# Status of relativistic 2-body problem

HeavyWater
I'd like to get back into theoretical physics as a retiree. I last worked on the relativistic 2-body problem over 25 years ago. I've been reading Trump and Schieve's text on classical relativistic dynamics and I'm wondering has the classical 2-body relativistic problem been solved. I realize the word "solved" has different meanings for different people. I'd like to hear your comments on the status of this problem. Perhaps you can suggest a more recent text than this one.

Thanks,
Heavywater

Dickfore
Please define what is the relativistic 2-body problem. As you may know, the meaning of potential energy as a function of the positions of the particles is meaningless in SR, because the relative position of the particles is not Lorentz invariant.

A Caltech-Cornell group led by Lee Lindblom and Saul Teukolsky have made excellent progress in the numerical simulation of black hole collisions. For a list of publications, see http://www.tapir.caltech.edu/~lindblom/Content/PublicationsSubject.html.

Passionflower
I'd like to get back into theoretical physics as a retiree. I last worked on the relativistic 2-body problem over 25 years ago. I've been reading Trump and Schieve's text on classical relativistic dynamics and I'm wondering has the classical 2-body relativistic problem been solved.
If you mean by solved that someone provided an analytical solution then the answer is no, an analytical solution is impossible if we assume GR is correct.

Adding to Bill_K's pointer, the following is a nice public website (that include the full paper list as well) for the collaboration he refers to:

http://www.black-holes.org/

Going from two body black holes to two bodies with equations of state, the best result I've been able to locate is the following:

http://arxiv.org/abs/1103.3874

Going to the 3 body problem, I have not found any attempt to deal with this for bodies with equations of state, but there is progress for 3 black holes. Following is a sample of work:

http://arxiv.org/abs/0711.1165
http://arxiv.org/abs/astro-ph/0509814
http://arxiv.org/abs/gr-qc/0702076
http://arxiv.org/abs/1012.4423
http://arxiv.org/abs/1108.4485

[EDIT: I guess I should add the general articles in the Numerical Relativity category of:

http://relativity.livingreviews.org/Articles/subject.html
]

Last edited:
If you mean by solved that someone provided an analytical solution then the answer is no, an analytical solution is impossible if we assume GR is correct.

I know it has never been solved. Has anyone proved it is impossible to find an exact analytical solution? I suppose this would depend on the definition of the bodies: point particles, classical fluid bodies, orbiting wormhole throats.

Skippy

Passionflower
I know it has never been solved. Has anyone proved it is impossible to find an exact analytical solution? I suppose this would depend on the definition of the bodies: point particles, classical fluid bodies, orbiting wormhole throats.

Skippy
All we know is that in the end we have the two bodies come together in a singularity.
But any setup does not only require a position and momentum (which is already next to impossible to uniquely specify as we have no background) but also an infinite number of waves over the whole spacetime.

juanrga
I'd like to get back into theoretical physics as a retiree. I last worked on the relativistic 2-body problem over 25 years ago. I've been reading Trump and Schieve's text on classical relativistic dynamics and I'm wondering has the classical 2-body relativistic problem been solved. I realize the word "solved" has different meanings for different people. I'd like to hear your comments on the status of this problem. Perhaps you can suggest a more recent text than this one.

Thanks,
Heavywater

No, it is still an open problem. We know sure that 2-body problem has not solution in ordinary relativity and little more.

Their extension of relativity to deal with many-body dynamics is the more popular in theoretical physics but still open to many objections.

Even if were to ignore the objections, the generalized Hamiltonian equations that they work only deal with simplest case.

HeavyWater
Thank you for such a quick response. I'm sure a lot of theoretical physicists would like to see this problem(or should I say problems) resolved. Can you or anyone else point me to 1 or 2 current references. The latest text I have found is by Trump and Schieve and it is dated 1998. I am more interested in the relativistic classical mechanical problem than the relativistic quantum mechanical problem--but any reference after 1999 will be helpful.

Heavywater

HeavyWater
Thank you Dickfore,
I am thinking about the Classical relativistic 2 body problem where both bodies are point particles AND I was only thinking about special relativity. I should have clarified my problem statement.

I am aware of the famous Currie, Jordan, Sudharshan "No Interaction" or "No Go" theorem from around 1963. I am aware there were some issues associated with a world line being an invariant and/or observable. Also I'm aware of some uncertainties associated with whether their position variable was a canonical variable.

