It is known that radius of black hole is 2GM/c^2. And second radius is 3GM/c^2. (light ray at the second radius is not eaten but it circulate around black hole.)

But, let us assume that we have body which moves almost with speed of light.
1. Are these two radiuses the same or approximately the same?

Let as assume Light-ray which come from infinity and come close to one body from distance (27)^(1/2) GM/c^2. If this body is changed to black hole, black hole will affect this light-ray that he wil come 3GM/c^2 close to it. And after this it will stay close to black hole, as described above.

2. question, if this is very fast body (not light-ray) is calculation the same?
BR

pervect
Staff Emeritus
It is known that radius of black hole is 2GM/c^2.
Let's be a little more precise and use some technical jargon - the Schwarzschild radius is 2GM/c^2. We could alternatively say that the "event horizon" is at 2GM/c^2.

And second radius is 3GM/c^2. (light ray at the second radius is not eaten but it circulate around black hole.)

That's known as the photon sphere. So we say, concisely, that the photon sphere is at 3GM/c^2.

We should probably mention something. When we say that the event horizon is at r=2GM/c^2, and the photon sphere is at r=3GM/c^2, we are giving the location of the event horizon and the photon sphere in terms of Schwarzschild coordinates. These are not distances. We are saying "The Schwarzschild r coordinate is such-and-such" not "the distance from the black hole is such-and-such" when we make the above statements.

But, let us assume that we have body which moves almost with speed of light.
1. Are these two radiuses the same or approximately the same?

For an observer moving along with the black hole, a co-moving observer, nothing changes if the black hole changes speed, as all speed is relative.

You probably want to ask "what sort of coordinate system would an observer moving relative to a black hole use".

Unfortunately, there isn't any obvious answer to this. In special relativity, we can transform to a moving frame of reference via the Lorentz transform.

In general relativity, we can't do this in general. So there's nothing like a "Lorentz boost" that we can do to change the Schwarzschild coordinates globally into "moving coordinates".

While we can do a certain amount with performing a Lorentz boost on what is known as "frame fields", those do not apply globally, but only apply near the observer.

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I really wanted to know, if it is any difference between photon in gravitational field and elementary particle in gravitational field, which behaves almost similarly as photon this means its velocity is almost velocity of light.
I suppose that here is no difference.
This means: if we have particles with v =c/2, their "event horizon" and "photon sphere" (or "particle sphere") are 4 times larger. Is this true?

p.s.
Of course we have here electromagnetic waves and matter, not photons and elementary particles, which are from quantum mechanics, but the sentence is shorter.

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