# Is Maxwellian Gravitoelectromagnetism a useful analogy?

• I

## Main Question or Discussion Point

I'm referring to the stuff in here:
https://en.wikipedia.org/wiki/Gravitoelectromagnetism

Is it a good analogy to conceptualize for beginners who have just started GR?

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Simon Bridge
Homework Helper
Depends on your measure for "good"ness of an analogy.
I don't think it is used in actual physics courses though - by the time someone is just starting GR, they have covered a lot of algebra and physics already so they should be ready just to go right to the basic tensor stuff and learn it directly rather than through an analogy.
In general, learning by analogy is pretty bad.

Depends on your measure for "good"ness of an analogy.
Does it help students understand GR concepts, that's what I'm thinking of. Does it, or will they still have to throw out all of the Maxwellian ideas?

Simon Bridge
Homework Helper
Analogies bot help and hinder students understanding ideas in areas of physics unfamiliar to them ... in the long run they hinder more than help.
They will end up having to unlearn some stciky ideas - again: the fact it is not normally used in routing teaching should be a clue.
I think GEM is actually more complicated than GR ... what is the conceptual hurdle you are trying to overcome?

• Dale
haushofer
It can help to understand e.g. the non-relativistic limit of GR, but I wouldn't recommend it when you start learning it. Usually, in the newtonian limit you assume slowly moving testparticles and weak, static grav. fields. This leaves only one scalar component in the metric: the Newton potential. However, the non-relativistic limit drops the assumptions of the grav.field. In that case you end up with more than just a Newton potential which couples to your particle. GEM is then a nice tool to compare this situation with a particle in an electromagnetic field, and gives you a good intuition what all the different metric components do in the non-relativistic limit.

But again, it's only nice when you already know GR well.

ShayanJ
Gold Member
I'm referring to the stuff in here:
https://en.wikipedia.org/wiki/Gravitoelectromagnetism

Is it a good analogy to conceptualize for beginners who have just started GR?
Regardless of the fact that its good to use it in a course or not, GEM is not an analogy for GR, its an approximation to GR in some regime of parameters. An analogy is supposed to help people understand some concept using a similar more familiar concept but GEM simply doesn't have some important features of GR.

stevendaryl
Staff Emeritus
Depends on your measure for "good"ness of an analogy.
I don't think it is used in actual physics courses though - by the time someone is just starting GR, they have covered a lot of algebra and physics already so they should be ready just to go right to the basic tensor stuff and learn it directly rather than through an analogy.
In general, learning by analogy is pretty bad.
I don't think that learning about GR rigorously is an alternative to learning about ways to think of particular GR effects. If you just hand someone the Einstein field equations and the geodesic equation, it certainly isn't obvious that

https://en.wikipedia.org/wiki/Gravitoelectromagnetism#Higher-order_effects

...if two wheels are spun on a common axis, the mutual gravitational attraction between the two wheels will be greater if they spin in opposite directions than in the same direction...
Using the tensor equations, you can derive such an effect, but only if you have a reason to look for it. Analogies with other topics in physics can be a guide to know what to explore in a theory.

robphy
Homework Helper
Gold Member
While GEM is an approximation to GR,
there is a Maxwell-like formulation of GR called the "Quasi-Maxwellian equations".

Here's a really old post of mine (from 2004) quoting from Hawking&Ellis

[Still] On my to-do list... How are these (GEM and Quasi-Maxwell) related?
What are the Quasi-Maxwell equations telling us?
Could they be useful for solving the initial-value problem analytically and numerically?
Could they be "quantized"?

Possibly useful... but I haven't looked at them in detail...
http://arxiv.org/abs/1302.7248 "The Quasi-Maxwellian Equations of General Relativity: Applications to the Perturbation Theory" (Novello, et al)
https://arxiv.org/abs/1207.0465 "Gravito-electromagnetic analogies" (Costa & Natario)

Jonathan Scott
Gold Member
I like the way that the magnetic side of GEM illustrates how gravitational effects can mimic rotation, and the way that the induced field due to accelerating sources mimics inertia. However, the GEM equations can definitely give wrong impressions; for a start, there are various factors of two which don't match the original equations (and for which there are a confusing number of different conventions), and they fail to include important terms from the tensor form.

• PeterDonis
Jonathan Scott
Gold Member
For me, the most obvious difference between electromagnetic and gravitational equations is that the rate of change of four-momentum in gravity involves the square of the four-velocity (at least approximately) where in electromagnetism it just involves the four-velocity.

