What are the implications of GR that are not captured by gravitomagnetism?

[ohwilleke]

Many physicists agree that GR implies, under certain conditions, that gravity has a component, which if gravity is viewed as the analog of the electric force, is analogous to the magnetic force under a set of equations similar to Maxwell's equations. See e.g. http://arxiv.org/abs/gr-qc/0207065 Thus, a moving mass has additional effects not predicted by the simple Newtonian approximation. This gravitomagnetism effect is also know as "frame dragging." It also seems to imply that there are some clock effects to gravity even under this quasi-Maxwellian formulation.

My question is, what are the most important implications of GR which are not captured by gravitomagnetism?

One that seems obvious is that GR would apply to photons, while gravitomagnetism would seem not to, but it isn't intuitively obvious how much else the more sophisticated by classical gravitomagnetic expression fails to capture, and that one deviation, by itself, is a fairly trivial correction.

The referenced article is:

Matteo Luca Ruggiero and Angelo Tartaglia, "Gravitomagnetic Effects", To appear in Nuovo Cim. 117B (2002) 743-768

Abstract:

This paper contains a review of the theory and practice of gravitomagnetism, with particular attention to the different and numerous proposals which have been put forward to experimentally or observationally verify its effects. The basics of the gravitoelectromagnetic form of the Einstein equations is expounded. Then the Lense-Thirring and clock effects are described, reviewing the essentials of the theory. Space based and Earth based experiments are listed. Other effects, such as the coupling of gravitomagnetism with spin, are described and orders of magnitude are considered to give an idea of the feasibility of actual experiments.

 

[pmb_phy]

Many physicists agree that GR implies, under certain conditions, that gravity has a component, which if gravity is viewed as the analog of the electric force, is analogous to the magnetic force under a set of equations similar to Maxwell's equations.

I know of nobody who accepts GR who doesn't think that there is an analogy with EM. This analogy is strongest in the weak field approximation. A nice aticle on this is

Analogy between general relativity and electromagnetism for slowly moving particles in weak gravitational fields, Edward G. Harris, Am. J. Phys. 59, 421 (1991)

My question is, what are the most important implications of GR which are not captured by gravitomagnetism?

One that comes to mind is that pressure is a source of gravity in full-fledged GR. In gravitomagnetism the contributions of T^jk (stress/pressure) are assumed to be neglegible.

One that seems obvious is that GR would apply to photons, while gravitomagnetism would seem not to, ..

That is incorrect. Mass is considered to be a gravitational charge and since light has mass (since it has energy) it has a gravitational charge. For details on this point see Rindler's new SR/GR text.

Thanks for the reference!

Pete

 

[robphy]

On a possibly related note,
there is a curious equation in Hawking/Ellis (p. 85) that has always intrigued me. Sometimes these are called "quasi-maxwellian equations".

...the Bianchi Identities
$$R_{ab[cd;e]}=0$$
They can be rewritten as
$$C^{abcd}{}_{;d}=J^{abc}$$ (4.28)
where $$J^{abc}=R^{c[a;b]}+\frac{1}{6}g^{c[b}R^{;a]}.$$ (4.29)
These equations are rather similar to Maxwell's equations in electrodynamics:
$$F^{ab}{}_{;b}=J^a$$
where $$F^{ab}$$ is the electromagnetic field tensor and $$J^a$$ is the source current. Thus in a sense one could regard the Bianchi Identities (4.28) as field equations for the Weyl tensor giving that part of the curvature at a point that depends on the matter distribution at other points.

I've been toying around with that J-tensor but haven't found a satisfactory physical and geometric interpretation for it. Has anyone enountered this J-tensor or the quasi-Maxwellian equations?

 

[ohwilleke]

One that comes to mind is that pressure is a source of gravity in full-fledged GR. In gravitomagnetism the contributions of T^jk (stress/pressure) are assumed to be neglegible.

Yes, this is what flows from the equations, but what observable consequences of this difference flow from the difference?

 

[pmb_phy]

Yes, this is what flows from the equations, but what observable consequences of this difference flow from the difference?

