The Schwarzschild Mass in General Relativity

Praveen Singh Negi

doi:doi:10.11648/j.ijass.20190706.13

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The Schwarzschild Mass in General Relativity

Praveen Singh Negi

Published in International Journal of Astrophysics and Space Science (Volume 7, Issue 6)

Received: 23 November 2019 Accepted: 18 December 2019 Published: 30 December 2019

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Abstract

The central (surface) energy-density, E₀ (E_R), which appears in the expression of total static and spherical mass, M (corresponding to the total radius R) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, M, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein’s field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, M. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density, (3M/4πR³), of the configuration which turns out to be independent of the radial coordinate r and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be independent of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate r and the proper distance from the centre of the sphere to its surface [1]. In the light of this reasoning, the various EOSs and analytic solutions of Einstein’s field equations in which the central and the surface density are interdependent can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration.

Published in	International Journal of Astrophysics and Space Science (Volume 7, Issue 6)
DOI	10.11648/j.ijass.20190706.13
Page(s)	75-78
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Static Spherical Structures, Analytic Solutions, Neutron Stars, Dense Matter, Equation of State

References

[1]	Schutz, B., F.: A first course in general relativity. Cambridge University Press, UK (1990).
[2]	Tolman, R., C. Phys. Rev. 55, 364 (1939).
[3]	Oppenheimer, J. R., Volkoff, G., M. Phys. Rev. 55, 374 (1939).
[4]	Stephani, H, Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations, 2nd edn., Cambridge, UK (2003).
[5]	Zeldovich, Ya., B., and Novikov, I. D.: Stars and Relativity (Relativistic Astrophysics), Vol I, University of Chicago Press, Chicago (1971).
[6]	Shapiro, S. L., and Teukolsky, S. A.: Black Holes, White Dwarfs, And Neutron Stars: The Physics of Compact Objects. Wiley, New York (1983).
[7]	Haensel, P., Potekhin, A. Y., and Yakovlev, D. G.: Neutron Stars 1. Equation of State and Structure. Spinger, Berlin (2006).
[8]	Tooper, R. F. Astrophys. J. 140, 434 (1964); Tooper, R. F. Astrophys. J. 142, 1541 (1965).
[9]	Buchdahl, H. A., Astrophys., J. 147, 310 (1967).
[10]	Adams, R., C., Cohen, J., M. Astrophys. J. 198, 507 (1975).
[11]	Durgapal, M., C., Bannerji, R. Phys. Rev. D 27, 328 (1983); ibid. D 28, 2695 (1984).
[12]	Durgapal, M., C., Fuloria, R. S. Gen. Rel. Grav. 17, 671 (1985).
[13]	Negi, P. S. Int. J. Theort. Phys., 45, 1695 (2006).
[14]	Brecher, K, Caporaso, G. Nature 259, 377 (1976).
[15]	Negi, P. S. Int. J. Mod. Phys. D, 20, 1171 (2011).
[16]	Negi, P. S. Mod. Phys. Lett. A, 19, 2941 (2004).
[17]	Durgapal, M. C., Pande, A. K. Ind. J. Pure Appl. Phys. 18, 171 (1980).

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Praveen Singh Negi. (2019). The Schwarzschild Mass in General Relativity. International Journal of Astrophysics and Space Science, 7(6), 75-78. https://doi.org/10.11648/j.ijass.20190706.13

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ACS Style

Praveen Singh Negi. The Schwarzschild Mass in General Relativity. Int. J. Astrophys. Space Sci. 2019, 7(6), 75-78. doi: 10.11648/j.ijass.20190706.13

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AMA Style

Praveen Singh Negi. The Schwarzschild Mass in General Relativity. Int J Astrophys Space Sci. 2019;7(6):75-78. doi: 10.11648/j.ijass.20190706.13

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@article{10.11648/j.ijass.20190706.13,
  author = {Praveen Singh Negi},
  title = {The Schwarzschild Mass in General Relativity},
  journal = {International Journal of Astrophysics and Space Science},
  volume = {7},
  number = {6},
  pages = {75-78},
  doi = {10.11648/j.ijass.20190706.13},
  url = {https://doi.org/10.11648/j.ijass.20190706.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijass.20190706.13},
  abstract = {The central (surface) energy-density, E0 (ER), which appears in the expression of total static and spherical mass, M (corresponding to the total radius R) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, M, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein’s field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, M. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density, (3M/4πR3), of the configuration which turns out to be independent of the radial coordinate r and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be independent of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate r and the proper distance from the centre of the sphere to its surface [1]. In the light of this reasoning, the various EOSs and analytic solutions of Einstein’s field equations in which the central and the surface density are interdependent can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration.},
 year = {2019}
}

