The external spacetime field produced by an object of mass M, the Schwarzschild spacetime metric solution, is usually obtained as follows [1]:
1) Assumes a spherically symmetric spacetime metric, and is static and time invariant;
2) Assumes a vacuum conditions outside, with Tµν = 0;
3) Solve the Einstein field equation, Rµν - (1/2)gµνR=Tµν...... (EQ.1)
4) Utilize the boundary condition: the Newtonian potential ф = -GM/r, which introduces the mass M. Obtain the result:
ds2 = -(1-2GM/r)dt2 + (1-2GM/r)-1dr2 + r2dΩ2...... (EQ.2)
Overall, the Schwarzschild metric employs a priori derivation steps. The solution is unique according to Birkhoff's theorem.
Einstein does not explain why M leads to ds2, our questions are:
a) The spacetime metric is containing the energy-momentum Tspacetime , which can only originate from Tµν and is conserved. Why then must spacetime receive, store, and transmit energy-momentum by curvature* ?
b) The implication of condition 2) is that the spacetime field energy-momentum is independent of M or can be regarded as such. Comparing this to the electric field of an electron is equivalent to the fact that the energy contained in the electron's electric field is independent of the electron itself. Since Tspacetime is also bound to M, is it not part of M?
c) For complex scenarios, in the Tµν of Einstein's field equation EQ.1, should one include the spacetime energy momentum at the location? With the above Schwarzschild solution, it seems that there is none, otherwise both sides of the equation (EQ.1) become a deadly circle. So, should there be or should there not be? Does the field equation have a provision or treatment that Tµν can only contain non-spacetime energy momentum?
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Notes
* “How the view of space-time is unified (3)-If GR's space-time is not curved, what should it be?” https://www.researchgate.net/post/NO17How_the_view_of_space-time_is_unified_3-If_GRs_space-time_is_not_curved_what_should_it_be
** "Doubts about General Relativity (1) - Is the Geometry Interpretation of Gravity a Paradox?" https://www.researchgate.net/post/NO36_Doubts_about_General_Relativity_1-Is_the_Geometry_Interpretation_of_Gravity_a_Paradox
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References
[1] Grøn, Ø., & Hervik, S. (2007). Einstein's Field Equations. In Einstein's General Theory of Relativity: With Modern Applications in Cosmology (pp. 179-194). Springer New York. https://doi.org/10.1007/978-0-387-69200-5_8
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2024-04-26
Additional information*:
1) In his Karl Schwarzschild Memorial Lecture, Einstein summarized the many scientific contributions of his short life, stating [1], in commenting on Schwarzschild's solution, that “he was the first to succeed in accurately calculating the gravitational field of the new theory”.
(2) Einstein emphasized in his article “Foundations of General Relativity” [1], “We will make a distinction between 'gravitational field' and 'matter', and we will call everything outside the gravitational field matter. Thus the term 'matter' includes not only matter in the usual sense, but also electromagnetic fields.” ; “Gravitational fields and matter together must satisfy the law of conservation of energy (and momentum).”
(3) Einstein, in his article “Description based on the variational principle” [1], “In order to correspond to the fact of the free superposition of the independent existence of matter and gravitational fields in the field theory, we further set up (Hamilton): H=G+M
4) Einstein's choice of Riemannian spacetime as the basis for the fundamental spacetime of the universe, which I have repeatedly searched for in The Collected Papers of Albert Einstein, still leads to the conclusion that he had no arguments, even if only descriptions. In his search for a geometrical description, he emphasized that “This problem was unsolved until 1912, when I hit upon the idea that the surface theory of Karl Friedrich Gauss might be the key to this mystery. I found that Gauss' surface coordinates were very meaningful for understanding this problem.”[2] And, although many physicists also do not understand what Space-Time Curvature is all about, everyone accepted this setup. This concept of `internal curvature', which cannot be mapped to physical reality, is at least a suitable choice from a modeling point of view.
