Undivided Looking

In my last post about spacetime, I explained how the geometry of spacetime is determined at each spacetime point by a set of 10 numbers.  These 10 numbers are packaged together into a $4 \times 4$ matrix called the metric, which is written as $g_{ab}$.  The subscripts $a$ and $b$ stand in for any of the 4 coordinate directions (in a 4-dimensional spacetime).  Since the metric is symmetric, i.e. $g_{ab} = g_{ba}$, there are 10 possible numbers in this matrix.  The value of these 10 numbers depends on your position and time,which makes them a field, specifically the gravitational field.

However, there is an important caveat in all this.  The coordinates which you use to describe a given spacetime are totally arbitrary.  For example, a flat 2-dimensional Euclidean plane can be described using Cartesian coordinates $-\infty < x < +\infty$ and $-\infty < y < +\infty$.  In this coordinate system, the distance-squared is given by the Pythagorean formula

$$(ds)^2 = (dx)^2 + (dy)^2,$$

which can be written in terms of the metric as

$$g_{xx} = 1; \qquad g_{yy} = 1; \qquad g_{xy} = 0.$$

On the other hand, for applications involving rotations, it's often useful to use polar coordinates: $0 \le r < +\infty$ (the distance from the origin) and $0 \le \theta < 2\pi$ (the angle around the origin, measured in radians).  They're related to the original coordinate system by

$$x = r \sin \theta;\\ y = r \cos \theta.$$

  In polar coordinates, the distance-squared is given by

$$(ds)^2 = (dr)^2 + r^2 (d\theta)^2,$$

where the extra $r^2$ factor comes in because circles that are a greater distance from the origin have a larger circumference, so there's more space as you move outwards.  This can be written in terms of the metric like this:

$$g_{rr} = 1; \qquad g_{\theta\theta} = r^2; \qquad g_{r\theta} = 0.$$

(Note: I've given these coordinate systems their traditional coordinate names to make them look more familiar, but this is actually just a redundancy to make it easier for humans to think about it.  I could have written the two coordinates as $(x^0, x^1)$—the superscript being a coordinate index, not an exponent—and then you could tell whether it was Cartesian or polar coordinates just by inspecting the formula for the metric.)

Now the point is, these two coordinate systems describe the same geometry in a different coordinate system.  If we were playing pool (or billiards) on a planar surface, and you wanted to describe how billiard balls bounce off of each other, you could equally well describe it using either coordinate system.  The physics would be the same.

Of course, the language you use to describe the system differs.  Suppose that I analyze a collision using Cartesian coordinates, while you use polar coordinates.  And suppose we had to communicate to each other what happened.  If you say to me, "The cue ball had a velocity in the $x^1$ direction", then I'll get confused because $x^1$ means something different to me than it does to you.  These kind of statements vary under a change of coordinate system, they are "relative" to your coordinate-perspective.  So if you want to communicate with me, you have to find a way to describe what's going on which does not refer to coordinates in any way.  For example, you could say "The cue ball hit the 3 ball, which knocked the 8 ball into a pocket."  Since the two balls and the pocket are unique physical objects, we can all agree on whether or not this happened, no matter what coordinate system we use.  These kind of statements are invariant under a change of coordinate system.  The goal of coordinate-invariant physics is to describe everything in this sort of way.

Here's another way in which coordinate systems can let you down: when you use polar coordinates, there are places where the coordinates go kind of funny.  For example, when you're going around the origin clockwise in the direction of increasing $\theta$, and you arrive at $\theta = 2\pi$, you immediately teleport back to $\theta = 0$ since you've come full circle.  Even stranger, space seems to come to an end at $r = 0$ (the origin) since there's no such thing as negative $r$.  And if you're sitting right at $r = 0$, the different values of $\theta$ all refer to the same point as each other.  However, in reality we know that nothing weird is happening to the geometry at any of these points, since nothing strange happens in Cartesian coordinates.  (A similar issue comes up in black hole physics.  The original set of coordinates found by Schwarzschild blow up at the event horizon, but actually nothing unusual happens there in classical general relativity.)

