same time" assumes what's not true - that the passage of time is universal.
• second, you need to examine exactly what's being measured and how. Each case uses 2 pairs of measurements taken from 3 sensor clocks. There's no contradiction because the measurements that show the slowing-down are completely different.

To see this, just look at it carefully from both perspectives, and think about how the measurements are made.

In the following diagram, first we have clocks X,Y moving past clocks A,B. Then we have the opposite perspective on the same situation: clocks A,B moving past clocks X,Y (in the opposite direction, of course). There are 4 clocks in total, 2 attached to each grid.

To measure the time dilation effect in a given grid, you use only 3 of the 4 clocks, and you ignore the remaining one. Schematically, you can write it like this, where the numerator is the difference in readings on 2 stationary clocks, and the denominator is the difference in readings on 1 moving clock. The main point is that both of these ratios are greater than 1.0:

(B - A) / ΔX > 1.0

(Y - X ) / ΔA > 1.0

(In the first case, we choose to ignore Y, in the second we choose to ignore B.) SR simply says that these two ratios are equal.

(B - A) / ΔX = (Y - X) / ΔA

But note that the two sides come from different measurements. That's why there's no contradiction between the two perspectives.

## Warp Factor

The above ratio of time intervals 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 γ. 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.

βγ
0.001.0
0.751.5
0.872.0
0.985.0
0.997.1
0.99510.0

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:

The formula for γ is simply: ## 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:

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. The 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 an object moving freely through a sensor grid. How can you measure the interval between any two events on the object's history? Well, there are two ways:
• 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.
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.

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 these sensors 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 = 0.99999998958
γ = 6928
travel-time on the proton's wristwatch:
= 385 years / 6928
= 20 days, 7 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.

## 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:
• the elapsed time for the parent is the greatest
• the elapsed time for the children varies, but is always less than the parent
This is just the effect of time dilation, of course. A moving clock runs slowly with respect to two stationary clocks. 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 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 specific to a grid, not universal.

## 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:

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 aging 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

The above idea of wiggle histories, along with one history corresponding to a maximum value of some quantity, is an example of an action principle. Fundamental 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:
• classical mechanics
• special relativity
• electrodynamics
• geometrodynamics
• quantum field theories

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.
In the case of the maximum aging 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 c and the space-time squared interval s2. 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
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:

We use two grids, grid G with unprimed coordinates (ct,x), and grid G' with primed coordinates (ct',x').

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

`(ct,x,y,z)=(ct',x',y',z')=(0,0,0,0)`
The only difference is that grid G' (in red) moves along the x-axis, in either the positive or negative direction, with respect to grid G (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 G, the task is to find the coordinates for the same event in grid G' (the boosted grid), which is moving in this specific way with respect to grid G.

As always, each event must keep the same squared interval s2 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:

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
• the flow is always along an interval shell (the hyperbolas)
• when β changes sign, the flow reverses

## Boost Transformation - Details

Here are some animations for exploring boost transformations in more detail:
• 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 G':

• 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 G' moving to the right, in the +x direction), and β=-0.5 (grid G' moving to the left, in the -x direction):

These space-time diagrams are a bit unusual, since they have two sets of axes. The red set of axes for G' is at an oblique angle with respect to the black set of axes for G. Don't attach too much importance to the distorted appearance of the axes for the G' grid. The distortion isn't "real", it's just how the G' grid looks with respect to the G grid. From the G' grid's point of view, the G grid 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 red 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 red dots, as projected on the ct axis, 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 the G' grid, 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 G' grid, since it doesn't have perpendicular axes.

Instead of this geometric construction, you can also easily calculate the event coordinates in the G' grid. 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: Let's call these equations the boost transformation (also called the Lorentz transformation). These equations describe how an event changes coordinates during a boost from one inertial grid to another.

Remember that -1<β<1. That is, β is the velocity along the x axis, not just the speed. It can be either positive (G' grid moving in the +x direction) or negative (G' grid moving in the -x direction).

If you know about matrices, then you can view the boost transformation as a matrix equation: When a square matrix multiplies a basis vector like (1,0), the result is simply one of the columns of the matrix itself. In our case, the event co-ordinates (1,0) and (0,1) are mapped to the two columns of the boost transformation. This provides another way of finding the axes of a boosted grid.

