Minkowskian Solution of General Relativity with Cosmological Constant and the Accelerating Universe

ABSTRACT

A Minkowskian solution of the equation of General Relativity (as written by Einstein in 1915) is trivial because it simply means that both members of the equation are equal to zero. However, if alternatively, one considers the complete equation with a non-zero constant Λ (Einstein 1917), a Minkowskian solution is no longer trivial because it amounts to impose a constraint on the right hand side of the equation (*i*.*e*. a non-null
stress-energy tensor). If furthermore one identifies (as usual) this tensor to
the one of a perfect fluid, one finds that this fluid has a positive energy
density and a negative pressure that depend on the three constants of the
equation (*i*.*e*. gravitational constant *G*,
cosmological constant Λ and velocity of light c). When doing that (§1), one has
to consider the “Minkowskian Vacuum” as a physical object of GR (an enigmatic
non-baryonic Minkowskian fluid). Can one build a model of this object on the
basis of a dynamical equilibrium between the effective gravitational attraction
due to the positive energy density versus the negative pressure repulsion? We
propose to study such a model, where the (enigmatic) fluid is assumed to exist
only in a limited sphere whose surface acts like a “test body” sensitive to the
gravitational field created by the fluid. No static equilibrium exists, but a
pseudoNewtonian “dynamical equilibrium” (§2) can be reached if the
pseudoEuclidean fluid is in state of expansion. Up to there, we have simply
constructed a model of an “abstract Universe” (*i*.*e*.* *the limited sphere: There is no fluid outside this sphere!) that
gives to a (purely mathematical) constant Λ a concrete physical meaning. We
discover finally that our expanding fluid has not only dynamical
(gravitational) properties (§3) but also optical properties that are connected
with Doppler Redshift (§4). Remembering that recent observations in Cosmology
indicate that the “real Universe” seems to be “Flat” and in “Accelerated
Expansion”; remembering also (after all) that the archetypal Flat Universe is
simply a Minkowskian Universe, we logically wonder if the unexpected
Minkowskian global solution, could not be also a significant cosmological model
(conclusion).

A Minkowskian solution of the equation of General Relativity (as written by Einstein in 1915) is trivial because it simply means that both members of the equation are equal to zero. However, if alternatively, one considers the complete equation with a non-zero constant Λ (Einstein 1917), a Minkowskian solution is no longer trivial because it amounts to impose a constraint on the right hand side of the equation (

Cite this paper

Pierseaux, Y. (2014) Minkowskian Solution of General Relativity with Cosmological Constant and the Accelerating Universe.*Journal of Modern Physics*, **5**, 1725-1732. doi: 10.4236/jmp.2014.516172.

Pierseaux, Y. (2014) Minkowskian Solution of General Relativity with Cosmological Constant and the Accelerating Universe.

References

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[2] Einstein, A. (1917) Kosmologische Betrachtungen zur Allgemeinen Relativitatstheorie. In: Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften, VI, Berlin, 142-152.

[3] Poincare, H. (1906) Rendicontidel Circolo Matematico di Palermo, 21, 129-175.

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http://dx.doi.org/10.1093/mnras/78.1.3

[5] Pierseaux, Y. (2010) From Unexpected Minkowskian Solution of Einstein’s Equation of General Relativity with Cosmological Constant to the Accelerating Universe. Revue IIHE.

http://arxiv.org/abs/1009.1375

[6] Rindler, W. (2001) Relativity. Special, General and Cosmological. Oxford University Press, Oxford.

[7] Einstein, A. (1905) Annalen der Physik, 17, 891-921.

[8] Bondi, H. (1962) Relativity and Common Sense. A New Approach to Einstein. Dover Publications Inc., New York.

[9] Pierseaux, Y. (2013) Annales de la Fondation Louis de Broglie, 38, 41-55.

[10] Riess, A., et al. (1998) Astronomical Journal, 116, 1009-1038.

http://dx.doi.org/10.1086/300499

[11] Peebles, P.J.E. (2002) The Cosmological Constant and Dark Energy.

http://arxiv.org/abs/astro-ph/0207347

[12] Turner, M. (2002) The New Cosmology. World Scientific, Singapore.

[1] Einstein, A. (1916) Annalen der Physik, 354, 769-822.

[2] Einstein, A. (1917) Kosmologische Betrachtungen zur Allgemeinen Relativitatstheorie. In: Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften, VI, Berlin, 142-152.

[3] Poincare, H. (1906) Rendicontidel Circolo Matematico di Palermo, 21, 129-175.

[4] de Sitter, W. (1917) Royal Astronomical Society Monthly Notices, LXXVIII, 3-28.

http://dx.doi.org/10.1093/mnras/78.1.3

[5] Pierseaux, Y. (2010) From Unexpected Minkowskian Solution of Einstein’s Equation of General Relativity with Cosmological Constant to the Accelerating Universe. Revue IIHE.

http://arxiv.org/abs/1009.1375

[6] Rindler, W. (2001) Relativity. Special, General and Cosmological. Oxford University Press, Oxford.

[7] Einstein, A. (1905) Annalen der Physik, 17, 891-921.

[8] Bondi, H. (1962) Relativity and Common Sense. A New Approach to Einstein. Dover Publications Inc., New York.

[9] Pierseaux, Y. (2013) Annales de la Fondation Louis de Broglie, 38, 41-55.

[10] Riess, A., et al. (1998) Astronomical Journal, 116, 1009-1038.

http://dx.doi.org/10.1086/300499

[11] Peebles, P.J.E. (2002) The Cosmological Constant and Dark Energy.

http://arxiv.org/abs/astro-ph/0207347

[12] Turner, M. (2002) The New Cosmology. World Scientific, Singapore.