original by Philip Gibbs 5-July-1996

## If you go too fast do you become a black hole?

According to relativity the following are true facts:

1. As an object approaches the speed of light it's kinetic energy increases without limit.
2. Energy is related to mass by the formula E = m c2.
3. As an object approaches the speed of light it's length contracts towards zero.
4. If enough mass is squeezed into a sufficiently small space it will form a black hole

Put these facts together and it looks like we should be able to conclude that an object which moves at a speed sufficiently close to the speed of light should collapse to form a black hole. We could even argue that if you move fast enough relative to a star then that star must appear as a black hole to you because of its increased energy as observed by you. This would be paradoxical since we would expect things to appear very differently to an observer who is stationary relative to the star. So what has gone wrong?

In fact objects do not have any increased tendency to form black holes due to their extra energy of motion. In a frame of reference stationary with respect to the object, it has only rest mass energy and will not form a black hole unless it's rest mass is sufficient. If it is not a black hole in one reference frame then it is not a black hole in any reference frame.

In part the misunderstanding arises because of the use of the concept of relativistic mass in the equation E=mc2. Relativistic mass, which increases with the velocity and kinetic energy of an object, can not be blindly substituted into formulae such as the one which gives the radius for a black hole in terms of its mass. One way to avoid this is to not speak about relativistic mass and think only in terms of invariant rest mass (see Relativity FAQ Does mass change with velocity?).

The statement that "If enough mass is squeezed into a sufficiently small space it will form a black hole" is rather vague. Crudely speaking we would say that if an amount of mass, M is contained within a sphere of radius 2GM/c2 (The Schwarzschild radius) then it must be a black hole. But this is based on a particular static solution to the Einstein field equations of general relativity and ignores momentum and angular momentum as well as the dynamics of space-time itself. In general relativity, gravity does not simply couple to mass as it does in the Newtonian theory of gravity. It also couples to momentum and momentum flow, the gravitational field is even coupled to itself. It is actually quite difficult to define the correct conditions for a black hole to form. Hawking and Penrose proved a number of useful singularities theorems about the formation of black holes, and from astrophysics we know that the theorems should apply to sufficiently massive stars when they reach the end of their life and collapse into a small volume.