5) Einstein's initial assumptions for the field equations were also very vague, as evidenced by his use of terms such as “nine times out of ten” and “it seems”. He was hoping to obtain the gravitational field equation by analogy with the Poisson equation. Thus, the second-order derivative of the spacetime metric is assumed on the left side of the equation, and the energy-momentum density is assumed on the right side.
-----------------------------------------
* The citations therein are translated from Chinese and may differ from the original text.
[1] University, P. (1997). The Collected Papers of Albert Einstein. Volume 6: The Berlin Years: Writings, 1914-1917. In. Chinese: 湖南科学技术出版社.
[2] Einstein, A. (1982). How I created the theory of relativity(1922). Physics Today, 35(8), 45-47.
The curvature memory structure is formed at the same time as the matter energy substance accumulates. Thus, there is no information exchange to maintain the energy density-variant evolved space-time for the observation sample.
"Why then must spacetime receive, store, and transmit energy-momentum by curvature* ?"
I share your "doubts" -- to put it mildly :
In my view we have no warping of space & time. The alteration was with the light, not time and space.
Preprint Further Animadversion upon the wiki figure purporting to dem...
Preprint Animadversions upon the wiki figure purporting to demonstrat...
This thread question is scinetfically answered in a couple of SS posts in
https://www.researchgate.net/post/NO36_Doubts_about_General_Relativity_1-Is_the_Geometry_Interpretation_of_Gravity_a_Paradox/1
Cheers
There is an alternate derivation of the Schwarzschild metric for which Tµν is nonvanishing, but instead contains a central mass M. The mass M appears as a mass density described by a continuous thin-shell broadened Gaussian delta function, whose radius can approach zero while holding M constant, thus emulating a point mass. The derivation requires assumptions 1 and 3, but not 2 or 4.
Upon solving Einstein's field equations, the Schwarzschild metric pops out automatically with the potential ф=-GM/r. Thus 2M is not a constant of integration, but is intrinsic to the field equations. More specifically, it is the constant κ=8πG/c2 in front of Tµν that fixes the value 2M. I am not sure this directly answers the question, but it may be relevant to the discussion.
Dear Kathleen Rosser
Indeed, your answer is exactly what I was seeking.
If spherically symmetric solution to Einsteins equations is unique in order to Birkhoff's theorem[1], then when we determine two constraints, The weak field limit(r→∞) and the potential ф=-GM/r, the answer should also be unique. In that case different derivations that use, or meet, these two conditions will give the same result. However, we cannot consider this as an alternate derivation, because the physical meaning of the same result is completely different. One is a vacuum solution and the other is a non-vacuum solution, and they describe completely different physical realities. The spacetime metric becomes constitutive of M when we consider M as distributed (it is field, usually endless). This must be the most physically rational explanation. The next step is how to eliminate the point particle assumption.
I guessed that this would be your own answer, so I found your paper on the subject [2]. I hope I can benefit more from reading it.
Best Regards, Chian Fan
-------------------------------
[1] Carroll, S. M. (1999). Lecture Notes on General Relativity. https://www.researchgate.net/publication/2354635_Lecture_Notes_on_General_Relativity
[2] https://www.researchgate.net/publication/336240837_Thin_shells_in_general_relativity_without_junction_conditions_A_model_for_galactic_rotation_and_the_discrete_sampling_of_fields。
https://www.researchgate.net/post/Should_General_Relativity_now_be_considered_an_established_fiction_What_about_Silbersteins_objection3
Dear Chian Fan
Yes, my derivation differs from the vacuum derivation in that mass density ρ(r) is nowhere exactly zero, or as you say, the mass M is distributed. However, outside the thin shell, the mass density may be considered infinitesimal (infinitely close to a vacuum). Thus, the exterior metric is exactly the Schwarzschild solution to infinitesimal accuracy. Indeed, there is no limit to how close to exact the solution is. In the paper, I call this "asymptotically exact".
