The Equations of Motion

So far in my explanation of General Relativity, I've discussed the metric $g_{ab}$, from which one can calculate the curvature tensor $R^{a}_{bcd}$ by way of the connection $\Gamma^{a}_{bc}$.

In practical astrophysical contexts:

• The metric is related to the gravitational potential at a point, i.e. how much ``potential energy'' a unit mass will have sitting in the gravitational field.  But I haven't said anything about energy yet, so you're entitled to ignore this remark...
• The connection (which involves a derivative of the metric) tells you the gravitational force at a point, i..e the amount by which freely-falling objects will accelerate in a given coordinate system.
• Finally, the curvature (which involves a derivative of the connection) tells you the tidal forces at a point, i.e. a difference in the force acting on a nearby object.  Yes, the ocean tides happen because the moon's gravitational field has nonzero curvature at the Earth's location.  That's why it's called that.

So far this is just the kinematics of general relativity—that is, what kind of entities are involved, and the basic outline of their behavior.  For example, If I wanted to tell you the kinematics for basic Newtonian mechanics (what you learn in high school physics), I'd say that A) there are a bunch of objects which have masses and positions (and orientations if you want things to get complicated...), B) the position of an object can change with time, but its mass is ``conserved'' and therefore doesn't, and C) if you want to work out the ``force'' of an object, you can do so using $F = ma$.

OK, so I've told you all about Newtonian Mechanics, and now you can go use it to solve problems, right?  No, of course not!  You can recite "the time-derivative of the position is the velocity, the time-derivative of the velocity is the acceleration, and the acceleration equals the force over the mass" over and over again, but it's totally useless until I tell you what the forces actually are!  Without that, you can't make any predictions at all about what the objects are doing.

Unless you count boring predictions like "the object will be somewhere", you need to know something else.  This something else is called the dynamics, which means the rules for how things actually change with time.  (For example, if I told you that any two objects with mass $m_1$ and $m_2$ at a distance $r$ are gravitationally attracted towards each other's positions, with a force that is proportional to $F = Gm_1m_2/r^2$, and if you know the initial positions and velocities, then you can work out their orbits!  At least, you can if you're clever at math, like Newton was.)

So we need to write down an equation which says how things can change with time.  We call this the equations of motion.  Ever since Newton wrote down $F = m{\ddot x}$ (each dot being a time derivative, so that his archnemesis Leibnitz would have written $F = m (d^2x/dt^2)$ to say the same thing) we've realized that these equations typically involve taking two derivatives.  So we shouldn't be surprised that the equation of motion for general relativity involves the curvature tensor $R^{a}_{bcd}$, since it's a double derivative of the metric, which is the basic field of General Relativity.

To write down the equations of motion, we need to massage the curvature tensor a little bit.  If you've forgotten the ground rules for tensors, click on the link.  We start with the the Riemann curvature tensor $R^{a}_{bcd}$.  Since each of the letters is a spacetime vector index with four possible values, it looks like this has $4 \times 4 \times 4 \times 4 = 256$ components.  Fortunately there are a lot of symmetries and constraints, so there's actually only 20 independent components per spacetime point.  We can define the Ricci tensor $R_{ab}$ by contracting the top index with the middle index on the bottom, like so:

$$R_{ab} = R^c_{acb};$$

Recall that the Einstein summation convention says that if you ever see the same letter as both a subscript and as a superscript, you've got to 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).  Since the Ricci tensor is symmetric ($R_{ab} = R_{ba}$), it only represents 10 out of the 20 curvature components.  If this is not enough simplification for you, we can go further by contracting again using the inverse metric:

$$R = R_{ab} g^{ab}.$$

$R$ is called the Ricci scalar, because it has just one component.

Whew!  Without further ado, here's the equation of motion for General Relativity, called "the Einstein equation" after you know who:

$$R_{ab} - \tfrac{1}{2} g_{ab} R = 8\pi G T_{ab}.$$

Compact, beautiful, and probably completely incomprehensible since I haven't explained all of the symbols yet!