The upshot of all this for general relativity is the following: I told you above that you can describe general relativity using the metric $g_{ab}$, which involves 10 numbers at each point.  But this description actually has some redundancy in it, since there's infinitely many possible coordinates systems you could use (one for each way of labelling the points uniquely with four numbers), and the metric looks different in each one—it isn't an invariant object.

When a theory has redundancy like this, we say there is a gauge symmetry.  A regular symmetry says that two different states (i.e. configurations) of a system behave in the exact same way as each other.  A gauge symmetry is stronger than a regular symmetry: it says that the two configurations are actually the same physical state of affairs.  In general relativity, the choice of coordinates is a gauge-symmetry.  It is a mere human convention which doesn't correspond to any actual physical thing in Nature.

Of course, even if you aren't doing general relativity, you can still use whatever coordinate system you like!  Most games of billiards can be understood in the approximation where space is flat (unless you like to spice up your games with black holes and gravity waves, like the cool kids do!)  In flat space time, all coordinates are equal, but some are more equal than others.  Although nothing stops you from calculating in horrible coordinates, the laws of physics look especially simple in ordinary Minkowski coordinates, where the symmetries of spacetime look especially simple.  Since Newton's First Law of motion holds in these coordinates, we call it an inertial frame.  (Here I'm ignoring the downward pull of gravity, since in billards we're only interested in horizontal motions.)

However, if you're doing general relativity, then there's a property of spacetime which forces you to describe physics in a coordinate-invariant way; at least if you want the equations of the theory to look elegant and lovely instead of like horrendous cludge.  This property is called curvature—but we're out of time for today.

In Time as the Fourth Dimension?, I explained how to calculate the distance (or duration) squared between any two points of spacetime, using a spin-off of the Pythagorean theorem:

$$s^2= (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - (\Delta t)^2.$$

Then I explained the Ten Symmetries of Spacetime, i.e. ways to shift or rotate the coordinate system $(t,\,x,\,y,\,z)$ that don't change the formula for $s^2$.

Well, it turns out that I lied.  The formula isn't actually true, except in the special case that there is nothing in the universe.  A significant reservation, I know.  Instead, what's true is that the geometry of space is a field, meaning that it varies from place to place, depending on where you are!  However, if you zoom in really close at any particular point, it looks similar to the formula I told you.

The field that says what geometry is like at any given place and time is called (brace yourself) the gravitational field.  In order to describe it, we use something called the metric, which indicates what the geometry of spacetime looks like at any given point.  The way this works is, suppose we have two points $p$ and $q$ which are very close to each other.  Suppose we want to know the distance between these points.

Since the points are really close to each other, we call the distance between them $ds$, where the $d$ is just a reminder that we're using Calculus to study infinitesimal quantities.  If you don't know Calculus, just pretend these are really small numbers.  We want to figure out what $ds$ is, if we know the infinitesimal coordinate differences $(dx,\, dy,\,dz,\,dt)$.  The way we do this is by generalizing the heck out of the Pythagorean theorem.  I'll write it down, and then explain what it means:

$$(ds)^2 = g_{xx}\,(dx)^2 + g_{yy}\,(dy)^2 + g_{zz}\,(dz)^2 + g_{tt} \,(dt)^2 + \\ 2[ g_{xy}\,dx\,dy + g_{xz}\,dx\,dz + g_{xt}\,dx\,dt + 2_{yz}\,dy\,dz + g_{yt}\,dy\,dt + g_{zt}\,dz\,dt].$$

The right-hand side of the equation consists of every possible way of multiplying two of the coordinate distances $(dx,\, dy,\,dz,\,dt)$.  There are 4 different ways to pick the first $(dx,\, dy,\,dz,\,dt)$, and 4 different ways to pick the second, which gives $4 \times 4 = 16$ possible combinations in all.  However, multiplication is commutative so e.g. $dx\,dy = dy\,dx$.  So I added terms like that together; that's where the factor of 2 came from.  Taking that into account, there's 10 terms in all.

