Previously, I described the main formula of Special Relativity: $$s^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 – (\Delta t)^2.$$This formula tells us the amount of distance squared between two points (if \(s^2 > 0\)) or the amount of duration squared (if \(s^2 < 0\)).  (By using some trigonometry we can also use this formula to figure out the size of angles, so this encodes everything about the geometry).  All the crazy time dilation and distance contraction effects you’ve probably heard about are encoded in this formula.

Today I want to talk about the symmetries of spacetime.  What I mean by a symmetry is this: a way to change the coordinates \((t,\,x,\,y,\,z)\) of spacetime in a way that leaves the laws of physics the same.  Now I haven’t told you what the laws of physics are, but the important thing is that they depend on the geometry of spacetime.  So that means that we need to check in what ways we can change the coordinates of spacetime without changing the formula for \(s^2\).

The first kind of symmetry is called a translation.  This consists of simply shifting the coordinate system e.g. one meter to the right, or one second to the future.  This doesn’t affect the formula for \(s^2\) since it only depends on the coordinate differences \(\Delta t\), \(\Delta x\) etc.  We can write a time translation like this: $$ t^\prime = t + a,$$ i.e. the new time parameter \(t^\prime\) equals the old one plus some number \(a\).  Similarly, the three possible kinds of spatial translations are:$$ x^\prime = x + b; \\ y^\prime = y + c; \\ z^\prime = z + d.$$By choosing the numbers a, b, c, d, arbitrarily, one obtains a four dimensional space of possible translation symmetries.

The second kind of symmetry is more complicated, but you’ve certainly heard of it before—it’s called a rotation.  If we have two spatial coordinates, then we can rotate them by some angle \(\theta\) (measured in radians), which leaves all the distances the same.  The algebraic formula for a rotation looks like this:$$x^\prime = \phantom{-}\cos (\theta) \, x + \sin (\theta) \,y; \\ y^\prime = -\sin (\theta) \,x + \cos (\theta) \,y.$$That involves some trigonometry, but things look a bit simpler if we take the angle \(\theta\) to be a really tiny parameter \(\epsilon\), and just consider the resulting infinitesimal coordinate changes \(\delta x \approx (x^\prime – x)\): $$\delta x = \phantom{-}\epsilon\,y; \\ \delta y = -\epsilon\,x.$$ Translated into English, that says that if you rotate the y-axis of your coordinate chart a little bit towards the x-axis, you have to rotate the x-axis a little bit away from the y-axis (or vice versa if \(\epsilon\) is negative).  I’m too lazy to draw this, but if for some reason you can’t visualize it, a little bit of fidgeting with any rigid flat object should convince you.

Now actually we have three different spatial coordinates: x, y, and z.  That means that you can actually rotate in 3 different ways: along the x-y plane, the y-z plane, and the z-x plane.  Of course there are other angles you can rotate at as well, but they are all just combinations of those three; in other words the space of possible rotations is 3-dimensional.

But now, what about the time direction?  It would feel terribly lonely if it were left out, and in fact it is also possible to rotate spacetime about the t-x plane, the t-y plane, and the t-z plane.  However, remember how time is not quite the same as space?  Instead, it’s just like space except for a funny minus sign.  So not surprisingly, the formula for a rotation also has a funny minus sign—or rather, a funny absence of a minus sign: $$\delta t = \epsilon\,x; \\ \delta x = \epsilon\,t.$$So if you rotate the t-axis towards the x-axis (which corresponds to changing your coordinate system so that you are travelling at a constant speed), then the x-axis has to rotate towards the t-axis (which means that your notion of simultaneity has to change as well).  If you know how to integrate this with calculus, you can get the effects of a finite “rotation” in space (called a Lorentz boost) through an “angle” \(\chi\):$$t^\prime = \cosh (\chi) \, t + \sinh (\chi) \,x; \\ x^\prime = \sinh (\chi) \, t + \cosh (\chi) \,x.$$In the above, cosh and sinh are functions similar to cosine and sine but defined using hyperbolas instead of circles.

So this rotation has some wierd properties: It describes a crazy world (ours!) in which things rotate in hyperbolas instead of circles.  That’s because of the minus sign in the formula for \(s^2\) above, which makes it so the points of equal distance (or duration) correspond to hyperbolas instead of circles.  This has some additional consequences: 1) Because hyperbolas are infinitely long, the “hyperbolic angle” \(\chi\) ranges from \(-\infty\) to \(+\infty\), unlike circular angles which come back to where you started after you rotate through \(2\pi\) radians.  2) Because the two axes both move towards (or both move away) from each other, when you do a really big rotation it scrunches everything up towards \(t = x\) or \(t = -x\).  What this means is that when you accelerate objects more and more, they don’t go arbitrarily fast.  Instead they just get closer and closer to the speed of light.

In conclusion, spacetime has 10 kinds of symmetry: 4 kinds of translations and 6 kinds of rotations.  The space of possible symmetries is 10 dimensional.  It is called the Poincaré group.

