Kate rosser

No, I meant nonvacuum. Let me try to straighten out this confusion. I derived an exact solution to EFE for the static (yes, as you defined it) spherical NON-vacuum, with a non-zero mass density distribution rho(r). The presence of the massive shell of course requires a nonvacuum solution. However, outside the shell, there is a vacuum, and so outside the shell, the total solution looks exactly like the Schwarzschild solution. To solve the resulting pair of simultaneous differential equations, I used a generalization of the method Dirac used for finding the Schwarzschild solution. The latter is of course a vacuum solution. I generalized Dirac's method to the non-vacuum. The reason I mention Dirac's method is, it is so elegant it is easily generalized. Once he's calculated all the Christoffel symbols and curvature tensors, he derives the Schwarzschild solution in 4 easy steps. Other than that one point of confusion, everything you said looks perfectly rigorous. Thanks for responding.I highly recommend Dirac's 69 page texbook on GR, which makes it easy to learn to solve Einstein's Field Equations from scratch. You can probably find a used copy. I will check out your link too. Your point about the continuity of the manifold is well taken. This topic currently being debated at the highest levels of physics. Some theorists freely manipulate singularities in GR, eg for black holes, the big bang, etc. Others argue as you do, that GR breaks down at these singularities. I see useful points on both sides. First, singularities are indeed probably non-physical. So if we can find a cosmological theory, for example, that avoids the big bang singularity, we are likely better off. On the other hand, singularities are an invaluable mathematical technique, and do give some insight. How this applies to what I have done, I used a spherical Dirac delta function to model the massive sphere. This is an idealization that involves a singularity. Some physicist would argue as you do that GR thus breaks down. However, I would point out that the general solution is well nigh intractible if I had used something else, say a Gaussian, to model the shell. So I took the liberty, as many do, to use a delta function. Now, at the practical level, the delta function can be spread out over some region of r to achieve continuity. This would not affect the solution outside that region. About authors freely using singularities, note the double Schwarzschild black hole solution I referenced above. The authors used a singular strut to keep the black holes rigid. I agree with you that this singularity can hardly be physical. Yet the method produces interesting results. And ... it was published in Physical Review D. The latter is a very open-minded journal. Which is why I like it. About matching Newtonian results, yes, this is important for calculating forces etc. The problem here is, the original question was about time dilation, for which Newtonian gravity has no answer. So we're on our own.

I think my method is right. I found an exact solution to EFE. No solutions were patched together, and there were no boundary conditions, just constants of integration. The Schwarzschild metric did indeed appear automatically. However its dependence on m, the mass of the shell, fixed one of the constants of integration (like Newtonian gravity fixes the constant of integration m in the Schwarzschild solution). My method was as follows: I used Einstein's Field Equations for the static spherical nonvacuum. This takes the form of two simple differential equations; kappa*rho(r) = 1/g11*r**2 + 1/r**2 - g11'/r*g11**2 kappa*p(r) = 1/g11*r**2 + 1/r**2 + g00'/r*g11*g00 where rho(r) is a spherically symmetric density distribution, and p(r) is the pressure, primes denote differentiation with respect to r, and kappa is -8piG/c**2. I generalized to the nonvacuum case Dirac's method for the Schwarzschild solution (see General Theory of Relativity, by P.A.M. Dirac, Wiley 1975, p31). Assuming the Equation of State rho=p, at this stage of the calculation, this produces a g00 that looks exactly like the Schwarzschild-de Sitter solution, except that rho(r) is not constant, but appears inside an integral. Next, I modeled the density function rho(r) for a hollow sphere of radius R as a Dirac delta function: rho(r)=mu*del(r-R), where mu is the mass area density of the shell. This model is an idealization, but the discontinuity can be smoothed out by broadening the delta function in such a way that the mass of the shell remains constant. Taking the integral over rho(r)*r**2, one obtains a step function, which again can be smoothed out by broadening the delta function. Smoothing the delta function would not change the overall result. The only caveat, as I said in a previous comment, is the Equation of State. Your method of patching solutions together using boundary conditions is definitely wrong in General Relativity. It is sometimes done, for example in J.A. Wheeler's black hole lattic metric, but it is always an approximation. What I found was an exact solution to EFE. Do you have any references when you speak of the concensus? If so, that would have been the answer I was seeking when I first entered this forum.

