Can the geodesic equation be derived from the EFE in a certain limit?

[john baez]

TL;DR Summary

Einstein, Infeld and Hoffman tried to derive the equations of motion of point particles in GR from Einstein's field equations

On another thread, now closed, Intrastellar asked:

Since the EFE describes the shape of spacetime, it describes the way black holes, for example, evolve. Can one derive the geodesic equation from it in some limit ?

Unfortunately it seems that thread is closed before anyone pointed out Einstein's papers on this question:

• A. Einstein, L. Infeld and B. Hoffman, The gravitational equations and the problem of motion, Annals of Mathematics 39 (1938), 65-100.
• A. Einstein and L. Infeld, The gravitational equations and the problem of motion, II, Annals of Mathematics 41 (1940), 455–464.
• A. Einstein and L. Infeld, On the motion of particles in general relativity theory, Canadian Journal of Physics 1 (1949), 209–241.

Einstein had been working on this problem for a long time even before the first of these papers came out! In a letter to Max Born on Dec. 4, 1926 he wrote:

“I am plaguing myself with the derivation of the equations of motion of material points, conceived of as singularities [in the gravitational field], from the equations of general relativity.”

In 2019 Kiessling and Tahvildar-Zadeh argued that Einstein, Infeld and Hoffman made some mistakes in their treatment of charged particles, and they tried to fix those mistakes here:

• M. K.-H. Kiessling and A. S. Tahvildar-Zadeh, The Einstein-Infeld-Hoffmann legacy in mathematical relativity I: the classical motion of charged point particles.

However, I'm pretty sure the Einstein-Infeld-Hoffman approach works (within in its limitations) for uncharged particles. A lot of people have used it and built on it.

Kiessling and Tahvildar-Zadeh say some interesting things about Einstein's original work on this:

We don’t know when Einstein first conceived of the notion of point particles as singularities in relativistic fields, but his letter to Max Born makes it plain that by the end of 1926 his ideas had matured to the point where he pursued a dynamical theory for such point singularities, expecting that their law of motion could be extracted from his gravitational field equations. Already a month later Einstein & Grommer announced that “the law of motion is completely determined by the field equations, though shown in this work only for the case of equilibrium.” In that paper the case of a static, spherically symmetric spacetime with a single time-like singularity was studied. The truly dynamical many-body problem was treated a decade later by Einstein, Infeld, and Hoffmann in their famous paper, with follow-ups in 1940 and 1949. They argued explicitly that the field equations of general relativity theory alone determine the equations of motion of neutral matter particles, viewed as point singularities in space-like slices of spacetime. They also claimed that they had generalized their results to charged point-singularities, with the details written up in a set of notes deposited with the secretary of the IAS. In 1941 the motion of charged point-singularities was revisited by Infeld’s student P. R.Wallace, who presented the details of the calculations in 1941. Here is the gist of the Einstein-Infeld-Hoffmann argument [...]

 

[PeterDonis]

Einstein, Infeld and Hoffman did this in 1938

To be clear, what they did was to derive equations of motion based on what we would now call a post-Newtonian expansion, correct? In other words, their method, strictly speaking, only works for solutions of the EFE in which the fields are weak and the relative velocities are small compared to the speed of light.

Also, do you know under what conditions the equations of motion so derived would predict that the resulting trajectories would be geodesics of the spacetime geometry formed by all of the particles taken together? The Einstein-Infeld Hoffman equations are not, in themselves, the same as the geodesic equation, and deriving the former has, in itself, nothing to do with deriving the latter.

 

[Intrastellar]

Thank you very much!

I will check out the papers.

 

[john baez]

To be clear, what they did was to derive equations of motion based on what we would now call a post-Newtonian expansion, correct? In other words, their method, strictly speaking, only works for solutions of the EFE in which the fields are weak and the relative velocities are small compared to the speed of light.

I think that's at least one of the things they did. Einstein and Infeld wrote three papers on this stuff, and I haven't read them.

