Relativistic Stars and the Schwarzschild Solution

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Metric in diagonal Form with orthonormal Coordinates • get rid of cross term dt‘ dr‘ with transformation most general static line element with all the required symmetries, r´ just shell label
Stellar Metric Schwarzschild Coordinates • choose a radial coordinate r (not a “radius”!) as R 2 (r‘) = r 2 � measures the area of a 2-sphere • this choice is called geometrical coordinates or Schwarzschild coordinates (t, r, θ, φ) • now the angle dependence is written as r 2 (sin 2 ө dφ 2 + dө 2 ) • for a curve with t =cnst, r= cnst, in the equatorial plane ө = π/2 and parameter φ = 0 … 2π ds 2 = r 2 dφ 2
Page 1 and 2: Gravitational Field of non-Rotating
Page 3 and 4: “It is like a hole in the sky…
Page 5 and 6: BLACK HOLE Solutions Karl Schwarzsc
Page 7 and 8: Birkhoff‘s Theorem • Birkhoff's
Page 9: Find Stellar Metric using the Symme
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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