The funny $g$ things with subscripts are just functions of spacetime, i.e. they are just numbers that depend on where you are, i.e. they are fields.  In the special case where we pick these numbers to be $g_{xx} = g_{yy} = g_{zz} = +1,\,g_{tt} = -1$ and the rest zero, we get the geometry I told you about, which goes by the aliases "Minkowski space", "flat spacetime", and "Special Relativity".  In all other cases we have what is colloquially called "curved spacetime" which is the province of "General Relativity".

The formula above looks kind of ugly, but we can prettify it by choosing good notation.  We collectively refer to all ten of these gravitational fields as the metric, denoted $g_{ab}$, where subscripts like $a$ and $b$ can refer to any of the four coordinate labels.  (People often call these labels $(0,\,1,\,2,\,3)$ instead of $(x,\,y,\,z,\,t)$ to avoid confusion, since the metric itself says which of the coordinate directions behave more like space, and which behave more like time, and this can vary from place to place!)  Then we write the four coordinate differences $(dx,\, dy,\,dz,\,dt)$ collectively as $dx^a$, where the superscript says which of the four it is.  Finally, we make up a rule called the Einstein summation convention, that if we ever see the same letter as both a subscript and as a superscript, we add up all of the four possible ways for them to be the same (i.e. both 0, both 1, both 2, or both 3).  These are just changes in how we write things, not substantive changes, but they let us rewrite that long ugly equation like this:

$$ds^2 = g_{ab}\,dx^a\,dx^b.$$

There, isn't that much prettier?

Suppose we want to find the distance (or duration) between two points which are NOT infinitesimally close to each other.  In that case, we have to choose a path between the two points, since the amount of distance (or duration) depends on which path you choose, and in a curved spacetime there's not necessarily one "best" path.  This shouldn't seem that strange, since even in everyday life we know perfectly well that the distance between San Francisco and L.A. depends on which highway you take, and the distance between Tokyo and New York depends on which way around the globe you fly.  (It's totally intuitive for distances, but when the duration depends on the route you take through spacetime, people call it the Twin Paradox and act all shocked!)

So this is the first main idea of General Relativity: the geometry of spacetime is a field which varies from place to place.  This field affects matter by determining the paths that things take through space and time, but it also is affected by matter—we call this gravity.  The second main idea is that coordinates are an arbitrary choice; I'll tell you about this later.  The third main idea is the Einstein equation which says how matter affects the metric.  I haven't told you anything about this equation yet, but once I do, you would in principle be able to calculate everything about the gravitational field from that one equation.

There can also be distortions of the spacetime geometry which exist independently of matter.  These gravity waves are to gravity what light is to electromagnetism, ripples in the field which travel through empty space, and can be emitted and absorbed.  The propagation of these waves is also determined by the Einstein equation.  Since gravity comes from massive objects, gravity waves are emitted when extremely large masses oscillate, for example when two neutron stars orbit each other.  We know gravity waves are there, but we haven't detected them directly.  However, we hope to detect them soon with the LIGO experiment.

UPDATE: I realized that I never said how you would calculate the distance between two points, once you choose a path.  The answer is that you chop the path into lots of tiny little line segments, and find the length of each line segment using the metric.  Then you add them all up.  If you know Calculus, this can be done using an integral.

There's been a huge kerfuffle in the quantum gravity community since this summer, when some people here at UCSB published a paper arguing that (old enough) black holes may actually be surrounded by a wall of fire which burns people up when they cross the event horizon.  This is huge, because if it were true it would upset everything we thought we knew about black holes.

