# Is gravity Lorentz-invariant?

If I go whizzing past the Sun in my spaceship at 0.8c, I will see that the Sun looks like it has been squished. That is, it will be Lorentz-contracted along the direction of my motion (by a factor of 0.6). Since there is no such thing as absolute motion, I am at liberty to regard myself as at rest, and the Sun as moving. Certainly, if the Sun were at rest with respect to me, I would expect it to be (nearly) spherical, and I would be at a loss to explain its being an oblate spheroid instead. Is there some aspect of the theory of gravitation, either Newtonian or Einsteinian, by which an object's gravitational field is modified by its motion, so as to produce this effect?

In Chapter 6 of EM Fields and Waves, by Lorrain and Corson, it is demonstrated that when the Lorentz transformation is applied to the force between two charges, considered in a frame where they are at rest vs one where they are comoving, the resulting modification of the force gives rise to terms that are mathematically identical to that which results from the magnetic field induced by the motion. In their words; "..the magnetic field in frame 1 has appeared as a result of the application of a relativistic transformation to the electric force in frame 2." That is, an observer in the moving frame would find the behavior of the charges inexplicable if he did not include magnetism in his theory of EM forces. Thus, magnetic forces are necessary to make electromagnetism Lorentz invariant. Since I am not aware of any aspect of gravitational theory which corresponds to magnetism, I do not see how gravitational theory can account for the effects of motion.

• I don't understand your question. Things that are moving relative to you are Lorentz-contracted, yes. What does that have to do with "Is there some aspect of the theory of gravitation, either Newtonian or Einsteinian, by which an object's gravitational field is modified by its motion, so as to produce this effect? "? Which "effect" do you mean? Jan 18, 2016 at 1:48
• motion is not that subjective : symmetry only works for inertial frames
– sure
Jan 18, 2016 at 8:59
• I have added a clarification to my question. Jan 18, 2016 at 20:16

## 3 Answers

Since I am not aware of any aspect of gravitational theory which corresponds to magnetism, I do not see how gravitational theory can account for the effects of motion.

Gravitomagnetism is in fact a known and measured phenomenon. It emerges from Einstein's general relativity, rather than Newtonian gravity (which, as has been noted in other answers, is definitely not Lorentz-invariant.) The derivation itself is rather involved, but basically, one makes the following assumptions:

1. The metric is very close to flat, i.e., $g_{ab} = \eta_{ab} + \gamma_{ab}$, where $\gamma_{ab}$ is "small";

2. The matter sources that are responsible for the curvature are moving slowly relative to light; and

3. The test particles that are accelerating due to this curvature are moving slowly relative to light.

(All statements above concerning velocities and accelerations are implicitly made with respect to the reference frame used to construct the flat background metric.) You then figure out the equations obeyed by the components of $\gamma_{ab}$ given the known distribution of the source's energy density and energy flux. Finally, you use the Christoffel symbols for this perturbed metric to look at the acceleration of a test particle moving under the gravitational influence of this source.

Under the above assumptions, you do in fact get a magnetic analog of the gravitational force! The Newtonian force on the test particle is the term that you obtain when you set the two velocities (source and test particle) to zero. The term that's linear in the source & test particle velocities, meanwhile, looks for all the world like a magnetic Lorentz force. It's parallel to $\vec{v} \times \vec{B}$, where $\vec{v}$ is the test particle velocity and $\vec{B}$ depends on the energy flux distribution of the source in a way that looks for all the world like the Biot-Savart Law.

These gravitomagnetic effects do, however, lead to measurable phenomena, of which the best-known is frame-dragging. The idea here is that the rotation of a central body will cause orbiting bodies to behave differently than if the body were not rotating, just as a configuration of moving charges will exert different forces on a test charge than if the same configuration of charges was stationary. This effect was finally confirmed a few years ago (after many, many years of trials and tribulations) by the Gravity Probe B experiment. The effect they were looking for was basically the gravitational analog of two magnetic dipoles exerting a torque on one another; in their case, one of the "dipoles" was a gyroscope on the satellite, and the other was the Earth.

Unfortunately, I am unaware of any good sources that go through this derivation explicitly (if anyone puts one in the comments, I'll revise this sentence.) Wald's General Relativity discusses the phenomenon briefly in Section 4.4, but he relegates most of the calculation to an end-of-chapter problem.

