parenrightbigg 1 2 GM rc 2 1 r 2 1 r 2 sin 2 \u03b8 Since the coefficients are

# Parenrightbigg 1 2 gm rc 2 1 r 2 1 r 2 sin 2 θ since

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parenrightbigg 0 0 0 0 1 2 GM rc 2 0 0 0 0 1 r 2 0 0 0 0 1 r 2 sin 2 θ Since the coefficients are independent of coordinate time t , stationary ob- servers will measure physical properties of spacetime to be independent of time (note: a “static” or “stationary” observer” in a Schwarzschild gravita- tional field is one who is at a constant ( r,θ,φ )) . At this point, one may also notice that the metric diverges at r = R s , where R s = 2 GM c 2 The quantity R s is called the “Schwarzschild radius” or “gravitational radius” or “Event Horizon” of an object of mass M . We can calculate its value for different types of objects found in the universe. A brief list is given below: M/M R s R star 1 2 . 86 km 7 × 10 5 km Sun 1 2 . 86 km 7 × 10 3 km WD 1 2 . 86 km 10 km NS 2 5 . 72 km Typical stellar mass black hole 10 8 2 . 86 × 10 8 km Black hole in nuclei of some galaxies Ordinary nuclear burning stars (e.g. the Sun), white dwarfs, and neutron stars, have radii that are larger than the corresponding Schwarzschild radius. Note, however, that the radius of a one solar mass neutron star is only slightly above its Schwarzschild radius. That is, this star is already so compressed that it will only take a small reduction in radius to make it collapse into a black hole. Birkhoff’s Theorem (1923) Birkhoff’s theorem states that “any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat”. It follows that the exterior solution must be given by the Schwarzschild metric, which, at large distances, will reduce to the Newtonian limit.
130 SPACETIME NEAR A BLACK HOLE: SCHWARZSCHILD METRIC A corollary states that “the metric inside a spherical cavity inside a spher- ical mass distribution” is the Minkowski metric. Spatial sections Spatial sections: t section Take a t = constant slice, so that dt = 0. The proper distance dl will satisfy dl 2 = ds 2 = dr 2 1 2 GM c 2 r + r 2 d 2 If θ and φ are constant, then dl 2 = ds 2 = dr 2 1 2 GM c 2 r The proper distance between spheres of radial coortdinates r 1 and r 2 is D = integraldisplay r 2 r 1 dl = integraldisplay r 2 r 1 dr radicalbigg 1 2 GM c 2 r negationslash = r 2 r 1 thus r is not the radial distance! D r r 2 1 r - r 2 1 / D = Approximately we can write D = integraldisplay r 2 r 1 bracketleftBigg 1 + 1 2 2 GM c 2 r + o parenleftbigg 2 GM c 2 r parenrightbigg 2 bracketrightBigg dr ( r 2 r 1 ) + GM c 2 ln parenleftbigg r 2 r 1 parenrightbigg
THE TIME COORDINATE T 131 At large distances (e.g. r 1 = 100 R s ) the radial coordinate distance d = r 2 r 1 takes approximately the same value as the proper distance ( D ) (see figure below). However, near the Schwarzschild radius, the proper distance D and the coordinate distance d = r 2 r 1 take quite different values, as shown in the next figure.