Definiton of Schwarzschild radius clarification request

In summary, the Schwarzschild radius is a term used in the context of black holes to refer to the distance from the center of a dense mass where light cannot escape. It is a small distance compared to the cosmological horizon, which is the distance at which the expansion of space equals the speed of light. The Schwarzschild radius is also used to calculate a critical density for an object to be a black hole, but it is not applicable in cases where there is matter distributed throughout space.
  • #1
liometopum
127
24
I am looking for some input on the proper use of the definition of the Schwarzschild radius.

Case 1: In the world of black holes, the Schwarzschild radius is used in reference to the distance from the center of a mass dense enough to not allow light to exit. It is a short distance compared to Case 2's value.

Case 2: Every point in space has a Schwarzschild sphere of points around it at a distance at which the accumulated expansion of space equals the speed of light. No mass is necessary. These points are presumably 13.8 billion light years away from the point, or earth, or whatever.

So, is it common, or proper, to use the term Schwarzschild distance, or radius, for case 2?

(the math definition does not seem to help. In Case 1, a large mass in the 2GM/c^2 equation gives a large radius. The tiny (or almost zero mass) in Case 2 gives a tiny radius.
 
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  • #2
The Schwarzschild radius is in fact the location of the event horizon around a dense, electrically neutral mass. We can also use ##2GM/c^2## to compute a characteristic scale for any massive object. For objects that are not black holes, this scale is smaller than the spatial extent of the object. Conversely, we can use the Schwarzschild radius to compute a critical density for an object to be a black hole.

One could also ask about the Schwarzschild radius of elementary point particles, like the electron. In those cases, the Compton radius is larger than the Schwarzschild radius, so these are not black holes either. The Compton wavelength can be used to set a quantum limit that a black hole must have a mass greater than a characteristic mass that is proportional to the Planck mass.

The distance that you are referring to in case 2 is called the cosmological horizon. It is related to the cosmological event horizon, as explained at the wiki. This cosmological event horizon is analogous to the one around a black hole, since they are fundamentally defined in the same way. However, we don't use the term Schwarzschild in this context, since the cosmological solutions are described by different metrics from black holes.
 
  • #3
liometopum said:
Case 2: Every point in space has a Schwarzschild sphere of points around it at a distance at which the accumulated expansion of space equals the speed of light. No mass is necessary. These points are presumably 13.8 billion light years away from the point, or earth, or whatever.
This isn't really correct. The Schwarzschild metric describes a space-time where there is no matter anywhere but at the center of the black hole. As a result, terms used to describe this space-time (such as the Schwarzschild radius) are only valid when it is a reasonable approximation to reality. That is, it's only valid if nearly all of the matter, at least within a few Schwarzschild radii of the center, is concentrated within the black hole. Once this is no longer a reasonable approximation (as it is not for most of the universe), the term is no longer sensible.
 

1. What is the Schwarzschild radius?

The Schwarzschild radius is a characteristic radius of a non-rotating black hole, named after German physicist Karl Schwarzschild. It is the distance from the center of the black hole at which the gravitational pull becomes so strong that not even light can escape.

2. How is the Schwarzschild radius calculated?

The formula for calculating the Schwarzschild radius is Rs = 2GM/c^2, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.

3. What is the significance of the Schwarzschild radius?

The Schwarzschild radius is important because it marks the boundary of the event horizon, the point of no return for an object falling into a black hole. It also determines the size of a black hole's shadow, which can be observed in images captured by telescopes.

4. Can the Schwarzschild radius differ for different black holes?

Yes, the Schwarzschild radius depends on the mass of the black hole. The more massive the black hole, the larger its Schwarzschild radius will be. For example, a black hole with 10 times the mass of the sun would have a Schwarzschild radius of about 30 kilometers.

5. Is the Schwarzschild radius the same for all types of black holes?

No, the Schwarzschild radius is specific to non-rotating black holes. Rotating black holes have a different characteristic radius called the Kerr radius, which takes into account their angular momentum.

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