Also, do you know under what conditions the equations of motion so derived would predict that the resulting trajectories would be geodesics of the spacetime geometry formed by all of the particles taken together?

That sounds hard except in the limit where the masses of the particles are very small, since otherwise they are finite-sized black holes and you have to worry about all the complexities of interacting black holes: gravitational radiation, etc. Anyway, I don't know much about this issue.

The Einstein-Infeld Hoffman equations are not, in themselves, the same as the geodesic equation, and deriving the former has, in itself, nothing to do with deriving the latter.

I like the case of "dust" - that is, a continuous fluid of point particles with no pressure, just energy density:

• Wikipedia, Dust solution.

The stress-energy tensor $T_{\alpha\beta}$ is then related to the velocity vector field $u_\alpha$ of these point particles via

$T_{\alpha\beta} = \rho u_\alpha u_\beta$

where $\rho$ is the energy density of the dust in its own local rest frame.

Einstein's field equations then imply that the dust moves along geodesics! In fact you can show this starting from local conservation of energy-momentum,

${T^{\alpha\beta}}_{; \beta} = 0$

which is a consequence of Einstein's field equations.

 

[atyy]

Is the Einstein-Infeld-Hoffmann work different from that of Mino-Sasaki-Tanaka, about which the review https://arxiv.org/abs/1102.0529 by Poisson, Pound and Vega says:

"In this alternative derivation, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: it must move according to Eq. (1.48). This alternative derivation (which was given a different implementation in Ref. [15]) is entirely free of singularities (except deep within the black hole), and it suggests that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored). This derivation, however, was limited to the case of a non-rotating black hole, and it relied on a number of unjustified and sometimes unstated assumptions [16–18]. The conclusion was made firm by the more rigorous analysis of Gralla and Wald [16] (as extended by Pound [17]), who showed that the MiSaTaQuWa equations apply to any sufficiently compact body of arbitrary internal structure."

 

[PeterDonis]

Einstein's field equations then imply that the dust moves along geodesics!

Yes, agreed; the FLRW solution for a matter-dominated universe (since "matter" in that context means "dust"--zero pressure) is of course the most commonly enountered example.

 

[strangerep]

Einstein and Infeld wrote three papers on this stuff, and I haven't read them.

Heh, heh, you've obviously forgotten the improved version of this approach by Roy Kerr, which I mentioned to you maybe 15-20 yrs ago when this came up on spr.

I recently mentioned Kerr's papers again here on PF, in a different context. And yes, I have read them.

Kerr's papers are actually easier to follow than the original EIH papers, imho.

[But I should also say: thanks for making the effort to reply helpfully to @Intrastellar's question in the other now-closed thread. ]

 

[Intrastellar]

Since I am reading the papers and trying to make sense of them, I will meanwhile ask a question:

Do we expect that particles are related in any way to singularities in quantum gravity ?

 

[strangerep]

Do we expect that particles are related in any way to singularities in quantum gravity?

No. Those papers deal with classical GR, using perturbative calculations.

 

[Intrastellar]

Those papers deal with classical GR, using perturbative calculations.

I understand that the derivation in the paper is entirely classical, my question is in regards to the use of singularities generally to substitute for point particles following geodesics. Is that something that we expect to carry through to a theory of quantum gravity ?

 

[strangerep]

I understand that the derivation in the paper is entirely classical, my question is in regards to the use of singularities generally to substitute for point particles following geodesics. Is that something that we expect to carry through to a theory of quantum gravity ?

Until we have a satisfactory theory of QG,... who knows? The classical notion of "particle" is already problematic in ordinary QFT, and we must think of particles in terms of asymptotic states and an S-matrix. In the semiclassical theory of QFT on curved spacetime, different observers cannot, in general, even agree on a definition of "particle".

