Abstract
We present a novel physical interpretation of the level sets of the (canonical) lapse function in static isolated general relativistic spacetimes. Our interpretation uses a notion of constrained test particles. It leads to a definition of gravitational force on test particles and to a previously unknown uniqueness result for the lapse function. In Sect. 5, we discuss photon spheres in static isolated relativistic spacetimes and relate them to the level sets of the lapse function.
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Notes
- 1.
- 2.
This follows from the Newtonian limit analysis combined with a more detailed study of the geometry of pseudo-Newtonian gravity, cf. [2].
- 3.
Our definition of gravitational force can be extended to general timelike curves that are not necessarily geodesics. These can be interpreted as test particles that are subject to not only gravitational but also to non-gravitational forces (e.g. electro-magnetic ones).
- 4.
The author has some first ideas how to generalize the notion of gravitational force presented here to extended bodies. A second Newtonian law of motion for extended bodies, however, seems more difficult at the moment.
- 5.
The acceleration of the test particle is induced from the Lorentzian metric \(ds^2\) and thus naturally refers to the spatial metric \(g\) and not to the conformally transformed metric \(\gamma \) because the definition of test particles relies on the “dynamics” of the spacetime. On the other hand, the definition of gravitational force uses the analogy of pseudo-Newtonian and Newtonian effects and thus should be formulated in pseudo-Newtonian terms.
- 6.
However, we do assume that the lapse function exists in the first place.
- 7.
See [4] for more information.
- 8.
- 9.
cf. [8] for an exposition of this analysis.
- 10.
Making the same mild technical assumption as Israel [10] that the lapse function foliates the region exterior to the photon sphere.
- 11.
This relationship can actually be pursued much further; it even constitutes a central tool for showing that the Newtonian limit of the ADM-mass of a static isolated spacetime “is” the Newtonian mass of “its” Newtonian limit. For more details, see [2].
References
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Cederbaum, C. (2015). Level Sets of the Lapse Function in Static GR. In: Puetzfeld, D., Lämmerzahl, C., Schutz, B. (eds) Equations of Motion in Relativistic Gravity. Fundamental Theories of Physics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-18335-0_24
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