On the formation of a black hole due to high kinetic energy

[Geometry_dude]

Do you mean that the kinetic energy is $$T^{00}$$?
If so, wouldn't every component of the energy-momentum tensor contribute to the formation of the black hole?

No, but here lies the origin of your confusion. See below.

As for your question if I can see how the energy is contant, just multiply both sides by $$ g_{ij}$$, right?

I don't know what you are aiming at here.Anyway, in case anyone stumbles on this thread later, I would like to point out what I believe to be the flaw in the argument. PAllen has already said it and Orodruin has hinted at it:

Kinetic energy in Newtonian mechanics is a relative concept, i.e. it depends on the chosen frame of reference. The `energy' that curves spacetime in general relativity - and here you have to be very careful what you mean, the enery-momentum tensor field (not it's components!) is but one possible choice - cannot be a relative, but must be an absolute concept in the sense that one has to be able to express it in terms of geometric invariants. This is dictated by the general principle of relativity, which is most conveniently expressed as "Fundamental laws of nature have to be formulated in a coordinate-independent manner" or more colloquially "Physics shouldn't depend on how you chose to describe it".

It is also worth pointing out that the concept of energy is an excellent example of where general relativity and Newtonian mechanics are incommensurable. This is a concept popularized by the philosopher Thomas Kuhn and generally means that concepts in two different scientific theories cannot be compared, because the conceptual framework is fundamentally different. So even though you may use the word energy in GR and Newtonian mechanics in a sloppy way, they mean fundamentally different things in the respective theories.

 

[Joker93]

No, but here lies the origin of your confusion. See below.
I don't know what you are aiming at here.Anyway, in case anyone stumbles on this thread later, I would like to point out what I believe to be the flaw in the argument. PAllen has already said it and Orodruin has hinted at it:

Kinetic energy in Newtonian mechanics is a relative concept, i.e. it depends on the chosen frame of reference. The `energy' that curves spacetime in general relativity - and here you have to be very careful what you mean, the enery-momentum tensor field (not it's components!) is but one possible choice - cannot be a relative, but must be an absolute concept in the sense that one has to be able to express it in terms of geometric invariants. This is dictated by the general principle of relativity, which is most conveniently expressed as "Fundamental laws of nature have to be formulated in a coordinate-independent manner" or more colloquially "Physics shouldn't depend on how you chose to describe it".

It is also worth pointing out that the concept of energy is an excellent example of where general relativity and Newtonian mechanics are incommensurable. This is a concept popularized by the philosopher Thomas Kuhn and generally means that concepts in two different scientific theories cannot be compared, because the conceptual framework is fundamentally different. So even though you may use the word energy in GR and Newtonian mechanics in a sloppy way, they mean fundamentally different things in the respective theories.

"I don't know what you are aiming at here"
Then how would one prove it?

 

[Geometry_dude]

Then how would one prove it?

There was a misunderstanding here. I did not intend to ask you whether you know why the expression is constant - obviously $c^2$ is constant -, but rather why I would say that (modulo the factor) I would call this the relativistic kinetic energy.

The reason is that it is the geometric invariant in GR that comes 'closest' to $\vec v ^2 /2$ in Newtonian mechanics.