Since I left the scene, 25+ years ago, I've found lots of interesting work was done in classical relativistic dynamics with the many-body problem. My trail of references seems to have stopped with Trump and Schieve's text.

Are you aware of any references after Trump and Schieve's (1998) text?

Thanks for your inputs and encouragement.

HeavyWater

HeavyWater
A Caltech-Cornell group led by Lee Lindblom and Saul Teukolsky have made excellent progress in the numerical simulation of black hole collisions. For a list of publications, see http://www.tapir.caltech.edu/~lindblom/Content/PublicationsSubject.html.

Bill-K,
Your references will definitely help me. I'll have to wait until Monday until I can get to a library. I am not interested in the problem as it relates to GR, only as it relates to SR. Feel free to make any other suggestion. I will definitely check out the work by Linblom and Teukolsky.

Thanks, Heavywater

HeavyWater
If you mean by solved that someone provided an analytical solution then the answer is no, an analytical solution is impossible if we assume GR is correct.

Thank you for your insights Passionflower. I am currently only interested in the SR aspect of the problem. I am going to check out the references that some of the others identified. If you have any suggestions as to the the SR classical problem, please feel free to identify a reference or two.

Thanks, Heavywater

HeavyWater
I know it has never been solved. Has anyone proved it is impossible to find an exact analytical solution? I suppose this would depend on the definition of the bodies: point particles, classical fluid bodies, orbiting wormhole throats.

Skippy

Skippy,
Thank you for your insights and response. I was thinking of a much simpler problem, one that only deals with point particles, and only special relativity. In particular, I was thinking about the "No Go" or "No Interaction Theorem" of Currie, Jordan, and Sudharshan from way back in 1963.

Thanks, Heavywater

HeavyWater
Adding to Bill_K's pointer, the following is a nice public website (that include the full paper list as well) for the collaboration he refers to:

http://www.black-holes.org/

Going from two body black holes to two bodies with equations of state, the best result I've been able to locate is the following:

http://arxiv.org/abs/1103.3874

Going to the 3 body problem, I have not found any attempt to deal with this for bodies with equations of state, but there is progress for 3 black holes. Following is a sample of work:

http://arxiv.org/abs/0711.1165
http://arxiv.org/abs/astro-ph/0509814
http://arxiv.org/abs/gr-qc/0702076
http://arxiv.org/abs/1012.4423
http://arxiv.org/abs/1108.4485

[EDIT: I guess I should add the general articles in the Numerical Relativity category of:

http://relativity.livingreviews.org/Articles/subject.html
]

PAllen,
Thank you for letting me know about the references. I just spent an enjoyable 1/2 hour on the website.

Let me clarify, I am working on a simpler problem--that being SR with the classical mechanics of 2 point particles. A frequently identified reference in this area is the "No Go" theorem by Currie, Jordan and Sudharshan.

Thank you for you coments and references,
HeavyWater

Passionflower
Thank you for your insights Passionflower. I am currently only interested in the SR aspect of the problem. I am going to check out the references that some of the others identified. If you have any suggestions as to the the SR classical problem, please feel free to identify a reference or two.
Frankly I would not know what you mean by the two body problem in SR.
If you are not talking about gravitation then what are you talking about?

Dickfore
So, in the first paragraph of the Introduction it says:
"...But the combined requirements of relativistic symmetry and manifest invariance may restrict the theory so severly that it is capable only of describing non interacting particles. We will show that this is in fact the case in a Lorentz symmetric classical mechanical theory of the motion of a pair of particles..."

So, I guess this goes in favor of my first post in this thread. The point is that, due to the finite speed of propagation of interactions, one ought to consider a field as a physical object carrying the interaction. A field has (innumerably) infinitely many degrees of freedom, and the Lorentz invariant two-body problem turns into a problem in continuum mechanics.

juanrga
Thank you for such a quick response. I'm sure a lot of theoretical physicists would like to see this problem(or should I say problems) resolved. Can you or anyone else point me to 1 or 2 current references. The latest text I have found is by Trump and Schieve and it is dated 1998. I am more interested in the relativistic classical mechanical problem than the relativistic quantum mechanical problem--but any reference after 1999 will be helpful.

