Schwarzchild Radius: By definition must be dependent on distance.

[SarcasticSully]

The Schwarzschild Radius of an object is the length such that if the object is shrunk down that small, the escape velocity becomes equal to the speed of light. That being said, however, the escape velocity of any gravitational body matters where it is measured relative to the center of mass because the idea of the escape speed is that the kinetic energy plus the potential energy equals 0 and gravitational potential energy is dependent on position. So my question is what distance from the center of mass of the object does the escape speed have to be equal to the speed of light for the distance the object is contracted to to fit the criteria of the Schwarzschild Radius?

 

[DennisN]

Hi SarcasticSully, and welcome to PF!

Any mass has an associated specific Schwarzschild radius - this is not depending on any distance. So pick any mass you like, and calculate its Schwarzschild radius. Organize the mass into a sphere (for simplicity) and start shrinking the sphere. When the radius of the massive sphere reaches the Schwarzschild radius, the escape speed at the surface will be equal to the speed of light. When the radius of the massive sphere becomes less than the Schwarzschild radius, it will become a black hole.

See also The Schwarzschild Radius (HyperPhysics).

 

[Dale]

As DennisN mentioned the Schwarzschild radius is simply 2MG/c². It is a factor which shows up in the spherically symmetric vacuum solution of the Einstein field equations, aka, the Schwarzschild spacetime. That solution is an important solution so that factor is important and is given its own name. Because of the curvature of spacetime you cannot think of the Schwarzschild radius as simply representing some radial distance, it is just a factor in the equations.

 

[SarcasticSully]

Hi SarcasticSully, and welcome to PF!

When the radius of the massive sphere reaches the Schwarzschild radius, the escape speed at the surface will be equal to the speed of light.

See also The Schwarzschild Radius (HyperPhysics).

So if I understood this correctly, that means when a mass is shrunk to its Schwarzschild radius, the escape speed is the speed of light at the surface of the object.

 

[pervect]

There are certain technical issues with the seemingly simple task of measuring the radius of a very massive body such as a black hole, due to the way it warps space and time. In particular, there's no guarantee that the r coordinate measures a spacelike interval (and using Schwarzschild coordinates, in particular, it turns out that in the interior region dr does NOT measure a spacelike interval, but rather a timelike one).

Even for massive bodies that are not black holes, the radial distance is NOT equal to the change in the r coordinate value, one must use the metric to calculate the former.

Fortunately, there is an easy solution. We don't measure the radius, we measure the circumference of a circle of constant radius - or equivalently, the surface area of a 2-sphere of constant radius. This is in fact the way r is defined in the Schwarzschild geometry.

Kip Thorne's famous "hoop conjecture" https://en.wikipedia.org/wiki/Hoop_Conjecture states that an object becomes a black hole when you cannot place a hoop of circumference 2 pi G M / c^2 around it.

This is currently a conjecture, it is known to work for spherically symmetric and (I think) axissymmetric objects, but it hasn't been shown to work for objects lacking these symmetries.