If a 1kg mass was accelerated close to the speed of light would it turn into a black hole?

I'm a big fan of the podcast Astronomy Cast and a while back I was listening to a Q&A episode they did. A listener sent in a question that I found fascinating and have been wondering about ever since.

From the show transcript:

Arunus Gidgowdusk from Lithuania asks: “If you took a one kilogram mass and accelerated it close to the speed of light would it form into a black hole? Would it stay a black hole if you then decreased the speed?”

Dr. Gay, an astrophysicist and one of the hosts, explained that she'd asked a number of her colleagues and that none of them could provide a satisfactory answer. I asked her more recently on Facebook if anyone had come forward with one and she said they had not. So I thought maybe this would be a good place to ask.

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Yet another reason why the concept of relativistic mass is not a good one. ;-) physics.stackexchange.com/q/2516/724#2518 – Bruce Connor Jan 20 '11 at 20:25

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Yeah, though for the purposes of this question, it's mostly just a vocabulary issue: for matter, gravity does couple to relativistic mass density (i.e., energy density), so the question still makes sense in terms of rel. mass. But gravity also couples to momentum (and pressure, etc.), which tends to cancel the effect of higher energy density. – Stan Liou Jan 20 '11 at 20:53

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– MBN Jan 20 '11 at 22:26

4 Answers

The answer is no.

The simplest proof is just the principle of relativity: the laws of physics are the same in all reference frames. So you can look at that 1-kg mass in a reference frame that's moving along with it. In that frame, it's just the same 1-kg mass it always was; it's not a black hole.

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An addendum: It's worth pausing to ask why one might have thought it would form a black hole, and why those reasons are incorrect. Presumably the thought is that a combination of Lorentz contraction and relativistic "mass increase" squeeze the object below its Schwarzschild radius. So what's wrong with that reasoning? The main thing is just that the derivation of the Schwarzschild radius only applies under certain conditions. At the very least, it only applies in the object's rest frame (since it assumes spherical symmetry -- i.e., no preferred direction). – Ted Bunn Jan 20 '11 at 20:27
but if 1kg mass is rotated? – voix Feb 19 '11 at 13:24

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The gravitational field of a rotating 1-kg mass is different from that of a non-rotating mass. I don't remember the details, which are complicated, but the gravitational pull probably does get stronger because the rotational kinetic energy gravitates. If you start with a mass that's larger than its Schwarzschild radius, I don't know whether you can make it turn into a black hole by supplying rotational kinetic energy. – Ted Bunn Feb 19 '11 at 14:48
@TedBunn "If you start with a mass that's larger than its Schwarzschild radius, I don't know whether you can make it turn into a black hole by supplying rotational kinetic energy." But even kinetic energy is relative, if I accelerate to the same velocity as your object, your object don't have any kinetic energy relative to me. – Calmarius Apr 22 at 15:59

No, a 1kg mass would not turn into a black hole, even if it were zipping past you at very close to the speed of light.

The principle of relativity is a fundamental idea in physics, and one consequence of it is that we can understand the physics of something that's moving by imagining we're moving alongside it.

For example, you are watching people play pool on a train as it rushes past you. You want to know whether a certain shot that's just been made will sink the 8-ball. You figure it out by imagining you're inside the train and calculating everything you'd expect to happen from that simpler viewpoint where the pool table is stationary. If the 8-ball goes into a certain pocket from that point of view, you can rest assured it will go into the same pocket if you analyze the situation again from your original vantage point on terra firma.

Applying the same principle to the 1kg mass, we see that moving along side it, it just looks like a normal mass, not a black hole. Hence, from another point of view in which it moves close to the speed of light, it still looks like a normal mass, not a black hole.

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So does it then follow that the relative speed of a mass has no bearing on the gravitational force felt by a nearby stationary mass? That is if a mass flew by me at .1 c, would I feel the same tug as if it flew by at .999c? Would there be some sort of equivalence given the time it takes the object to pass? That is, would the total force felt over time be the same; sort of like how the area covered by an orbit is the same over a given time? – shopsinc Jan 20 '11 at 20:39
@shops Your question can't be answered using simply the principle of relativity because it's asking about different types of relative motion.  You might try asking it as a separate question on the main site.  I don't have a good, concise answer to that question. – Mark Eichenlaub Jan 20 '11 at 20:50

While good, I think the other answers are currently missing one ingredient, so I'll post this answer.

For particles traveling at constant velocity there is no event horizon, and so they act nothing like a black hole. Light from other regions of space will eventually reach it, unlike a black hole. Further, the forces between atoms in what ever matter constitutes the mass are co-moving and so there is no increased gravitational interaction between them. While the distances between them appear to change to an outside observer (as the mass is accelerated) once it reaches constant velocity they are fixed.

What has not been mentioned in other answers is the effect of acceleration. When a particle is continuously accelerated there is an apparent event horizon. See the relevant Wikipedia page here. So this has some features that we associate with a black hole, however there are still major differences. An object undergoing constant acceleration does indeed behave like it is static in a constant gravitational field. However, in the case of such an object the direction of the equivalent field is constant (and in a constant direction) throughout the object. This is not true for the gravitational field of a black hole, which is spherically symmetric.

Of course once the particle stops accelerating the apparent horizon disappears.

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True, but the apparent horizon in this situation is very different from a black hole horizon. In the case of the accelerating particle, the stuff that's "behind" the apparent horizon is far away from the particle -- just the opposite of the black hole's event horizon. That is, heuristically, a black hole's event horizon says that once you're sufficiently close to the black hole you can't get far away, whereas the accelerated particle's horizon says that once you're sufficiently far from the particle you can't get close. – Ted Bunn Jan 21 '11 at 15:03

I am presuming the idea is the 1kg mass will length contract to below the Planck length. It is either that or the relativistic energy (mass) $E~=~\gamma mc^2$ would be so large it would gravitationally implode. The question though can be thought of according to what would happen to an observer on the mass. The question could be turned around: Would the universe implode? If a mass $M$ passes by a smaller mass $m~<<~M$ then one might think that $M$ could become a black hole and the small mass $m$ if close enough would become trapped in the black hole. However, from the frame of the big mass $M$ the small mass is not a black hole. This is a contradiction.

A ultra-relativistic mass will behave similar to a gravity wave as it passes another reference point. This Aichelburg-Sexl ultraboost has a plane wave pulse of spacetime. The relativistic mass will result in a gravity wave pulse as detected by a stationary observer. So there is a gravitational implication to such extreme relativistic boosts.

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