State of the art re the geodesic hypothesis

[bcrowell]

Stated loosely, the geodesic hypothesis says that test particles follow geodesics, where "test particle" means it has to be small in some sense (size, mass, ...), and there is an ambiguity in the word "geodesics" because we want to talk about geodesics of the spacetime that would have existed without the test particle. If we don't make any assumption about the matter field that the test particle is built out of, then there are counterexamples to the geodesic hypothesis.

The best references I know of for the current state of the art on this hypothesis are:

Yang, http://arxiv.org/abs/1209.3985

Ehlers and Geroch, http://arxiv.org/abs/gr-qc/0309074v1

Ehlers assumes an energy condition. Yang assumes that the matter field works according to the Klein-Gordon equation. I wrote up a presentation of the Ehlers-Geroch ideas, using crayons, here: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3).

Is one of these results weaker or stronger than the other? Naively, I would imagine that if something is a solution of the Klein-Gordon equation with mass m, then ... well, m is positive, so it satisfies energy conditions, doesn't it? Which energy conditions,...er, I don't know.

 

[pervect]

Wald gives:

Fock (1939) [not English]
Geroch and Jang (1975) "Motion of a body in General Relativity"

as references, and for the next level, which regards departure from geodesic motion for bodies that aren't quite "small" enough

Papapetrou(1951) "Spinning test particles in General Relativity"
Dixon(1974) "Dynamics of Extended bodies In General Relativity III Equations of Motion"

It's probably worth noting that spinning test bodies couple to "gravitomagnetism". (The gravitomagnetic part of the Riemann). So no body with spin can be "small enough", which makes studying the next level interesting to make sure one isn't missing anything by over-idealizing.

Solutions of the Klein Gordon equation aren't going to have any spin, one way around this issue but perhaps a bit misleading.

I'm not quite sure what the geodesic hypothesis says about the emission of gravitational waves, I would think that it implies "small enough" bodies can't do that (because if they did, the gravitational waves would carry momentum and make the bodies depart from geodesic motion).

 

[bcrowell]

Wald gives:

Fock (1939) [not English]
Geroch and Jang (1975) "Motion of a body in General Relativity"

as references, and for the next level, which regards departure from geodesic motion for bodies that aren't quite "small" enough

Papapetrou(1951) "Spinning test particles in General Relativity"
Dixon(1974) "Dynamics of Extended bodies In General Relativity III Equations of Motion"

Thanks -- but these are all quite a bit older than the 2003 and 2012 references I gave in #1.

It's probably worth noting that spinning test bodies couple to "gravitomagnetism". (The gravitomagnetic part of the Riemann). So no body with spin can be "small enough", which makes studying the next level interesting to make sure one isn't missing anything by over-idealizing.

I don't think this is quite right. The Ehlers-Geroch result applies regardless of spin, as long as the dominant energy condition holds. Basically the dominant energy condition rules out counterexamples involving spin because it keeps you from maintaining a fixed spin in the limit of small particles.

Solutions of the Klein Gordon equation aren't going to have any spin, one way around this issue but perhaps a bit misleading.

Hmm...well, they won't have any intrinsic spin, but they could have "spin" in the sense of non-intrinsic angular momentum.

I'm not quite sure what the geodesic hypothesis says about the emission of gravitational waves, I would think that it implies "small enough" bodies can't do that (because if they did, the gravitational waves would carry momentum and make the bodies depart from geodesic motion).

Right, this is the main reason that it requires the test particle to be small.

 

[pervect]

I don't think this is quite right. The Ehlers-Geroch result applies regardless of spin, as long as the dominant energy condition holds. Basically the dominant energy condition rules out counterexamples involving spin because it keeps you from maintaining a fixed spin in the limit of small particles.

Hmm...well, they won't have any intrinsic spin, but they could have "spin" in the sense of non-intrinsic angular momentum.

To stay classical, I suppose we do have to restrict the notion of spin to angular momentum.

But it's well known that angular momentum couples to the curvature tensor. It's a bit difficult to find much on the topic that isn't behind a paywall, I did find http://arxiv.org/pdf/gr-qc/0701080.pdf

Thus, a spinning body has a recognizable interaction with the gravitational curvature,
through its spin tensor, and it does not follow a geodesics motion; in this scheme, the
particle retains a test character, but it acquires an internal structure, due to its spin,
which prevents it a free motion even in absence of “external” fields.

I'm not 100% sure how this "gets along" with the EG result, which is why I suggest that it's worth investigating the issue of departure from geodesic motion as well as the limit.

 

[bcrowell]

I'm not 100% sure how this "gets along" with the EG result, which is why I suggest that it's worth investigating the issue of departure from geodesic motion as well as the limit.

I see...hmm...the paper you linked to is about intrinsic spin rather than classical "spin" (i.e., angular momentum). In the case of classical spin, it's pretty straightforward to work out how rapidly spin has to scale down with the linear size of an object. I've written this up here: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3). But that's a purely classical scaling argument. It certainly can't be expected to hold for a quantum-mechanical spin. So I guess the Ehlers-Geroch result really needs a bright red warning label on it say, "Hey, I'm only classical."

Without digging through the quantum stuff (which I don't think I have the chops for), I think it's fairly easy to estimate the size of the possible effect for a quantum spin. The Papapetrou result in the classical case is F~LR, where F is the anomalous force, L is the particle's angular momentum, and R is the Riemann curvature. I think this just has to be this way because of units, so it should be valid for a quantum mechanical spin as well, up to unitless factors of order unity. When L is on the order of Planck's constant, I suspect you're simply not going to get a force that's anywhere close to measurable.

 

[stevendaryl]

I don't think this is quite right. The Ehlers-Geroch result applies regardless of spin, as long as the dominant energy condition holds. Basically the dominant energy condition rules out counterexamples involving spin because it keeps you from maintaining a fixed spin in the limit of small particles.

Are you saying that a point-mass with nonzero spin (such as an electron) is disallowed by the dominant energy condition?

 

[bcrowell]

Are you saying that a point-mass with nonzero spin (such as an electron) is disallowed by the dominant energy condition?

No. The Ehlers-Geroch theorem is written in such a way that it doesn't apply if the test particle is a singularity. Therefore it doesn't say anything about point masses, only about a certain limiting process in which the test particle is taken to be smaller and smaller.

A secondary issue is that GR can't describe fundamental particles as point particles. A point particle in GR has to be either a black hole or a naked singularity, and the actual properties of particles like electrons can't be reconciled with either of those possibilities.

Of course you also can't explain half-integer spin using a classical theory, so in that sense there's no way GR can even talk about electrons.