My results were almost but not exactly the same as yours. I did solve EFE for the global metric that spans the entire spacetime. I understand why you say the metric should be {-1,1,1,1} at infinity and {-a,1,1,1} inside the hollow shell. I considered this situation when I found my solution. It does depened on the constants of integration, so we follow each other up to that point (which in my experience is a pretty detailed understanding as communication goes). However, when I evaluated the constants of integration, I found the metric to be Minkowski inside the shell, with g00=1, g11=-1 (or in your notation {-1,1,1,1} for rectangular coordinates). If I set the constant of integration differently, I would not have gotten a Schwarzschild metric outside the shell. (I'll check again.) Have you calculated the solution yourself? So my result has the potential inside the shell as identically zero, as it is at infinity, with no time dilation. I get g00=1-(2m/r)S(R;r), where S(R;r) is a step function S=0 for r<R, S=1 for r>R. The only way to iron out these mathematical details is to show the entire derivation. Which leads to my next comment:' ViXra, like Twitter or the telephone, has no good or bad status in and of itself. It's just a communication vehicle for anyone to publish anything they want, instantly, in beautiful PDF format, at no charge, that is free to read. On viXra, I could show you my full derivation in perfect rigorous notation, and present it as a formal paper. Cranks? Crackpots? You should read Physical Review D! The wildest ideas of all fly there!

I modeled the shell as a Dirac delta function del(r-R). This is equivalent to decreasing the thickness while increasing the density in such a way that the mass m stays constant. For the Schwarzschild metric, I'm assuming the gravitational time dilation for a stationary clock is ds=sqrt(1-2m/r)dt=g00dt, where s is proper time TAU. So an interval ds of proper time goes to zero at r-2m. In other words, clocks stop at the event horizon. Yes, so if one is talking about the tt component of the metric in the form g00=(1+2PHI/c**2), then the potential PHI determines the time dilation. (Assuming dr=dOMEGA=0, ie no velocity.)

Your intuition seems logical. I'm baffled by this too. But my solution to Einstein's Field Equations (if correct) is exact, and it shows everything zero everywhere inside the shell. (g00=-g11=1 for r<R). One caveat however: To get g00=1, you have to assume the Equation of State that density = pressure (the most common GR Equation of State, and one which is consistent with the Schwarzschild metric). But the question is: Is this EoS valid for a rigid sphere held together say by eletromagnetic, strong and Pauli forces? That's a deeper problem. If the sphere were made of dust, it would not be stable. Now I need confirmation that my solution is correct. So I plan to post an article on viXra to get feedback. (I'll put a link here, but it may take a few weeks.) The solution is elegant and simple, but there's plenty of room for mistakes with this much algebra.

Good question. I don't understand this double black hole metric very well. How I got g00=1 was by setting the masses equal and doing the algebra for g00 at the origin. (Of course, I may have made a mistake.) For the equations, see Section 2 of the arXiv article at Strange's link above. The authors don't show the singular strut as having mass, nor do they explain what what it really is. BTW, I looked at your website. Looks like you've done a lot of good work. Impressed About time dilation inside a sphere: I managed to solve Einstein's Field Equations for the general static spherical metric with non-zero mass density. To my surprise (I was expecting something more like your answer), the metric for a hollow shell is exactly Schwarzschild outside the shell, and exactly Minkowski inside the shell. So g00=g11=1 inside the shell. Not only is there no time dilation in the interior of a hollow massive sphere, the potential itself is identically zero, just as it is at infinity. (Unless I made a mistake.) I'm surprised because I agree with the logic of J.C. MacSwell above that the sphere would be deeper in a gravitational well. I will see about writing an article for viXra to get feedback on my solution, which was exact. The solution was elegant and easy, which also surprised me.

Yes, it's zero infinitely far away, and also zero between the two masses, which seems counterintuitive considering the gravity well concept. (By time dilation being zero I mean g00=1). I did some math on the double Schwarzschild metric, and it has unexpected properties, so maybe it's not valid for answering the question of time dilation between two masses. For example, in the limit as one mass goes to zero, the double S metric approaches g00=1-2m/r for large r, but not for small r. So the double Schwarzschild metric does not approach the Schwarzschild metric in the limit of a single mass. This raises the question of just what IS the double Schwarschild metric. Maybe the singular strut that connect the two masses throws the solution off.

