Gravitational time dilation

[PAllen]

Here is a reference which discusses, in passing, stationary double Schwarzschild solutions with a so called Weyl strut supporting them:

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0909.4413

 

[PAllen]

Still can't find a good online accessible discussion of axisymmetric solutions at the level I'm looking for. A complete treatment is in:

https://www.amazon.com/dp/0521467020/?tag=pfamazon01-20

chapter 20. Select portions of this can be viewed in Google books.

One thing I find, on review, is that the struts joining singularities always have a singularity of some type (e.g. conical singularity). This helps address yuiop's doubts: what kind of strut can support singularities? Another type of singularity.

Another point is generality: It is proven that a static solution of the EFE with two masses must also have a support of some type.

Getting back to one desire of the OP: if you want to explore time dilation in the vicinity of more than one mass, but have the situation be static so you can consider all time differences gravitatational, this class of solution it the only way I can conceive of going about it.

 

[yuiop]

One thing I find, on review, is that the struts joining singularities always have a singularity of some type (e.g. conical singularity). This helps address yuiop's doubts: what kind of strut can support singularities? Another type of singularity.

Another point is generality: It is proven that a static solution of the EFE with two masses must also have a support of some type.

Getting back to one desire of the OP: if you want to explore time dilation in the vicinity of more than one mass, but have the situation be static so you can consider all time differences gravitatational, this class of solution it the only way I can conceive of going about it.

Could we use two equally charged black holes to provide a static solution that does not need a strut?

 

[George Jones]

A complete treatment is in:

https://www.amazon.com/dp/0521467020/?tag=pfamazon01-20

chapter 20. Select portions of this can be viewed in Google books.

This is very useful, comprehensive book, but I also like the book that I reference below, which is less comprehensive, but a little more detailed.

The paper at atty's link is about magnetically charged objects that have opposite magnetic charges, and that are held in equilibrium by an external magnetic field. In 1966, Bonner considered a similar situation, but with a "strut" instead of an external magnetic field. In 1947, Majumdar and Papapetrou (independently) found equilibrium solutions for an arbitrary number of extremal electrically charged objects that are distributed randomly. The Majumdar-Papapetrou solutions require neither struts nor an external field, and so represent a balance between gravity and electrostatic repulsion.

All of these, the paper to which atyy links, Bonner's work, the Majumdar-Papapetrou solutions, and other equilibrium solutions, are referenced and discussed briefly in an interesting book that I picked up a few weeks ago, Exact Space-Times in General Relativity by Jerry B, Griffiths and Jury Podolsky. Particularly relevant are subsection 10.8.1 Equilibrium configurations with distinct sources of section 10.8 Axially symmetric electrovacuum space-times and section 22.5 Majumdar-Papapetrou solutions.

Could we use two equally charged black holes to provide a static solution that does not need a strut?

 

[PAllen]

Could we use two equally charged black holes to provide a static solution that does not need a strut?

Yes, such solutions are possible and are discussed in chapter 21 of the book I mentioned in my prior post (the simplest are a type of superposition of two Reissner-Nordstrom solutions). I was fixated on neutral matter, as the real universe contains no large bodies with substantial charge.

 

[PAllen]

This is very useful, comprehensive book, but I also like the book that I reference below, which is less comprehensive, but a little more detailed.

Thanks George! From that, on Google books, I then find the following freely available paper:

http://arxiv.org/abs/0706.1981

[Edit: The importance of the above is that the equilibrium is between non-extreme sources. Extreme sources have the maximum charge/mass ratio allowed by GR. The spacetime is dominated more by the EM field than the mass. However, the above paper also shows that for non-extreme sources, one must be a naked singularity, the other a black hole with horizon. ]

[Edit 2: News not so good. To produce a naked singularity, you need a super-extremal source, and the geometry is influenced by EM field of the charge at least as much as by mass. Further, there is strong reason to believe the super-extremal sources are impossible in nature. The upshot is that this type of solution is not a good model for a static configuration of mass. If it were me, I would be more interested in solutions with a strut as a better exact, simplified model of something physically plausible. ]