What is the relation between a black hole's mass and its kinetic energy?

[#Thomas#]

I've been scratching me' head a little. For curiosity's sake I've been trying to calculate what "would" be the kinetic energy of a black hole moving across the galaxy at a certain velocity, but whenever I tried assigning any value to the equation it spells disaster:

E_kin=(2D∏r³|v|²)/3

Based on this equation there are 2 possible outcomes:

1) the radius of the black hole is zero, the most widely accepted belief in that case the black hole will allways no kinetic energy whatsoever regardless of its velocity (E_kin=0)

2) Some believe that the radius of the black hole is infinetely small, but yet not zero, in that case the black holes have infinite kinetic energy (E_kin=∞)

In either case it got me into a bewilderment. You propably noticed that I took the classic equation apart because I was unsatisfied with the black holes mass being the same while its components went into the extreme.

Is it futile to even approach black hole phisics from this perspective or is there another way?

 

[Nabeshin]

You're trying to rewrite m in terms of an average density multiplied by a radius, which is OK, but then you need to be careful. If you take your radius to be the schwarzschild radius, the average density is known (in fact, you know the total mass much easier, so this whole procedure is a little silly to begin with). If you try and apply this to an arbitrarily small sphere (say, approaching the singularity), you'll have to say the density goes to infinity, while the radius goes to zero. The product Infinity*0 isn't well defined, so you cannot proceed in such a way.

In reality, the black hole has a well defined total mass, so simply use mv^2/2 (or the relativistic version, if necessary).

 

[#Thomas#]

I know its silly but I don't know the phisicists manage sumarily dismiss the radical changes that happen to the components that define the collapsed black hole's mass. They simply declare the mass of the black hole based on the stuff that's crashed down into a singularity, but how can you define the mass at all in the same way if in the factor DxV one variable is infinite and the other is absolute zero?

Even if you approach it from a relativistic point of view the values must be finite in order to account for different black hole sizes at all!

 

[Nabeshin]

I know its silly but I don't know the phisicists manage sumarily dismiss the radical changes that happen to the components that define the collapsed black hole's mass. They simply declare the mass of the black hole based on the stuff that's crashed down into a singularity, but how can you define the mass at all in the same way if in the factor DxV one variable is infinite and the other is absolute zero?

Even if you approach it from a relativistic point of view the values must be finite in order to account for different black hole sizes at all!

If you look at the solution to Einstein's equations which corresponds to a black hole (see: http://en.wikipedia.org/wiki/Schwarzschild_metric ), it is defined by a single parameter: M. This M parameter corresponds to exactly what we would call a mass. So there you have it, to a black hole we assign a mass, no need to futz around with densities or volumes -- such concepts do not have well defined meanings for a black hole, especially if you are talking about the singularity. I'm not sure how I can make it any clearer than that.