 

[PAllen]

I think the best work to date on motion of small bodies derived from only the field equation is that of Gralla and Wald:

https://arxiv.org/abs/0806.3293

A major innovation of their approach is that prior treatments of material bodies (rather than singularities) required the assumption of energy conditions to get geodesic motion in the limit (e.g. all of Geroch et.al. results required the dominant energy condition). Given the modern doubts about the status of energy conditions, Gralla and Wald substitute the assumption of timelike motion for a small body. Given this, they the show geodesic motion in the appropriate limit. They note, in a footnote, that the only known way to derive timelike motion for a body from the EFE is to assume the dominant energy condition. However, the beauty of their approach is to allow adoption of an assumption of timelike motion instead of energy conditions. I suspect that almost all physicists who have doubts about energy conditions would be comfortable with an axiom of timelike motion for material bodies. In particular, the Gralla Wald result applies to any form of exotic matter as long as you add this assumption.

 

[PeterDonis]

the use of singularities generally to substitute for point particles following geodesics

What do you mean by this?

 

[Intrastellar]

Unfortunately it seems that thread is closed before anyone pointed out Einstein's papers on this question:

• A. Einstein, L. Infeld and B. Hoffman, The gravitational equations and the problem of motion, Annals of Mathematics 39 (1938), 65-100.
• A. Einstein and L. Infeld, The gravitational equations and the problem of motion, II, Annals of Mathematics 41 (1940), 455–464.
• A. Einstein and L. Infeld, On the motion of particles in general relativity theory, Canadian Journal of Physics 1 (1949), 209–241.

I looked at the first paper, but didn't quite spot the Aha! moment where the equation was derived. As I understood it, the derivation was left outside the paper for being too complex. Is my understanding correct ?

What do you mean by this?

Just that point particles were modeled as singularities

 

[PeterDonis]

Just that point particles were modeled as singularities

Meaning in the Kerr paper that @strangerep referenced? I haven't had a chance to read it yet.

 

[Intrastellar]

Meaning in the Kerr paper that @strangerep referenced? I haven't had a chance to read it yet.

I haven't look at that yet. I meant in the original papers in the OP.

 

[PeterDonis]

I meant in the original papers in the OP.

I don't think those papers model point particles as singularities. As I understand it, Einstein, Infeld, and Hoffman were using a post-Newtonian expansion, in which the gravitating masses are just modeled as point particles; they are not given any internal structure and their internal spacetime geometry is not modeled at all.

 

[PAllen]

I don't think those papers model point particles as singularities. As I understand it, Einstein, Infeld, and Hoffman were using a post-Newtonian expansion, in which the gravitating masses are just modeled as point particles; they are not given any internal structure and their internal spacetime geometry is not modeled at all.

The quote by :

Kiessling and Tahvildar-Zadeh say some interesting things about Einstein's original work on this:

that was given in the OP, reviewing Einstein's work in detail, definitely claims singularities were used. Of course, it would help to see the original papers, but I don't know how to find those online or get copies without paying too much.

 

[PeterDonis]

The quote by :

Kiessling and Tahvildar-Zadeh say some interesting things about Einstein's original work on this:

that was given in the OP, reviewing Einstein's work in detail, definitely claims singularities were used.

Yes, I see that, but from description of the approximation method that EIH used, given at the top of p. 3 of that paper, it looks like the "point singularities" were actually removed from the model before it was used to make calculations: the actual model works with closed surfaces around the point particles, so that effectively the solution is for a 4-d spacetime with a finite number of "world tubes" of finite radius removed, and appropriate boundary conditions enforced on the surfaces of the world tubes.

The paper then goes on to point out what it says is a problem of consistency with that approach; I haven't digested that part fully yet.

 

[john baez]

I don't think those papers model point particles as singularities. As I understand it, Einstein, Infeld, and Hoffman were using a post-Newtonian expansion [...]

I haven't actually read the Einstein-Infeld-Hoffman papers, but in my post I quoted Einstein as saying

“I am plaguing myself with the derivation of the equations of motion of material points, conceived of as singularities [in the gravitational field], from the equations of general relativity.”

So that was certainly his goal: to model particles as singular solutions of Einstein's equations.