• As I understand it (see Wikipedia, for instance) "gravitomagnetics" refers specifically to "frame dragging", which is in turn related to rotation. Nothing about the Sun being squished by linear velocity. Jan 18, 2016 at 22:40
• @JeromeBerryhill Gravitomagnetism is not a synonym for frame dragging. The Gravitoelectric force points towards where the sun used to be and the Gravitomagnetic force supplies an additional force so that the total force on a mass at rest to the sun can be a force pointing towards where the sun is now if the sun moves at a steady velocity. Just like in electromagnetism. Jan 19, 2016 at 4:02
• OK, I'll buy that. The wikipedia article seems to imply that gravitomagnetism only applies to rotating masses, but it is fairly clear from the equations that linear velocity would also give rise to a g-m firld. Jan 19, 2016 at 4:42
• The correct description of the behavior of charged particles requires magnetism. Coulomb's law is inadequate to the description of moving charges. A second phenomenon, magnetism, must be introduced in order to account for the effects of motion. Is the same true of GR? If not, why must it be supplemented by gravitomagnetism? Jan 20, 2016 at 3:58
• @JeromeBerryhill: The analogous statement for gravity would go something like this: Newtonian gravity doesn't view moving masses any differently from stationary ones. This means that it is incompatible with Lorentz invariance (in the same sense that "electric forces only" is incompatible with Lorentz invariance.) Therefore, if we want to modify Newtonian gravity to be Lorentz-invariant, there must be some gravitational analog of magnetic forces. And, in fact, if you look carefully at general relativity (a Lorentz-invariant theory of gravity), you find that such forces do in fact exist. Jan 20, 2016 at 16:39

Yes.

General Relativity is perfectly capable of showing how a large oblate spheroid of gas moving in a particular direction contracts to form an oblate spheroid shaped star moving in that same direction that shines and emits light.

And it is perfectly capable of making models that predict all the observations that any observer would make.

Frankly, its a bit hard to imagine that you'd think otherwise. The physics doesn't depend on your frame. So you make the same predictions for what every observer detects regardless of any choice of frame that may or may not be used to describe it. So if you can describe the collapse process in the frame of the star then you can make a model of the collapse process. And that model can be used for any observer.

The model is a geometric structure that has everything you need to make predictions. Here is an easier example. In the center of momentum frame an elastic collision has two particles come in and each reverse there velocity and then head out. That is a description in a frame, and it is particularly easy. But the model of such a process is a geometric structure.

The geometric structure has the tangent to the worldline be proportional to the energy-momentum four vector. So each incoming particle has its own worldline, its own unit tangent, and its own energy-momentum vector (call them p1 and p2). And at the point of collision the sum of those energy momentum vectors exists, call it s.

To find the outgoing tangent, you take the incoming tangent v1 and rotate it into s then repeat that rotation to get v1'. And similarly you take v2 and rotate it into s then repeat that rotation to get v2'.

This description is about lines and curves and tangents and such and so doesn't depend on what frame you pick or even whether you pick a frame. So we can literally predict what happens during an elastic collision without having to pick some frame or talk about who is an observer. Because we just make a geometric model of the events

Sure, in the center of momentum inertial frame it looks like a simple bounce where each has its velocity sent to its negative, but the geometric description never required you to pick a frame.

Picking a frame is a mere convenience, like picking an x axis or a y axis. The vector equations hold and work and don't care which frame or even whether you pick a frame.

Stars cause curvature outside themselves because the kinds of vacuum curvature that was outside the star has a natural dynamic (governed by the speed of light) that fills itself into any vacuum region as the collapse leaves more vacuum outside. So that is how the spacetime outside the gas cloud becomes curved as the gas contracts and becomes the star. And General Relativity describes this fine, without ever having to pick a frame.

So there is no issue. General Relativity is a geometric theory that doesn't require that you pick a frame. It just requires that its models satisfy Einstein's Equation.

if the Sun were shaped by electrical forces, a person who observed it from a frame with large relative velocity would note that it looked squished.

It wouldn't just looked squished. The shape and even volume of extended objects (and even whether it is a single connected object) depends on your frame.

He would attribute this departure from spherical symmetry to the magnetic field generated by the motion he perceives.

