Undergrad Differences b/w Schwarzschild Radius & Event Horizon

[accdd]

I understood that the event horizon is a null surface and not a place in space, what is the relationship between it and the Schwarzschild radius? Also, what does the Schwarzschild radius physically represent for example for an object such as a star?

 

[Dale]

what does the Schwarzschild radius physically represent for example for an object such as a star?

The Schwarzschild radius is simply the length scale for the curvature. That length scale is directly proportional to the mass by $$ R_s=\frac{2G}{c^2}M$$
For our sun that is about 3 km, so even right at the surface of the sun is more than 200000 times that length scale. Meaning that spacetime curvature is fairly small even right at its surface. This is why it took so long to notice it and even longer to accurately measure it.

 

[accdd]

Thank you.
So very close to the center of the sun what's going on?
And at a distance of less than 3km from the center?
Is it something measurable?

 

[PeroK]

So very close to the center of the sun what's going on?

Nuclear fusion.

And at a distance of less than 3km from the center?

The Sun is not a Schwarzschild black hole!

 

[accdd]

So is the schwarzschild radius only important for black holes?
Besides indicating that spacetime is significantly curved near it?
Is it a place in space?

 

[PeroK]

So is the schwarzschild radius only important for black holes?
Besides indicating that spacetime is significantly curved near it?
Is it a place in space?

From what source are you learning GR? I ask because these seem fundamental questions that a textbook should cover.

 

[accdd]

Carroll, I'm studying the chapter on Schwarzschild metrics. Maybe I was wrong to ask before finishing it. Sorry.

 

[Dale]

So is the schwarzschild radius only important for black holes?
Besides indicating that spacetime is significantly curved near it?

That is pretty much it’s only use besides being a length scale for curvature. Remember, the Schwarzschild metric is a vacuum metric. So it only applies in the space outside a gravitating body. Only a black hole has a Schwarzschild radius that is in the vacuum region.

For other objects, the ratio of their Schwarzschild radius and their physical radius indicates the curvature at the surface. Or the ratio of the Schwarzschild radius to the orbital distance indicates the curvature at that orbital distance.

 

[PeroK]

Carroll, I'm studying the chapter on Schwarzschild metrics. Maybe I was wrong to ask before finishing it. Sorry.

The opening of chapter 5 states that the Schwarzschild solution applies to any spherically symmetric gravitational field in vacuum. E.g. the Sun or Earth, as well as to black holes.

The difference, of course, is that the black hole is entirely a vacuum solution; whereas, for the Earth and Sun the vacuum solution extends only to the object's surface.

 

[Ibix]

I understood that the event horizon is a null surface and not a place in space, what is the relationship between it and the Schwarzschild radius?

An event horizon is a boundary between regions of spacetime that can send signals to infinity and regions that cannot. In the specific case of a non-rotating uncharged black hole (a Schwarzschild black hole) this is a spherically symmetric null surface with area $4\pi R_s^2$.

Also, what does the Schwarzschild radius physically represent for example for an object such as a star?

It is the distance that corresponds to the mass of the star in geometrised units so you'd tend to suspect that interesting stuff would happen when the radius of the star was similar in scale to the Schwarzschild radius. Nothing special happens inside a normal star (one with radius $R_*\gg R_s$) at that radius, though, because the interior of the star is not a vacuum so is not described by Schwarzschild spacetime. It's much the same as expecting the gravitational force in Newtonian physics to go to infinity at $r=0$ - it doesn't because modelling the star as a fixed point mass is wildly incorrect inside the star.

 

[strangerep]

Carroll, I'm studying the chapter on Schwarzschild metrics. Maybe I was wrong to ask before finishing it.

Not "wrong" perhaps, but rather a suboptimum study technique. I've found it's better to read a chapter through once, fairly quickly, then read it again slower and more carefully, completing for yourself any steps between equations which are not explicitly shown in the book. Then do the exercises at the end of the chapter. (PF has a homework forum where you can get help if/when you get stuck on any of the exercises, provided you've made a conscientious attempt first.)

 

[PeterDonis]

Carroll, I'm studying the chapter on Schwarzschild metrics.

Have you worked your way through the book and are now to that point? Or did you just start right in at that chapter?

The latter is far less likely to be helpful to your learning.