Kruskal Dynamics for Radial Geodesics

ABSTRACT

The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-spectively. The relationship between r and t for a test particle moving along a radial or non-radial geodesic is well known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of motion, is well known. However, the same is not true for the Kruskal coordinates; and, we derive here the expression for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value of ) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in sharp contrast with the Schwarzschild derivative, , at the Event Horizon. We also explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion for a test particle on a radial geodesic. The physical implications of this result will be discussed elsewhere.

The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-spectively. The relationship between r and t for a test particle moving along a radial or non-radial geodesic is well known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of motion, is well known. However, the same is not true for the Kruskal coordinates; and, we derive here the expression for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value of ) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in sharp contrast with the Schwarzschild derivative, , at the Event Horizon. We also explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion for a test particle on a radial geodesic. The physical implications of this result will be discussed elsewhere.

Cite this paper

A. Mitra, "Kruskal Dynamics for Radial Geodesics,"*International Journal of Astronomy and Astrophysics*, Vol. 2 No. 3, 2012, pp. 174-179. doi: 10.4236/ijaa.2012.23021.

A. Mitra, "Kruskal Dynamics for Radial Geodesics,"

References

[1] C.W. Misner, K. S. Thorne, and J. Wheeler, “Gravitation”, (Freeman, San Fransisco, 1973)

[2] S.L. Shapiro and S.A. Teukolsky, “Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Ob-jects,” Wiley, New York, 1983. doi:10.1002/9783527617661

[3] A. Mitra, “Black Holes or Eternally Collapsing Objects: A Review of 90 Years of Misconceptions,” in “Focus on Black Hole Research”, ed. Paul V. Kreitler. ISBN 1-59454-460-3, Nova, New York, 2006.

[4] A. Mitra, “On the non-occurrence of Type I X-ray bursts from the black hole candidates,” Advances in Space Re-search, Vol. 38, 2006, p. 2917-2919.

[5] doi:10.1016/j.asr.2006.02.074 A. Mitra, “Quantum Information Paradox: Real or Ficti-tious,” Pramana, Vol. 73, No. 3, 2009, pp. 615-620. doi:10.1007/s12043-009-0113-9

[6] A. Mitra, “Comments on `The Euclidean Gravitational Action as Black Hole Entropy, Singularities, and Space Time Voids',” Journal of Mathematical Physics, Vol. 50, No. 4, 2009.

[7] A. Mitra, “The fallacy of Oppenheimer Snyder Collapse: No General Relativistic Collapse at All, No Black Hole, No Physical Singularity,” Astrophysics and Space Science, Vol. 332, No. 1, 2011 pp. 43-48. doi:10.1007/s10509-010-0578-5

[8] M.D. Kruskal, “Maximal Extension of Schwarzschild Metric,” Phys. Rev. Vol. 119, Vol. 119, Issue 5, 1960, pp. 1743-1745.

[9] P. Szekeres, “On the singularities of a Riemannian mani-fold. Math. Debreca.,” Vol. 7, 1960, 285.

[10] S. Chandrasekhar, “The Mathematical Theory of Black Holes,” Clarendron, Oxford, 1983.

[1] C.W. Misner, K. S. Thorne, and J. Wheeler, “Gravitation”, (Freeman, San Fransisco, 1973)

[2] S.L. Shapiro and S.A. Teukolsky, “Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Ob-jects,” Wiley, New York, 1983. doi:10.1002/9783527617661

[3] A. Mitra, “Black Holes or Eternally Collapsing Objects: A Review of 90 Years of Misconceptions,” in “Focus on Black Hole Research”, ed. Paul V. Kreitler. ISBN 1-59454-460-3, Nova, New York, 2006.

[4] A. Mitra, “On the non-occurrence of Type I X-ray bursts from the black hole candidates,” Advances in Space Re-search, Vol. 38, 2006, p. 2917-2919.

[5] doi:10.1016/j.asr.2006.02.074 A. Mitra, “Quantum Information Paradox: Real or Ficti-tious,” Pramana, Vol. 73, No. 3, 2009, pp. 615-620. doi:10.1007/s12043-009-0113-9

[6] A. Mitra, “Comments on `The Euclidean Gravitational Action as Black Hole Entropy, Singularities, and Space Time Voids',” Journal of Mathematical Physics, Vol. 50, No. 4, 2009.

[7] A. Mitra, “The fallacy of Oppenheimer Snyder Collapse: No General Relativistic Collapse at All, No Black Hole, No Physical Singularity,” Astrophysics and Space Science, Vol. 332, No. 1, 2011 pp. 43-48. doi:10.1007/s10509-010-0578-5

[8] M.D. Kruskal, “Maximal Extension of Schwarzschild Metric,” Phys. Rev. Vol. 119, Vol. 119, Issue 5, 1960, pp. 1743-1745.

[9] P. Szekeres, “On the singularities of a Riemannian mani-fold. Math. Debreca.,” Vol. 7, 1960, 285.

[10] S. Chandrasekhar, “The Mathematical Theory of Black Holes,” Clarendron, Oxford, 1983.