Static gravity, i.e. the Schwarzschild metric, depends mostly on the vacuum field equation: Ricci tensor = 0. Using notation from https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution we get one thing from symmetry and the definition of metrics, a form in which there are two relevant coefficients, A(r) and B(r) (the time and space coefficients, roughly speaking).
From the Ricci tensor = 0 constraint, which has been given only vague physical interpretations, we get two things:
As a friend showed a few months back, this leads to gravity which for a distant observer with a tether amounts to an inverse square force law, oddly enough. But potential, obviously, is not Newtonian for any observer.
Suppose we want one of the two variations of potential to be the time dilation factor. These are both approximately equal to the Schwarzschild factor for large r. They are 1/(1-GM/rc2) and (1+GM/rc2). Take the second one. It implies, according to another question I posed https://www.researchgate.net/post/What_differential_equation_has_a_solution_of_the_form_Fx1_1_Kx-2 , that the differential equation in step 2 above must be something like rA'=2A(1-A).
That factor of 2 when backed through to the Ricci tensor means that it cannot be quite zero for small radii. At that point I get lost trying to draw more definite conclusions. For example, if we have a different equation:
Ricci tensor = X
Then what is X (probably a complicated function) such that we still have A=1/B, but we get rA'=2A(1-A). It is a matter for someone who knows all the little summations and conventions by heart and is good at fudging. Any takers?
The
R_{ij}=0
(all components of the Ricci tensor are zero) equation is derived from the Einstein field equations for gravitation, assuming an empty space and zero cosmological constant.
If you have matter (e.g., you want to derive the solution for the interior of a star), then you have to solve Einstein's equations, R_{ij} -1/2 g_{ik}R = T_{ik}, where T_{ik} is the stress-energy tensor of matter (for a fluid, it contains density and pressure).
Note, that you have a field equation, which you need to solve. You get a solution, which contains some constants of integration (here, M). You also see, that far away from the center, it is almost the metric of flat space-time, so there is a weak field there. That is why you can compare the relevant components to the Newtonian potential, and the comparison tells you that the integration constant is the mass of the black hole or the central body.
I hope this helps. I do not know, if you are looking for another solution of the Einstein equation, which looks like you described (but in that case, it has to be a solution), or an alternative gravitation theory where the black hole solution is the one you described. But in the latter case, be aware, that the measurements supporting general relativity are rather precise, meaning that if your theory is not GR, it still has to be bloody close to it
Arpad, if you look at the link to the short Wiki on the Schwarzschild derivation, you will see that it does not use the energy-momentum tensor. It does not solve the field equation for any mass. Instead, it solves the vacuum equation, which would apply only outside of a mass, and then uses the Newtonian limit to re-introduce a mass, the gravitational constant, etc. There is only one possible solution given the Newtonian constraint.
I want to come up with a different equation. Instead of Ricci = 0, it would be Ricci = something-non-zero, to match alternate solutions I already have.
So it is a reverse problem. I have the solution. Write the equation which would produce it.
The matter is non-trivial because not only is Ricci=X an array of 16 equations (only three of which are significant for Ricci=0), but the Ricci tensor itself is formed in a somewhat complex way from the Christoffel symbols, which in tern are formed from the metric.
Well, you know, I have the metric don't I? Just thought of that. So one can start with this metric, write the Christoffel symbols, and the Ricci tensor, and evaluate the Ricci tensor.
ds2 = -c2dt2(1+GM/rc2)-2 + dr2(1+GM/rc2)2 + r2(dθ2 + sin2θdφ2)
Why didn't I think of that before? Just write the Christoffel symbols and Ricci tensor for that. I'm afraid it would take me a week and I would get some of the indices backward. I was hoping someone could do it easily.
Then it might be necessary to generalize it. That is pretty easy. Just substitute K=GM/c2.
Speaking of "close to GR," the whole idea is to justify a family of theories so close to GR that they are indistinguishable by current observations, but might be distinguishable in the next few decades by detailed observations of black holes, like in the galactic center. For example, the above metric differs from GR at 2 million miles from the sun by only one part in 1013, far beyond current measurements.
Further justification and derivation of two alternate metrics is given in the paper below.
OK. So, in the GR case, you have all the components of R_{ik}=0 to solve. The reason you only need to solve 3 is that the symmetry of the Ansatz reduces the number of independent components.
In your case, what you can try is to calculate the components of the Ricci tensor for your assumed metric (I would definitely use computer algebra, e.g., enter the formulae for the Christoffel symbols and the Riemann and Ricci tensors in Reduce (that is free) or Mathematica). Then you get the components of the stress-energy tensor, however, as a function of the coordinates. I think that due to symmetries, it will remain diagonal, so you can separate energy density and pressure. If you can cancel "r" from the two, then you have an equation of state.
It would be also useful to verify, if this stress-energy tensor satisfies some energy conditions.
However, the physical interpretation of this matter is not clear to me.
Arpad, thanks for the suggestion of computer algebra. I'm not familiar with it though so there is a learning curve. Can you recommend a system or website? Would perhaps something already have definitions for Christoffel symbols and so forth?
For the solution space I have developed, I used a separate means of satisfying energy conditions. This is the problem I'm trying to split apart - the energy constraint as a separate constraint from the field strength function, which are rolled into one in the vacuum equation.
If your institute has a licence, I would recommend Mathematica (from Wolfram), and for the tensor manipulations, xAct.
Otherwise, I recommend Reduce, which is now free. Here you'll have to type the expressions for the Christoffel symbols, etc..
www.xact.es
www.reduce-algebra.com
Thank you Arpad. Looks interesting. I will have to collect up enough free time to delve into learning a new and I'm guessing probably complex tool, so it might be a while before I can report back. Have a big work project due mid October at present.
