Why do parallel light beams not attract gravitationally?

[Danyon]

Say that two electrons travel in parallel at Ultra relativistic speeds, such that the observed energy-mass of said electrons (from a resting observer) is sufficient to generate a gravitational field great enough to overcome the repulsive coulomb force between the two electrons. This implies the two electrons are attracted together in the rest frame of the Stationary observer, however, in the rest frame of the two electrons, the electrons are stationary and the repulsive force is sufficient to repel them apart

 

[jfizzix]

So do beams of ultra relativistic electrons spread apart due to repulsion, or converge due to gravitational attraction? Different reference frames seem to imply different results.
It's a good question to be sure.

To extend this a bit further:
You have a rocket that is propelled so fast that it's relativistic mass is large enough that the size of the object is within one Schwarzschild radius, making it a projectile black hole.
In the moving reference frame of the rocket, it would not seem like you're in a black hole.
Is there one, or isn't there?

I'd like to hear an expert on General relativity on this.
It might not be correct to think of relativistic mass as generating gravity, in which case, there's no problem.

 

[Danyon]

So do beams of ultra relativistic electrons spread apart due to repulsion, or converge due to gravitational attraction? Different reference frames seem to imply different results.
It's a good question to be sure.

To extend this a bit further:
You have a rocket that is propelled so fast that it's relativistic mass is large enough that the size of the object is within one Schwarzschild radius, making it a projectile black hole.
In the moving reference frame of the rocket, it would not seem like you're in a black hole.
Is there one, or isn't there?

I'd like to hear an expert on General relativity on this.
It might not be correct to think of relativistic mass as generating gravity, in which case, there's no problem.

What if you had two projectile black holes that are within each-others event horizon and are also charged, such that they converge in one reference frame and exit each others event horizons in another due to electrostatic repulsion

 

[Ibix]

Baez: http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html.

Short answer: changing frames can't change observables like that. The source of gravity in GR is the stress-energy tensor. Both mass and kinetic energy contribute to different components of the tensor, but KE does not contribute to "increasing the gravity" in that way. The beams will never be attracted; a fast moving spaceship will never become a black hole.

 

[bahamagreen]

Isn't it all about c? It seems to me that it does not matter what changes are observed or inferred about the electrons, the mutual influence of those changes on each electron would be calculated to be coming from the past and from a distance behind and with diminishing lateral component.

My thinking is that to the inertial observer at rest wrt the two ultra relativistic electrons, c would appear not to be fast enough to support a lateral mutual communication perpendicular to the direction of motion... similar to the light clock, the apparent communication path is diagonalized from lateral such that the mutual influence is observed to progressively originate from further behind for greater scenario speeds, the lateral component of that influence diminishing quickly for greater speeds, and approaching the limit, the influence approaches becoming all longitudinal component and the source of mutual influence receding infinitely.

 

[Nugatory]

Isn't it all about c? It seems to me that it does not matter what changes are observed or inferred about the electrons, the mutual influence of those changes on each electron would be calculated to be coming from the past and from a distance behind and with diminishing lateral component.

Are you talking about the electromagnetic repulsion here, or the gravitational attraction?

We handle them both the same way. We work in a frame in which the two electrons are at rest, because it's very easy to calculate both the gravitational and electrical forces between the electrons in that frame. We use these forces to calculate whether the electrons move towards one another or apart, and find that they move apart because the electrical repulsion is stronger. Once we have our answer, we can apply the appropriate coordinate transform to find the motion of the electrons in any other frame (they move apart in all frames, and you'll find the details of the calculation in the wikipedia article on relativistic velocity addition).

The apparent paradox that Danyon and Jfizzix describe is the result of:
1) Starting work in a frame in which the two electrons are not at rest. All inertial frames are equivalent so there's nothing wrong with this choice... But it is doing things the hard way.
2) The very tempting but incorrect assumption that the gravitational field of a moving body looks like the gravitational field of a stationary body, but with $m$ replaced by $\gamma{m_0}$. As Ibix said above... It does not.

It's worth noting that this problem is much more difficult if the two electrons are traveling in opposite directions. In that case, there is no frame in which both electrons are at rest, so we're stuck with frames in which one or both electrons are moving. If the gravitational effects are not negligible (fortunately they nearly always are) we're forced to work with the GR solution for two masses, at least one of which is moving.

 

[pervect]

Say that two electrons travel in parallel at Ultra relativistic speeds, such that the observed energy-mass of said electrons (from a resting observer) is sufficient to generate a gravitational field great enough to overcome the repulsive coulomb force between the two electrons. This implies the two electrons are attracted together in the rest frame of the Stationary observer, however, in the rest frame of the two electrons, the electrons are stationary and the repulsive force is sufficient to repel them apart

The explanation of this is a bit technical, but to give a simple summary expressed in familiar Newtonian terms, to get a consistent understanding of how gravity works relataivistically, you need to include the "magnetic" component of gravity. In the weak-field approximation (the only regime in which quasi-Newtonian explanations of gravity as a "force" works well), this effect is known as "gravitoelectromagnetism", or GEM. An article on this can be found in the wiki,
http://en.wikipedia.org/w/index.php?title=Gravitoelectromagnetism&oldid=649447664

When one includes the gravitomagnetic effects, one finds that there is a gravitational equivalent to magnetism which causes a gravitational "replusion" between moving electrons that mostly cancels the increased attraction you'd expect considering only the coulomb force. (By coulomb force, I mean the force given by the gravitational force due to coulomb's law, usually written as GmM/ r^2, the force analogous to the electrostatic force between charges in electromagnetism.) Another way of saying this is that the coulomb force, in isolation, is not relativisitically covariant. One needs to include a magnetic force along with the coulomb force to get a relativistically covariant force law.

[add]
I wanted to expand on this point in more detail. Informally, the main point of covariance is to say that you can analyze motion in any frame of reference you like, and get consistent result. If you look at the problem statement, the question raised is about that very point . That is to say, we can rephrase the question as "How is gravity covariant? The Newtonian force law doesn't apear to be covariant", because you seem to get different results in a moving reference frame versus a fixed one. The answer to the question is that the Newtonian gravity, with only the F=gmm/r^2 force law, is in fact NOT relativistically covariant, for the reasons mentioned. The answer to the conondrum is to use a relativistic theory of gravity, such as GEM (which is only an approximation), or the full theory. BUt in the full theory, gravity is generally not regarded as a force at all, which makes it hard to explain to a lay audience, so I am giving the explanation in terms of an approximate relativistic theory of gravitation.

If one take the limit of the electrons approaching the speed of light, the cancellation between the coulomb force and the gravitomagnetic force becomes exact, and you get the well-known result that parallel light beams do not attract.

For a reference, see "Simple Explanation for why Parallel-Propagating Photons do not Gravitationally Attract" by Raymond Jensenin in "Progress in Physics", oct 2013, or one of the refereces therein. You can find this online currently at http://www.ptep-online.com/index_files/2013/PP-35-L3.PDF

In 1931 Tolman, Ehrenfest and Podolsky [1] were first to pub-lish studies on how light interacts with light gravitationally. Among other things, they found that when photons move in parallel beams, there is no gravitational attraction between them. The authors did not give a physical explanation for this peculiarity. In 1999, Faraoni and Dumse [2] studied the problem of gravitational attraction between photons and concluded that for photons moving in parallel, the reason for the lack of gravitational attraction is due to an exact cancellation of the gravitomagnetic and gravitoelectric forces between them.