Relativistic Stars and the Schwarzschild Solution

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Topics - Schwarzschild • What is Birkhoff´s Theorem ? • Ansatz for the metric of relativistic stars. • The essence of the TOV equations. • Sequences of degenerate compact stars and maximum mass. • Profiles of metric functions. • Gravitational redshift and Black Hole solution (Schwarzschild 1916). • Stellar and photon orbits near compact objects. • Innermost stable circular orbit (ISCO).
Birkhoff‘s Theorem • Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric. • This means that all spherical gravitational fields, whether from a star or from a black hole, are indistinguishable at large distances. • A consequence of this is that purely radial changes in a spherical star do not affect its external gravitational field � no scalar modes of gravitational waves!
Page 1 and 2: Gravitational Field of non-Rotating
Page 3 and 4: “It is like a hole in the sky…
Page 5: BLACK HOLE Solutions Karl Schwarzsc
Page 9 and 10: Find Stellar Metric using the Symme
Page 11 and 12: Stellar Metric Schwarzschild Coordi
Page 13 and 14: Other Coordinate Choices are possib
Page 15 and 16: Symmetries: Killing Vectors Static
Page 17 and 18: Derivation of Field Equations • V
Page 19 and 20: Riemann Tensor in Static Frames Tid
Page 21 and 22: � Einstein´s Equations (3)
Page 23 and 24: Tolman-Oppenheimer-Volkoff Eqs 2 Pr
Page 25 and 26: Exterior Schwarzschild Solution For
Page 27 and 28: White Dwarfs and Neutron Stars Brow
Page 29 and 30: Mass ~ 1.35 M S Radius = 10 - 12 km
Page 31 and 32: Hadron Stars - Mass-Radius Based on
Page 33 and 34: Frequencies of radial oscillations
Page 35 and 36: Masses of Compact Stars Black Holes
Page 37 and 38: U µ Gravitational Redshift Photon
Page 39 and 40: Galactic Center Black Hole 3.4 Mio.
Page 41 and 42: Kruskal-Szekeres Solution
Page 43 and 44: The Singularity at r = 0 • The Sc
Page 45 and 46: LQ Black Hole is non-singular Non-s
Page 47 and 48: Canonical Momenta
Page 49 and 50: Effective Potential Equatorial plan
Page 51 and 52: Effective Potential ISCO Energy E o
Page 53 and 54: Example Orbits Bound orbits Unbound
Page 55 and 56: Innermost Stable Orbit (ISCO) = 0
Page 57 and 58:
Accretion Disks around Black Holes
Page 59:
Roots: Nature of Orbits Bound orbit
Page 63 and 64:
Ansatz: ~ Keplerian Orbits χ: rela
Page 65 and 66:
Conserved Quantities in p and e
Page 67 and 68:
Mercury Perihelion Shift
Page 69 and 70:
Mercury´s Perihelion Shift Is trea
Page 71 and 72:
Appendix What is a Penrose Diagram
Page 73 and 74:
Penrose Diagrams: Minkowski Map inf
Page 75 and 76:
Penrose Diagram Schwarzschild
Page 77 and 78:
Reissner-Nordström Metric • The
Page 79 and 80:
= 0 approach in infinite time � r
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Relativistic Stars and the Schwarzschild Solution

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