152
6 de Sitter Cosmology
The Newman–Penrose components of the Weyl conformal curvature tensor are
given by
0 = −
l(1 + k 2 u 2
+ )
1 − k 2 u 2
+
δ(v) , , 4 = −
k(1 + l 2 v 2
+ )
1 − l 2 v 2
+
δ(u) , , 1 = 2 = 3 = 0 .
(6.168)
Thus the boundaries v = 0, 0 ≤ k 2 u 2 < 1 and u = 0, 0 ≤ l 2 v 2 < 1 are the
histories of impulsive gravitational waves corresponding to the delta function terms
here. The post collision region u > 0, v > 0 is conformally flat and is a space-time
of constant curvature with Riemann curvature tensor components given by
R abcd = −2 k l(g ac g bd − g ad g bc ) .
(6.169)
Hence this region of space-time does not possess a curvature singularity, in striking
contrast to the post collision region of the Khan–Penrose space-time.
For this model collision the energy in the incoming impulsive gravitational waves
is re-distributed after the collision into two light-like shells of matter and two
impulsive gravitational waves moving away from each other followed by a spacetime of constant curvature (i.e. de Sitter or anti-de Sitter space-time). When the
surface energy densities of the post collision light-like shells of matter are required
to be positive the space-time of constant curvature must be anti-de Sitter space-time.
References
1. R. Penrose, General Relativity, papers in honour of J.L. Synge (Clarendon Press, Oxford,
1972), p. 112
2. P.A. Hogan, Int. J. Mod. Phys. D 27, 1850045 (2018)
3. I. Robinson, A. Trautman, J. Math. Phys. 24, 1425 (1983)
4. I. Ozsváth, I. Robinson, K. Rózga, J. Math. Phys. 26, 1755 (1985)
5. H.V. Tran, Ph.D. Thesis, University of Texas at Dallas (1988)
6. I. Robinson, University of Texas at Dallas Internal Report (1983)
7. A. Trautman, Recent Developments in General Relativity (Pergamon Press Inc., New York,
1962), p. 459
8. R.P. Kerr, A. Schild, Atti del convegno sulla relatività generale: problemi dell’energia e ondi
gravitationali (Barbèra, Firenze, 1965), p. 222
9. R.P. Kerr, A. Schild, Proc. Symp. Appl. Math. 17, 199 (1965)
10. I. Ozsvàth, J. Math. Phys. 6, 590 (1965)
11. I. Ozsvàth, Gravitation and Geometry, ed. by A. Trautman, W. Rindler (Bibliopolis, Naples,
1987), p. 309
12. K.A. Khan, R. Penrose, Nature 229, 185 (1971)
13. P. Bell, P. Szekeres, Gen. Rel. Grav. 5, 275 (1974)
14. C. Barrabès, P.A. Hogan, Phys. Rev. D 92, 044032 (2015)
15. P. Szekeres, Nature 228, 1183 (1970)
16. P. Szekeres, J. Math. Phys. 13, 286 (1972)
17. C. Barrabès, P.A. Hogan, Singular Null Hypersurfaces in General Relativity (World Scientific,
Singapore, 2003)
18. C. Barrabès, W. Israel, Phys. Rev. D 43, 1129 (1991)
6 de Sitter Cosmology
The Newman–Penrose components of the Weyl conformal curvature tensor are
given by
0 = −
l(1 + k 2 u 2
+ )
1 − k 2 u 2
+
δ(v) , , 4 = −
k(1 + l 2 v 2
+ )
1 − l 2 v 2
+
δ(u) , , 1 = 2 = 3 = 0 .
(6.168)
Thus the boundaries v = 0, 0 ≤ k 2 u 2 < 1 and u = 0, 0 ≤ l 2 v 2 < 1 are the
histories of impulsive gravitational waves corresponding to the delta function terms
here. The post collision region u > 0, v > 0 is conformally flat and is a space-time
of constant curvature with Riemann curvature tensor components given by
R abcd = −2 k l(g ac g bd − g ad g bc ) .
(6.169)
Hence this region of space-time does not possess a curvature singularity, in striking
contrast to the post collision region of the Khan–Penrose space-time.
For this model collision the energy in the incoming impulsive gravitational waves
is re-distributed after the collision into two light-like shells of matter and two
impulsive gravitational waves moving away from each other followed by a spacetime of constant curvature (i.e. de Sitter or anti-de Sitter space-time). When the
surface energy densities of the post collision light-like shells of matter are required
to be positive the space-time of constant curvature must be anti-de Sitter space-time.
References
1. R. Penrose, General Relativity, papers in honour of J.L. Synge (Clarendon Press, Oxford,
1972), p. 112
2. P.A. Hogan, Int. J. Mod. Phys. D 27, 1850045 (2018)
3. I. Robinson, A. Trautman, J. Math. Phys. 24, 1425 (1983)
4. I. Ozsváth, I. Robinson, K. Rózga, J. Math. Phys. 26, 1755 (1985)
5. H.V. Tran, Ph.D. Thesis, University of Texas at Dallas (1988)
6. I. Robinson, University of Texas at Dallas Internal Report (1983)
7. A. Trautman, Recent Developments in General Relativity (Pergamon Press Inc., New York,
1962), p. 459
8. R.P. Kerr, A. Schild, Atti del convegno sulla relatività generale: problemi dell’energia e ondi
gravitationali (Barbèra, Firenze, 1965), p. 222
9. R.P. Kerr, A. Schild, Proc. Symp. Appl. Math. 17, 199 (1965)
10. I. Ozsvàth, J. Math. Phys. 6, 590 (1965)
11. I. Ozsvàth, Gravitation and Geometry, ed. by A. Trautman, W. Rindler (Bibliopolis, Naples,
1987), p. 309
12. K.A. Khan, R. Penrose, Nature 229, 185 (1971)
13. P. Bell, P. Szekeres, Gen. Rel. Grav. 5, 275 (1974)
14. C. Barrabès, P.A. Hogan, Phys. Rev. D 92, 044032 (2015)
15. P. Szekeres, Nature 228, 1183 (1970)
16. P. Szekeres, J. Math. Phys. 13, 286 (1972)
17. C. Barrabès, P.A. Hogan, Singular Null Hypersurfaces in General Relativity (World Scientific,
Singapore, 2003)
18. C. Barrabès, W. Israel, Phys. Rev. D 43, 1129 (1991)