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

Imagine, a rod of length L is moving with velocity approaching the speed of light with respect to a human observer on Earth. Due to Lorentz contraction, the rod will observed to be very short. And since all laws of physic hold true in every frame of references, the gravitation law which state the force acting is inversely proportional to square of distance between them, will be acting between various part of the rod. Now, as velocity approach c , L will approach 0. This should cause an enormous gravitational force enough to form a black hole. Isn't this suggesting a black hole can be formed when the object velocity comes near the speed of light?

According to the rod's frame of reference, it will see itself as stationary and a man who observe it is moving, so according to the rod, the human must be a black hole. Isn't this a paradox?

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marked as duplicate by Mark Eichenlaub, Manishearth, Ron Maimon, Kostya, Qmechanic May 16 '12 at 14:11

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Not really a duplicate. This question doesn't mention the "mass increase" but instead only mentions the length contraction. –  John Rennie May 16 '12 at 6:01
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@JohnRennie: But the answers to the first completely answer the second, this is a dup, you don't want to make people repeat text verbatim. –  Ron Maimon May 16 '12 at 8:03
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I don't think the answers to the previous question actually answer the question. They all just basically say "it would contradict the principle of relativity" and that's no answer. I was tempted to post saying "it's because the Riemann tensor is co-ordinate invarient" but that's a glib answer as well. What I would like to see are some calculations showing me why a black hole can't form (obviously it can't!). I've so far given it one five minute tea break's thought but without any significant progress - maybe my lunch hour will be enough :-) –  John Rennie May 16 '12 at 8:44
    
Fundamentally the problem is that you are trying to apply simultaneously Newtonian gravity, special relativity, and general relativity. The paradox basically says that your assumption that all three can be applied to the same system at the same time is daft. –  Willie Wong May 16 '12 at 11:39
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BTW--@John, if you can't raise enough interest to re-open this question, you can flag it to be considered for merging with the possible duplicate. Or you can just leave your answer here: the question will still show up in search, people can still vote on it, and the OP can still accept your answer if (s)he is happy with it. My take is that this is the better version of the question but that they are duplicates. –  dmckee May 16 '12 at 14:38

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up vote 6 down vote accepted

OK, having devoted my lunch hour to this (the sacrifices I make for Physics!) I have an answer for you. I'm not sure this is the best possible answer, so if anyone can improve on it please jump in.

Firstly you're absolutely correct to say that the density of your object increases as it Lorentz contracts. This isn't an illusion: the RHIC observes this every day. Note how the illustrations on the RHIC page I've linked to show the colliding nuclei flattened into disks. However the contracted object can't form a black hole because this violates one of the principles of relativity i.e. the presence or otherwise of the black hole could be used to tell who was moving and who was standing still. So what's going on?

The paradox arises from your assumption that it's the mass/density of the object that determines whether it will be a black hole or not, because this isn't true, or rather it's only true in special cases. The Einstein equation that gives us the curvature, and therefore whether a black hole will form, is:

$$G_{\alpha\beta} = 8\pi T_{\alpha\beta}$$

$G_{\alpha\beta}$ is the Einstein tensor that describes the curvature, while $T_{\alpha\beta}$ is the stress-energy tensor. So it's not the mass/density of the object that determines the curvature, it's the stress-energy tensor.

There's a shortcut here, because the stress-energy tensor is an invarient i.e. it is the same in all co-ordinate systems. That means the stress-energy tensor we observe is the same as the stress-energy tensor observed in the rest frame of your test object. So if the test object doesn't form a black hole in it's rest frame it will not form a black hole in any other co-ordinate system, even the one you describe in which the object is moving at almost the speed of light.

However it's at this point that I run out of steam a bit, which is why I think there's scope for this answer to be improved. It would be nice to give an intuitive feel for what the stress-energy tensor is, and why it doesn't change when we see the object moving at almost the speed of light. We normally write the stress-energy tensor as a 4 x 4 matrix, and with a few approximations about your test object the tensor only has one non-zero value, $T_{00}$, which is indeed the density. If we write the stress-energy tensor in our frame, where the object is moving, our value for $T_{00}$ will increase as the density increases, and if nothing else changed this would eventually form a black hole. However in our frame the other entries in the matrix are no longer zero. The changes in the other entries balance out the change in the density, so when we plug our stress-energy tensor into the Einstein equation we get the same curvature as in the test object's rest frame. No black hole!

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The intuitive stuff: the time-space components are the momentum, and in the limit you are discussing, they are equal in magnitude to the time-time component. Two parallel pencils of light (or two masses highly boosted in the same direction) don't attract--- they repel by gravito-magnetic force to balance the attraction. This can be understood from SR (since if you boost two attracting objects perpendicular to the line of attraction, the time dilation makes the time to collision go to infinity). The same is true in EM, where two stationary charges boosted to near c don't repel (E and B cancel). –  Ron Maimon May 17 '12 at 1:49
    
Thank you for sacrifices your lunch to give me clear picture on the subject :). very good answer... thank you very much :) –  someone_ smiley May 17 '12 at 3:30
    
Actually i was carried away with idea that mass-energy tell how space-tijme to curve. so i thought its only mass-energy that responsible for gravity which is obviously increase with speed of object. will explore more on stress-energy tensor. thanks again :) –  someone_ smiley May 17 '12 at 3:38
    
Thanks Ron. My (admittedly slightly childish) outburst was because I'd love to see you, and others, expand on your comment as an answer to this question. Maybe there'll be a future opportunity. –  John Rennie May 17 '12 at 5:44

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