Gravitational time dilation

Right, it would only follow if all of the masses and observers are static. That would not be so with an orbitting smaller body, true, where kinetic properties would also need to be accounted for, but I am not considering that, only static contributions. Perhaps you could think of the smaller body as accelerated upon a constantly accelerating platform. If a smaller body were added to the larger one, say a star, then it would be held up by the same process that holds up the star, the internal pressures. The overall reason I am considering only static bodies is not to determine some multiple body orbital problem, which would be extremely complex, but rather to integrate all of the point masses of a static non-rotating sphere of uniform density to find the overall time dilation at some r by considering the contributions of the time dilations for all of the static point masses that make it up.
That might be an interesting exercise. I think the best approximation so far for time dilation of multiple masses is the equation I gave in #16 along with the provisos and limitations mentioned. Simply sum or integrate all the all the potentials of the points masses and insert the total potential into the gravitational time dilation equation.

PAllen
2019 Award
If you are really looking for an exact solution even for a static, simple case (even two masses), you cannot just declare them static. If you think rockets, your stress energy tensor must include the thrust. Alternatively, you can posit idealized struts. This is probably the way to go. There are exact solutions of the EFE for two black holes separated by a strut. At the moment, I can't spend the time to track down a reference, but they are pretty well known. Using such a solution, you could follow your program of computing exact results for a very simple case.

If you are really looking for an exact solution even for a static, simple case (even two masses), you cannot just declare them static. If you think rockets, your stress energy tensor must include the thrust. Alternatively, you can posit idealized struts. This is probably the way to go. There are exact solutions of the EFE for two black holes separated by a strut. At the moment, I can't spend the time to track down a reference, but they are pretty well known. Using such a solution, you could follow your program of computing exact results for a very simple case.
Except that you cannot, even in principle, attach a rocket or strut to a black hole (because there is no surface). Better to use two neutron stars

That might be an interesting exercise. I think the best approximation so far for time dilation of multiple masses is the equation I gave in #16 along with the provisos and limitations mentioned. Simply sum or integrate all the all the potentials of the points masses and insert the total potential into the gravitational time dilation equation.
The equation you wrote in post #16 would in fact give the same formula for a uniformly dense non-rotating sphere as it would for a point mass, wouldn't it, since the integration of the potential of the point masses is the same as that for the entire mass, at least for the Newtonian potential. Is the external Schwarzschild metric and gravitational time dilation involved derived for that of a uniformly dense non-rotating sphere? I read that it is derived as a fluid model, but I'm not sure what that means exactly.

PeterDonis
Mentor
2019 Award
The overall reason I am considering only static bodies is not to determine some multiple body orbital problem, which would be extremely complex, but rather to integrate all of the point masses of a static non-rotating sphere of uniform density to find the overall time dilation at some r by considering the contributions of the time dilations for all of the static point masses that make it up.
In other words, you are trying to understand how the field of a single spherical object is "composed" out of the fields of the individual pieces of matter that make it up? The answer is, it isn't. GR is not Newtonian gravity; the "field" (i.e., spacetime curvature) is not "built up" out of individual pieces. There is a single solution for the curvature of the spacetime as a whole, which is obtained by looking at the distribution of stress-energy in the spacetime as a whole.

When I was talking about the weak field case before, I was assuming we were talking about two bodies only, and we had to make a number of simplifying assumptions, the chief of which was that gravity was weak: that translates into all of the distances involved being much larger than any of the masses (in geometric units). But if you are trying to understand the total field due to a single spherical body as somehow being "built" out of the fields due to all the individual pieces of matter that make up the body, that assumption no longer holds; two individual pieces can be arbitrarily close. Instead, as I said above, you have to derive a single solution for the entire spacetime, using the distribution of stress-energy in the entire spacetime. See below.

Is the external Schwarzschild metric and gravitational time dilation involved derived for that of a uniformly dense non-rotating sphere? I read that it is derived as a fluid model, but I'm not sure what that means exactly.
It means that "the stress-energy in the entire spacetime" is modeled as a perfect fluid occupying the volume from the center at r = 0 out to some nonzero radius R. Then the Einstein Field Equation is solved with this stress-energy tensor on the RHS to obtain the metric, which tells you the "potential", the time dilation, the "acceleration due to gravity", and everything else. But it's all one single solution; it is not "adding up" the individual solutions due to individual pieces of matter in the object.

