Black hole electron: How can we drop the geodesic equation?

[greypilgrim]

Hi,

Einstein once showed that if we assume elementary particles to be singularities in spacetime (e.g. black hole electrons), then it is unnecessary to postulate geodesic motion, which in standard GR has to be introduced somewhat inelegantly by the geodesic equation. I don't have access to those papers (and probably wouldn't understand half of it), but I'm just curious how it can be possible to derive a motion simply from the field equations, without assuming any dynamic equation?

 

[PAllen]

One doesn't need anything like the black hole electron hypothesis to do away with the geodesic equation. Einstein was a pioneer (with collaborators) in showing this as well. The modern view is that that the geodesic hypothesis is logically superfluous, and merely an excellent approximation anyway (for any real body, the motion will not be exactly geodesic). Over the years, many different methods have been used to derive motion of bodies directly from the field equations. The multiplicity of methods yielding geodesic motion in an appropriate limit add weight to the view that the geodesic hypothesis is logically superfluous.

In general, the ability to derive geodesic motion becomes less surprising when GR is given the ADM formalism, with evolution from initial conditions on a 3-surface provided by the field equations. Numerical relativity based on this approach solves binary inspiral problems where, due to intense gravitational radiations, you cannot talk about either body even remotely following a geodesic - yet GR field equations determine their motion.

I would be happy to provide a sampling of papers showing several methods of deriving the geodesic equation in the limit of small bodies, along with references on the ADM formalism for more general motion. However, it seems likely not worth if you won't understand any of it. If you reply that you are interested, I will provide them.

 

[Dale]

I would be interested in ones describing the numerical approaches, particularly if they use some easily available code, e.g. in python or something similar.

 

[Ibix]

I would be interested in ones describing the numerical approaches, particularly if they use some easily available code, e.g. in python or something similar.

Me too. Not sure I'll understand yet, but I'd be interested.

 

[PAllen]

Well, the easier papers to follow are derivations of geodesic motion from field equations (which is also done less rigorously in most GR textbooks). However, what you both asked about was getting started with Numerical relativity. I am near retirement, and have just begun a 'rest of my life' fantasy project to do a simulation of inspiraling BH with ray tracing of the star background. I am doing this very, very, slowly. At this moment I am slowly trying to work through:

http://arxiv.org/pdf/gr-qc/0703035.pdf
and
http://www.black-holes.org/for-researchers-main

 

[ShayanJ]

Well, the easier papers to follow are derivations of geodesic motion from field equations (which is also done less rigorously in most GR textbooks). However, what you both asked about was getting started with Numerical relativity. I am near retirement, and have just begun a 'rest of my life' fantasy project to do a simulation of inspiraling BH with ray tracing of the star background. I am doing this very, very, slowly. At this moment I am slowly trying to work through:

http://arxiv.org/pdf/gr-qc/0703035.pdf
and
http://www.black-holes.org/for-researchers-main

I'm interested in seeing papers that get the geodesic motion analytically(can I use this word for them?). But the fact that you said most GR textbooks do it the easy way makes me think that you're talking about getting the geodesic equation by varying the proper time integral. Is this what you mean?

 

[PAllen]

I'm interested in seeing papers that get the geodesic motion analytically(can I use this word for them?). But the fact that you said most GR textbooks do it the easy way makes me think that you're talking about getting the geodesic equation by varying the proper time integral. Is this what you mean?

No, not at all. That requires that the geodesic assumption. However, Synge, Wald, and MTW (haven't checked Carroll) give reasonable derivations of geodesic motion from the field equations. Synge's is more in the spirit of the Einstein/Infeld/Hoffman approach of the 1940s, while MTW uses a then modern approach. I would say the following is the 'latest and greatest' on the topic:

section two of: http://arxiv.org/abs/1002.5045
all of: http://arxiv.org/abs/0806.3293

[One thing to be aware of in the above, is that to arrive at geodesic motion, it is necessary to assume matter follows a timelike path, which, as noted in a footnote in one of the above papers (I don't remember which), implicitly means the dominant energy condition is assumed. I am not in the mood to dig up other papers that demonstrate that assuming that it is possible to arbitrarily violate the dominant energy condition, you can arrive at motion for bodies that is not only non geodesic, but also not even timelike - i.e. tachyonic motion.]

[edit: It is the second of the above links that discusses the need to assume the dominant energy condition. Section 2 of the first link is a more streamlined derivation, but leaves out such technical details. It is no less rigorous because it does explicitly list as an assumption that a body must follow a timelike path, without further commenting on the implication or justification of this assumption.]