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Finding Equations of Motion from the Stress Energy Tensor

  1. Nov 5, 2011 #1


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    So, I'm reading Wald, and in it he talks about how the divergence-free nature of the stress-energy tensor implies "a lot" of knowledge about how matter moves in a curved space time. I'm wondering, how much is "a lot"? Can we obtain the full equations of motion from this? Wald gives the example of a perfect fluid in which you can; however, I'm wondering if you can obtain the full equations of motion of an arbitrary distribution of matter from this condition.

    Wald says that this condition implies that small masses move on geodesics (so that the "geodesic hypothesis" is actually present within Einstein's equation itself.), he goes on to say that large masses which feel tidal forces do not move exactly on geodesics, but move according to divT=0.

    Thinking back, it's natural that conservation of energy (and momentum and stress) would imply a condition on the motion of particles. After all, in classical mechanics, one often uses conservation of energy and momentum to restrict a particle's motion. However, in classical mechanics, conservation of energy (and momentum, and angular momentum, etc) by itself is usually not sufficient to determine the full trajectory of a particle, it usually only gives 1 or 2 constants of integration (making the problem easier). But to get the full equations of motion, one must usually just solve the differential equations (Euler Lagrange eqns, or some such).

    In GR, is divT=0 sufficient to find the FULL trajectory of particles moving in curved space-time?

  2. jcsd
  3. Nov 5, 2011 #2
  4. Nov 5, 2011 #3
    Wald is right that one can find the {*} equation of motion for some simple cases from [itex]\nabla_b T_{ab}[/itex], but it is not possible when there are more than four degree of freedom.

    {*} Not the more general equation of motion, but one that ignores dissipation, noise, quantum corrections...
  5. Nov 5, 2011 #4


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    So in the case that divT=0 doesn't determine fully the motion, what are we to do? We must set up the full Lagrangian and solve it that way?

    Like: L=L_H+L_M+L_F+L_MF

    Where L_M is the Lagrangian corresponding to matter (-mc^2sqrt(1-v^2/c^2)), L_H is the Lagrangian corresponding to the Hilbert Action, L_F is the Lagrangian corresponding to the external field (e.g. EM field), and L_MF would be the coupling between the matter terms and field terms?

    That seems like a huge mess.
  6. Nov 6, 2011 #5


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    Going from Larangian to Hamiltonan mechanics one constructs H[f] as a functional of the fields f and calculates the canonical equations of motion using the Poisson brackets {H,f}.

    H and T°° are related somehow, and therefore it may be possible to derive H from T°° as well. But usually the canonical and the covariant densities are not identical and already their construction is different.
  7. Nov 6, 2011 #6
    Sorry but GR cannot provide a consistent and complete Lagrangian valid for any system of masses because GR is only a geometric theory.

    The best than general and numerical relativists can do is to postulate some approximate Lagrangian and next derive approximated equations of motion from it.

    Luckily the approx. equations are enough for many astrophysical applications but a complete many-body equation does not exist in GR due to its geometrical deficiencies.

    This is not a problem exclusive of GR alone. A similar problem exists in Maxwell electrodynamics. Jackson devotes a section of his standard textbook to explain why no full relativistic Lagrangian for a two-body system exists within the scope of field theory.

    The best that Jackson and others can do is to propose an approx. Lagrangian valid up to c^2 order and derive approx. equations of motion from it.
    Last edited: Nov 6, 2011
  8. Nov 8, 2011 #7


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    Yes, it can be done. I will try to do it for you and (if I managed to do it) will post the details soon.

  9. Nov 10, 2011 #8
    Think of it this way: a model for a kind of matter consists of variables representing the matter and differential equations relating the variables. These equations will also involve the metric of spacetime. Now, to couple this matter to gravity described by general relativity, you need to say what the energy-momentum tensor of the matter is, in terms of the variables and the spacetime metric. Then you add [itex]G_{ab}=T_{ab}[/itex] to your list of equations. However, now take the divergence of both sides of this equation. By the Bianchi identity, the left side vanishes, implying [itex]\nabla_a T^{ab}=0[/itex] must be satisfied in order for solutions to your coupled gravity-matter system to exist. The point is you can make up whatever matter model you want, but if you want to consistently couple it to gravity, its equations must satisfy conservation of stress-energy. In very special cases (such as a dust fluid), conservation of stress-energy entirely dictates what the equations must be. In more general cases, this won't be true.
  10. Nov 10, 2011 #9


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    A little reminder: use "itex" instead of "latex": [itex]G_{ab}=T_{ab}[/itex] :smile:
  11. Nov 10, 2011 #10
    Very interesting viewpoint.

    How would you describe those deficiencies?
  12. Nov 10, 2011 #11


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    Seems I'm getting some conflicting answers here, but it seems the general idea is that this cannot be done for generic matter fields, but only for special cases such as a perfect fluid? o.o
  13. Nov 10, 2011 #12


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    Since GR can be written as a Lagrangian field theory, the only thing I can imagine is that the exact Lagrangian has to be written as a field, i.e. a Lagrangian density over space-time, and that the difficulties mentioned are for "solving for the equations of motion", i.e. making a lumped parameter model for a Lagrangian as a finite number of variables (rather than a continuous field), ie writing a Lagrangian as a fuction of, say, 3*n position variables, coordinate time, and the 3*n derivatives of the position variables with respect to coordinate time.

    I'd attribute these difficulties to finding a lumped approximation for gravitational radiation.