PeterDonis
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2019 Award
The equation you wrote in post #16 would in fact give the same formula for a uniformly dense non-rotating sphere as it would for a point mass, wouldn't it
I should point out that the equation in post #16 is *not* the correct equation for the "potential" inside a uniformly dense non-rotating sphere. The "potential" *outside* the sphere, in the exterior vacuum region, is given by the Schwarzschild metric (which is where the formula in post #16 comes from); but *inside* the sphere, in the interior non-vacuum region, the potential is given by a different formula, which is *not* the sum of contributions from individual pieces of the object, as I said in my previous post. The correct formula for the "potential" at radius r inside a non-rotating spherical object with uniform density is this:

$$\frac{d\tau}{dt} = \frac{3}{2} \sqrt{1 - \frac{2M}{R}} - \frac{1}{2} \sqrt{\frac{1 - 2M r^{2}}{R^{3}}}$$

where M is the total mass of the object and R is the radius at its outer surface. You will note that at r = R, i.e., at the object's surface, this becomes

$$\frac{d\tau}{dt} = \sqrt{1 - \frac{2M}{R}}$$

which is the same as the "potential" from the exterior Schwarzschild metric at r = R; this shows that the "potential" changes smoothly as the surface of the object is crossed.

PAllen
2019 Award
Except that you cannot, even in principle, attach a rocket or strut to a black hole (because there is no surface). Better to use two neutron stars
No. So far as I know, there are no exact solutions for two idealized fluid bodies joined by a strut. There are for two black holes. The strut has implausible properties, but it is a a complete, exact solution of the EFE.

The equation you wrote in post #16 would in fact give the same formula for a uniformly dense non-rotating sphere as it would for a point mass, wouldn't it, since the integration of the potential of the point masses is the same as that for the entire mass, at least for the Newtonian potential.
That is my guess too. That guarantees it will give the same result as the external Schwarzschild solution. It probably won't work for the internal solution for all the reasons Peter has just given.

Is the external Schwarzschild metric and gravitational time dilation involved derived for that of a uniformly dense non-rotating sphere? I read that it is derived as a fluid model, but I'm not sure what that means exactly.
The external metric only concerns itself with the total mass, so it does not care if it is a uniformly dense sphere or a point mass. Additionally the external metric does not care if the sphere is collapsing or pulsating. (See Birkhoff's theorem.) The internal non rotating metric on the other hand, does depend on how the density is distributed. It may or may not depend on the radial motion of the particles. Before Birkhoff it was not immediately apparent that the external solution is independent of the radial motion of the massive bodies constituent parts (as long as the object remains spherically symmetric).

No. So far as I know, there are no exact solutions for two idealized fluid bodies joined by a strut. There are for two black holes. The strut has implausible properties, but it is a a complete, exact solution of the EFE.
If I may borrow your strut material for a while, I can construct a body with a radius of 9/8 Rs with time stopped at the centre. If I use slightly less implausible material that is not infinitely rigid, the 9/8 Rs sphere can collapse very slowly towards Rs, such that the collapsing motion is negligible and I will have my sphere with proper time running backwards at the centre.

PeterDonis
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2019 Award
If I may borrow your strut material for a while, I can construct a body with a radius of 9/8 Rs with time stopped at the centre.
No, you can't, because the pressure and pressure gradient at the center will be infinite. The strut material PAllen talks about does not allow infinite pressure; no material does. The strut is "unphysical" because it violates energy conditions, not because any of the SET components are infinite.

If I use slightly less implausible material that is not infinitely rigid, the 9/8 Rs sphere can collapse very slowly towards Rs, such that the collapsing motion is negligible and I will have my sphere with proper time running backwards at the centre.
No, you can't; see above. There is not even an approximate "pressure balance" possible once R is less than or equal to 9/8 R_s; the object will collapse in an accelerating fashion, so collapsing motion will not be negligible.

PAllen
2019 Award
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 [Broken]

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PAllen
2019 Award
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.

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
Staff Emeritus
Gold Member
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?

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PAllen
2019 Award
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
2019 Award
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. ]

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