Einsteins Theory of Relativity 10 Internal Schwarzschild Metric

Einstein‘s Theory of Relativity 10. Internal Schwarzschild Metric, Stellar Structure Equations Ulrich R. M. E. Geppert 2/21/2021 U. R. M. E. Zielona Gora 1

remark on Schwarzschild metric proof 2/21/2021 U. R. M. E. Zielona Gora 2

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classical, nonrelativistic stellar structure equations 2/21/2021 U. R. M. E. Zielona Gora 4

Schwarzschild metric in the form Einstein equations in the form 2/21/2021 ⇐ relativistic Euler and continuity equation of a liquid in a gravitational field U. R. M. E. Zielona Gora 5

remark on Euler equation: in detail: 2/21/2021 U. R. M. E. Zielona Gora 6

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proof: 2/21/2021 now U. R. M. E. Zielona Gora 8

in more detail analogue 2/21/2021 U. R. M. E. Zielona Gora 9

f 2/21/2021 U. R. M. E. Zielona Gora 10

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insertion: contraction of Christoffel symbols, determinant of metric tensor 2/21/2021 U. R. M. E. Zielona Gora 12

why: ? 2/21/2021 U. R. M. E. Zielona Gora 13

antisymmetric tensor, see GR_1 2/21/2021 U. R. M. E. Zielona Gora 14

two identical indices 2/21/2021 U. R. M. E. Zielona Gora 15

back to determination of metric coefficient: 2/21/2021 U. R. M. E. Zielona Gora 16

(see GR_9, slide 31, Eq. B) 2/21/2021 U. R. M. E. Zielona Gora 17

insert this, and this Tolman, Oppenheimer, Volkoff 2/21/2021 U. R. M. E. Zielona Gora 18

slide 6: hydrostatic stellar equilibrium, derived from Newton‘s field equation 2/21/2021 U. R. M. E. Zielona Gora 19

joins this solution smoothly to the external Schwarzschild solution? 2/21/2021 U. R. M. E. Zielona Gora 20

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up to now 2 stellar structure equations: 3 fields but only 2 equations 2/21/2021 U. R. M. E. Zielona Gora 22

e. g. : General relativistic treatment of thermal, magnetic and rotational evolution of isolated neutron stars with crustal magnetic fields D. Page 1, U. Geppert 2, and T. Zannias 3 2/21/2021 U. R. M. E. Zielona Gora 23

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relativistic stars with constant density simplest case: - spherical symmetric, i. e. no rotation no magnetic field 2/21/2021 U. R. M. E. Zielona Gora 27

TOV-equation with (1) and (2): 2/21/2021 U. R. M. E. Zielona Gora 28

R r 2/21/2021 U. R. M. E. Zielona Gora 29

(little exercise? ) 2/21/2021 U. R. M. E. Zielona Gora 30

stability of a constant density star: gravitational collapse 2/21/2021 U. R. M. E. Zielona Gora → time dependent metric 31

⇒ central pressure in non-relativistic stars remains finite 2/21/2021 U. R. M. E. Zielona Gora 32

reminder GR_6, slide 17 2/21/2021 U. R. M. E. Zielona Gora 33

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volume in internal Schwarzschildmetric 2/21/2021 U. R. M. E. Zielona Gora 35

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