(ct,x) in G(ct',x') in G'
(1,0)(γ, -γβ)
(0,1)(-γβ, γ)

## Active Versus Passive Boosts

A space-time diagram can show the effect of a boost transformation in two different ways, called active and passive:

NameDescriptionNum EventsNum Sets of Axes
ActiveThe boost "moves" the event21
PassiveThe boost "moves" the axes12

This web site usually uses the active style.

These two kinds of diagrams are simply different ways of describing the same physics. Both styles have their uses. Use whichever style makes sense to you in a given case. The names active and passive shouldn't be taken too literally. They are just convenient names for distinguishing the two styles.

The following examples show how a boost transforms the event (ct,x)=(1,0) from the unprimed grid (in black) to the primed grid (in red). The primed grid G' moves with respect to the unprimed grid G along the positive x-axis. The example shows both the active and passive styles of showing a boost in a space-time diagram.

This example also illustrates the effect of time dilation, and the source of its γ factor.

In the unprimed grid G (black), two ticks of a clock are represented by:

• tick #1: (ct,x)=(0,0), the origin event
• tick #2: (ct,x)=(1,0)

The important point is that in the primed grid G' (red), the second tick has ct' = γ.

## Time Order Can Change

We're usually concerned with time-like intervals, since only time-like intervals occur along the history of an object having mass. 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:

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.

If two events with a space-like interval are simultaneous in one grid, then they will not be simultaneous in a boosted grid. That is, the idea of two events being simultaneous is relative, not invariant. In other words, there is no universal now.

In a given grid, you can consider the events having ct=constant. Let's call that a time-slice. Those events are simultaneous in that grid. But time is relative, and a time-slice in one grid G is not a time-slice in a boosted grid G'. That is, events that are simultaneous in grid G will not be simultaneous in a boosted grid G'.

Remember that in a boost, the axes of the boosted grid are at an angle with respect to the unboosted grid:

In grid G, a time-slice has ct=constant, parallel to the x axis. In grid G', a time-slice has ct'=constant, parallel to the x' axis. The time-slices in the two grids are never parallel, and can never represent the same set of events.

From It's About Time, by David Mermin:

That no inherent meaning can be assigned to the simultaneity of distant events is the single most important lesson to be learned from relativity.

From General Relativity, by Robert Wald:

The notion that there is absolute simultaneity is a deeply ingrained one. The fact that there is no such notion is one of the most difficult ideas to adjust to in the theory of special relativity.

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:

Separation of two events:Time-likeSpace-like
Time order is invariant:yesno
Cause-effect relation allowed:yesno
In each other's light cone:yesno

## Pancake Effect (Length Contraction)

Here, it's important to distinguish between these two ideas:

• the distance between two events in a given grid (always relative, for both Newton and Einstein)
• the distance between two histories at a given time in a given grid (invariant for Newton, but relative for Einstein)

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 space-time histories. Then you pick a time, and draw a corresponding line (or flat surface) ct=constant in space-time. This line (or surface) will cut across the two histories. Let's call this a time-slice. The distance between the two objects at that time, and in that grid, is then just the Δx along that line (or surface).

More generally, a time-slice is used to define the spatial geometry of an object at a given time, in a given grid. Since time runs differently in different grids, the geometry of an object is relative to a grid, and not invariant.

Let's return to the Zoomer and Home Base spacecraft, and look at the spatial geometry (especially the size) of the Zoomer spacecraft according to a grid attached to the Home Base spacecraft.

In the Zoomer grid the spacecraft is at rest, and 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
Does the Home Base grid measure the same dimensions for the moving Zoomer spacecraft? No. It measures these dimensions:
• Δx=20/γ meters long (in the direction of motion)
• Δy=8 meters wide
• Δz=5 meters high
In summary, the dimensions of the Zoomer spacecraft are:

Dimension Zoomer GridHome Base Grid
length 20 m (20/γ) m
width 8 m 8 m
height 5 m 5 m

For y and z, the dimensions are the same. The dimension of an object in the direction of motion is divided by γ (with γ calculated using the speed of Zoomer with respect to Home Base). Since γ is greater than or equal to 1, that dimension is smaller (contracted). The same can be said of the volume of the moving object as well, with the volume also divided by the same number γ.

Let's call this the pancake effect. It is highly directional. It takes place only in the direction of relative motion. The more usual name given to this effect is the Lorentz-Fitgerald length contraction.

An ultra-relativistic ball is flattened into a pancake shape. When measured in the lab, its size along its direction of motion will be less than that of an identical, stationary ball. Of course, in the rest frame of the ball, the ball remains round. The effect comes from the geometry of space-time, not from physical changes in the objects themselves. This means that the geometry of an object is relative, not invariant. The geometry of an object depends on the grid that's measuring it.