Heavywater

So far as I know that classical theory has not advanced since Trump and Schieve published their monograph and, as they admit in the monograph, the current theory only can solve some weak-field defects of general relativity.

There is some interest in the application to many-body quantum bound problems, where quantum field theory also fails, and some recent attempts to apply the new relativistic theory to an extension of string/brane theory (which inherits the defects of both quantum field theory and general relativity): The Landscape of Theoretical Physics, A Global View: From Point Particles to the Brane World and Beyond in Search of a Unifying Principle. But as said before the whole theory is still open to many technical objections.

Another attempt to solve the defects of field theory is by Chubykalo and Smirnov-Rueda

See specially the Physical Review E paper correcting the defects of Maxwell theory of electrodynamics and their experimental application in Journal of applied physics.

Chubykalo and Smirnov-Rueda approach has been recently extended to many-body gravitation, with the bonus that the new potentials solve the dark matter problem (dark matter is fictitious). See Modified Newtonian Dynamics and Dark Matter from a generalized gravitational theory

Last edited:
HeavyWater
Regarding Currie, Jordan, Sudarshan's reference, here's a link:
http://dx.doi.org/10.1103/RevModPhys.35.350

Thank you Dickfore,
That article is a "classic". I've read it numerous times and I'm embarrassed to say that I still don't understand all of it. Now that I'm retiring, I'll have plenty of time to continue to study it.

HeavyWater

HeavyWater
So, in the first paragraph of the Introduction it says:
"...But the combined requirements of relativistic symmetry and manifest invariance may restrict the theory so severly that it is capable only of describing non interacting particles. We will show that this is in fact the case in a Lorentz symmetric classical mechanical theory of the motion of a pair of particles..."

So, I guess this goes in favor of my first post in this thread. The point is that, due to the finite speed of propagation of interactions, one ought to consider a field as a physical object carrying the interaction. A field has (innumerably) infinitely many degrees of freedom, and the Lorentz invariant two-body problem turns into a problem in continuum mechanics.

Thank you for taking the time and for your thoughtful response. I am looking at a copy of the paper right now. Let me spend the rest of the day reviewing it and collecting my thoughts and questions and I will get back with you tomorrow.

Heavywater

HeavyWater
So far as I know that classical theory has not advanced since Trump and Schieve published their monograph and, as they admit in the monograph, the current theory only can solve some weak-field defects of general relativity.

There is some interest in the application to many-body quantum bound problems, where quantum field theory also fails, and some recent attempts to apply the new relativistic theory to an extension of string/brane theory (which inherits the defects of both quantum field theory and general relativity): The Landscape of Theoretical Physics, A Global View: From Point Particles to the Brane World and Beyond in Search of a Unifying Principle. But as said before the whole theory is still open to many technical objections.

Another attempt to solve the defects of field theory is by Chubykalo and Smirnov-Rueda

See specially the Physical Review E paper correcting the defects of Maxwell theory of electrodynamics and their experimental application in Journal of applied physics.

Chubykalo and Smirnov-Rueda approach has been recently extended to many-body gravitation, with the bonus that the new potentials solve the dark matter problem (dark matter is fictitious). See Modified Newtonian Dynamics and Dark Matter from a generalized gravitational theory

Thanks Juanrga,
I'll have to get copies of these papers on Tuesday, to give you an intelligent reply. Thanks for your inputs and I'll get back with you as soon as I can read these documents. I do appreciate your comments about the status of this "problem". It looks like I will have plenty of things to think about in retirement. You will hear from me in about a week.
Heavywater

HeavyWater
Thank you for taking the time and for your thoughtful response. I am looking at a copy of the paper right now. Let me spend the rest of the day reviewing it and collecting my thoughts and questions and I will get back with you tomorrow.

Heavywater

Dickfore,
I got out my re-read your quote above. Before I get into my questions, I have got to ask you what do CJS mean by "manifest invariance"? Why the word, "manifest". Does "manifest invariance" differ from "invariance" in some way? I note that CJS defines "manifest invariance" a sentence or two below your quote. They also define it on p351 of CJS, " ... property that the world line of a particle transforms as a sequence of space time events according to the usual Lorentz transofmation formula." And then below that (also on p351), " ... be formulated in terms of a set of equations involving these Lie brackets".

So, is Manifest invariance different than invariance? Is it a "stronger" form of invariance?

Thanks,
Heavywater