Astronomer107 and the surface gravity of a black hole

[marcus]

Astronomer107 asked:

"How can I figure out the g of a 10 solar mass black hole? If the sun is a certain "g" then can I multiply it by 10 to get the g of a 10 solar mass black hole?"

Many people offered comment and suggestion but no one addressed this question.

The radius of the event horizon of an ordinary (nonrotating uncharged) black hole is given by Schwarzschild's formula

R = 2GM/c²

The acceleration of gravity at that radius is given by the classical formula GM/R².

This simplifies to

c⁴/4GM.

If you plug in values for c, G, and M, this will allow you to find out.

 

[marcus]

Astronomer 107, you were also asking about the surface gravity of the sun. It is NOT true that you can multiply the surface gravity of the sun by 10 and get the surface gravity of a 10-solar-mass black hole.

The way to find the survace gravity of the black hole is the formula in the preceding post:

c⁴/4GM.

The way to find the surface gravity of the sun is to evaluate
the classical formula

GM/R²

where M is the mass of the sun and R is the radius.
The radius is 7E8 meters.
You must square this which gives 49E16 square meters.
The figure of 7E8 meters which you say you got from some
book is just fine! It is sufficiently accurate. Go ahead and calculate. Before, you simply forgot to square the radius.

You are to be congratulated for wanting to know the surface
gravity g of a black hole. It is a famous result of Stephen Hawking that the temperature of a black hole is proportional to its surface gravity. In fact in the units people like Hawking use the temp is simply (and amazingly enough)
g/2pi

So you are right on the threshold of finding out for yourself the temperature of the socalled "Hawking radiation" of a 10 solar mass black hole (however do not get excited the large mass holes have low temp and it is the very little ones that are really hot so the temp of a 10 solar mass one is maybe not all that interesting)

Originally posted by marcus
Astronomer107 asked:

"How can I figure out the g of a 10 solar mass black hole? If the sun is a certain "g" then can I multiply it by 10 to get the g of a 10 solar mass black hole?"

Many people offered comment and suggestion but no one addressed this question.

The radius of the event horizon of an ordinary (nonrotating uncharged) black hole is given by Schwarzschild's formula

R = 2GM/c²

The acceleration of gravity at that radius is given by the classical formula GM/R².

This simplifies to

c⁴/4GM.

If you plug in values for c, G, and M, this will allow you to find out.