(Pure) GR don't know nuthin' 'bout charge, temperature, etc. I suspect that the answer to my question is yes -- the local curvature is greater at a distance of 1 AU if m(Sun')=2*m(Sun), so it should be possible to detect this even without calculating the period of revolution. What is true (according to Susskind), is that an Earth observer can't tell if the Sun is a black hole or not by simple local measurements of curvature. I.e., Newton's theorem holds even in GR: a spherical shell of matter still affects outside observers exactly the same as if all of its mass were located at the center of the sphere. At 09:23 AM 10/20/2013, Fred Lunnon wrote:

<< A non-rotating black hole is completely characterized by its mass & therefore its Schwarzschild radius. >>

What about its charge? And (quantum-theoretically) temperature? See http://en.wikipedia.org/wiki/No-hair_theorem

WFL

On 10/20/13, Henry Baker <hbaker1@pipeline.com> wrote:

...

I've been watching Leonard Susskind's lectures on GR (available on the Internet) & had a question.

A non-rotating black hole is completely characterized by its mass & therefore its Schwarzschild radius.

A bigger black hole has a bigger Schwarzschild radius, and space in the vicinity of the Schwarzschild radius of a very large black hole is relatively flat.

Q: Can a point observer outside the Schwarzschild radius of a black hole tell how big the black hole is by examining the curvature of space very near the observer?

I.e., suppose the Sun were a black hole, whose Schwarzschild radius is quite small, so the Earth is very far from this radius. Now consider a Sun' whose mass is, e.g., twice as big as the Sun. Its Schwarzschild radius is bigger than before, but if we are still at 1 AU, would we be able to tell _just from the local curvature_ how much mass is in the center of the solar system?

Of course, in order to stay in a "stable" orbit, the Earth would have to speed up to match the new mass in the center, but other than the period of the orbit, is there any way for Earth-bound scientists to measure the mass of the Sun' via curvature of space alone?