    There's also the usual difficulty of how to handle time in a relativistic multi-body Lagrangian, with a single body you can use proper time to describe the state of the system, but there isn't an obvious equivalent for proper time for a multi-body system.

    I seem to recall a section on Goldstein on the topic, and that the easiest route was to basically use coordinate time as I suggested above (but you loose some element of explicit covariance that way, though it works out all right in the end). I should probably re-read Goldstein, but I don't have the time at the moment. Maybe someone else might post some corrections, this is from memory,.

    But some more clarification by the OP would be in order, , perhaps I've missed or misunderstood the point.
  14. Nov 11, 2011 #13
    GR is a geometrical gravity theory based in the existence of a spacetime metric [itex]g_{\mu\nu}^{GR} = g_{\mu\nu}^{GR}(r,t)[/itex], which differs from the effective metric [itex]g_{\mu\nu}^{FTG}[/itex], associated to a physical gravitational field (gravitons), by terms roughly of the order of the strength of the field. Geometrical gravity is an approximation to physical gravity (Field Theory of Gravity) somewhat as geometrical optics is an approximation to physical optics.

    Approximating [itex]g_{\mu\nu}^{FTG}[/itex] by [itex]g_{\mu\nu}^{GR}[/itex] in the basic equations of FTG gives GR with all the well-known geometric deficiencies of GR such as lack of conservation laws, spacetime singularities, the problem of systems of reference, gravitational energy problem (no positive tensor), and others.

    Moreover, both [itex]g_{\mu\nu}^{FTG}[/itex] and [itex]g_{\mu\nu}^{GR}[/itex] are local metrics, which means that cannot represent the non-local correlations in a general interacting N-body system. As a consequence, does not exist a Lagrangian (Hamiltonian) from the which obtain the equations of motion of the system.

    As emphasized above. This is not a problem exclusive of GR alone. A similar problem exists in Maxwell electrodynamics, which is also a local theory.

    Precisely Jackson devotes a section of his standard textbook on classical electrodynamics to explain why no full relativistic Lagrangian for a two-body system exists within the scope of the field theory of charged particles. The only proposes equations valid up to order c^2.

    Eric Poisson was cited above in #2. Precisely he has another monograph about the equations of motion in GR and the best that he obtains are coupled one-body equations of motion within the c^2 approximation. And I would object for several flaws in his 'derivation'.

    That GR is unable to provide the equations of motion for an arbitrary system of N-particles is well-known. And several alternatives are currently under research and discussed in specialized literature.
    Last edited: Nov 11, 2011
  15. Nov 11, 2011 #14
    All I am seeing here is a proposition that a field theory of gravity is better than GR, but the irony is that we have no working field theory of gravity. Sounds to me more like wishful thinking than science.
  16. Nov 11, 2011 #15
    I see something different. I can see a list of geometric deficiencies of GR (discussed in textbooks), the fact that those deficiencies are absent in non-geometrical theories as FTG (used by astronomers {*}), the well-known fact that neither GR nor classical electrodynamics (see Jackson textbook) can provide a Lagrangian (Hamiltonian) for a generic N-body system, and therefore cannot give the equations of motion in the general case...

    {*} I do not know from where you got your «we have no working field theory of gravity».
  17. Nov 11, 2011 #16
    And how is that proof the theory is incorrect?
  18. Nov 11, 2011 #17

    George Jones

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    Please provide some references.
  19. Nov 11, 2011 #18
  20. Nov 11, 2011 #19


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    I took a look at Jackson, and Jackson's point does seem to be relevant to the original question of finding a Lagrangian for a multi-body system as a function of a finite number of position coordinates. Approximations exist that work at low velocities, but there isn't an exact solution, even in E&M.

    What I don't see in Jackson is any mention of it being a "limit due to E&M being a geometrical theory". The comments I found on the matter were as follows:

    So I think there is some agreement on there not being an exact finite dimensional Lagrangian even in E&M, but the remarks about it being due to any "geometrical nature" of the theories still seem strange to me and not well supported. I could belive a remark that it was due to the "infinite dimensonal" nature of E&M, though I didn't see any specific remarks in Jackson to support that viewpoint directly, but I don't quite follow the logic that ascribes it to "geometry".

    I took a quick look at the first cited reference with respect to FTG theories. It was interesting, though I'm not sure if it was peer-reviewed. I viewed the remarks near the end that the FTG theories didn't have black holes or event horizons as being a likely a problem for the FTG's, however.

    See http://iopscience.iop.org/0004-637X/615/1/402/60056.text.html for evidence that energy is going into black holes and not coming out, i.e. that black holes are black.

    If there isn't any event horizon in FTG theories, some other explanation for why black holes seem to be black needs to be found. I didn't see any mention of this in the cited paper, (though I did read it rather hastily), and I'm not familar enough with this class of theories to know if the issues can be fixed. I think it might be an interesting test of this class of theories, though.
  21. Nov 11, 2011 #20


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    My understanding is that classically, gravity as a special relativistic theory is equivalent to GR provided spacetime can be covered by harmonic coordinates (maybe that can be relaxed a bit). Perhaps one advantage of treating GR as a field theory in flat spacetime is that "Weinberg's low energy theorem" derives the equivalence principle (http://arxiv.org/abs/1105.3735, http://arxiv.org/abs/1007.0435).

    I do believe that there are harmonic coordinates that penetrate the event horizon (http://prd.aps.org/abstract/PRD/v56/i8/p4775_1, and probably earlier papers too).
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