It is now clear, from his latest remarks, that David has no intention of discussion, since he refused to answer my very simple questions in reply to his challenge. Therefore, there is nothing to discuss. For all other readers I will let Einstein and his followers answer the four simple questions I put to David. Here are Einstein’s field equations once again, for easy reference:
R
uv – (Rg
uv)/2 + /\g
uv = -kT
uv
Recall that T
uv is the energy-momentum tensor which describes all the material sources of Einstein’s alleged gravitational field, /\ is the ‘cosmological constant’, R
uv is the Ricci tensor, R is the Ricci scalar, g
uv is the metric tensor, and k is just a constant. The left side describes Einstein’s spacetime and its curvature (geometry). It is the curvature of spacetime that Einstein alleges is gravity. This curvature is induced by the right side of the equations (material sources), since material sources are the cause of Einstein’s gravitational field.
(a) Does T
uv = 0 in the equations R
uv = 0?
ANSWER: Yes.
“
Einstein’s equation, (7.26), should be exactly valid. Therefore it is interesting to search for exact solutions. The simplest and most important one is empty space surrounding a static star or planet. There, one has
Tμν = 0.”
‘t Hooft, G., Introduction to General Relativity, online lecture notes, 8/4/2002,
http://www.phys.uu.nl/thooft/lectures/genrel.pdf
http://www.staff.science.uu.nl/~hooft10 ... l_2013.pdf
“
The Einstein equations in the absence of matter are Ruv = 0.”
Dirac, P.A.M., General Theory of Relativity, Princeton Landmarks in Physics Series, Princeton
University Press, Princeton, NJ, 1996.
“
The vacuum field equations describe the metric structure of empty space surrounding a massive body. In the consideration of empty space where no matter or energy is present, we set Tab = 0. In this case, the field equations become Rab = 0.”
McMahon, D., Relativity Demystified, A Self teaching Guide, McGraw-Hill, New York, 2006.
“
the empty spacetime field equations are Ruv = 0.”
Foster, J., and Nightingale, J. D., A Short Course in General Relativity, 2nd ed., Springer-Verlag, 1994.
“
The law Guv = 0 in empty space is chosen by Einstein as his law of gravitation.”
Eddington, A. S., The mathematical theory of relativity, Cambridge University Press, Cambridge, 2nd edition, 1960. (
NOTE: Eddington uses the kernel G for the Ricci tensor and Ricci scalar, not R.)
(b) Do Einstein and his followers asset that a material source for his gravitational field is present in the universe described by R
uv = 0?
ANSWER: Yes.
See ‘t Hooft, and McMahon, in (a) above, for example. Also, Einstein refers to the solution for R
uv =0 as follows:
“
ds2 = (1 – A/r)dl2 – [dr2/(1 – A/r) + r2(sin2θ dφ2 + dθ2)] (109a)
A = kM/4π
M denotes the sun’s mass centrally symmetrically placed about the origin of co-ordinates; the solution (109a) is valid only outside this mass, where all the Tμν vanish.”
Einstein, A., The Meaning of Relativity, expanded Princeton Science Library Edition, 2005
(c) Does T
uv = 0 in the equations R
uv= /\g
uv where /\ is the so-called ‘cosmological constant’?
ANSWER: Yes.
“
the more general possibility for the field equations in empty space Ruv = /\guv.”
Tolman, R. C., Relativity Thermodynamics and Cosmology, Dover Publications Inc., New York, (1987)
“
We take the latter form of Einstein’s law Guv = /\guv in empty space, /\ being a universal constant …”
Eddington, A. S., The mathematical theory of relativity, Cambridge University Press, Cambridge, 2nd edition, 1960. (
NOTE: Eddington uses the kernel G for the Ricci tensor and Ricci scalar, not R.)
(d) Is de Sitter’s empty universe a solution for R
uv = /\g
uv?
ANSWER: Yes.
“
De Sitter’s world thus corresponds to the revised form of the law of gravitation Guv = /\guv”
Eddington, A. S., The mathematical theory of relativity, Cambridge University Press, Cambridge, 2nd edition, 1960. (
NOTE: Eddington uses the kernel G for the Ricci tensor, not R)
“
the de Sitter line element corresponds to a model which must strictly be taken as completely empty.”
Tolman, R. C., Relativity Thermodynamics and Cosmology, Dover Publications Inc., New York, 1987.
“
This is not a model of relativistic cosmology because it is devoid of matter.” [3]
d’Inverno, R., Introducing Einstein’s Relativity, Oxford University Press, 1992.
“
the solution for an entirely empty world.”
Eddington, A. S., The mathematical theory of relativity, Cambridge University Press, Cambridge, 2nd edition, 1960.
“
there is no matter at all!”
Weinberg, S., Gravitation and Cosmology: Principles and Applications of the General theory of Relativity, John Wiley & Sons, Inc., 1972.
CONCLUSIONS
Thus, as explained in my article,
Crothers, S. J., To Have and Not to Have - the Paradox of Black Hole Mass, 12 August, 2015,
http://vixra.org/pdf/1508.0106v1.pdf
according to Einstein and his followers a material source is both present and absent, in different universes respectively, by means of the very same mathematical constraint for material sources, namely, T
uv = 0. Since that is impossible, and since de Sitter’s empty universe contains no material sources because T
uv = 0, the universe described by R
uv = 0 also contains no material sources. But it is from the solution for R
uv = 0 that the black hole was generated by the cosmologists. Hence, the black hole is a fallacy. Einstein’s claim that his R
uv = 0 describes his gravitational outside a body such as a star is false, owing to contradiction of de Sitter’s obviously empty universe, and the very nature of his 'field equations'.
Stephen J. Crothers