In general, the pancake effect distorts the geometry of objects in their direction of motion. An object's spatial geometry, defined as the intersection of a ct=constant time-slice with its history, is relative to the grid that's measuring it.

The pancake effect flattens out angles. Take for example a grid in which a stick moves rapidly, and the stick is oriented at an angle to its direction of motion. The faster the stick moves, the more transverse it becomes to its direction of motion.

A quote from Relativistic Mechanics, by R. D. Sard:

If the rod's orientation is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are different in the rod's rest frame and the laboratory frame.

The inequality of [the angles of a stick with its direction of relative motion, in two different grids] is as much of an affront to pre-Einsteinian habit as is the length contraction and the slowing down of clocks. A weathervane pointing northeast on a house flying east with speed such that γ=2.4, points north-northeast in the ground frame!

Both time dilation and length contraction 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 Δx are specific to a grid. 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 distance of 385 light years from the Earth is a statement attached to a specific grid comoving with the center of mass of the solar system. In other grids, moving with high γ with respect to the solar system along the line joining the solar system to the Pleiades, it's a different story: the pancake effect changes the distance to the Pleiades, and divides it by the value of γ.

## Why Size Decreases

The following space-time diagram shows why the pancake effect makes the object smaller and not larger in the direction of relative motion.

The diagram shows the history of a stick, as the shaded area in grey.

• in grid G, the stick is stationary and parallel to the x axis.
• grid G measures the length of the stick using a time-slice AB having ct=constant, parallel to the x axis.
• grid G' is moving to the right with respect to G, along the x axis.
• grid G' measures the length of the stick using a time-slice AC having ct'=constant, parallel to the x' axis. From above, we know that the x' axis is at an angle with respect to the x axis in the space-time diagram. So, AC makes an angle with AB.
• the two time-slices across the history of the stick use different events for measuring the length of the stick. In the diagram below, the events A and B are used in grid G, and the events A and C are used in grid G'.

Grid G Grid G'
Stick is: stationary moving
Time slice: AB (ΔtAB = 0) AC (Δt'AC = 0)
Stick length: L = ΔxAB L' = Δx'AC
-s2AC: L2 - c2(ΔtBC)2 L'2 - 0

The last row has the squared-interval between events A and C as measured in both grids. Since the value of the squared-interval is invariant, those two values must be equal:

``` L'2 = L2 - c2(ΔtBC)2 ```

Which means the stick length in grid G' is smaller than the stick length in grid G:

