The derivaton I have seen (for a Schwarschild black hole) is this(adsbygoogle = window.adsbygoogle || []).push({});

One imagines a mass m attached to a long (well infinitely long) inextensible rope (of negligible mass!). One raises the mass a distance dl.

A local observer (hovering above the black hole at a coordinate r) will say that the work required is [itex] m g dl [/itex] , where this g is the local acceleration given by (setting G=1)

[tex] g= \frac{M }{r^2} (1 - \frac{2M}{r})^{-1/2} [/tex] where M is the mass of the black hole.

On the other hand, they say that the work as measured by a distant observer is

[tex] m g_{\infty} dl [/tex] which is taken to define the surface gravity [itex] g_{\infty}[/itex].

Then one says that the two expressions are related by a redshift factor and this directly leads to the answer for [itex] g_{\infty}[/itex], 1/4M.

What bothers me is that we usually say that the rest mass energy of an object with mass m located at a coordinate r is measured from the observer at infinity to be [itex] m (1-2M/r^{1/2} [/itex] On the other hand, the derivation for the surface gravity made above uses the rest mass m to write down the work done by the remote observer.

So why is that the correct thing to do?

Thanks in advance

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# Question about derivation of surface gravity of a black hole

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