# Understanding solutions of GR

## Main Question or Discussion Point

What I'm actually trying to find is:
The metric for a massless string coming in from +infinity on the z axis, holding a spherical mass (with angular momentum) a fixed distance from the origin, along with another string coming in from -infinity on the z axis also holding a spherical mass. In addition to the metric, I'm curious what the tension in the strings are.

Is there already an exact solution like that?

A friend recommended Plebanski–Demianski solutions which can involve broken strings, so could possibly contain the solutions I am curious about. This paper,
http://arxiv.org/PS_cache/gr-qc/pdf/0511/0511091v1.pdf
discusses some very general solutions. Unfortunately, I don't understand the terms well enough to follow the discussion.

What physically is a NUT parameter?
Also, I must be missing something because the metric in equation 5 looks time independent, so I don't understand how it could describe accelerating blackholes. What do expanding and twist mean in this setting? is it just accelerating and angular momentum respectively?

Any help understanding that article would be much appreciated.

## Answers and Replies

Related Special and General Relativity News on Phys.org
George Jones
Staff Emeritus
Gold Member
Unfortunately, I don't have any time to think about these things right now.

You should have a look (possibly by interlibrary loan) at the excellent new book Exact Space-Times in Einstein's General Relativity by Jerry B. Griffiths and Jiri Podolsky,

https://www.amazon.com/dp/0521889278/?tag=pfamazon01-20.

Chapter 16 is based largely on the article for which you gave a link. For an explanation of contraction, twist, and shear for a congruence of lightlike worldlines for "test photons", see 2.1.3 Geodesics and Geometric Optics. See also chapters 12 Taub-NUT Space-Time, 14 Accelerating Black Holes, 15 Further Solutions fo Uniformly Accelerating Particles, and 16 Plebanski-Demianski Solutions.

Use the LOOK INSIDE feature at amazon to see a detailed Table of Contents.

I'll have more time in about a week.

While you're waiting for GJ, this may be of interest. It's a summary of known solutions in astrophysics and includes Taub-NUT.

1. arXiv:gr-qc/0004016v1 5 Apr 2000

Regarding two gravitating objects and strings or struts - there is an axial solution which has a 'strut' along the z-axis holding the objects apart. It sounds like you have the compementary picture, with a singularity on the outside, rather than between the objects. The strut solution is in

2. Letelier P.S. and Oliveira S.R., Class. Quantum Grav., 15, 421 (1998).

I found the refence here

3. arXiv:gr-qc/0502062v1

I've been looking a spacetime with 2 singlarities held apart by a field of some sort ( not quite a solution) which is why I've been reading around the subject.

George,
It looks like the library system I have access to just picked up that book. I put a request in, so we'll see I guess. Thanks for the suggestion.

I picked up another book at the library which is Exact Solutions of Einstein's Field Equations, by Kramer and others. Unfortunately, they assume I understand way too much. If someone hasn't already solved an exact solution, I am well aware I don't have the math capabilities to compete with the experts and so I won't be able to solve for it. So it is frustrating that so much of the book is spent discussing features of the solutions (categorizing by "algebraicly special" features I don't understand, and different methods of obtaining the solutions) which are beyond me.

What I'd love to find, is a large table or book which lists solutions (by giving some coordinates and the metric in those coordinates) and explaining what the parameters in the solutions mean physically. I'll keep checking books. Hopefully I'll come across something.

While you're waiting for GJ, this may be of interest. It's a summary of known solutions in astrophysics and includes Taub-NUT.
I found some other NUT parameter papers. It sounds like, in analogy to electromagnetism, if you consider mass a "gravito" monopole then the NUT parameter is a "gravitomagnetic" monopole term.

It sounds like in some cases it can't be consider as such and is related to the "twist", which is similar, but due to how the spacetime is connected, as opposed to the "source" itself? Hopefully I am at least getting close to understanding.

Regarding two gravitating objects and strings or struts - there is an axial solution which has a 'strut' along the z-axis holding the objects apart. It sounds like you have the compementary picture, with a singularity on the outside, rather than between the objects. The strut solution is in

2. Letelier P.S. and Oliveira S.R., Class. Quantum Grav., 15, 421 (1998).

I found the refence here

3. arXiv:gr-qc/0502062v1

I've been looking a spacetime with 2 singlarities held apart by a field of some sort ( not quite a solution) which is why I've been reading around the subject.
Wow, that reference has some interesting stuff in it. Thanks!

This is much more approachable than the other papers. The "Superposition of two Chazy-Curzon particles" section sounds like it might even contain the solution I was originally looking for above when choosing $$\gamma_{CCb}=0$$ in the "strut" region (and therefore appears to have some "strand" like defect in z<0 and z>b). I'll need to play with this some more.

Since this is a static metric, is it okay to use the Komar mass relations to get what the energy/length of a string defect is, and claim this is the tension? How do I distinguish between the mass/length of a string and the tension in the string?

Also, does anyone know of a spinning ring of string solution? It seems like this should be possible with the formalism presented in that paper, but it is unclear to me how they knew what functions to choose (I mean, a Newtonian bar -> Schwarzschild black hole... that is not obvious to me at all). I realize Kerr has a "ring" singularity at r=0, but you can't go "inside" the ring r<0, right? So the Kerr is more of a ring "edge" of spacetime, than a spinning ring of string solution ... correct?