Deriving the Schwarzchild radius?

[21joanna12]

I'm a bit confused about the derivation of the Schwarzschild radius. I can do it quite easily using Newton's Law of gravitation, but this law is only an approximation, so I am wondering whether the result I obtain,
r_{s}=\frac{2GM}{c^{2}}, is an approximation or not. It seems to me that it should be, however Wikipedia insists that Schwarzschild derived this result from the Einstein field equations. Is this just a special case where Newton's Equations happen to give exactly the same answer as Einstein's field equations, or am I missing something?

Thanks in advance :)

 

[Nugatory]

You can read the value of the Schwarzschild radius directly out of the Schwarzschild metric, which is a solution of the Einstein field equations (google for "schwarzchild metric derivation").

When you say you can derive the Scwarzchild radius "quite easily using Newton's law of gravitation" do you mean that's when you get when you calculate the radius at which the escape velocity would be $c$? If so, you aren't calculating the Schwarzschild radius, you're calculating something else that by interesting coincidence comes out to that value.

 

[A.T.]

You can read the value of the Schwarzschild radius directly out of the Schwarzschild metric,

I have a question about this: The Schwarzschild metric describes space-time for a uniform density sphere in vacuum. And the Schwarzschild radius derived from it defines the radius (or Schwarzschild radial coordinate) below which an event horizon would form. What if the sphere is not of uniform density? Would you still get the same solution for the radial coordinate below which an event horizon would form?

When you say you can derive the Scwarzchild radius "quite easily using Newton's law of gravitation" do you mean that's when you get when you calculate the radius at which the escape velocity would be c? If so, you aren't calculating the Schwarzschild radius...

Wiki defines "Schwarzschild radius" exactly like this: http://en.wikipedia.org/wiki/Schwarzschild_radius

"The Schwarzschild radius (...) is the radius of a sphere such that, if all the mass of an object is compressed within that sphere, the escape speed from the surface of the sphere would equal the speed of light."

Is there a better definition somewhere, that applies generally, not just to the special case of a uniform density sphere in vacuum?

...you're calculating something else that by interesting coincidence comes out to that value.

I'm always wondering if the result of a mathematical derivation can be considered a "coincidence". If two mathematical models lead to the exactly the same symbolical result, isn't this a mathematical connection, rather than just "coincidence".

 

[21joanna12]

Thank you for your reply! Yes- I calculated the radius at which the escape velocity is c. I thought that this was the definition of the Schwarzschild radius (Wikipedia defines it as "The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is the radius of a sphere such that, if all the mass of an object is compressed within that sphere, the escape speed from the surface of the sphere would equal the speed of light"). If I am not calculating the Schwarzschild radius, what am I calculating?
Thanks again ^_^

 

[21joanna12]

Sorry, just realized A.T. posted similar questions just before me...

 

[Nugatory]

I have a question about this: The Schwarzschild metric describes space-time for a uniform density sphere in vacuum. And the Schwarzschild radius derived from it defines the radius (or Schwarzschild radial coordinate) below which an event horizon would form. What if the sphere is not of uniform density? Would you still get the same solution for the radial coordinate below which an event horizon would form?

As long as the density distribution is spherically symmetric and and all the mass is inside the Schwarzschild radius, the metric above the surface will be Schwarzschild - it's the only vacuum solution for a spherically symmetric and static mass distribution.

Wiki defines "Schwarzschild radius" exactly like this: http://en.wikipedia.org/wiki/Schwarzschild_radius

"The Schwarzschild radius (...) is the radius of a sphere such that, if all the mass of an object is compressed within that sphere, the escape speed from the surface of the sphere would equal the speed of light."

Is there a better definition somewhere, that applies generally, not just to the special case of a uniform density sphere in vacuum?

This is not, IMO, one of wikipedia's better moments :)
It's not at all clear what is meant by "escape speed" when nothing can escape. Thinking in those terms also hides the geometrical structure of the Schwarzschild spacetime and assigns too much weight to the coordinate singularity that appears when you write the Schwarzschild metric in Scwarzchild coordinates.

I'm always wondering if the result of a mathematical derivation can be considered a "coincidence". If two mathematical models lead to the exactly the same symbolical result, isn't this a mathematical connection, rather than just "coincidence".

That's fair, and if someone wanted to correct "coincidence" to "mathematical connection that doesn't contribute a lot of deep insight to our understanding" or even "mathematical connection that we kinda expect to come out that way because that's how we chose the boundary conditions" I wouldn't argue.

 

[A.T.]

It's not at all clear what is meant by "escape speed" when nothing can escape.

Isn't that just a matter a limit?

Thinking in those terms also hides the geometrical structure of the Schwarzschild spacetime and assigns too much weight to the coordinate singularity that appears when you write the Schwarzschild metric in Scwarzchild coordinates.

But how would you define it alternatively? Is the "escape speed = c" definition inconsistent with GR?

 

[21joanna12]

As long as the density distribution is spherically symmetric and and all the mass is inside the Schwarzschild radius, the metric above the surface will be Schwarzschild - it's the only vacuum solution for a spherically symmetric and static mass distribution.



This is not, IMO, one of wikipedia's better moments :)
It's not at all clear what is meant by "escape speed" when nothing can escape. Thinking in those terms also hides the geometrical structure of the Schwarzschild spacetime and assigns too much weight to the coordinate singularity that appears when you write the Schwarzschild metric in Scwarzchild coordinates.That's fair, and if someone wanted to correct "coincidence" to "mathematical connection that doesn't contribute a lot of deep insight to our understanding" or even "mathematical connection that we kinda expect to come out that way because that's how we chose the boundary conditions" I wouldn't argue.

Nugatory, you've been so helpful! I apologise for being slow, but just wanted to check that I understand the distinction between what I calculated and what the Schwarzschild radius really is. It seems I calculated the event horizon radius using the idea of an escape speed from the energy requires, but the the Schwarzschild radius should really be defined as radius where you have a critical degree of curvature of spacetime, and the extent of this curvature means that light cannot escape. Hence why the Schwarzschild radius was derived from the Schwarzschild metric, and Schwarzschild simply took the value of r that gives this critical curvature. Am I on the right track?

 

[21joanna12]

Isn't that just a matter a limit?

But how would you define it alternatively? Is the "escape speed = c" definition inconsistent with GR?

Hi A.T. If I understood correctly, I think I may have figured out how you would define the Schwarzschild radius alternatively in my post above... better wait for Nugatory though! I could be really wrong! :)

 

[Orodruin]

Isn't that just a matter a limit?

But how would you define it alternatively? Is the "escape speed = c" definition inconsistent with GR?

Escape velocity needs you to define what the velocity is relative to, so it will not be as easy. The typical definition of a horizon is the boundary for what is observable for a given observer, in this case for an observer at infinity. Also note that r does not necessarily has an interpretation as a particular distance from the center, it is merely a coordinate. For r inside the horizon, it is even a time-like coordinate (and t a space-like).

Also to everyone: It is Schwarzschild ("black shield" in German), not Schwarzschild.

 

[Nugatory]

Also to everyone: It is Schwarzschild ("black shield" in German), not Schwarzschild.

Good catch, thank you.