Iterative Tangents

A curiosity journal of math, physics, programming, astronomy, and more.

Physics

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

m Subscript r e l a t i v i s t i c Baseline equals StartFraction m Over StartRoot 1 minus v squared slash c squared EndRoot EndFraction

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:

r Subscript upper S c h w a r z c h i l d Baseline equals StartFraction 2 upper G m Over c squared EndFraction

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,

StartLayout 1st Row 1st Column r Subscript upper S 2nd Column equals StartFraction 2 upper G m Subscript r e l Baseline Over c squared EndFraction 2nd Row 1st Column Blank 2nd Column equals StartFraction 2 upper G m Over c squared StartRoot 1 minus v squared slash c squared EndRoot EndFraction 3rd Row 1st Column r Subscript upper S Baseline Superscript 2 2nd Column equals StartFraction 4 upper G squared m squared Over c Superscript 4 Baseline left-parenthesis 1 minus v squared slash c squared right-parenthesis EndFraction 4th Row 1st Column 1 minus StartFraction v squared Over c squared EndFraction 2nd Column equals StartFraction 4 upper G squared m squared Over c Superscript 4 Baseline r Subscript upper S Baseline Superscript 2 Baseline EndFraction 5th Row 1st Column StartFraction v squared Over c squared EndFraction 2nd Column equals 1 minus StartFraction 4 upper G squared m squared Over c Superscript 4 Baseline r Subscript upper S Baseline Superscript 2 Baseline EndFraction 6th Row 1st Column v squared 2nd Column equals c squared minus StartFraction 4 upper G squared m squared Over c squared r Subscript upper S Baseline Superscript 2 Baseline EndFraction 7th Row 1st Column v 2nd Column equals StartRoot c squared minus StartFraction 4 upper G squared m squared Over c squared r Subscript upper S Baseline Superscript 2 Baseline EndFraction EndRoot EndLayout

Doing some order-of-magnitude analysis,

StartLayout 1st Row 1st Column v 2nd Column equals StartRoot 10 Superscript 8 Baseline squared minus StartFraction 10 Superscript negative 11 Baseline squared Over 10 Superscript 8 Baseline squared EndFraction StartFraction m squared Over r Subscript upper S Baseline Superscript 2 Baseline EndFraction EndRoot 2nd Row 1st Column Blank 2nd Column equals StartRoot 10 Superscript 16 Baseline minus StartFraction 10 Superscript negative 22 Baseline Over 10 Superscript 16 Baseline EndFraction StartFraction m squared Over r Subscript upper S Baseline Superscript 2 Baseline EndFraction EndRoot 3rd Row 1st Column Blank 2nd Column equals StartRoot 10 Superscript 16 Baseline minus 10 Superscript negative 38 Baseline StartFraction m squared Over r Subscript upper S Baseline Superscript 2 Baseline EndFraction EndRoot 4th Row 1st Column Blank 2nd Column equals StartRoot 10 Superscript 16 Baseline minus left-parenthesis 10 Superscript negative 19 Baseline StartFraction m Over r Subscript upper S Baseline EndFraction right-parenthesis squared EndRoot EndLayout

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.