The Schwarzschild radius is simply the length scale for the curvature. That length scale is directly proportional to the mass by $$ R_s=\frac{2G}{c^2}M$$accdd said:what does the Schwarzschild radius physically represent for example for an object such as a star?
Nuclear fusion.accdd said:So very close to the center of the sun what's going on?
The Sun is not a Schwarzschild black hole!accdd said:And at a distance of less than 3km from the center?
From what source are you learning GR? I ask because these seem fundamental questions that a textbook should cover.accdd said: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?
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.accdd said:So is the schwarzschild radius only important for black holes?
Besides indicating that spacetime is significantly curved near it?
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.accdd said:Carroll, I'm studying the chapter on Schwarzschild metrics. Maybe I was wrong to ask before finishing it. Sorry.
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##.accdd said: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?
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.accdd said:Also, what does the Schwarzschild radius physically represent for example for an object such as a star?
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.)accdd said:Carroll, I'm studying the chapter on Schwarzschild metrics. Maybe I was wrong to ask before finishing it.
Have you worked your way through the book and are now to that point? Or did you just start right in at that chapter?accdd said:Carroll, I'm studying the chapter on Schwarzschild metrics.
The Schwarzschild Radius is a characteristic distance that defines the size of a non-rotating, uncharged black hole. It is the distance from the center of the black hole at which the escape velocity exceeds the speed of light. Anything within this radius is considered to be within the event horizon and cannot escape the gravitational pull of the black hole.
The Event Horizon is the boundary around a black hole beyond which nothing, not even light, can escape. It is the point of no return for anything that enters the black hole. The size of the event horizon is directly related to the mass of the black hole, with larger black holes having larger event horizons.
The Schwarzschild Radius and Event Horizon are both measures of the size of a black hole. The Schwarzschild Radius defines the size of the event horizon, while the Event Horizon is the physical boundary itself. In other words, the Schwarzschild Radius is the distance from the center of the black hole at which the Event Horizon is located.
Yes, the Schwarzschild Radius and Event Horizon can be different if the black hole is rotating or has an electric charge. In these cases, the size and shape of the event horizon may be altered, resulting in a different Schwarzschild Radius. However, for a non-rotating, uncharged black hole, the Schwarzschild Radius and Event Horizon will be the same.
Understanding the differences between the Schwarzschild Radius and Event Horizon is important because it helps us to better understand the nature of black holes and their effects on the surrounding space. It also has implications for our understanding of gravity and the laws of physics in extreme environments. Additionally, knowing the size of the event horizon is crucial for determining the properties and behavior of black holes, such as their mass and spin.