General relativity is our best theory of gravity to date, discovered by Einstein.  This is a  classical theory.  (In the secret code that we physicists use, classical is our code-word for "doesn't take into account quantum mechanics".  Don't tell anyone I told you.)

In my other posts on physics, I've been trying to explain the fundamentals of physics in the minimum number of blog posts.  This post is out of sequence, since I haven't described general relativity yet!  But I wanted to say something about exciting current events.

In classical general relativity, a black hole is a region of space where the gravity is so strong that not even light can escape.  They tend to form at the center of galaxies, and from the collapse of sufficiently large stars when they run out of fuel to hold them up.  A black hole has an event horizon, which is the surface beyond which if you fall in, you can't ever escape without travelling faster than light. The information of anything falling into the black hole is lost forever, at least in classical physics.

In the case of a non-rotating black hole, without anything falling into it, the event horizon is a perfect sphere.  (If the black hole is rotating, it bulges out at the equator.)  If you fall past the event horizon, you will inevitably fall towards the center, just as in ordinary places you inevitably move towards the future.

At the center is the singularity.  As you approach the singularity, you get stretched out infinitely in one direction of space, and squashed to zero size in the other two directions of space, and then at the singularity time comes to an end!  Actually, just before time comes to an end, we know that the theory is wrong, since things get compressed to such tiny distances that we really ought to take quantum mechanics into account.  Since we don't have a satisfactory theory of quantum gravity yet, we don't really know for sure what happens.

Now it's important to realize that the event horizon is not a physical object.  Nothing strange happens there.  It's just an imaginary line between the place where you can get out by accelerating really hard, and the place where you can never get out.  Someone falling into the black hole just sees a vacuum.  If the black hole was formed from the collapse of a star, the matter from the star quickly falls into the singularity and disappears.  The black hole is empty inside, except for the gravitational field itself.

We don't know how to describe full-blown quantum gravity, but we have something called semiclassical gravity which is supposed to work well when the gravitational effects of the quantum fields are small.  In semiclassical gravity, one finds that black holes slowly lose energy from thermal "Hawking" radiation.  This radiation looks exactly like the random "blackbody radiation" coming from an ordinary object when you heat it up. Here's the important fact: You can prove that the radiation is thermal (i.e. random) just using the fact that someone falling across the horizon sees a vacuum (i.e. empty space) there.

The Hawking radiation comes from just outside the event horizon.  It does not come from inside the black hole, so in Hawking's original calculation it doesn't carry any information out from the inside.   Nevertheless, for various reasons I can't go into right now, most black hole physicists have convinced themselves that the information eventually does come out.

As the black hole radiates into space, it slowly evaporates, and eventually probably disappears entirely (although knowing what happens at the very end requires full-blown quantum gravity).  If the outgoing Hawking radiation carries all the radiation out, then for a black hole at a late enough stage in its evaporation, the radiation must not be completely random, because it actually encodes all the information about what fell in.

The gist of what Almheiri, Marolf, Polchinski, and Sully argued, is that if we take both of these statements in bold seriously, then it follows that the black holes are NOT in the vacuum state from the perspective of someone who falls in.  Instead you would get incinerated by a "firewall" as you cross the horizon.  (It's not clear yet whether this is only for really old black holes, or if it applies to younger ones too.)  That's if we still believe there is an "inside" at all.  The argument shows that semiclassical gravity is completely wrong in situations where we would have expected it to work great.

If this is right, then it's devastating to the ideas of many of us who have been thinking about black holes for a long time.  As a reluctant convert to the idea that information is not lost, I'm wondering if I should reconsider.  At the end of this month, I'm going to Stanford for a weekend, since Lenny Susskind has invited a bunch of us to try to get this worked out.  Exciting times!

 

What is the world made out of?  In the most usual formulations of our current best theories of physics, the answer is fields.  What are those?

