Quantitative properties of the Schwarzschild metric

0488531 - MÚ 2019 RIV BG eng J - Journal Article
Křížek, Michal - Křížek, Filip
Quantitative properties of the Schwarzschild metric.
Publications of the Astronomical Society of Bulgaria. Roč. 2018, č. 1 (2018), s. 1-10
R&D Projects: GA MŠMT(CZ) LG15052
Institutional support: RVO:67985840 ; RVO:61389005
Keywords : exterior and interior Schwarzschild metric * proper radius * coordinate radius
OECD category: Applied mathematics; Astronomy (including astrophysics,space science) (UJF-V)
http://astro.shu-bg.net/pasb/index_files/Papers/2018/SCHWARZ8.pdf

In this paper we show that the di erence between the Euclidean geometry and Schwarzschild geometry curved by a tiny mass ball can be quite large on galactic and cosmological scales. We also provide formulas for the proper (relativistic) radius and volume of a homogeneous mass ball. For instance, the homogeneous ball, whose mass and radius is the same as that of the Earth, has relativistic volume about 457 km3 larger than its Euclidean volume. Similarly, the Euclidean circumference of the Sun is about 3 km shorter than its relativistic circumference, provided the Sun would be homogeneous. Finally, we give some cosmological applications. In particular, the most probable model of a homogeneous and isotropic universe for a xed time is a three-dimensional hypersphere, since a homogeneous distribution of mass yields a positive curvature.
Permanent Link: http://hdl.handle.net/11104/0283113