The 8 and the $\pi$ are the same numbers which you learned about in school.  $G$ is Newton's constant, which I sneakily introduced earlier in this post.  Note that the $8\pi$ isn't really just there for backwards compatibility with Newton's force law.  If Einstein's equation had been discovered first, we would have left out the $8\pi$ from it, and then we would have written the force law as $F = G m_1 m_2/ 8\pi r^2$.  But as it is, Newton got his $G$ before Einstein did, so we're stuck with it.

But the really important symbol here is $T_{ab}$.  This is the energy-momentum tensor, or (because why should anything have only one name!) the stress-energy tensor.  It's a $4 \times 4$ symmetric matrix which tells you how the energy and momentum of matter (stuff) are flowing through a given point.  Now if you are a true Israelite in whom there is no guile, you should be asking: "What on earth (or in the heavens) are energy and momentum!  You haven't explained that yet!"  No I haven't.  For now, let's just say it's a property of matter, but we will get to it in a later post.

The combination of curvatures $R_{ab} - \tfrac{1}{2} g_{ab} R$ which appears on the left-hand-side is also known as the Einstein tensor.  It has the same 10 components as the Ricci tensor $R_{ab}$; they're just repackaged a bit differently.  So the Einstein equation is actually 10 equations.

So, if you know what the matter is doing, you can figure out something about the geometry of matter.  At least, you can figure out the 10 of the components of the curvature which correspond to the Ricci tensor $R_{ab}$.  Since the full Riemann tensor $R_{abcd}$ has 20 components, there are 10 components left which are undetermined.  The remaining 10 components are called the Weyl tensor, and can be nonzero even in regions in which there is no matter.  That's why there can be tidal forces outside of the surface of the sun or moon, even though there isn't any solar or lunar matter there.  It's the Weyl tensor which does that.  Also, as I wrote in Geometry is a Field:

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.

It's also the Weyl tensor which allows for gravity waves.

Clever readers may notice that I never wrote down what the Weyl tensor actually is.  There's a clever formula where you start with $R^a_{bcd}$, and then cleverly suck out all of the information about $R_{ab}$, and end up with the Weyl tensor $C^a_{bcd}$.  But it's a bit complicated, so don't ask.  The important thing is even when all of the components of $R_{ab}$ are zero, $R^a_{bcd}$ doesn't have to be zero.

When we say that the Einstein equation is the ``equation of motion'' for General Relativity, we mean that you can use it to work out how the metric changes with time.   So, if you know the metric everywhere at some ``time'' which we will call $t = 0$ (think of this as being like the position of the gravitational field), and if you also know its first derivative $\dot{g}_{ab}$ (think of this as being like the velocity), and if you know what the matter is doing, then the Einstein equation (which is like the force law) lets you work out the second derivative $\ddot{g}_{ab}$.  By continuing to apply the Einstein equation, you can work out the value of the metric for all time!

Well, not quite.  Remember that coordinates don't matter!  This means that we can't actually hope to totally determine the metric, since if we start with a metric which obeys the Einstein equation, and distort it by changing the coordinate system, we get an equally good solution to Einstein's equation.  So what we should really say, is that if you know the metric and its first derivative at $t=0$ (and you know how matter behaves so you can figure out $T_{ab}$), then you can determine the fields at $t > 0$ or $t < 0$ up to coordinate transformations.

So we can actually only need to figure out $\ddot{g}_{ab}$ up to coordinate transformations.  There are 10 components of  $g_{ab}$, but there are also 4 spacetime coordinates $(t,\,x,\,y,\,z)$ whose values can be freely determined.  As a result, we actually only need to use 10 - 4 = 6 of the Einstein equations in order to figure out how the metric changes with time.

The remaining 4 equations are called constraints, because they don't involve second derivatives of the metric.  Instead, they restrict which values of $(g_{ab}(x,y,z),\,\dot{g}_{ab}(x,y,z))$ you are allowed to start with.  These constraints are one of the most subtle features of General Relativity, because they ensure that the total energy and momentum of an object (like the sun) are encoded in the gravitational field coming out from it.  However, since I haven't yet explained what energy and momentum are, I should probably say something about that first, before going into this.