There has been much speculation that the Schwarzschild field harbors regions of gravitational repulsion, witnessed by a positive acceleration. The criterion is that the radial velocity should be greater than B/31/2 , where B=(1-R/r), and R is the Schwarzschild radius. The radial velocity can, therefore, surpass the velocity of light. To avoid this, one usually claims that "local observers cannot detect gravitational repulsion, apparently because their measuring instruments are affected by gravity, whereas distant observers, whose instruments are not affected by the gravitational field, can measure a positive value for the acceleration of gravity." (McGruder, Phys. Rev. D 25 (1982) 3191) Apart from the fact that this would provide a means for the observers to distinguish their position through measurement (something outlawed in non-Euclidean geometries), the gravitational field of a distant observer is unaffected by gravity why should he measure a positive acceleration?
The assumption that the metric coefficients depend only r and not on t implies that the fields are completely static. So how do accelerations and velocities come in the description?
Dear Bernard,
actually the solution gets repulsive as soon as the infall speed overcomes c/sqrt(3)...that is why it is quite controversial that the Schw solution can be used for infalling objects.
Dear Stefano,
I thought so too, but, then I realized that the geodesics of the Schwarzschild metric and the metric that can be used to derive Weber's force are the same! (The vanishing of Weber's force give the geodesics
d^2r/dt^2=-(1/r)[c^2-(dr/dt)^2/2]
Dear Bernard,
That is well known and the condition is that being rs Schwarzschild radius
dr /dt >( c/ √ 3) (1 − rs/ r )
This is not a real property of gravity which cannot be a repulsion force and in fact only speaks on non realistic conditions for such a metric. This happens in many cases more in RG, for instance look at the paper
https://arxiv.org/pdf/gr-qc/0205066.pdf
Dear Daniel,
Your condition means that dr/dt can be as large as it wants for gravitational repulsion. Yet everything is built into the condition
Adr^2-B(cdt)^2
I don’t have access to McGruder’s article. Does anyone here have a pdf they can send to me?
[From the abstract and the comments I see here I suspect that non-physical coordinate-based quantities are being misunderstood as physically meaningful − a common error in the literature of Relativity. Coordinate systems contain no physics; predictions deduced from the theory must be expressed in terms of invariants if they are to be physically meaningful. “Schwartschild spacetime” is not a coordinate system! I cannot say more without seeing the article...]
I found it on the internet. But here is what seems is a more recent one
https://arxiv.org/pdf/1802.08546.pdf
It repeats the same mistakes.
Dear Eric,
The paper of McGrude is here
Article Acceleration of particles to high energy via gravitational r...
Dear Daniel,
I have read the manuscript you referred to. I think they are right in that inconsistencies have to appear when the very general (G) is equated to the very specific (T). That questions the "very general", doesn't it? Even Einstein had his reservation when he equated a magnificent marble palace to a wooden shack. Regarding TEC, rho-3p>0, doesn't a perfect gas, rho=(3/2)p violate it?
By the way, if you put A=(1+R/r)=1/B into the expression for the acceleration given above you get exactly to second order in the Schwarzschild radius, R, Gerber's orbital equation by replacing c^2 by c^2/3 and the angular momentum L by 3^{1/2}L. The first condition was obtained by Schrodinger in 1925 when he suggested that the relativity requirement could be fulfilled by classical mechanics.
Dear Bernard,
Let me to start with a suggestion. Perhaps you could put one message to McGruder for entering in your thread. I agree with your considerations.
Schwarzschild solution was studied very exhaustevely for the black holes and this anomaly is well known. It is obvious that there is a place where the conditions of a Schwarzschild solution are broken. Gravitation cannot pass from being attractive to repulsive physically. The main ingredient to examine is the potential and its change of sign and it is easy to see that you have one effective potential
V(r) proportional to (-mM/r)(1+v2/c2)
which tell us that the effective mass of a particle is increased by a factor 2v2/c2. This tell you that this solution introduce something extra in the motion of one particle.
The "physical explanation" that I know, perhaps there are more, is that this result can only be explained quantum mechanically within a black hole. Where a particle is studied close to one horizon.