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TY  - JOUR
T1  - The Schwarzschild Mass in General Relativity
AU  - Praveen Singh Negi
Y1  - 2019/12/30
PY  - 2019
N1  - https://doi.org/10.11648/j.ijass.20190706.13
DO  - 10.11648/j.ijass.20190706.13
T2  - International Journal of Astrophysics and Space Science
JF  - International Journal of Astrophysics and Space Science
JO  - International Journal of Astrophysics and Space Science
SP  - 75
EP  - 78
PB  - Science Publishing Group
SN  - 2376-7022
UR  - https://doi.org/10.11648/j.ijass.20190706.13
AB  - The central (surface) energy-density, E0 (ER), which appears in the expression of total static and spherical mass, M (corresponding to the total radius R) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, M, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein’s field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, M. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density, (3M/4πR3), of the configuration which turns out to be independent of the radial coordinate r and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be independent of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate r and the proper distance from the centre of the sphere to its surface [1]. In the light of this reasoning, the various EOSs and analytic solutions of Einstein’s field equations in which the central and the surface density are interdependent can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration.
VL  - 7
IS  - 6
ER  -

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Author Information

Praveen Singh Negi

Department of Physics, Faculty of Science, Kumaun Universsity, Nainital, India

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Praveen Singh Negi. (2019). The Schwarzschild Mass in General Relativity. International Journal of Astrophysics and Space Science, 7(6), 75-78. https://doi.org/10.11648/j.ijass.20190706.13

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ACS Style

Praveen Singh Negi. The Schwarzschild Mass in General Relativity. Int. J. Astrophys. Space Sci. 2019, 7(6), 75-78. doi: 10.11648/j.ijass.20190706.13

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AMA Style

Praveen Singh Negi. The Schwarzschild Mass in General Relativity. Int J Astrophys Space Sci. 2019;7(6):75-78. doi: 10.11648/j.ijass.20190706.13

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@article{10.11648/j.ijass.20190706.13,
  author = {Praveen Singh Negi},
  title = {The Schwarzschild Mass in General Relativity},
  journal = {International Journal of Astrophysics and Space Science},
  volume = {7},
  number = {6},
  pages = {75-78},
  doi = {10.11648/j.ijass.20190706.13},
  url = {https://doi.org/10.11648/j.ijass.20190706.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijass.20190706.13},
  abstract = {The central (surface) energy-density, E0 (ER), which appears in the expression of total static and spherical mass, M (corresponding to the total radius R) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, M, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein’s field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, M. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density, (3M/4πR3), of the configuration which turns out to be independent of the radial coordinate r and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be independent of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate r and the proper distance from the centre of the sphere to its surface [1]. In the light of this reasoning, the various EOSs and analytic solutions of Einstein’s field equations in which the central and the surface density are interdependent can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - The Schwarzschild Mass in General Relativity
AU  - Praveen Singh Negi
Y1  - 2019/12/30
PY  - 2019
N1  - https://doi.org/10.11648/j.ijass.20190706.13
DO  - 10.11648/j.ijass.20190706.13
T2  - International Journal of Astrophysics and Space Science
JF  - International Journal of Astrophysics and Space Science
JO  - International Journal of Astrophysics and Space Science
SP  - 75
EP  - 78
PB  - Science Publishing Group
SN  - 2376-7022
UR  - https://doi.org/10.11648/j.ijass.20190706.13
AB  - The central (surface) energy-density, E0 (ER), which appears in the expression of total static and spherical mass, M (corresponding to the total radius R) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, M, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein’s field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, M. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density, (3M/4πR3), of the configuration which turns out to be independent of the radial coordinate r and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be independent of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate r and the proper distance from the centre of the sphere to its surface [1]. In the light of this reasoning, the various EOSs and analytic solutions of Einstein’s field equations in which the central and the surface density are interdependent can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration.
VL  - 7
IS  - 6
ER  -

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