As you may see from the paper, the solution would be exact if the shell thickness ε were allowed to reach zero, but we cannot allow shell thickness ε to be exactly zero. This stems from the requirement that the mass density ρ(r) must be continuous from r=0 to ∞. For if ρ(r) were not continuous, we would need junction conditions. And when junction conditions are used, we must assume the Schwarzschild metric outside the shell based on Birkhoff's theorem, and we are back where we started: the vacuum solution with 2M as a constant of integration.
While the shell thickness ε is considered infinitesimal, the shell radius r0 need not to go to zero. The (infinitesimally exact) Schwarzschild solution holds outside the shell no matter how big its radius. I had only said in my above answer that M can approach a point mass because that is how it is usually visualized for the Schwarzschild solution.
Thank you very much for your interest.
Kate
General relativity does not account for the acceleration of the Universe. Acceleration is supposed to be equivalent to gravity, but is discounted in this key case. If the acceleration of the Universe is identical to gravity, everything depends on this particular acceleration. This is accounted for in general relativity by the cosmological constant -- the tail of the (GR) dog. Might the tail be wagging the dog? The energy tensor in GR could just be part of a mathematical formalism, and nothing more.
General Relativity has no real content, because the introduction of the new time concept by Einstein is in error. Cf. "Animadversions upon the wiki page purporting to demonstration the "Relativity of Simultaneity" in the link below below. See also Kılıç's objection, ibid.
https://www.researchgate.net/post/Should_General_Relativity_now_be_considered_an_established_fiction_What_about_Silbersteins_objection3
Additional information*:
1) In his Karl Schwarzschild Memorial Lecture, Einstein summarized the many scientific contributions of his short life, stating [1], in commenting on Schwarzschild's solution, that “he was the first to succeed in accurately calculating the gravitational field of the new theory”.
(2) Einstein emphasized in his article “Foundations of General Relativity” [1], “We will make a distinction between 'gravitational field' and 'matter', and we will call everything outside the gravitational field matter. Thus the term 'matter' includes not only matter in the usual sense, but also electromagnetic fields.” ; “Gravitational fields and matter together must satisfy the law of conservation of energy (and momentum).”
(3) Einstein, in his article “Description based on the variational principle” [1], “In order to correspond to the fact of the free superposition of the independent existence of matter and gravitational fields in the field theory, we further set up (Hamilton): H=G+M
4) Einstein's choice of Riemannian spacetime as the basis for the fundamental spacetime of the universe, which I have repeatedly searched for in The Collected Papers of Albert Einstein, still leads to the conclusion that he had no arguments, even if only descriptions. In his search for a geometrical description, he emphasized that “This problem was unsolved until 1912, when I hit upon the idea that the surface theory of Karl Friedrich Gauss might be the key to this mystery. I found that Gauss' surface coordinates were very meaningful for understanding this problem.”[2] And, although many physicists also do not understand what Space-Time Curvature is all about, everyone accepted this setup. This concept of `internal curvature', which cannot be mapped to physical reality, is at least a suitable choice from a modeling point of view.
5) Einstein's initial assumptions for the field equations were also very vague, as evidenced by his use of terms such as “nine times out of ten” and “it seems”. He was hoping to obtain the gravitational field equation by analogy with the Poisson equation. Thus, the second-order derivative of the spacetime metric is assumed on the left side of the equation, and the energy-momentum density is assumed on the right side.
-----------------------------------------
* The citations therein are translated from Chinese and may differ from the original text.
[1] University, P. (1997). The Collected Papers of Albert Einstein. Volume 6: The Berlin Years: Writings, 1914-1917. In. Chinese: 湖南科学技术出版社.
[2] Einstein, A. (1982). How I created the theory of relativity(1922). Physics Today, 35(8), 45-47.
This thread question is scientifically answered on page 1, i.e. in a couple of SS posts in
https://www.researchgate.net/post/NO36_Doubts_about_General_Relativity_1-Is_the_Geometry_Interpretation_of_Gravity_a_Paradox/1
Cheers
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