Definiton of Schwarzschild radius clarification request

[liometopum]

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.

 

[fzero]

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.

 

[Chalnoth]

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.