Dear Daniel and Stefano,
What I couldn't write down (for lack of symbols) I have put on
https://bernardhlavenda.com/node/185
Daniel, your effective potential brings to mind Gerber's
-mM/r(1+2v/c+3v^2/c^2+...)
This gives exactly the Weber force to second order, as is well-known ( Assis's "Weber's Electrodynamics"), but it's origin still remains a deep mystery (Roseveare, "The Perihelion of Mercury: From Le Verrier to Einstein").
Dear Bernard,
You are right, it has certain similiarity
V(r,v)=(mM/r)(1-v/c)-2
This Gerber classical potential (older than GRT) is very different to the effective one associated to Schwarzschild solution
(-mM/r)(1+v2/c2)
The fantastic thing is that both gives a good result for the Mercury's perihelion.
Dear Bernard,
I was reading your interesting comment
https://bernardhlavenda.com/node/185
and I have not time enough but I think that you have lost some solutions. Let us to give you one reference trying this problem close to a singularity where the values of the gravitational field can be very high
https://arxiv.org/pdf/1803.08346.pdf
Dear Daniel,
Thanks very much for the reference. I have a problem with his equations (4) and (5), taken from Schutz, his ref. 1. His equation (4) reads
(d/dtau)(p_0)=(d/dtau)(g_00 p^0)=g_00(d/dtau)p^0=0.
This implies that
(d/dtau)p^0=0, d^2t/dtau^2 t=0 or dt/dtau=const.
and not g_00 dt/tau=const. so that when you eliminate dt/dtau you don't get a const but rather 1/g_00^2.
The fact that he writes g_00 dt/dtau=const., implies g_00 is a function of tau! The geodesic equation should read
g_00 d^2t/dtau^2+2g(dr/d\tau)(dt/dtau)=0
Have I missed something?
Dear Bernard,
The fact that he writes g_00 dt/dtau=const., implies g_00 is a function of tau!
Yes. That seems to be logic for me, the only that you have is that r is r(tau), i.e. the distance is function of the time.
Dear Daniel,
It's not only him, but the whole caboodle!
If what you say is true (which I firmly believe is) how do you explain the vanishing of rR_01=-(dg_rr/dt)/g_rr because you have g_00(dt/dtau)=const.? If R_01 doesn't vanish, then all the other components of the Ricci tensor that contain time derivatives will not vanish either. What then does the vanishing of the Ricci tensor mean?
Dear Bernard,
I think that I'm not understanding you. The Schwarzschild solution is assumed to be a solution of the vacuum Einstein's equations. So the Ricci tensor must be null for r higher than zero (singularity).
Dear Daniel,
All the components of the Ricci tensor may vanish if g_00 and g_rr are not functions of time--whatever time you are talking about because they are related by g_00(dt/dtau)=const.Schutz calls this energy, but there is no conservation here. Actually tau is any affine parameter. The one component where the time derivative alone is present is in R_01. If this doesn't vanish, the other components of the Ricci tensor will not vanish either. It has nothing to do with "emptiness"!
No No No No
1. Modern physicists consider graviton as a hypothetical particle . But participatory science has proved its existence by explaining the effects. Our instruments are unable to see these particles but our consciousness sees it (biggest and finest measuring instrument of the universe).
1. Energized Gravitons and fall of object.
2. Energized Gravitons and planetary motion.
3. Energized Gravitons and bending of star light.
4. Energized Gravitons and red shift.
5. Energized Gravitons and atomic structure and spectra.
6. Energized Gravitons and atomic clock.
7. Energized Gravitons and weight of object.
8. Energized Gravitons and structure of visible universe.
9. Energized Gravitons and galaxy structure.
10.Energized Gravitons and perihelion of mercury.
11. Energized Gravitons and dying stars.
12. Energized Gravitons and inter orbital shift.
13. Energized Gravitons and merging of galaxies.
14. Energized Gravitons and Binary stars system.
15. Energized Gravitons and Nuclear fusion on stars ( would be discussed separately)
16. Energized Gravitons and Life sciences
17 . Energized Gravitons and pulsar ( would be discussed separately)
18. Energized gravitons and earth quack . ( would be discussed separately)
19. Energized gravitons and spin and magnetic property of Earth and particles, nucleus . ( would be discussed separately)
20. Energized gravitons and ignition of earth inside . ( would be discussed separately)
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