# I Relativistic Effect on Attraction between Moving Masses?

1. Oct 31, 2016

### ObjectivelyRational

Magnetism has be explained as a relativistic effect of moving charges. This was something we were shown in undergrad.

Is there a similar effect when masses move relative to each other?

1. Is there any difference between the gravitational force between a single massive particle and a very long "line" of mass at rest wrt the particle versus when the "line" of mass is travelling at relativistic speeds (in the same direction as the line) in relation to the particle?

2. Is there any difference between the gravitational force between two parallel lines of mass at rest wrt each other versus the lines of mass travelling at relativistic speeds (along the same directions as the lines) in relation to each other?

Why?

2. Oct 31, 2016

### Jonathan Scott

In General Relativity, there are similar effects when masses move relative to each other, although they are a bit more complicated than for electromagnetism.

In many simple cases (provided that fields are not extremely strong), you can simply use Special Relativity. If you transform the system to a frame where the source masses are at rest and calculate the acceleration in that frame, you can then use a Lorentz transformation on the acceleration to find the acceleration in the original frame. This applies in particular to your case (2). You will find that the acceleration gets time dilated more and more and hence appears to be slower when the lines of mass are moving faster.

When a test particle is moving at relativistic speed relative to a gravitational source, the path differs somewhat from Newtonian theory mainly because of curvature with respect to space. However, if one calculates the acceleration taking that into account, it can then be transformed using Special Relativity to a new frame.

I must admit I don't recall off hand what effect a very long line of mass has on the shape of space, so I don't know exactly what acceleration a relativistic test particle would experience in that case, but if you can calculate the acceleration when the line is at rest but the particle is relativistic and then transform it to bring the particle to rest then that would give the answer to your case (1).

3. Oct 31, 2016

### Staff: Mentor

Yes. It is called "gravitomagnetism" and is discussed in most major relativity textbooks.

4. Oct 31, 2016

Thank you!

5. Nov 1, 2016

### pervect

Staff Emeritus
Yes, there is an analogous effect, as some other posters have already mentioned. See https://en.wikipedia.org/wiki/Gravitoelectromagnetism for some of the technical details

"Gravitational force" turns out to be trickier to rigorously define than one might expect. The following factoids might help illustrate some of the principles while sidestepping the issues arising from discussing "gravitational force".

Two parallel light beams won't attract. Left to themselves, they'll remain parallel and at a constant distance from each other as they propagate.

Two anti-parallel light beams will attract. Left to themselves this attraction means that if they start out anti-parallel, they won't remain so. In an extreme example, one can imagine two anti-parallel light beams orbiting around a common center with no other forces or gravitational influences, though this situation turns out to be unstable.

A motionless test particle will be attracted to a nearby passing laser pulse.

A light beam passing by a large mass will be deflected by twice the amount that one would expect from a naive calculation.

These factoids can be qualitatively integrated into a coherent picture by imagining that the energy in the parallel light beams causes an attraction analogous to the electrostatic coulomb force due to charge. The analogy here is between the electromagnetic forces between charges, and the gravitational attraction between sources of energy. This "coulomb" attraction is counterbalanced a a "magnetic-like" gravitational repulsion when the light beams are parallel. When they are anti-parallel, the "coulomb-like and "magnetic-like" forces point in the same direction rather than opposite directions.

The factor of two deflection of a light beam by a test mass illustrates a need for caution with Newtonian analogies. You'll get a qualitative understanding of what happens by such analogies, but to really get accurate answers and to understand where this factor of 2 comes from will require deeper study, and I won't attempt to describe "why" in this short post. If you pursue the GEM (GravitoElectroMagnetism) analogy (see the wiki reference), which compares weak-field gravity to electromagnetism, you'll see this factor of 2 in the GEM equations.

For an old and rather confusingly worded reference on the behavior of parallel and anti-parallel light beams, see for instance http://journals.aps.org/pr/abstract/10.1103/PhysRev.37.602