Particles (not?) following geodesics in GR

[Michael_1812]

particles (not??) following geodesics in GR

In a three-month old thread
https://www.physicsforums.com/showthread.php?p=2557522&posted=1#post2557522
one of the tutors ("atyy") said:
"And GR in full form does not have particles traveling on geodesics..."

What does that mean? How can a free particle deviate from its geodesic?
(Or was it implied here that particles were interacting with fields other than gravity?)

Many thanks,
Michael

 

[atyy]

atyy is not a tutor - he is a clueless biologist!

Anyway, what I meant was that the equations of motion should come out from the Einstein field equations and the equation of state of whatever matter fields exist. I think it's the Einstein-Infeld-Hoffmann equations (not sure, doing this off the top of my head).

Try http://arxiv.org/abs/gr-qc/9912051, I believe I read this in Rindler's text too.

 

[JesseM]

atyy, did you mean that "particles follow geodesics" is not part of the basic axioms of GR but instead has to be derived, or did you mean that there are actually situations in GR where 'particles' (which I assume implies negligible mass, though maybe not) which aren't being acted on by any non-gravitational forces will be predicted to have non-geodesic paths?

 

[atyy]

atyy, did you mean that "particles follow geodesics" is not part of the basic axioms of GR but instead has to be derived, or did you mean that there are actually situations in GR where 'particles' (which I assume implies negligible mass, though maybe not) which aren't being acted on by any non-gravitational forces will be predicted to have non-geodesic paths?

I meant they can be derived. When we use the geodesic equation, we exclude one "particle" from the total mass-energy that produces spacetime curvature, then we postulate that this particle that has been left out follows the geodesic equation. It turns out that if we include this "particle" in the total mass-energy that produces spacetime curvature, we get the geodesic equation (to very good approximation).

 

[bcrowell]

IMO atyy's earlier statement ("...GR in full form does not have particles traveling on geodesics...") was either incorrect or not expressed well. But anyway, thanks, atyy, for the pointer to the Anderson paper, which I thought was very interesting. Even if we sometimes disagree, I think your posts are always interesting and informative.

I don't think anyone, including Anderson, questions the idea that in the limit of low mass, test particles in GR follow geodesics. Anderson is just arguing that the metrical aspect of GR is not fundamental.

The EIH equations are a series approximation to GR in which a system of gravitationally interacting particles is treated as interacting via instantaneous action at a distance, with a velocity-dependent interaction. The lowest order terms are the same as Newton's laws. I don't think it's really correct to say that the motion of test particles along geodesics has to come from the EIH equations.

With Anderson's arguments in mind, it's actually kind of interesting to ask why the geodesics in GR have to be the world-lines of test particles. I think the basic answer is that in the limit of low-mass test particles, GR can in some sense be treated in a linear approximation, so that you can't get any funky effects like the back-reaction of a particle's own gravitational waves on the particle itself. In this approximation, there is simply nothing else in the structure of spacetime that *could* serve as the world-lines of particles, so we kind of have to interpret the geodesics that way. If a test particle did deviate from a geodesic, then we would have a strange situation. We could then go into a local Minkowski frame in which the particle was initially at rest. The particle would accelerate off in some random direction. What would physically determine this direction? If the particle had a lot of mass, then we could imagine that the direction was determined by the back-reaction of the particle's gravitational interaction from its own past motion. But since we're considering the limit of low mass, this can't be the case, and the particle's acceleration violates Lorentz invariance.

 

[atyy]

IMO atyy's earlier statement ("...GR in full form does not have particles traveling on geodesics...") was either incorrect or not expressed well. But anyway, thanks, atyy, for the pointer to the Anderson paper, which I thought was very interesting. Even if we sometimes disagree, I think your posts are always interesting and informative.

That's too kind. I'm just a biologist who find physics interesting - and besides I'm quite often wrong even on stuff on which I'm supposed to be an expert!

I don't think anyone, including Anderson, questions the idea that in the limit of low mass, test particles in GR follow geodesics. Anderson is just arguing that the metrical aspect of GR is not fundamental.