To Robert Shuler
"From the Ricci tensor = 0 constraint, which has been given only vague physical interpretations"
Permission by Einstein in 1915
the argument it gave a correct answer when the solution of the motion of Mercury. Main led to the covariance of the equations of the gravitational field.
...but there was uncertainty in the local field of energy.
@Valery, yes I realize the argument that "it gave a correct answer" was floating in the background, but curve fitting does not justify the strong and widely believed prediction of esoteric phenomena like event horizons, singularities, and according to some physicists even closed time-like loops.
The strongest argument I have found for Ricci=0 is the "conservation of curvature" (from MTW), which is often not explained well other than by analogy to the conservation of flux through area in the case of ordinary electric fields and Newtonian gravity. Sometimes the argument is also given that Ricci=0 is the "simplest" vacuum equation that agrees with the classical limit, and I would not quibble with that. I would only quibble with whether any of these are strong enough to justify assuming the esoteric strong field predictions are true, taking an all or nothing approach to verification without theoretical alternatives that provide a realistic basis for comparison.
- "What is the non-zero Ricci tensor equation for a metric field slightly non-Schwarzschild?"
- I think I found the answer, even two. It was found that one of them is wrong
- https://www.researchgate.net/publication/287980615_Solution_of_the_Einstein_equation_and_metric_of_accelerated_system?ev=prf_pub
- https://www.researchgate.net/publication/298412453_On_problem_of_homogeneous_reference_system?ev=prf_pub
- https://www.researchgate.net/publication/303844514_Gravitational_field_equations_and_relativistic_homogeneous_gravitational_field?ev=prf_pub
Preprint Solution of the Einstein equation and metric of accelerated system
Research Proposal On problem of homogeneous reference system
Preprint Gravitational field equations and relativistic homogeneous g...
Valery, thanks. But can you distill this down a little bit, perhaps suggest just one paper to look at?
The first is history. Many superfluous.
The second short story that it can not be done.
The third is a solution, but there is doubt. Someone like, someone does not understand.
@Valery,
I looked over the papers. While a bit to digest in detail, I conclude that your motivation and setup are somewhat different than mine. However, the qualitative results are similar, a gravity less strong than Schwarzschild. The image below is from "Isotropy of Inertia Revisited" which is forthcoming in J. Advances in Astrophysics.
In a paper I'm still working on, a framework for very strong field metrics, I come to the conclusion that the Schwarzschild is an extremal case, in which it is "assumed" that ALL of the causation of the gravitational field remains inside any radius as r -> 0. This is after all another way of saying the vacuum equation is zero. Any mechanism at all which allows some mass-energy to remain outside zero radius maintains a distant field in perfect agreement with Newton-Einstein, while being lower in short range strength than Schwarzschild. When I originally formulated this RG question thread, I did not have this insight. I had other reasons for thinking the field strength less, having to do with the uncertainty principle and Mach's principle. However the existence of non-singular solutions, as you find, has been known apparently for some time. Someone pointed them out to me a few months ago, though I don't recall who or the reference at the moment.
The bottom line is, Schwarzschild is not compelled, even if one assumes GR is absolutely perfect (which for strong fields is not as yet verified, despite hand waving articles on gravitational waves from collapsing binary pairs, which only demonstrate that they decay in a reasonable way, not the particulars of their final decay state).
It would be an equally valid interpretation of Schwarzschild (perhaps his own had he lived, as it is implied by the coordinates he used) to say that collapse to less than what's now called the Schwarzschild radius (zero in his coordinates) is not permitted by the equations. This is very different than saying that it is inevitable. There are several other threads on RG that discuss this, and most of the participants stick to their opinions without much give.
I do not think that the GH needs to be replaced or corrected.
The problem in the gravitational field equations. Einstein tried to find the equation of the gravitational field of about two years. The result is a compromise Einstein equation, which uses little energy field. I've been doing this for more than two years, but will not fit Einstein's equation. We had to do without the equation.
Now I finish the job. In some extent, the clarity is achieved
The problem is much simpler than solving the field equations. Just take your metric, plug it into the definition of the Ricci tensor to obtain the components of the latter, calculate the scalar curvature and the Einstein tensor, then you have, up to a factor, the stress-energy tensor that will produce your metric.
What might happen though is that the so-obtained energy density becomes negative, in which case realization of the metric would require "exotic" energy. This could then lead to instability and things such as worm holes and closed time-like loops, i.e. causality violations. However, this is not a necessity. You will only know it after doing the calculation.
Hello Klaus, nice to hear from you. This question is very old and essentially obsolete. I have one paper accepted and one pending that shed a great deal of light on this (and which do not have any energy anomalies) and will upload them when published. There is still a need for some work on dynamics like grav waves. I haven't committed to doing that. We'll see how retirement goes.
To K. Kassner
"Calculate the scalar curvature and the Einstein tensor, then you have, up to a factor, the stress-energy tensor that will produce your metric."
That's exactly what I did.
"What might happen but is that the so-obtained energy density becomes negative, in which case the realization of metric would require" exotic "energy."
The energy must be negative. This is not an exotic, but a classic result.
I have not only negative energy, but also exactly the same as the classical solution.
"This could then lead to instability and things such as worm holes and closed time-like loops, i.e. causality violations." However, this is not a necessity.
There is no such problem. The solution of the Schwarzschild problem has no horizon. However, a second solution appeared with an ejecting field.
Https://www.researchgate.net/profile/Valery_Morozov/publication/314142053_Solutions_of_the_Nordstrom_equation_instead_of_the_Schwarzschild_solution/data-preview/58b6ed12aca27261e51a3a37/Grafik-vsegif?origin=fileInline
Data Solutions of the Nordstrom equation instead of the Schwarzsc...
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