 

[john baez]

Heh, heh, you've obviously forgotten the improved version of this approach by Roy Kerr, which I mentioned to you maybe 15-20 yrs ago when this came up on spr.

Wow, I forgot after only 15-20 years? My memory must be getting really bad!

Kerr's papers are actually easier to follow than the original EIH papers, imho.

Thanks! That's not surprising - things often get clearer after a few tries. I'll check out those Kerr papers someday. Maybe in 15-20 years.

Just to help myself remember:

• Roy Kerr, The Lorentz-Covariant Approximation Method in General Relativity - I, Nuovo Cimento, vol XIII, no 3 (1959), 469.

 

[PeterDonis]

that was certainly his goal: to model particles as singular solutions of Einstein's equations.

I agree that was his original goal; it just seems to me that the actual Einstein-Infeld Hoffman equations ended up not quite doing that. The point singularities started out being there, but then got eliminated from the model that was actually used to derive the equations, as I described in an earlier post.

 

[strangerep]

[...] from description of the approximation method that EIH used, given at the top of p. 3 of that paper, it looks like the "point singularities" were actually removed from the model before it was used to make calculations: the actual model works with closed surfaces around the point particles, so that effectively the solution is for a 4-d spacetime with a finite number of "world tubes" of finite radius removed, and appropriate boundary conditions enforced on the surfaces of the world tubes.

Yes, exactly. Kerr's method is similar. (Indeed, how else could one work consistently with singularities except by excluding an infintesimal region around them?)

The paper then goes on to point out what it says is a problem of consistency with that approach; I haven't digested that part fully yet.

Yes. Kerr, also Moffat & Kerr, (ref not known?) point out some errors in the EIH treatment, which Kerr's method improves upon, yielding equations of spin as well as motion, by a now-consistent treatment (or so he claims).

 

[samalkhaiat]

“There are actually indications that the field equations for the $g_{\mu\nu}$ cannot be satisfied in the space surrounding the point singularities unless these singularities move along those world lines which are determined by the law of motion”
P.G. Bergmann, Ch. XV, “Introduction to the THEORY OF RELATIVITY”.
In that chapter, Bergmann explains in full details the methods of Einstein-Infield-Hoffmann. At the end of that chapter, he also explain why Maxwell theory does not have that property.
Since the 4 contracted Bianchi identities are the reasons why the methods work, one can “equivalently” derive the geodesic equation starting from the right-hand-side of the field equations: $G^{\mu\nu} = T^{\mu\nu}, \ \Rightarrow \ \nabla_{\mu}T^{\mu\nu} = 0$. Without specifying the form of $T^{\mu\nu}$, the equation $\nabla_{\mu}T^{\mu\nu} = 0$ leads to the geodesic equation. The method is described in

https://www.physicsforums.com/threa...-the-stress-energy-tensor.547502/post-3616065

Source https://www.physicsforums.com/threa...n-from-the-stress-energy-tensor.547502/page-2

 

[strangerep]

(Sigh)... Yet another @samalkhaiat post that deserves to be promoted to an Insight, but probably won't be...

 

[vanhees71]

So that was certainly his goal: to model particles as singular solutions of Einstein's equations.

Isn't the classical point-particle picture an idealization? Classical physics usually works well in the macroscopic domain, and there you have "particles" as a simplified effective model for extended bodies. I think there are no point particles in the literal sense. They make a lot of trouble if going further in the non-relativistic approximation already in the, I think, simpler case of electrodynamics: There's no fully self-consistent solution for the dynamics of classical charged point particles and the em. field. AFAIK the current status is that the best one can come up with is the Landau-Lifshitz formulation of the Abraham-Lorentz-Dirac equation, but that's only an approximation too. I'd rather expect that one finds a solution in continuum-mechanical models than trying to describe point particles.

 

[pervect]

Another noteworthy effort on how to deal with extended bodies in General relativity are the series of papers by Dixon on "Dynamics of extended bodies in general relativity", of which there are a series of three.