BTW, I think this is the Anderson of "no absolute objects" which we nowadays think of as "no prior geometry", which is certainly heuristically right, even if a bit problematic when it comes to tight definitions.

 

[yuiop]

In another thread it was strongly suggested that a charged particle would not follow the same geodesic as neutral particle. This in turn implies that a test particle's geodesic is not only determined by its location and velocity but also by its charge, which I have never heard mentioned before in the context of the Schwarzschild metric. Any ideas on this?

 

[Michael_1812]

Anderson's line of reasoning has long been explored in detail by many. See for example the theories by

V. I. Ogiyevetsky and I. V. Barinov. "Interaction field of spin two and the Einstein equations." Annals of Physics, 35:167-208, 1965.

and

S. Deser. "Self-interaction and gauge invariance." General Relativity and Gravitation. 1:9-18, 1970

who described gravity with a nonlinear tensor field on a flat background space-time.

The team

L. P. Grishchuk, A. N. Petrov, and A. D. Popova.
"Exact theory of the (Einstein) gravitational field in an arbitrary background space-time." Communications in Mathematical Physics, 94:379-396, 1984.

went further, demonstrating that, in principle, *any* background space-time, with the right signature of the metric, can be employed; and that a nonlinear tensor-field dynamical theory developed thereon turns out to be mathematically equivalent to
Einstein's general relativity.

 

[bcrowell]

In another thread it was strongly suggested that a charged particle would not follow the same geodesic as neutral particle. This in turn implies that a test particle's geodesic is not only determined by its location and velocity but also by its charge, which I have never heard mentioned before in the context of the Schwarzschild metric. Any ideas on this?

I think the answer to this is similar to the argument I made in #5. If the particle has a large mass (e.g., a member of the Hulse-Taylor binary pulsar system), then it suffers a back-reaction from its own gravitational radiation. If the particle has a nonvanishing charge, it suffers a back-reaction from its own electromagnetic radiation. So for an ideal test particle, you want zero charge and infinitesimal mass (or zero mass, if it's going to be a lightlike geodesic).

 

[Michael_1812]

I think the answer to this is similar to the argument I made in #5. If the particle has a large mass (e.g., a member of the Hulse-Taylor binary pulsar system), then it suffers a back-reaction from its own gravitational radiation. If the particle has a nonvanishing charge, it suffers a back-reaction from its own electromagnetic radiation. So for an ideal test particle, you want zero charge and infinitesimal mass (or zero mass, if it's going to be a lightlike geodesic).

Does this have anything to do with violation of the Strong Equivalence Principle?

 

[bcrowell]

Does this have anything to do with violation of the Strong Equivalence Principle?

Possibly, depending on how you interpret the SEP. See this thread: https://www.physicsforums.com/showthread.php?t=369612&highlight=to+radiate

The SEP is a local property, and depending on how you define locality, you can see this either as violating the SEP or not violating it. In any case, the back-reaction effect is much too small to detect in practice. The reason you really don't want to use a charged particle as a test particle for measuring geodesics is simply because it will be accelerated by ambient electromagnetic fields.

 

[atyy]

The discussion in section 5.5.4 of http://relativity.livingreviews.org/Articles/lrr-2004-6/ is also relevant.

 

[Altabeh]

To me it is so ridiculous that Anderson claims that geometry doesn't play a role in determining the EIH equations while he also states that EIH make use of an approximation procedure, or in my own terms, EIH perturbative method which is strongly geometrical because a perturbation from a flat spacetime has a geometrical explanation in terms of a perturbative metric of the form $$(1+f_{00},-1+f_{11},...,-1+f_{33})$$ where the components of the diagonal matrix $$f_{\mu \nu }$$ are so small. They make use of a perturbed metric in their original paper that you can find on the Internet and witness why Anderson thinks that if we don't directly drag the metric in our calculations, then we are not using a geometrical method to obtain the equations of motion!

By the way, can anybody direct me to an article\book containing a very simplified methold of obtaining EIH equations? I know of http://www.iop.org/EJ/abstract/0370-1298/64/1/310", but unfortunately I don't have access to his article!

Any help will be much appreciated.

AB