# Linear Momentum in General Relativity

My question is, does a particle moving in a straight line at constant velocity through empty space create "frame dragging" that would tend to entrain other bodies in the direction of its motion, via the $$T_{0k}$$ terms in the stress energy tensor? Is there a metric that describes this case?

My thought process was that this situation, from the point of view of a test particle, is indistinguishable from the Kerr metric very close to a rotating body, if we only consider an instant in time, when the test particle acquires linear momentum tangential to the spherical body.

Edit: I suppose in the frame of the "moving" body, that momentum does not exist. But that just changes my question to: how do the $$T_{0k}$$ terms affect the local spacetime metric in general? A related point, if a particle has a very high linear momentum, its total energy which can be thought of as "relativistic mass," should create gravitational affects. How does this manifest in different frames?
I know this seems like multiple questions, but I think they are all facets of the same question.

Is there a metric that describes this case?

Yes. It is the Schwarzschild metric (valid outside of the gravitating body if we are talking about something like a star). When written in the form where the linear motion of the gravitating body is explicit, the metric usually referred as boosted Schwarzschild metric. Of course, boosted Schwarzschild is related to the usual form of Schwarzschild metric via diffeomorphism, however this is a large diffeomorphism and the vector field generating it does not vanish at infinity, so arguably, this metric does describe a distinct physical situation.

For more details and references on the boosted Schwarzschild metric see this answer of mine.

… does a particle moving in a straight line at constant velocity through empty space create "frame dragging" that would tend to entrain other bodies in the direction of its motion, via the $$T_{0k}$$ terms in the stress energy tensor?

Consider the following situation: test particle is initially at rest and a mass $$M$$ (let's call it a star) flies by it. In the reference frame of the particle, gravitational field of a star has both gravitoelectric and gravitomagnetic components. The particle would accelerate toward the star and after gaining velocity component toward it would interact with the gravitomagnetic field of the star (via analog of Lorentz force) gaining velocity component parallel to the velocity of a star. This effect could be described as “entraining”. Also, if the test body is equipped with a gyroscope, it would “wobble” during the star's flyby, analogously to the Lense–Thirring precession. Keep in mind however, that too literal, mechanistic interpretation of this frame–dragging via some kind “ether” could be unsatisfactory (see Rindler, 97).

Additionally, the extreme manifestations of frame–dragging associated with the strong gravity of rotating black holes such as ergosphere, Penrose process, Blandford–Znajek process all have analogues with linearly moving nonrotating black hole (see Penna, 2015). The energy powering such processes is the kinetic energy of moving black hole. One could formulate the laws of black hole mechanics that include the linear momentum.

I suppose in the frame of the "moving" body, that momentum does not exist. But that just changes my question to: how do the $$T_{0k}$$ terms affect the local spacetime metric in general?

GR is a nonlinear theory, so generally one cannot attribute physical effects to specific components of a tensor in a given reference frame. Moreover, since there are constraints on stress–energy tensor (in the form of its conservation law), we cannot just take components $$T_{0k}$$ of, say, a star moving with constant velocity and plug them alone into Einstein equations (even linearized ones) since those components by themselves do not satisfy the constraints. Similarly in electromagnetism, one cannot ask, What is the EM field of a current at a given point? (but one can talk of EM field of a current loop or EM field of a charge moving along a given trajectory, since in these cases constraints are satisfied).

A related point, if a particle has a very high linear momentum, its total energy which can be thought of as "relativistic mass," should create gravitational affects. How does this manifest in different frames?

Tensor objects describing gravitational fields would transform according to the rules of general covariance.

When linearized approximation is valid we could organize our reference frames using (approximately) cartesian coordinates and just use special relativistic transformation rules for perturbations of the flat Minkowski background globally. For example, Weyl tensor under $$3+1$$ split of spacetime in space ant time parts is described by its electric $$E_{ij}$$ and magnetic $$B_{ij}$$ parts (similar to Faraday tensor of EM field, but with more indices). So if we have different reference frames moving relative to each other, there would be transformation laws of electric and magnetic parts between the frames, similar to transformation rules of electromagnetic field. For static spacetimes in static reference frame Weyl tensor has only electric part, but if expressed in a moving reference frame there would also be a magnetic part.

• Thank you! Excellent response. I appreciate the references for further study May 20 at 17:11