Abstract
We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian character of these formulas and show that they converge to Newtonian mass and center of mass in the Newtonian limit, respectively, using Ehlers’ frame theory. Moreover, we present a novel physical interpretation of the level sets of the canonical lapse function and apply it to prove uniqueness results. Finally, we suggest a notion of force on test particles in geometrostatic space-times.
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Notes
- 1.
Here, time-like curves and the time functional are taken with respect to the static space-time metric \(ds^2=-N^2c^2dt^2+g\) induced by \((M^3,g,N)\).
- 2.
This assumes that the lapse function exists in the first place.
- 3.
Albeit without explicit reference to the speed of light.
- 4.
We note that our discussion of center of mass only applies to systems with non-vanishing mass.
- 5.
Under precise fall-off conditions at spacelike infinity that are satisfied here.
References
Szabados, L.: Quasi-local energy-momentum and angular momentum in general relativity. Living Rev. Relativ. 12(4), lrr-2009-4 (2009). http://www.livingreviews.org/lrr-2009-4
Cederbaum, C.: The Newtonian limit of geometrostatics. Ph.D. thesis, Free University Berlin, Berlin (2011)
Müller zum Hagen, H.: On the analyticity of static vacuum solutions of Einstein’s equations. Proc. Camb. Phil. Soc. 67, 415 (1970). doi:10.1017/S0305004100001237
Choquet-Bruhat, Y., Jork, J.: The Cauchy problem. In: Held, A. (ed.) General Relativity and Gravitation: One Hundred Years after the Birth of Albert Einstein, vol. 1, Chap. 4, pp. 99–172. Plenum Press, New York (1980)
Kennefick, D., O’Murchadha, N.: Weakly decaying asymptotically flat static and stationary solutions to the Einstein equations. Class. Quantum Grav. 12(1), 149 (1995). doi:10.1088/0264-9381/12/1/013
Arnowitt, R., Deser, S., Misner, C.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(3), 997 (1961). doi:10.1103/PhysRev.122.997
Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661 (1986). doi:10.1002/cpa.3160390505
Chruściel, P.: On the invariant mass conjecture in general relativity. Commun. Math. Phys. 120, 233 (1988). doi:10.1007/BF01217963
Huang, L.H.: Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics. Comm. Math. Phys. 300(2), 331 (2008)
Huisken, G., Yau, S.T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124, 281 (1996). doi:10.1007/s002220050054
Metzger, J.: Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. J. Differ. Geom. 77(2), 201 (2007)
Ehlers, J.: Examples of Newtonian limits of relativistic spacetimes. Class. Quantum Grav. 14, A119 (1997). doi:10.1088/0264-9381/14/1A/010
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Cederbaum, C. (2014). Geometrostatics: The Geometry of Static Space-Times. In: Bičák, J., Ledvinka, T. (eds) Relativity and Gravitation. Springer Proceedings in Physics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-06761-2_5
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DOI: https://doi.org/10.1007/978-3-319-06761-2_5
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