Replying to Strange: Yes! Great. You found it. This arxiv version looks just about the same as the one at Physical Review D. See Section 2 for the equations for the double black hole solution.

Reply to Markus Hanke: Interesting idea about the Schwarzschild deSitter metric. I'll look into it. Here's a quote from the article (but I can't reproduce the equations): "The double Schwarzschild BH is a static Weyl solution with axial symmetry, featuring two nonrotating, neutral BHs supported in equilibrium by a conical singularity which can be chosen to take the form of either two strings or one strut (see e.g. [32]). The metric can be reduced to the form ds2 ¼ −e2Udt2 þ e−2Uðe2K½dρ2 þ dz2 þ ρ2dφ2Þ; where t, φ are connected respectively to staticity and axial symmetry." I haven't worked out g00 for this metric yet, but on the face of it, it looks like the time dilation midway between two equal masses is zero! If so, this goes completely against my original intuition. I parked some screen shots of the equations for the metric at this link (Can't post whole article, APS might get mad at me!): https://quemadojournal.org/double-schwarzschild-black-hole-solution/ Sorry this is posted on my rudimentary website. Couldn't think of another way to do it. (I have access to Physical Review D because I bought an individual subscription for $240.) Reply to YaDinghus: No kidding ...

[I would edit my answer above, but can't see how.] Maybe I've answered my own question at the start of this topic. By unbelievable coincidence, I just found out that the analytic solution for two static black holes exists. It's called the double Schwarzschild solution, also known as Bach-Weyl or two-center Israel-Khan solution. I stumbled upon it in the April 2018 issue of Physical Review D at the following link: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.97.084020 Now if I can untangle the math for the metric, I should have the gravitational time dilation for two masses! If anyone's interested, please comment and I'll give more details.

Replying to Markus Hanke about my question above: You're right that the solution is not just a linear combination of two Schwarzschild metrics. If it were, you would get the absurd result that the time dilation is sqrt(2-4m/r)dt, which is not even Minkowski as r approaches infinity. My two examples of possible methods did not of course involve linear combinations of the metric. One method used the sum of the potentials, the other, the product of the metrics. These two methods at least give the correct Minkowski limit. My question might have seemed more reasonable had I said I was looking for an approximation to the time dilation, since as you point out, the exact time dilation is analytically intractible in GR. To compute the exact time dilation, you would have to plug into the energy-momentum tensor T a density distribution with masses centered at r and -r, and then solve Einstein's Field Equataions for the metric. (Not in my lifetime!) My question was also seeking an approximate answer in that I deliberately neglected dynamic effects.. It's hard to imagine a case of two static masses, of course. But in many situations, the dynamics would only affect higher order terms. So the static solution would be fairly accurate. An example would be two galaxies unbound to each other gradually receding due to Hubble expansion. Since I'm working on a cosmological model involving Mach's principle, this type of situation is more what I envision. The Schwarzschild metric, in any event, should give us a clue to the right approximation, since in two limits the Schwarzschild metric must be obeyed exactly---first in the limit as one mass vanishes, second in the limit as the angle between the two masses goes to zero.

What is the gravitational time dilation at a point midway between two equal spherically symmetric masses m? For one mass, the time dilation is given by the tt component g00 of the Schwarzschild metric, and is equal to sqrt(1-2m/r)dt, where m/r is the potential. For two masses, however, I see two intuitive possibilities. First, you could simply take the sum of the potentials to get sqrt(1-4m/r)dt. Second, you could calculate first the time dilation due to one mass, then apply to that dilated clock rate the time dilation due to the other mass. This would be equivalent to taking the product of the two g00 components, with the result (1-2m/r)dt. These two values for proper time differ by only higher order terms in m/r. But I would think the difference could be measurable, for example in GPS applications. General Relativity does not give the answer explicitly, but I expect simulations, for instance of black hole mergers, would have to take this into account. Another interesting question is the time dilation inside a hollow massive sphere. Short of emailing a well-known physicist or former professor, I don't know where to find this information. Yet it could be pertinent in developing a modified gravity theory that incorporates Mach's Principle, which is something I would like to do.