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This question addresses the scope of general relativity.

Scientists and engineers solve all sorts of physical problems in all sorts of fields and often to solve these problems a 'system' is defined including inputs, outputs, inter-relationships of variables and constraints, and from this definition the 'equations of motion' are written - specific to that system.

So my question is can the Einstein Field Equations (EFE) of General Relativity lead to the same equations of motion for these specific systems in all cases (discounting for the moment quantum mechanical systems)? In other words:

Are the EFE of General Relativity the 'Mother' of all equations of motion in the universe - outside of quantum mechanical systems? - And therefore a more generalized way of writing equations of motion for all systems?

If this is true, then from the field equations, can one derive Maxwell's equations?

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  • $\begingroup$ You have the wrong idea of how physics works. It does not try to find "the mother of all equations" and never did. It's not even clear what that is supposed to mean under the standard experimental constraints of science. $\endgroup$ – CuriousOne Sep 3 '15 at 14:18
  • $\begingroup$ Ok so you've set me straight. But why is it a bad question? I'm NOT a physicist, but rather an engineer trying to get a broader understanding. Thanks $\endgroup$ – docscience Sep 3 '15 at 17:11
  • $\begingroup$ Fair. I'll take it back. "What physics is" requires a longer discussion. Let me know if you want to have my take on it and we can do a chat one of these days. $\endgroup$ – CuriousOne Sep 4 '15 at 4:20
  • $\begingroup$ @CuriousOne Thanks. That would be great. I know allot of classical physics, and have a close physicist friend, but also realize I have many 'gaps' and misconceptions in my understanding. So if you are able and willing to share your world view, then please let me know how to go about that. I've not used the chat on this site, and not sure how one goes about scheduling a session. $\endgroup$ – docscience Sep 4 '15 at 13:49
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No.

The Einstein field equations are the equation of motion for the metric (i.e. gravity) in the Einstein-Hilbert action.

If you add other dynamical fields to the action, you not only change the stress-energy tensor appearing in the EFE, but you also have to vary the action with respect to the new fields to obtain e.o.m. for them.

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  • $\begingroup$ See Cham's answer. Do you disagree? $\endgroup$ – docscience Nov 19 '15 at 21:23
  • $\begingroup$ @docscience: Yes. I don't think you can derive Maxwell's equations just from conservation of the electromagnetic stress-energy tensor (but I'm willing to be proven wrong). I'm fairly certain that the equations of motion of other fields are distinct from the EFE. $\endgroup$ – ACuriousMind Nov 19 '15 at 21:37
  • $\begingroup$ @ACuriousMind Under what circumstances does this become necessary? When coupling to hydrodyamics, certainly, one typically just plugs in the stress-energy tensor of the fluid in question. $\endgroup$ – AGML Nov 19 '15 at 23:50
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I don't agree with the previous answer.

Firstly, the OP's question isn't about the lagrangian formulation, it's about the Einstein equation : \begin{equation}\tag{1} G_{\mu \nu} + \Lambda \, g_{\mu \nu} = -\; \kappa \, T_{\mu \nu}. \end{equation} Secondly, there are stress-tensors that can't be derived from an action : fluids tensors (especially with viscosity), for example.

Once you have added all contributions to energy-momentum to the right part of equ. (1), then all the local geometrical properties of spacetime are determined. And since gravity weights everything, that equation contains local conservation of all forms of energy-momentum. From Bianchi identity, you can derive the local conservation of energy-momentum, which is built-in the Einstein equation : \begin{equation} \nabla_{\mu} \, T^{\mu \nu} = 0. \end{equation}

From that, you can deduce Maxwell's equation, or the Klein-Gordon equation, or the Navier-Stokes equation (the only case I know which may be problematic is the Dirac equation, because of some algebraic subtleties).

So in a sense, yes the Einstein equation is the "Mother of all equations".

The only things that the Einstein equation isn't able to predict are the global properties of spacetime : number of dimensions, topologic features (spatially open or closed universe, boundary conditions, etc).

There's a very good discussion about all these in the huge MTW book (Misner, Thorne & Wheeler).

EDIT : I'm adding a nice citation from MTW's book (page 475), which says alot :

The Maxwell field equations are so constructed that they automatically fulfill and demand the conservation of charge ; but not everything has charge. The Einstein field equation is so constructed that it automatically fulfills and demands the conservation of momentum-energy ; and everything does have energy. The Maxwell field equations are indifferent to the interposition of an "extrenal" force, because that force in no way threatens the principle of conservation of charge. The Einstein field equation cares about every force, because every force is a medium for the exchange of energy.

Electromagnetism has the motto, "I count all the electric charge that's here." All that bears no charge escape its gaze.

"I weigh all that's here" is the motto of spacetime curvature. No physical entity escapes this surveillance.

This is why, in a sense, Einstein equation is the Mother of all equations : everything is implied in it, even if it could be difficult to explicitely extract the equations from the "Mother".

I think that this MTW citation is a beautifull resume of relativistic gravitation. The pure beauty of the "Mother" field equation is all there !

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  • $\begingroup$ MTW book? Please elaborate $\endgroup$ – docscience Nov 19 '15 at 21:22
  • $\begingroup$ The claim that you can derive the Maxwell equations from the conservation of energy-momentum is bold, and I have not encountered it before. Can you substantiate the claim (either by showing it in the post or pointing to the specific chapter in MTW)? $\endgroup$ – ACuriousMind Nov 19 '15 at 21:29
  • $\begingroup$ Ah, you are talking about chapter 20 of MTW, right? See exercise 20.8 for the fact that you cannot, in general, derive the Maxwell equations from the field equations, but only under certain (specific, although quite natural) assumptions. $\endgroup$ – ACuriousMind Nov 19 '15 at 21:41
  • $\begingroup$ @docscience MTW = Misner, Thorne & Wheeler $\endgroup$ – Mark Mitchison Nov 19 '15 at 22:00
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    $\begingroup$ If we set aside the macroscopic effects of matter (fluids, thermodynamics, ...), we may also say that the principle of stationary action plus a proper lagrangian may be seen as THE "Mother of all equations", but a principle isn't exactly an equation. $\endgroup$ – Cham Nov 19 '15 at 23:30

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