Since you mentioned it, what the heck is a spacetime metric?

First of all, nice chinchilla.

The full explanation of the spacetime metric is VERY mathy. Let me try to summarize a bit.

The full form of Einstein's general relativity equation looks like... erm I have no idea how to do equations in this...

So it's this equation with something typically written as "g sub uv" on the left and "T sub uv" on the right.

We call the "T sub uv" the "stress tensor" and the "g sub uv" is the "spacetime metric" or just the "metric".

What the what now?

Very loosely, that stuff on the right-hand side represents the mass and energy in the system. The stuff on the left describes the movement of objects in space and time. Put very simply, if you can count up all the energy in a region of space and put it on the right-hand side of that equation, the left-hand side will tell you how everything in that region should move.

Going just a little deeper, the "spacetime metric" part of the equation describes the shape of space and time... and once again the what now?

There are 2 ways of looking at this...

1) The spacetime metric describes how objects move. We understand that objects in a gravity well are pulled towards the center of that gravity. This equation describes how the object's position in the future is affected by gravity.

2) The spacetime metric describes how space and time are actually curved so that an object moving in what appears to be a straight line from the object's perspective is curved to a perspective somewhere else.

That number 2 there is bizarre. I'm about to dig a little deeper into the math, so just quit reading wherever you get bored...

What's a "vector"? Let's describe motion in 2D. We'll call the direction from West to East the "x axis". We'll call South to North the "y axis". You can describe your position and velocity as some combination of x and y. So bundle the x and the y together and call them a "vector." Your velocity can be thought of as a single value v. We understand that v has an x component and a y component, and together v.x and v.y are collectively known as v. Simple. Add altitude and we have a 3D vector. There you go... 3D space.

How does time fit in? At some time t=t0, you were at location p0 (a vector in 3D space). At some later time t=t1, you had moved to position p1. We could add t to our bundle of data to make the 4D vector (p.x, p.y, p.z, p.t). Now we can put that together like so: (p0.x, p0.y, p0.z, t0) or maybe we'd just call that (p0.x, p0,y, p0.z, p0.t)... collectively p0 as a 4D vector.

Why? Honestly, let's just start with "Why not?" We can do this if we want, so we did it. Justification not needed. It's a mathematical abstraction.

We now have a 4D coordinate space... "spacetime". Space and time combined into a single mathematical abstraction. Simple as that.

Einstein was struggling with how to resolve relativity (now known as "special relativity") and gravity. He started with what we call the "equivalence principle." This states that acceleration due to gravity feels EXACTLY like the inertial force you feel under acceleration. Putting that another way, free-fall feels EXACTLY like zero-g.

When he combined this with the Lorentz transforms of special relativity, this didn't make any sense. How can light not tell the difference between an accelerating frame and a gravity well?

Through a lot of hard work and some pretty heavy math, he eventually realized that if you describe the world using 4D coordinates "spacetime", AND you allow for the "spacetime" to be curved arbitrarily, the paradox can be resolved.

Einstein described the equations of motion using 4D spacetime. The radical part comes in when we allow for what we consider to be the "x direction" (remember that was East) to vary. At any given place and time, x is fixed pointing East. But what if at some other location or time, x could point in some other direction? Remember how we said our 4D axis was (East, North, Altitude, Time)? To picture this, imagine your East, North, Altitude axes on a globe (or on an AE projection if you prefer). The direction of "East" is different at different locations. If you keep going East, you'll eventually get back where you started from. Einstein's equations allowed for this exact situation. He started with an arbitrary 4D coordinate system to describe "spacetime". He didn't decide up front that x must point East. Instead he solved the equations and allowed the equations to tell HIM which way the x axis pointed.

Does that make any sense? Einstein abstracted away the coordinate system in which he described the forces and motions. He constrained this equation with what we knew about the physical universe and used that to solve for what the coordinate system looked like. THAT is the "spacetime metric". That's what the coordinate system looks like based on the way the universe behaves.

What'd we discover? What we see is that the presence of a point-mass on the right-hand side of the equation (a high density energy), the coordinate system becomes curved. That really shouldn't sound all that shocking at this point. After all, we invented our coordinate system with North, East, and Up in the first place right? And we know that ended up being curved, so why shouldn't the "spacetime metric" end up curved too?

What's really interesting about this curved spacetime metric is what happens if we place a small, stationary object in space near a point mass. If we solve for the future location of our stationary object, we see that it will have moved towards the point mass. If we put in a small value for the point mass, the solution for the future location of our object precisely matches Newton's universal law of gravitation. Only this time, we don't get there using F=mA. We get there just by solving the 4D coordinate system. Neat right?

So what? Well, what if instead of a stationary object, we put a beam of light next to our point mass? Prior to 1915, we had no reason to expect that gravity affected light, but this new "spacetime metric" gave the same solution for light that it did for matter - the light should bend towards the point mass. This was an unexpected prediction made by this new equation, and in 1919 that prediction was confirmed to be accurate. We now know that light is affected by gravity.

There are several other very surprising results that fall out of this equation, and one by one, each one has been confirmed. We know this equation must still be incomplete in some small way, and theoretical physicists and mathematicians are in a race to the next major breakthrough.

Summary: The "spacetime metric" is the part of Einstein's field equations (general relativity) that describe how an object moves through space. We usually describe this as time and space being curved, but to be precise, we're really talking about the motion of an object that is curved in space and time. Is it the motion that's curved or spacetime that's curved? That's sort of a philosophical question really. We can be certain that the math says the thing's gonna curve, and on that we can all agree.

But is this all theoretical? I mean, it's just math right? Many times, the math can give us new insights into the behavior of the physical world, and this is the perfect example. The math said that light should bend, and guess what... now that we knew to look for it, we were able to show that light DOES bend. The math says time passes differently in the presence of gravity, and guess what... it does! We don't just take the math and say, "oh well... math says it's true so I guess we'll just accept it." No, we look at the math and say, "according to this math, this should be true... let's see if we can test that!" The math has taught us a lot about the realities of the physical world... it shows us what to look for.