``` L'2 < L2 ```

``` L' < L ```

## Does It Really Dilate/Contract?

There are two common questions:
• does the moving clock really slow down?
• does the moving stick really get shorter?

In SR, the short answer to these two does-it-really questions is yes, while the long answer is:

• the measurements depend both on the state of the clock/stick at rest and on how the clock/stick is moving with respect to the chosen grid
• measurements are always real; measurements made in one grid are just as real as measurements made in any other grid

The measurements are relative: they don't depend solely on the thing itself. The measurements depend both on the thing itself, and on how the thing is moving through a specific grid.

These two does-it-really questions are ambiguous. There are at least two possible points points of view the questioner may have:

• the measurements can't show different results (the absolute or Newtonian point of view)
• the measurements can show different results, but one is somehow real while the other is not
Both of these points of view are incorrect.

## 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. 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. This invariant is not as important as the other invariants described above, but it's useful to be aware of it.

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

## Time Machines and Wormholes

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 length contraction 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 changed by length contraction, 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.

Machine How To Build It
Wormhole high-γ travel, one-way trip
Time machine into the future high-γ travel, return trip (also: return trip to a black hole)
Time machine into the past strictly forbidden, reverses cause-and-effect

In practice, such machines don't exist, because it's extremely difficult to accelerate a large mass to such high speeds. But, the rules of physics allow for their construction in principle.

For Hollywood movies, the only problem is that time-travel into the past is strictly forbidden.

## 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:

• 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.
Because of this, many people refer to space-time as having a 1+3 structure (or 3+1), 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:

• 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).
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.

From Einstein Gravity in a Nutshell, by Anthony Zee:

I regard Einstein's understanding of how symmetry dictates design as one of the truly profound insights in the history of physics. Fundamental physics is now conducted largely according to Einstein's schema rather than that of 19th century physics.

Einstein taught us to deduce physics from symmetry, instead of symmetry from physics.... Instead of laboriously distilling this theory from a motley collection of experimental facts and then extracting a symmetry, he formulated a symmetry empowering him to to write down his theory of gravity in one fell swoop.

## 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.

The same sort of thing happens in Quantum Field Theory, which more or less pops out of these 3 core ideas (not explained here):
• Lorentz invariance (the standard name for invariance in SR)
• renormalizability (removing infinities from the theory)
• local gauge invariance
"From the beginning it seemed to me a very wonderful thing that very few quantum field theories are renormalizable. Limitations of this sort are, after all, what we most want; not mathematical methods which can make sense out of an infinite variety of physically irrelevant theories, but methods which carry constraints, because these constraints may point the way to the one true theory."

- Steven Weinberg [Nobel Prize lecture, 1979].

## Boosted 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:

• 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.

## Several Core Items Are Dimensionless

These core items in SR are all dimensionless (they have no units):   D is the Doppler factor, discussed in Part II.

The coefficients in the boost transformation (γ and -γβ) are also dimensionless, and define the transformation of all 4-vectors: Dimensionless quantities are infrequent in physics, so perhaps this fact is noteworthy.

## Avoid Common Pitfalls

The word observer is dangerous
The idea you have of an inertial grid (frame of reference) needs to be crystal clear. It's critical that you make the distinction between a grid (used in the core theory) and a single camera (used in optics). This isn't a small detail. You need to know about this distinction. If you don't know about it, then you don't understand the theory well.

Many authors use the term observer to refer to a grid or frame of reference. This is probably a bad idea. The word observer has exactly the wrong connotation for a frame of reference. It puts the wrong image into your head. A grid needs to fill space, with many meter sticks and synchronized clocks. The word observer does not push you toward that image. It pushes you in the exact opposite direction, toward a camera, not a grid. That's not good.

The term relativistic mass is usually deprecated
Many authors are uncomfortable with the term relativistic mass, since it can lead to confusion. Instead, they suggest that the mass of an object should be defined as its invariant rest mass. In this way, the mass also corresponds to the length of the object's energy-momentum 4-vector.

The minimal travel-time isn't limited by c
Many people have gone through the ideas of SR, but without fully internalizing what it means. The best example of this are remarks like the following: "The Pleiades are 400 light-years away. So, travelling at the speed limit, it will take a minimum of 400 years to get there." This is wrong, at least in its unqualified form. According the traveller's wristwatch time (which is what the traveller cares about), the time will be less than 400 years, if they are travelling at ultra-relativistic speeds, because of time dilation. Indeed, there's no lower limit to the travel-time, according to the traveller's wristwatch.

## How Special Relativity Relates to Other Parts of Physics

The diagram below shows how you might relate the core subjects in physics to each other. Each part depends on the items directly beneath it. Some parts are left out, such as Fluid Mechanics and Nuclear Physics. Classical Mechanics (CM) is based on ideas about space and time used in the time of Galileo and Newton (absolute time, no speed limit for signals, and so on), so you could say that CM in turn depends on those ideas.

You can see that a lot is based on Special Relativity (SR), and that it acts as a cornerstone.

 SR Special Relativity CM Classical Mechanics RM Relativistic Mechanics QM Quantum Mechanics (non-relativistic) SP Statistical Physics (and Thermodynamics) GR General Relativity EM Electrodynamics QFT Quantum Field Theory QED Quantum Electrodynamics QFD Quantum Flavordynamics QCD Quantum Chromodynamics QEW Electroweak Interaction

The hash marks between CM (classical mechanics) and QM (quantum mechanics) symbolize the drastic differences in the core ideas of those subjects. The transition from the deterministic world of classical mechanics to the probabilistic world of quantum mechanics is a deep conceptual schism in physics.

The arrow from Quantum Mechanics to Statistical Physics symbolizes that SP depends a bit on QM. Statistical Physics was invented before the creation of Quantum Mechanics. But Statistical Physics is simpler when it uses ideas from the quantum world, such as the enumeration of discrete quantum energy states.

In this context, it's interesting to note this quotation from Quantum Mechanics (Non-relativistic Theory), by Landau and Lifshitz:

"Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

A quotation from Statistical Physics, also by Landau and Lifshitz:

"Only the concept of the number of discrete quantum states, which necessarily involves a non-zero quantum constant [Planck's constant h], enables us to define a dimensionless statistical weight and so to give an unambiguous definition of the entropy."

## 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.