There's some general info at https://physics.stackexchange.com/q...dixons-extended-bodies-theory-for-fundamental

I have not read Dixon's paper in depth, but I can describe my understanding of it's importance, in particular with regards to modelling the solar system. It's been known for some time that the "figure", i.e. the non-spherical shape, of the planets and moons has important impacts on the dynamics of the solar system in general (and the Earth-Moon system in particular).

What Dixon's paper allow is a way to model these effects purely in the context of General relativity, based on Einstein's field equations, without appealing to Newtonian arguments based on spherical harmonic expansions of the Newtonian potentials.

Actually, that's probably not the only point made in the three papers, so this isn't a complete summary, but it's the point I think is the most interesting an applicable to this thread.

 

[PeterDonis]

the series of papers by Dixon on "Dynamics of extended bodies in general relativity", of which there are a series of three

I'm confused by equation (1.1) of the first paper. It says $\nabla_\beta T^{\alpha \beta} = - F^{\alpha \beta} J_\beta$, and Dixon says this equation is a consequence of the Einstein-Maxwell equations. But this seems to contradict the Bianchi identities satisfied by the Einstein Field Equation, which require $\nabla_\beta T^{\alpha \beta} = 0$.

 

[john baez]

So that was certainly his goal: to model particles as singular solutions of Einstein's equations.

Isn't the classical point-particle picture an idealization?

Yes, and that's why Einstein wanted to derive the geodesic equation for point particles as a limiting case of a deeper field-theoretic description: Einstein's equations describing the motion of "black holes" (though he didn't know they were black holes). Naively, it should be true that if you consider the motion of black holes obeying Einstein's equations, and take the mass $\to 0$ limit of these black holes, you get the geodesic equations for point particles. But proving this sort of thing is hard.

I think there are no point particles in the literal sense.

Right.

They make a lot of trouble if going further in the non-relativistic approximation already in the, I think, simpler case of electrodynamics: There's no fully self-consistent solution for the dynamics of classical charged point particles and the em. field.

Actually I think nobody even knows for sure! There are a lot of unsolved problems in this vicinity, and lots of contradictory claims in the literature. I talked about these problems here:

• Struggles with the continuum (part 3).

... but since then I've gotten an email pointing me to some new papers I should read. The situation is a mess.

 

[PeterDonis]

I'm confused by equation (1.1) of the first paper. It says $\nabla_\beta T^{\alpha \beta} = - F^{\alpha \beta} J_\beta$, and Dixon says this equation is a consequence of the Einstein-Maxwell equations. But this seems to contradict the Bianchi identities satisfied by the Einstein Field Equation, which require $\nabla_\beta T^{\alpha \beta} = 0$.

Even more confusing, Equation (1.2) of the third paper is $\nabla_\beta T^{\alpha \beta} = 0$. But then, on p. 66, he replaces this with $\nabla_\beta T^{\alpha \beta} = - F^{\alpha \beta} J_\beta$. The only reason he gives for this is that he is now treating "the more general case of a body moving under the influence of both gravitational and electromagnetic forces", but that doesn't allow the violation of the Bianchi identities; the way to include the effects of electromagnetic forces is to include the electromagnetic stress-energy in $T^{\alpha \beta}$, and you will still have $\nabla_\beta T^{\alpha \beta} = 0$. So I'm very confused about what's going on with Dixon's approach.

 

[martinbn]

Even more confusing, Equation (1.2) of the third paper is $\nabla_\beta T^{\alpha \beta} = 0$. But then, on p. 66, he replaces this with $\nabla_\beta T^{\alpha \beta} = - F^{\alpha \beta} J_\beta$. The only reason he gives for this is that he is now treating "the more general case of a body moving under the influence of both gravitational and electromagnetic forces", but that doesn't allow the violation of the Bianchi identities; the way to include the effects of electromagnetic forces is to include the electromagnetic stress-energy in $T^{\alpha \beta}$, and you will still have $\nabla_\beta T^{\alpha \beta} = 0$. So I'm very confused about what's going on with Dixon's approach.

