Why black hole appears red shift while material emitted spectral lines.

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Chapter 15, Problem 9TY

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Why black hole appears red shift while material emitted spectral lines.

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A black hole is a blackbody if ever there was one, so it should emit blackbody radiation, called Hawking radiation. A black hole of mass M has a total energy of MC2, a surface area of 16πG2M2 / c4 and a temperature of hc3 /16π2kGM. Imagine a black hole in empty space, where it emits radiation but absorbs nothing. As it loses energy, its mass must decrease; one could say it "evaporates." Derive a differential equation for the mass as a function of time, and solve this equation to obtain an expression for the lifetime of a black hole in terms of its initial mass.

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A gamma ray burst delivers approximately 5.0 X10-6 joules/m2 to a detector on an orbiting space telescope. Assuming that the red shift indicates that the source is 5 X109 light years away (1 yr =3.16X107 seconds) and that the energy at the detector has been corrected for the redshift, what is the energy output of the source in units of 1047 joules and how does it compare to the rest energy of the Sun. The speed of light is 3.0 X 108 m/sec. A year is 3.16 X 107 sec. The mass of the Sun is 2 X 1030 kg. Rest mass energy is E=mc2. The surface area of a sphere is 4πr2.