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Alice and Bob are both floating in free, empty space. Alice does not like Bob very much.

Thus one day, believing that there is no such thing as overkill, Alice takes her programmable hyper-energetic railgun and shoots Bob with a 1 kg projectile travelling at very, very high velocity.

However, she misses, and the projectile passes Bob 1 meter to his side.

  1. What would Bob experience, as the velocity of the projectile Alice shoots at him (and misses) approaches the speed of light?

  2. Is there some speed at which, regardless of missing, Alice kills Bob anyway?

  3. What if this was happening in a more realistic setting, like the interstellar medium?

More details:

These questions seem relevant:

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

If two ultra-relativistic billiard balls just miss, will they still form a black hole?

The former suggests no effect, but the latter suggests that Bob and the projectile would at sufficiently high velocity, turn into a spinning black hole and then explode. So, hmm. Obviously that is an extreme case, though.

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  • Are you thinking that the relativistic mass increase might affect Bob? If so, the answer is no it won't. – John Rennie Sep 04 '15 at 15:10
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    maybe his question is that if at high enough relativistic boost, the effective cross-section of the projectile will grow, possibly via some drag effect related with dielectric polarization of the vacuum. With high enough effective cross-section, there might be an effect, even if geometrically would seem to miss – diffeomorphism Sep 04 '15 at 15:12
  • I read that other question, John Rennie. It seemed very specific to the issue of whether it produces a black hole. But I don't think the arguments there rules out the question of whether the relativistic mass would affect Bob. Shifting to the perspective where the projectile is at rest would show Bob travelling at high relative velocity. Effects on the relationship between the two are therefore not ruled out. – Fhnuzoag Sep 04 '15 at 15:22
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    Indeed, the accepted answer to this other question suggests very bad things will indeed happen to Bob if the projectile is travelling fast enough. So how do we square the circle between the two? – Fhnuzoag Sep 04 '15 at 15:28
  • At first glance, I thought point three said, a realistic setting, like the interstellar movie :) – snulty Sep 04 '15 at 15:47
  • I think the question should better be formulated like "What amplitude gravitational wave can inflict lethal tissue damage on a human being and how much mass is needed to generate such a gravitational wave?". – CuriousOne Sep 04 '15 at 15:52
  • @CuriousOne well, the exact numbers aren't that important, more the rough orders of magnitude, and the qualitative question of what it would feel like, or indeed whether there would be any damage at all. I'm not super persuaded that there would be no effect, unless someone can back it up. – Fhnuzoag Sep 04 '15 at 15:57
  • A short gravitational wavefront traveling trough matter will cause a mechanical stress and a displacement. If that stress becomes large enough to separate tissue there will be damage and potential death. The required mechanical stresses and displacement are well known from work with mechanical shock waves from explosives. What I am suggesting is that one can go back from there and calculate the required relativistic stress-energy-tensors to cause those effects. Having said that... I haven't done any such calculation in almost 40 years, so I will abstains from more than a general suggestion. – CuriousOne Sep 04 '15 at 16:13
  • @CuriousOne, I don't understand enough about how gravitational waves would be created in this situation. It would be good to post an answer regarding that. – Fhnuzoag Sep 04 '15 at 16:16
  • https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity#Gravitational_radiation might be a good start. It actually gives you a formula for the emitted power. Approximate the relative motion of the projectile and Bob by a rotation and you can get an idea of how much power might be generated... if it's not enough to do damage to Bob even if his body were to absorb all of it, then nothing bad will happen. – CuriousOne Sep 04 '15 at 16:36
  • Although there isn't much matter in the interstellar gas, you might find relativistic baseball (https://what-if.xkcd.com/1/) interesting. – Prof. Legolasov Sep 04 '15 at 19:28

1 Answers1

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However, she misses, and the projectile passes Bob 1 meter to his side.

The $1kg$ projectile is not a black hole and Bob is not a black hole. That means each of them is larger in extent that the black hole each would form if the parts became a black hole.

That means there is no way to miss and have the horizons merge. This isn't obvious or simple. If you had matter coming in radially from all directions there could be no black hole until they finally form the critically small surface area.

But the point is that the critical surface area is inside one and also the other, so when the two get as close as they can get, there is no way to enclose all the mass into a smaller surface area when they aren't even touching. For the in-falling matter it can become efficient to surround all the radially in-falling parts when it is about twice the critical surface area if it is made up of a bunch of thin shelled pieces.

What would Bob experience, as the velocity of the projectile Alice shoots at him (and misses) approaches the speed of light?

So there won't be a black hole formed unless one of the two was already close to forming a black hole. But there is still an effect.

Is there some speed at which, regardless of missing, Alice kills Bob anyway?

Yes. Bob alone might not radiate. The $1kg$ projectile by itself might not radiate. Either as a test particle in the curved spacetime caused by the other might be fine. But you have two masses and the spacetime between them is curved from both of them.

This will create gravitational waves. When strong enough these can disrupts the normal processes Bob uses to self regulate the interactions in Bob's body and Bob will die.

Of course at such a high speed Alice would be dead too from the same projectile. So it isn't about whether Alice doesn't care for Bob, it's about whether Alice cares to live.

What if this was happening in a more realistic setting, like the interstellar medium?

The interstellar medium will slow down the $1kg$ mass. But the exact effect depends on the shape and construction of the projectile. It could just scoop up all the material and drag it along for the ride (and slow down in the process). So increase the mass a bit and decrease the speed a bit. It could have the medium bounce off it so that the medium parts bounce off at a faster speed than the $1kg$ mass is going (and the $1kg$ mass slows down even more) or something in between.

The precursors could cause more havoc if the much smaller damage they do happens to start a process in Bob that makes Bob more vulnerable to the main attraction. Otherwise it just requires a bigger initial speed for the $1kg$ device before killing Bob.

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  • Addendum: such a projectile traveling through a medium would, beyond a certain relative speed, fuse with the medium, releasing energy. I have no sense of the density or energy scales involved, but I do know that gravitational waves + gamma rays is worse than gravitational waves alone. – Asher Dec 14 '15 at 16:35