This is the convention in some places, for example Sachs and Wu, the electromagnetic part of the stress energy tensor is written seperately. Einstein's equations are $G=T+E$.

 

[vanhees71]

Even more confusing, Equation (1.2) of the third paper is $\nabla_\beta T^{\alpha \beta} = 0$. But then, on p. 66, he replaces this with $\nabla_\beta T^{\alpha \beta} = - F^{\alpha \beta} J_\beta$. The only reason he gives for this is that he is now treating "the more general case of a body moving under the influence of both gravitational and electromagnetic forces", but that doesn't allow the violation of the Bianchi identities; the way to include the effects of electromagnetic forces is to include the electromagnetic stress-energy in $T^{\alpha \beta}$, and you will still have $\nabla_\beta T^{\alpha \beta} = 0$. So I'm very confused about what's going on with Dixon's approach.

In the EFE
$$G_{\mu \nu}=\kappa T_{\mu \nu}$$
$T_{\mu \nu}$ is the total energy-momentum tensor of matter (and radiation) and thus must obey the local conservation equation
$$\nabla_{\mu} T^{\mu \nu}=0,$$
due to the Bianchi identity, as is characteristic for gauge theories like GR.

The equation
$$\nabla_{\alpha} \tilde{T}^{\alpha \beta}=-F^{\alpha \beta} J_{\beta},$$
indicates that $\tilde{T}^{\alpha \beta}$ is the pure electromagnetic-field part of the EM tensor, i.e., the part for at least the electrically charged particles making up the current density $J$ is left out.

Which paper by Dixon are we talking about concretely? I know, somewhat superficially though, some papers by Dixon concerning bodies with spin, which is also a fascinating and pretty complicated (again even in the special-relativistic SR context not completely sloved beyond approximations) topic.

 

[PeterDonis]

Which paper by Dixon are we talking about concretely?

The first and third of the ones linked to in the StackExchange thread that @pervect referenced. For convenience, I'll post the direct links here:

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1970.0020

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1970.0191

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1974.0046

 

[PeterDonis]

The equation
$$\nabla_{\alpha} \tilde{T}^{\alpha \beta}=-F^{\alpha \beta} J_{\beta},$$
indicates that $\tilde{T}^{\alpha \beta}$ is the pure electromagnetic-field part of the EM tensor, i.e., the part for at least the electrically charged particles making up the current density $J$ is left out.

This doesn't seem right, because Dixon is taking moments of $T^{\alpha \beta}$ and saying they correspond to things like the total energy/momentum, total angular momentum, etc., not just quantities corresponding to the fields. If anything, the equation $\nabla_{\alpha} \tilde{T}^{\alpha \beta}=-F^{\alpha \beta} J_{\beta}$ suggests to me something like the Lorentz force equation, which would mean $T^{\alpha \beta}$ on the LHS would be the stress-energy of the particles, not the fields.

 

[TSny]

This doesn't seem right, because Dixon is taking moments of $T^{\alpha \beta}$ and saying they correspond to things like the total energy/momentum, total angular momentum, etc., not just quantities corresponding to the fields. If anything, the equation $\nabla_{\alpha} \tilde{T}^{\alpha \beta}=-F^{\alpha \beta} J_{\beta}$ suggests to me something like the Lorentz force equation, which would mean $T^{\alpha \beta}$ on the LHS would be the stress-energy of the particles, not the fields.

I think that's right. If ${T}_{\text{emf}}^{\alpha \beta}$ is the energy-momentum tensor for the electromagnetic field, then it would satisfy $\nabla_{\alpha}{T}_{\text{emf}}^{\alpha \beta} = +F^{\alpha \beta}J_{\beta}$.

So, $\nabla_{\alpha}{T}_{\text{total}}^{\alpha \beta} = \nabla_{\alpha}\left({T}_{\text{Dixon}}^{\alpha \beta}+{T}_{\text{emf}}^{\alpha \beta} \right) = 0$

That's my guess. I believe this is @martinbn 's interpretation.