Undivided Looking

I've told you so far that the gravitational field is encoded in a $4 \times 4$ matrix known as the metric.  Here it is, displayed as a nice table:

$$g_{ab} = \left( \begin{array}{cccc} g_{00} & g_{01} & g_{02} & g_{03}\\ g_{01} & g_{11} & g_{12} & g_{13} \\ g_{02} & g_{12} & g_{22} & g_{23} \\ g_{03} & g_{13} & g_{23} & g_{33} \end{array} \right)$$

There's 10 components because the matrix is symmetric when reflected diagonally.  The 4 diagonal components $(g_{00}, g_{11}, g_{22}, g_{33})$ tell you how to measure length-squared along the four coordinate axes.  For example, the length along the $1$-axis is given by

$$\Delta s = \sqrt{g_{11}} \Delta x^1,$$

where $\Delta x^1$ is the coordinate difference in the $1$-direction.  The remaining 6 off-diagonal terms keep track of the spatial angle between the coordinate axes.  If you know enough Trigonometry, you can figure out that the angle $\theta$ between e.g. the $1$-axis and the $2$-axis is given by this formula:

$$\cos(\theta) = \frac{g_{12}} {\sqrt{g_{11} g_{22}}}$$

However, I've also said that the metric depends on the choice of coordinates, which is arbitrary.  We can use this freedom to choose a set of coordinates where the metric looks particularly simple at any given point.   We can start by choosing our four coordinate axes to be at right-angles to each other.  This gets rid of all those funky off-diagonal components of the metric, which involve two different directions:

$$g_{ab} = \left( \begin{array}{cccc} g_{00} & 0 & 0 & 0\\ 0 & g_{11} & 0 & 0 \\ 0 & 0 & g_{22} & 0 \\ 0 &0 & 0 & g_{33} \end{array} \right)$$

If any of the four remaining numbers happen to be 0, we say that the metric is degenerate.  This would correspond to a weird geometry in which you can move in one of the directions for free without it affecting your total distance travelled.  Since we all know that's not the way the real world works, we'll ignore this possibility.

We can also rescale the tick marks along any coordinate axis.  This allows us to multiply each diagonal component of the metric by a positive real number.  So if say $g_{22}$ is positive, we can choose coordinates where it's $+1$, and if it's negative, we can choose coordinates where it's $-1$.  This gives us:

$$g_{ab} = \left( \begin{array}{cccc} \pm 1 & 0 & 0 & 0\\ 0 & \pm 1 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 \\ 0 &0 & 0 & \pm 1 \end{array} \right)$$

Since it also doesn't matter what order we list the four coordinate directions, all that matters is the total number of $+$'s and $-$'s.  This choice is called the signature of the spacetime.

Now if you remember my very first post on spacetime geometry, $+$ directions in the metric correspond to spatial dimensions, while the funny $-$ sign is what makes for a time dimension.  But the real world has one time dimension, everywhere.  No matter how far you travel, you'll never find a place (so far as we know) where there isn't any time direction, or where there are extra time dimensions.  So that means that the correct signature for spacetime has $(-, +, +, +)$ along the diagonal, which is called Lorentzian (a.k.a. Minkowskian) signature.  (If we had wanted to describe a timeless four-dimensional space, we would instead select the Riemannian (a.k.a. Euclidean) signature $(+, +, +, +)$.)  We conclude that for any point of spacetime, you can always choose a set of coordinates such that the metric takes a special form that we'll call $\eta_{ab}$:

$$g_{ab} = \left( \begin{array}{cccc}-1 & 0 & 0 & 0\\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 &0 & 0 & +1 \end{array} \right) = \eta_{ab}.$$

In other words, if you zoom in on any point, you recover Special Relativity.  So after all this fidgeting around, we end up with a somewhat profound conclusion: in General Relativity, every point of spacetime looks the same as every other point.

This is related to what Einstein called the Equivalence Principle, which says that at short enough distances, the effects of acceleration are indistinguishable from being in a gravitational field.  We all know from personal experience that riding in an elevator can make us weigh more or less, and from TV that astronomers in the Space Shuttle are weightless when they're in free fall.  In other words, you can always choose a coordinate system in which there is no gravitational force at any given point.

(Lewis Carroll actually described this principle several decades before Einstein in Sylvie and Bruno, which includes a description of a tea party taking place in a freely-falling house.  Then he describes what happens if the house is being pulled down with a rope faster than gravity would accelerate it, and explains how you could have a normal tea party as long as you have it upside-down.  I like this book better than his more famous classics, but don't read it unless you can withstand LD20 of Victorian sentimentality about fairy children.  Also, Carroll didn't go on to discover a revolutionary theory of gravity based on this principle.)

It might seem now like everything has become too simple.  If the metric looks the same at every single point, then why did we even bother with it?  Where's the information in the gravitational field?  Well, it's true that for any one point, there's a coordinate system where the metric looks just like $\eta_{ab}$.  But there's no coordinate system for which the metric looks like $\eta_{ab}$ everywhere at once.  (Unless there's no gravitational field anywhere, in which case Special Relativity is true).  If you make the metric look simple in one place, it has to look complicated somewhere else.

So in order to describe the gravitational field properly, we have to find a way to compare the metric at different points.  We can do this using something called parallel transport.  I'll give more details later, but basically it tells us how an object moves in a gravitational field when we carry it along a path through spacetime.  When we carry the object around a tiny loop so that it returns to its original position, we might find that it comes back rotated compared to its original orientation.  If so, we say that the spacetime contains curvature.  If the spacetime contains curvature, this is a fact about the gravitational field which is invariant, i.e. objectively true.  You can't eliminate it just by changing your coordinates.

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.