exupero's blog

# Accelerating to oblivion?

Besides the view from a photon, another question about relativistic velocity that has preoccupied me regards the apparent increase in mass. The relationship between mass and velocity is usually expressed as

where m is rest mass, v is velocity, and c is the speed of light. Thus, as velocity approaches the speed of light, relativistic mass approaches infinity.

The intriguing part is that mass is a factor in an object's Schwarzchild radius, which characterizes the size of an object's event horizon:

If an object's mass is entirely within its Schwarzchild radius, the object becomes a black hole.

So can relativistic mass be substituted for m? Does an object's Schwarzchild radius grow as it accelerates? If so, how fast would it have to travel to become a black hole?

Solving for v,

Doing some order-of-magnitude analysis,

That suggests an object's rest mass in kilograms needs to be about 10 quintillion times its radius in meters to even start bringing the speed threshold down appreciably from the speed of light. If relativistic mass can create a black hole, an object with a 1-meter radius and as massive as the Earth would be within its event horizon only when it reached 99.98% the speed of light.

The caveat to all this, of course, is whether inertial mass and an object's resistance to acceleration is equivalent to gravitational mass that can warp spacetime. Wikipedia's article on mass in special relativity discusses whether thinking in terms of relativistic mass is a historical mistake and whether the pedagogical focus should be on relativistic energy instead. But that merely rephrases the question: does energy warp spacetime?

Even if relativistic travel can warp spacetime and cause objects to form black holes, the math suggests that for any object un-massive enough to even approach the speed of light (i.e., fundamental particles), the velocity at which a black hole would form is close enough to the speed of light to be indistinguishable from it.