Well, if you know what a function is, you're already most of the way there.  A function, you will recall, is a gadget where, for any number you input, you can get a number out as an output.  We can write $f(x)$ where $x$ is the number you input, and $f(x)$ is the number you output.  The function $f$ itself is the rule for going from one to the other, e.g.  For example $f(x) = \sqrt{\sin x^2 + 1}$.

Now, nothing stops you from having a function that depends on multiple numbers as input; for example the function $f(x,\,y) = xy^2 + x^3y$ depends on two input variables, $x$, and $y$.  If there are $D$ input numbers, then the $D$-dimensional space of possible combinations of input numbers is called the domain of the function.

Also nothing stops you from having the output be a set of several numbers.  In this case we would need some sort of subscript $i$ to refer to the different possible output numbers.  For example, if we had a function with one input number $x$ and three output numbers $y$, then we could write $f_i(x)$, where $i$ takes the values 1, 2, or 3.  Then $f_i(x)$ would really be just a package of three different functions: $f_1(x)$, $f_2(x)$, and $f_3(x)$.  So if you specify the input $x$, you get three output numbers $(f_1, f_2, f_3)$.  If there are $T$ different output numbers, the $T$ dimensional space of possible outputs is called the target space.

Now a field is just a function whose domain is the points of spacetime.  For example, the air temperature in a room may vary from place to place, and it may also change with time.  So if you imagine checking all possible points of space in the room at all possible times, you could describe this with a temperature field $T(t, x, y, z)$.  However, the temperature field isn't a fundamental entity that exists on its own.  It subsists in a medium (air) and describes its motion.  When the air molecules are moving around quickly in a random way, we say it's hot, and when they start to move around slower, we say it's getting chilly.  An example of a field which actually is fundamental (as far as we know) would be the electromagnetic field.  This has 6 output numbers, since the electric field can point in any of the 3 spatial directions, and the magnetic field also has 3 numbers.

For a while in the 19th century, scientists were confused about this.  They thought that electromagnetic waves had to be some sort of excitation of some sort of stuff, which they called the aether.  That's because they were assuming (based on physical intuitions filtered through Newtonian mechanics) that matter is something solid and massy, which interacts by striking or making contact with other things.  The 20th century scientific advances partly came from realizing that its okay to describe things with abstract math.  Any kind of mathematical object you write down satisfying logically consistent equations is OK, as long as it matches experiment.  So electromagnetic waves don't have to be made out of anything.  They just are, and other things are (partly) made out of them.

In our current best theory of particle physics, the Standard Model, there are a few dozen different kinds fields, and all matter is explained as configurations of these fields.  I can't tell you exactly how many fields there are, because it depends on how you count them.  Not counting the gravitational field, there are 52 different output numbers corresponding to bosons, and 192 different output numbers corresponding to fermions (Don't worry about what these terms mean yet).  So you could say that there are 244 different fields in Nature, each with one output number.

That sounds awfully complicated.  But there's also a lot of symmetries in the Standard Model which relate these output numbers to each other.  This includes not only the Poincaré group of spacetime symmetries, but also various internal symmetries related to the dynamics of the strong, weak, and electromagnetic forces.  They are called internal because they don't move the points of spacetime around.  Instead they just mutate the different kinds of output numbers into each other.

So normally, particle physicists just package the output numbers into sets, such that the numbers in each set are related by the various kinds of symmetry.  (For example, the 6 different numbers of the electromagnetic field are related by rotations and Lorentz boosts.)  Each of these sets is called a field.  In future posts I'll give more details about the different kinds of fields.  As always, questions are welcome.

UPDATE: I forgot to include the 4 vector components of the spin-1 gauge bosons, so the numbers of degrees of freedom of the bosons were wrong before.  Note to Experts: These are the "off-shell" degrees of freedom before taking into consideration constraints or gauge symmetry.  Note to Non-Experts: the numbers in this post are just for flavor, in order to give you the sense that there are a LOT of different fields in Nature.  You won't need to understand how I got these numbers in order to enjoy future posts!

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 figeting 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.