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parenrightbigg00001−2GMrc200001r200001r2sin2θSince the coefficients are independent of coordinate timet, stationary ob-servers will measure physical properties of spacetime to be independent oftime (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 atr=Rs,whereRs=2GMc2The quantityRsis called the “Schwarzschild radius” or “gravitational radius”or “Event Horizon” of an object of massM. We can calculate its value fordifferent types of objects found in the universe. A brief list is given below:M/M⊙RsRstar12.86 km7×105kmSun12.86 km7×103kmWD12.86 km10 kmNS25.72 kmTypical stellar mass black hole1082.86×108kmBlack hole in nuclei of some galaxiesOrdinary nuclear burning stars (e.g. the Sun), white dwarfs, and neutronstars, have radii that are larger than the corresponding Schwarzschild radius.Note, however, that the radius of a one solar mass neutron star is only slightlyabove its Schwarzschild radius. That is, this star is already so compressedthat it will only take a small reduction in radius to make it collapse into ablack hole.Birkhoff’s Theorem (1923)Birkhoff’s theorem states that “any spherically symmetric solution of thevacuum field equations must be static and asymptotically flat”.It followsthat the exterior solution must be given by the Schwarzschild metric, which,at large distances, will reduce to the Newtonian limit.
130SPACETIME NEAR A BLACK HOLE: SCHWARZSCHILD METRICA corollary states that “the metric inside a spherical cavity inside a spher-ical mass distribution” is the Minkowski metric.Spatial sectionsSpatial sections:tsectionTake at=constantslice, so thatdt= 0. The proper distancedlwill satisfydl2=ds2=dr21−2GMc2r+r2dΩ2Ifθandφare constant, thendl2=ds2=dr21−2GMc2rTheproper distancebetween spheres of radial coortdinatesr1andr2isD=integraldisplayr2r1dl=integraldisplayr2r1drradicalbigg1−2GMc2rnegationslash=r2−r1thusrisnotthe radial distance!Drr21r - r21/D =Approximately we can writeD=integraldisplayr2r1bracketleftBigg1 +122GMc2r+oparenleftbigg2GMc2rparenrightbigg2bracketrightBiggdr≈(r2−r1) +GMc2lnparenleftbiggr2r1parenrightbigg
THE TIME COORDINATET131At large distances (e.g.r1= 100Rs) the radial coordinate distanced=r2−r1takes approximately the same value as the proper distance (D) (see figurebelow). However, near the Schwarzschild radius, the proper distanceDandthe coordinate distanced=r2−r1take quite different values, as shown inthe next figure.