# Finding Equations of Motion from the Stress Energy Tensor

Gold Member

## Main Question or Discussion Point

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?

Thanks.

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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?

Thanks.
Wald is right that one can find the {*} equation of motion for some simple cases from $\nabla_b T_{ab}$, 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...

Gold Member
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.

tom.stoer
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.

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.
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.

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samalkhaiat

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

Thanks.
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.

Sam

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 $G_{ab}=T_{ab}$ 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 $\nabla_a T^{ab}=0$ 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.

DrGreg
Gold Member
$G_{ab}=T_{ab}$
A little reminder: use "itex" instead of "latex": $G_{ab}=T_{ab}$ 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.
Very interesting viewpoint.

How would you describe those deficiencies?

Gold Member
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

pervect
Staff Emeritus
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.

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.
Very interesting viewpoint.

How would you describe those deficiencies?
GR is a geometrical gravity theory based in the existence of a spacetime metric $g_{\mu\nu}^{GR} = g_{\mu\nu}^{GR}(r,t)$, which differs from the effective metric $g_{\mu\nu}^{FTG}$, 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 $g_{\mu\nu}^{FTG}$ by $g_{\mu\nu}^{GR}$ 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 $g_{\mu\nu}^{FTG}$ and $g_{\mu\nu}^{GR}$ 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.

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GR is a geometrical gravity theory based in the existence of a spacetime metric $g_{\mu\nu}^{GR} = g_{\mu\nu}^{GR}(r,t)$, which differs from the effective metric $g_{\mu\nu}^{FTG}$, 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 $g_{\mu\nu}^{FTG}$ by $g_{\mu\nu}^{GR}$ 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 $g_{\mu\nu}^{FTG}$ and $g_{\mu\nu}^{GR}$ 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.
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.

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.
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».

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...
And how is that proof the theory is incorrect?

George Jones
Staff Emeritus
Gold Member
non-geometrical theories as FTG (used by astronomers {*})

pervect
Staff Emeritus
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:

Jackson said:
In Section 12.1 we discussed the general Lagrangian formalism for a relativistic
particle in external electromagnetic fields described by the vector and scalar po-
tentials, A and Ф. The appropriate interaction Lagrangian was given by A2.11).
If we now consider the problem of a conventional Lagrangian description of the
interaction of two or more charged particles with each other, we see that it is
possible only at nonrelativistic velocities. The Lagrangian is supposed to be a
function of the instantaneous velocities and coordinates of all the particles. When
the finite velocity of propagation of electromagnetic fields is taken into account,
this is no longer possible, since the values of the potentials at one particle due to
the other particles depend on their state of motion at "retarded" times. Only
when the retardation effects can be neglected is a Lagrangian description in terms
of instantaneous positions and velocities possible.
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.

atyy
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.
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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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.
Nowhere in this thread I find your above quote «limit due to E&M being a geometrical theory». I have not even insinuated that CED is a geometrical theory.

What I said was that a local theory cannot describe the motion of an arbitrary system on N-bodies because cannot describe multiparticle correlations {*}.

GR is a geometric theory and geometric theories are local theories; therefore, GR suffers from the same problem than CED.You cannot write down a GR Lagrangian or Hamiltonian describing the motion of an arbitrary N-particle system.

Contrary to your claims regarding FTG, the absence of Black holes is one of the strengths of the theory, because the absence of event horizon is due to gravitons energy and spacetime singularities are eliminated from the physics.

As far as I know FTG is perfectly compatible with the reference that you cite. FTG predicts a surface with a very high redshift, doing it indistinguishable from the infinite redshift that corresponds to an event horizon. To be more clear the observations by McClintock, Narayan, and Rybicki do not prove the existence of an event horizon, because do not invalidate alternative models.

{*} Jackson is right on that the classical field theory of EM cannot describe the motion of a two particle system in the general case, but he is wrong on his claim that this problem is related to retardation. In rigor it can be showed that its Darwin Lagrangian (zero retardation) suffers from inconsistencies and cannot describe the general motion, but this is not the place for the discussion of this highly specialized topic.

PAllen
2019 Award
The referenced Baryshev paper does not have any discussion of observational differences in the event horizon radius versus GR (at least I didn't see any; in fact I found no discussion of horizon at all). The minimal radius they compute from conservation of energy is inside the classical event horizon (minimal radius GM/2c^2 versus horizon radius of 2GM/c^2). It would be very interesting to see any predictions about the event horizon radius in FTG. Plans are well underway to observe this directly in two galactic centers. They expect to be able to detect the horizon surface itself as well as the gravitational optical effects predicted by GR. Does anyone know if FTG has made any different predictions for observational features of supermassive 'black holes'?

Note that if observational predictions for the horizon radius are the same as GR, direct test of non-collapse will never be possible.

What I said was that a local theory cannot describe the motion of an arbitrary system on N-bodies because cannot describe multiparticle correlations {*}.

GR is a geometric theory and geometric theories are local theories; therefore, GR suffers from the same problem than CED.You cannot write down a GR Lagrangian or Hamiltonian describing the motion of an arbitrary N-particle system.
I will wait for you to supply a foundation for you statement.
An inexpressibility given our current set of mathematical tools does not automatically disqualify a theory.

PAllen
2019 Award
This paper by Baryshev argues there is no event horizon in FTG theory. Instead there is an observable surface smaller than the event horizon:

http://arxiv.org/abs/0809.2328

This should then be ruled in or out within a decade by the program to closely image supermassive galactic central objects. My prediction is the classical GR will be confirmed here. I have argued elsewhere why I think deviation from classical GR predictions will be related to energy density. Thus I would expect deviations before formation of event horizon for small collapsed objects, yet well inside the event horizon for supermassive objects.

This paper by Baryshev argues there is no event horizon in FTG theory. Instead there is an observable surface smaller than the event horizon:

http://arxiv.org/abs/0809.2328

This should then be ruled in or out within a decade by the program to closely image supermassive galactic central objects. My prediction is the classical GR will be confirmed here. I have argued elsewhere why I think deviation from classical GR predictions will be related to energy density. Thus I would expect deviations before formation of event horizon for small collapsed objects, yet well inside the event horizon for supermassive objects.
My prediction is that nature prefers a formulation without singularities and event horizons.