P.S. In this whole discussion I have ignored the possibility of reflection symmetries such as \(t \to -t\) or \(x \to -x\).  These are also symmetries of the formula for \(s^2\), but they are discrete rather than continuous—there’s no such thing as a “small” reflection the way you can have a small rotation.  Adding these in doesn’t change the fact that the Poincare group is 10 dimensional.  However, these transformations are actually NOT symmetries of Nature.  They are violated by our theory of the weak force.  The only discrete symmetry like this which is preserved by the weak force is CPT: the combination of time reflection, space reflection, and switching matter and antimatter.

You’ve probably heard that time is the fourth dimension.  What does it mean?  It should seem rather fishy that time should be the same sort of thing as a spatial dimension.  We all know that you can only go in one direction in time—towards the future!  Time is measured in seconds, space in meters, etc.  It turns out though, that time can be thought of as the same sort of thing as space, but not exactly: there’s just one tiny change that makes everything turn out different.

Let’s start with two dimensional boring old Euclidean geometry of the sort one learns about in high school. It’s called two dimensional because you can specify the location of a  point using two numbers \((x,\,y)\).  Now the most important equation—from which everything else about geometry follows—is the Pythagorean theorem.  This allows you to compute the distance \(r\) between any two points \((x_1,\,y_1)\) and \((x_2,\,y_2)\).  If we define \(\Delta x = x_2 – x_1\) and \(\Delta y = y_2 – y_1,\) then we can think of \(\Delta x\) and \(\Delta y\) as the sides of a right-angled triangle.  Then the Pythagorean theorem says that then the distance is $$r = \sqrt{(\Delta x)^2 + (\Delta y)^2}.$$Now it turns out that the exact same formula works in three dimensional Euclidean space as well; you just have to add in the \(z\) coordinate: $$r = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}.$$Now our world has only three dimensions of space, so far as we know (though that hasn’t stopped physicists from exploring ideas such as string theory where there are more dimensions of space).  Nevertheless, nothing stops us from imagining that there are more spatial directions, say four.  (If you haven’t read the classic Flatland, go do it now.  Geometry as social satire!)  In that case, one would simply give the extra dimension a new name (say \(w\) since we’re out of letters at the end of the alphabet) and then write down $$r = \sqrt{(\Delta w)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}.$$Voilá!  Four dimensional space has arrived!

What has not yet arrived is spacetime.  Each of these dimensions are all exactly the same (it doesn’t matter which one we call “first”, “second”, “third”, or “fourth” because of rotational symmetry).  There is no notion of past or future.  To get a time coordinate, we have to do something to make it special.  And what we do is very simple, we just put in a minus sign: $$s = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 – (\Delta t)^2}.$$In deference to custom, I’ve renamed \(r\) as the separation \(s\), but that’s just a change of notation.  What you should be noticing instead is how similar this equation looks to the last one.  In fact, just as all of Euclidean geometry essentially follows from the Pythagorean theorem, so all of what’s called Special Relativity, or the geometry of Minkowski space \((x,\,y,\,z,\,t)\) essentially follows from this one equation.  (Minkowski was the name of Einstein’s math teacher, who first pointed out this way of understanding relativity.)

Now, what is the effect of this minus sign?  It turns out that it changes things rather a lot.  In Euclidean geometry, the thing inside of the square-root is always positive.  But because of the minus sign, the \(s^2\) of two different points can be either positive, negative, or zero:

• If \(s^2 > 0\), then its square root represents the distance between the two points, just like in Euclidean geometry.  In this case we say that the two points are “spacelike” separated.
• If \(s^2 < 0\), then the thing under the square-root is negative, so when we take the square-root we get an imaginary answer.  Fortunately, that’s okay!  We just flip the sign of the thing under the square root, and intepret it as a duration instead of a distance.  In this case the two points are “timelike” separated.
• If \(s^2 = 0\), then there is neither distance nor time between the two points.  In this case, we say that the two points are “lightlike” separated.  That’s because rays of light (in a vacuum) travel along paths whose points are lightlike separated—in a sense, because they are travelling equally in space and time, but they experience no time or distance as they travel.

I should say that I’m using units here where the speed of light (normally called \(c\)) is equal to one.  That means that e.g. if we measure time in seconds, we have to measure space in light-seconds (the distance light travels in one second).  Otherwise, the beautiful equations would get all cluttered up with \(c\)’s flying all over the place.

We’re used to dividing up time into three parts relative to ourselves: past, present, and future. The present is just an infinitesimal sliver, so in a sense this division is into two parts: points to the past have \(\Delta t < 0\) compared to you, while points to the future have \(\Delta t > 0\) compared to you.

However, special relativity tells us you have to chop up spacetime in a more complicated way.  Bearing in mind that you each live in a particular place as well as a particular time, you can chop up spacetime into three different regions.  The future is points that are timelike separated to you and have \(\Delta t > 0\); these are the points of spacetime that you can affect.  The past is points that are timelike but have \(\Delta t < 0\); these are the points that can affect you.  Then there is elsewhere, the points that are spacelike separated.  These points can neither affect, nor be affected, by each other.  The three regions are separated by the “light cone”, which consists of the points that you could send a lightray to (or from).  I’m too lazy to draw a picture right now, but you can see a pretty good explanation here.

Next up, we’ll talk about rotations.  As always, readers are free to ask me questions in the comments box.