As I said, its source of gravity. Therefore more pressure - more gravity - stronger gravitational field

 

[pervect]

Many physicists agree that GR implies, under certain conditions, that gravity has a component, which if gravity is viewed as the analog of the electric force, is analogous to the magnetic force under a set of equations similar to Maxwell's equations. See e.g. http://arxiv.org/abs/gr-qc/0207065 Thus, a moving mass has additional effects not predicted by the simple Newtonian approximation. This gravitomagnetism effect is also know as "frame dragging." It also seems to imply that there are some clock effects to gravity even under this quasi-Maxwellian formulation.

My question is, what are the most important implications of GR which are not captured by gravitomagnetism?

I'm not really sure what you're looking for. Perhaps if you explained what the most important implications of electromagnetic theory that are not captured by magnetism were, I could give an answer to your question.

 

[ohwilleke]

I'm not really sure what you're looking for. Perhaps if you explained what the most important implications of electromagnetic theory that are not captured by magnetism were, I could give an answer to your question.

For example, Maxwell's equations do not predict tunnelling or observer dependent behavior which are the epitome of QED.

The sort of things I'm trying to sort out are e.g.:

Can you get black holes with just gravitomagnetism, or do you need GR to get that result?

I would think, that as long as energy was considered mass, that you would get lensing.

I have read, that gravitomagnetism is not sufficient, without GR, to explain why the precession of Mercury is incorrect under Newtonian theory, although it isn't entirely obvious to me why that is the case (I don't see an influence through the lack of pressure terms, but maybe I'm missing something).

I know that gravitomagnetism implies some kinds of clock effects, but I wonder if there aren't other clock effects which you need full GR to detect.

This is what I am getting at.

 

[pmb_phy]

I have read, that gravitomagnetism is not sufficient, without GR, to explain why the precession of Mercury is incorrect under Newtonian theory, ..

Where did you read that?

Pete

 

[pervect]

For example, Maxwell's equations do not predict tunnelling or observer dependent behavior which are the epitome of QED.

The closest anaology here is that GR doesn't predict quantum effects either, you need quantum gravity for that - unfortunately, we don't have a full theory of quantum gravity, though we can do some quantum theory in curved space-time (this does not fully quantize gravity, though).

The sort of things I'm trying to sort out are e.g.:

Can you get black holes with just gravitomagnetism, or do you need GR to get that result?

I don't quite understand how you've mentally split off gravitomagnetism from the rest of GR. It's a bit like splitting off magnetism from electromagnetism. If I pursue that anology to it's logical extent, you'd get rid of the columb force between stationary charges as not being "magnetic", so you'd likewise get rid of the gravitational force between stationary masses as not being gravitomagnetic. But that doesn't seem to be what you're doing...

 

[pmb_phy]

I don't quite understand how you've mentally split off gravitomagnetism from the rest of GR. It's a bit like splitting off magnetism from electromagnetism. If I pursue that anology to it's logical extent, you'd get rid of the columb force between stationary charges as not being "magnetic", so you'd likewise get rid of the gravitational force between stationary masses as not being gravitomagnetic. But that doesn't seem to be what you're doing...

There is a bit of an inconsistency between EM and gravitomagnetism. In EM q is charge and the analogous quantity to q in gravitomagnetism is relativistic mass, M. But the active gravitational charge of the moving body is not M, its $M(1+\beta^2)$.

Gravitomagnetism is merely an approximation to GR - Period. It is not a different theory. Its the same theory, but you're simply dealing with weaker fields, low pressure sources and 'slowly' moving bodies.

Note: Slowly only means that $\beta^2$ are omitted but not the $\beta$ terms. In the Newtonian approximation each is omitted. Thus in the Newtonian approximation the gravitational force is not a function of velocity. In gravitomagnetism the gravitational force (gravitomagnetic + gravitoelectric) is a linear function of velocity and in full GR the gravitational force is a quadradic function of velocity.

Pete

 

[ohwilleke]

Thus in the Newtonian approximation the gravitational force is not a function of velocity. In gravitomagnetism the gravitational force (gravitomagnetic + gravitoelectric) is a linear function of velocity and in full GR the gravitational force is a quadradic function of velocity.

This is really helpful.