# Are the Einstein field equations the Equations of Motion for matter?

I'm learning general relativity and eventually I hope to understand the Einstein field equations $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ in full mathematical detail.

I wonder whether they describe the equations of motion of matter, since I read the phrase "matter tells spacetime how to curve. spacetime tells matter how to move.". However, it puzzles me that the field equations have two unknowns: the Einstein tensor $$G$$ and the matter stress energy momentum tensor $$T$$, whereas a usual wave equation would be a differential equation for only one.

For a usual equation of motion (e.g. wave equation), you would have a single unknown quantity, a partial differential equation and some initial conditions at $$t=0$$. This allows you to calculate the state of the unknown at $$t>0$$.

How does it work with the field equations? Do you have to state $$T$$ in all of spacetime first? But then you've already provided the distribution of matter at all times and you needed an additional equation of motion for that? Is all of matter supposed to be in $$T$$ (electrons, photons, ...)? So the field equations do not tell you anything about how you should set up $$T$$?

I could imagine that you calculate $$T$$ from equations which are not the field equations. Then you calculate the curvature of spacetime by the field equations which enables you to predict the trajectory of a new test particle. However, this test particle should have been in $$T$$ in the first place and being solved with the actual equation of motion? This seems to make gravity redundant given that you have the proper EoM for $$T$$?

So, the statement is rather something like "a postulated background tells spacetime how to curve. spacetime tells an additional test point-particle how to move." and the field equations are not able to predict the physics of $$T$$?

• Dirac's 75 page brochure on General Relativity (1975) is an answer to this question. Commented Feb 19, 2023 at 9:15
• In electromagnetism you also have field and source terms. The fields have an influence on the sources. How does it differ? Commented Feb 19, 2023 at 9:42
• $\kappa T_{\mu\nu}$ is a specific function of fields and their derivatives, particles' paths etc., and equating that to $G_{\mu\nu}$ provides a PDE the matter distribution solves.
– J.G.
Commented Feb 19, 2023 at 11:49

Unlike the Lorentz force law, which has to be postulated separately from Maxwell's equations, Einstein's field equation automatically implies the geodesic equation for how a test particle will move. This is because it is "part of $$T$$" as you say.

But is it true that Einstein's equations "do not tell you anything about how you should setup $$T$$"? Yes because GR is not a theory of everything. In the example of how spacetime is curved by light and in turn affects light, you would have the Einstein-Hilbert action plus the Maxwell action $$S_{EH} + S_M$$. These would give you coupled differential equations... Einstein's equation but also Maxwell's equations in curved spacetime.

• Is it possible to say that if you had the proper equations for $T$, then the fields equations would not be needed anymore? I mean you could calculate curvature, but it would be useless, since all "test particles" are already included in $T$ as waves?
– Gere
Commented Feb 19, 2023 at 9:51
• And are plain Maxwell's equations a theory of light respecting all of what GR teaches us already? (i.e. they are the perfect EoM for $T$ if there was only light)
– Gere
Commented Feb 19, 2023 at 9:55
• Right, there are no changes that GR tells us to make to Maxwell's equations. But calculating curvature sounds essential to me. Otherwise, you would have the proper equations for $T$ (which I take to mean the equations of motion for the matter action) but not enough information to solve them. Commented Feb 20, 2023 at 1:17

If you have an initial distribution of fundamental particles (initial positions and velocities) then you know the initial $$T_{\mu\nu}$$ (at least, you do in principle - actually calculating $$T_{\mu\nu}$$ may be very difficult). The Einstein field equations then give you $$G_{\mu\nu}$$, which tells you the initial curvature of spacetime. This in turn tells you how the particles move, which tells you how $$T_{\mu\nu}$$ evolves, which in turn tells you how $$G_{\mu\nu}$$ evolves, and so on.

This seems to make gravity redundant

Yes, indeed it does.

The field equations are not able to predict the physics of T

Correct. Einstein's field equations do not tell you what types of fundamental particles exist in the first place. Neither do they tell you anything about the non-gravitational interactions between these fundamental particles (via electromagnetism, the strong force and the weak force, as far as we know), all of which contribute to $$T_{\mu\nu}$$. Any attempt at solving the field equations must first make some assumptions about what types of fundamental particles or fields exist and how they interact.

• Are you sure you can evolve $T$? I thought even defining a $t=0$ slice is tricky. And since you can arbitrarily choose $T$ on the whole spacetime and yet find a good $G$, I cannot see how a slice of $G$ would determine the future $T$.
– Gere
Commented Feb 19, 2023 at 11:38
• G gives you geodesics and so tells you how particles move in the absence of non-gravitational forces. Add in the effect of electromagnetic forces etc. and you now know how T evolves - in principle. In practice you get a bunch of interconnected non-linear differential equations, so not a practical approach unless you make some simplifying assumptions (such as ignoring non-gravitational forces). Commented Feb 19, 2023 at 11:50