Advertisement

Curved space, curved time, and curved space-time in Schwarzschild geodetic geometry

  • Rafael T. Eufrasio
  • Nicholas A. Mecholsky
  • Lorenzo RescaEmail author
Research Article

Abstract

We investigate geodesic orbits and manifolds for metrics associated with Schwarzschild geometry, considering space and time curvatures separately. For ‘a-temporal’ space, we solve a central geodesic orbit equation in terms of elliptic integrals. The intrinsic geometry of a two-sided equatorial plane corresponds to that of a full Flamm’s paraboloid. Two kinds of geodesics emerge. Both kinds may or may not encircle the hole region any number of times, crossing themselves correspondingly. Regular geodesics reach a periastron greater than the \(r_S\) Schwarzschild radius, thus remaining confined to a half of Flamm’s paraboloid. Singular or s-geodesics tangentially reach the \(r_S\) circle. These s-geodesics must then be regarded as funneling through the ‘belt’ of the full Flamm’s paraboloid. Infinitely many geodesics can possibly be drawn between any two points, but they must be of specific regular or singular types. A precise classification can be made in terms of impact parameters. Geodesic structure and completeness is conveyed by computer-generated figures depicting either Schwarzschild equatorial plane or Flamm’s paraboloid. For the ‘curved-time’ metric, devoid of any spatial curvature, geodesic orbits have the same apsides as in Schwarzschild space-time. We focus on null geodesics in particular. For the limit of light grazing the sun, asymptotic ‘spatial bending’ and ‘time bending’ become essentially equal, adding up to the total light deflection of 1.75 arc-seconds predicted by general relativity. However, for a much closer approach of the periastron to \(r_S\), ‘time bending’ largely exceeds ‘spatial bending’ of light, while their sum remains substantially below that of Schwarzschild space-time.

Keywords

General theory of relativity Gravitation Schwarzschild metric Space-time curvature Space curvature Geodesics 

Notes

Acknowledgements

The authors of this paper are listed in alphabetical order. We acknowledge financial support from NASA/ADAP Grants NNH11ZDA001N and NNX13AI48G from the Vitreous State Laboratory at the Catholic University of America. We dedicate our work to the memory of Maria Rita Soverchia Resca.

References

  1. 1.
    Grøn, Ø.: Celebrating the centenary of the Schwarzschild solutions. Am. J. Phys. 84(7), 537–541 (2016)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Weinberg, S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972)Google Scholar
  3. 3.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, New York (1973)Google Scholar
  4. 4.
    Schutz, B.F.: A First Course in General Relativity, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  5. 5.
    Schutz, B.F.: Gravity from the Ground Up. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  6. 6.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)CrossRefGoogle Scholar
  7. 7.
    Rindler, W.: Essential Relativity: Special, General, and Cosmological, Revised 2 edn. Springer, Berlin (1979)zbMATHGoogle Scholar
  8. 8.
    Berry, M.: Principles of Cosmology and Gravitation. Cambridge University Press, Cambridge (1976)zbMATHGoogle Scholar
  9. 9.
    Hobson, M.P., Efstathiou, G.P., Lasenby, A.N.: General Relativity: An Introduction for Physicists. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  10. 10.
    Hartle, J.B.: Gravity: An Introduction to Einstein’s General Relativity. Pearson, London (2003)Google Scholar
  11. 11.
    Frolov, V.P., Zelnikov, A.: Introduction to Black Hole Physics. Oxford University Press, Oxford (2011)CrossRefGoogle Scholar
  12. 12.
    Narlikar, J.V.: Lectures on General Relativity and Cosmology. Macmillan, New York (1979)Google Scholar
  13. 13.
    Price, R.H.: General relativity primer. Am. J. Phys. 50(4), 300–329 (1982)ADSCrossRefGoogle Scholar
  14. 14.
    Morris, M.S., Thorne, K.S.: Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity. Am. J. Phys. 56(5), 395–412 (1988)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dadhich, N.: Einstein is Newton with space curved. Curr. Sci. 109(2), 260–264 (2015)Google Scholar
  16. 16.
    Resca, L.: Spacetime and spatial geodesic orbits in Schwarzschild geometry. Eur. J. Phys. 39, 035602 (2018).  https://doi.org/10.1088/1361-6404/aab12f CrossRefzbMATHGoogle Scholar
  17. 17.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)zbMATHGoogle Scholar
  18. 18.
    Flamm, L.: Beiträge zur Einsteinischen gravitationtheory, Physikalische Zeitschrift 17, 448–454 (1916). Translation and Republication of: Contributions to Einstein’s theory of gravitation, by Ludwig Flamm (2015)  https://doi.org/10.1007/s10714-015-1908-2
  19. 19.
    Einstein, A., Rosen, N.: The particle problem in the general theory of relativity. Phys. Rev. 48, 73–77 (1935)ADSCrossRefGoogle Scholar
  20. 20.
    Price, R.H.: Spatial curvature, spacetime curvature, and gravity. Am. J. Phys. 84(8), 588–592 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    Ellingson, J.G.: The deflection of light by the Sun due to three-space curvature. Am. J. Phys. 55(8), 759–760 (1987)ADSCrossRefGoogle Scholar
  22. 22.
    Gruber, R.P., Gruber, A.D., Hamilton, R., Matthews, S.M.: Space curvature and the ‘heavy banana paradox’. Phys. Teach. 29, 147–149 (1991)ADSCrossRefGoogle Scholar
  23. 23.
    Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V.: Editors, NIST Digital Library of Mathematical Functions, Release 1.0.18 of 2018-03-27. https://dlmf.nist.gov/
  24. 24.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces: Revised and Updated, 2nd edn. Dover Publications, New York (2017)Google Scholar
  25. 25.
    Pressley, A.: Elementary Differential Geometry, 2nd edn. Springer, London, New York (2012)zbMATHGoogle Scholar
  26. 26.
    Will, C.M.: Henry Cavendish, Johann von Soldner, and the deflection of light. Am. J. Phys. 56(5), 413–415 (1988)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of ArkansasFayettevilleUSA
  2. 2.Department of Physics and Vitreous State LaboratoryThe Catholic University of AmericaWashingtonUSA

Personalised recommendations