That's a gross oversimplification. The person would note that it formed into the shape it was based on the past condition (past shape and past positions and past motions and past fields) and the dynamical laws. Since they disagreed about the shapes in the past, they end up disagreeing about the shapes now.

They agree on the dynamics, for instance on the proper accelerations felt at any specific event (they agree on the rate momentum changed per proper time, they agree on the worldline of the particle).

Disagreements consist of whether a worldline's tangent vector is "at rest" or not. And whether the propoer time agrees with an inertial time. And whether an electromagnetic field has a particular amount electric field or magnetic field. But they agree on the force.

But gravity isn't a force. So people don't take Gravitoelectromagnetism seriously as anything other than an approximation for some situations. So they don't worry about breaking a Gravitoelectromagnetic force into a Gravitoelectric force and a Gravitomagnetic force.

The sun is a squished moving star because it formed from the collapse of a squished moving ball of gas. And when you use GR to model the collapse of a squished moving ball of gas you get exactly what you expect.

Please don't pretend this is not known and that this isn't 100% exactly what GR predicts. Even collapsing dust would do this when there are zero forces whatsoever. A squished moving ball of dust collapses into a denser squished moving ball of dust, according to regular GR, with no forces.

• you should add "inertial" everywhere before "frame", or its just false
– sure
Jan 18, 2016 at 8:53
• @sure I added one "inertial" in the one case I was trying to describe a specific frame. But otherwise, no it's general relativity, if you know how to use a frame you can use it. Doesn't matter if its inertial or not. Jan 18, 2016 at 8:57
• @sure No. The physics doesn't depend on what frame a person may or may not use. But you do have to know how to use a frame and if you only know how to use an inertial frame then please please please don't use anything else. I wouldn't want anyone to do physics wrong. The physics is about the geometry. For example, whether the event of sending or receiving a signal happened between this clock tick event on the world line and this other clock tick event on the same world line. You don't even ever need to pick frames if you don't want to. There isn't a frame police that makes you pick one. Jan 18, 2016 at 9:02
• @sure Since Physics is science it is solely about the predictions. You are free to force your students to pick frames. And I'm free to to force my students to never pick frames. And if we are making the same predictions it is the same Physics. If either of us claimed our way was the only possible way then we'd be liars. And such restrictions probably aren't good. But if the predictions are the same then its the same Physics. You seem to not understand what I said about models. You can make a model without having to label any curve an observer, inertial or otherwise. And you can get predictions Jan 18, 2016 at 9:24
• @sure The actually predictions that you orbit it at a certain distance, that you see light from the star shine on you. That you see a certain ratio of the frequency of the light and the inverse period of the propoer time of your orbit. These things are all generated by the model. Not by a frame. You don't do GR by picking a frame and then describing things in terms of that frame. Jan 18, 2016 at 9:31

Is there some aspect of the theory of gravitation, either Newtonian

Newtonian gravitation has instantaneous transmission of the gravitational field

or Einsteinian, by which an object's gravitational field is modified by its motion, so as to produce this effect?

In general relativity Lorenz transformations are inherent at the flat limit, see this set of question and answers; and the Newtonian gravitational field is an emergent one, from the curvature of space induced by the masses.

One needs to go to the flat space approximation and use Lorenz transformations for a field emergent from the general relativity curvature, where the gravitational waves travel also with velocity c, to explain the particular effective forces at the spaceship. Look at how the GPS system uses general relativity.

• I don't quite see what quantum mechanics has to do here. The question is purely non-quantum. Jan 18, 2016 at 10:13
• @Ruslan I was thinking virtual gravitons and the velocity of light, but they are probably overkill. I will edit. Jan 18, 2016 at 11:56
• If I understand you correctly, you are saying that gravitational theory does not "automagically" include the effects of the Lorentz transformation, as EM theory does. So your answer to my question is "No". But maybe I am misunderstanding. Please see the clarification I have added. Jan 18, 2016 at 20:23
• @JeromeBerryhill I am certain that Anna is not saying no. I don't think any physicist would say no. Every relativistic theory of gravity hands Lorentz transformations just fine. Jan 19, 2016 at 4:03
• I had not seen your edits, and I edited out my quantum mechanical version, which of course, having gravitons , would have Lorenz transforamations. Lorenz transforamtions are built in General Relativity.. Jan 19, 2016 at 4:47