In an electrostatic field the rate of change of energy and momentum (Lorentz force law) is given in four-vector form (in momentum units) as follows:

$$\frac{dp}{dt} = q \mathbf E. \frac{(c, \mathbf v)}{c}$$

In gravity, using $\mathbf g$ to represent a static Newtonian field, $E$ to represent the test particle energy and $c$ temporarily to represent the coordinate speed of light in an isotropic coordinate system the equation comes out as follows (under the weak field assumption that the scale factor of space is the same as the reciprocal of the time factor):

$$\frac{dp}{dt} = \frac{E}{c^2} \mathbf g.( 1 + 2\mathbf v/c + v^2/c^2 ) = \frac{E}{c^2} \, \mathbf g.\left( \frac{(c, \mathbf v)}{c} \right)^2$$

The GEM equivalent effectively treats the squared four-velocity $(1 + 2\mathbf v/c + v^2/c^2)$ as being roughly the same as doubling the speed to $(1 + 2\mathbf v/c)$ but completely ignores the $v^2/c^2$ term.

(The squared four-velocity is an approximation for a product in the tensor formulation that involves both the relative velocity of the source and test object and the velocity of the test object relative to the observer, so this is not actually a law of nature, just a good approximation).

[Edited to correct some typos in the LaTeX]
[Edited again to fix up slips in powers of c]

[Another edit for clarification] For gravity, the timelike component of the coordinate four-momentum in momentum units is E/c, which varies with c. In a static field, the coordinate energy E in energy (or frequency) units is constant, so the whole equation is still valid if divided by E to remove the energy.

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Simon Bridge
Homework Helper
Using the tensor equations, you can derive such an effect, but only if you have a reason to look for it. Analogies with other topics in physics can be a guide to know what to explore in a theory.
I don't think the student has a reason to look for it from GEM either ... and GEM will also hide facets of GR. I agree students need a guide though - that is why the student is doing a course in GR ... the whole point of doing a course is to have a guide to your study.
The extent to how an analogy or an approximation (as in this case) helps students conceptually depends on what conceptual hurdles you are trying to overcome. Without identifying the problem we cannot properly assess the proposed strategy.

Physics by analogy causes more problems than it solves - do the physics by physics first, then do the analogies as temporary stepping stones where some sort of scaffolding is needed.

OTOH: If you now any resources for using GEM to teach GR more effectively, then please share.

The GEM equivalent effectively treats the squared four-velocity $(1 + 2\mathbf v/c + v^2/c^2)$ as being roughly the same as doubling the speed to $(1 + 2\mathbf v/c)$ but completely ignores the $v^2/c^2$ term.
... looks like a binomial approximation - so that (v/c)<<1. afict The idea is that GEM is for that narrow range when the relative speed is not quite low enough for galilean relativity but not quite high enough to need the full Einstein treatment. The hope is, presumably, to leverage the students existing understanding of EM to get them a "feel" for some of the effects of GR sooner and less painfully than may otherwise be the case. I don't think the existence of an approximation is fatal to a theory.

You can get to some neat stuff (i.e. http://arxiv.org/pdf/gr-qc/0311024.pdf ) and some people do advocate that drawing analogies between gravity and electromagnetism can be useful. It is commonly done between classical electrostatics and Newtonian gravity in (UK/US/NZ) secondary schools for eg.
GEM looks like an update on that approach for Maxwel and GR ... and it may be a conceptual tool for post-grad students wanting to think about possible paths to a theory of quantum gravity ... maybe. The other approach is to look for a geometric interpretation of electromagnetism, also by analogy.

It all boils down to what problem you are trying to solve.

Aside:
[Another edit for clarification] For gravity, the timelike component of the coordinate four-momentum in momentum units is E/c, which varies with c. In a static field, the coordinate energy E in energy (or frequency) units is constant, so the whole equation is still valid if divided by E to remove the energy.
I thought "c" was just the (invariant) scale factor to get the units to come out how you want which is why we usually pick units so that c=1.
Wouldn't the coordinate E usually vary with relative speed and mass?

stevendaryl
Staff Emeritus
Physics by analogy causes more problems than it solves - do the physics by physics first, then do the analogies as temporary stepping stones where some sort of scaffolding is needed.
I don't think that there is any problem with using analogies, as long as you remember that an analogy is at best a hint as to how something might work. You have to look at the details to see whether the analogy pans out.

• Simon Bridge
Simon Bridge
Homework Helper
I don't think that there is any problem with using analogies, as long as you remember that an analogy is at best a hint as to how something might work. You have to look at the details to see whether the analogy pans out.
I agree. As long as all those things happen.

Back to the topic:
Is GEM an effective way to introduce GR? Compared with current approaches for the same target group?
Do you know of anyone using GEM to teach "introduction to GR" courses?

Note: GEM is not an analogy itself - it's an approximate model whose motivation comes from drawing analogies between EM and GR.
GEM is where the underlying analogies take us, so maybe we should be thinking about whether the underlying analogies are a useful way to introduce GR?

I think we need to hear from OP before any further headway can be made.

I agree. As long as all those things happen.

Back to the topic:
Is GEM an effective way to introduce GR? Compared with current approaches for the same target group?
Do you know of anyone using GEM to teach "introduction to GR" courses?

Note: GEM is not an analogy itself - it's an approximate model whose motivation comes from drawing analogies between EM and GR.
GEM is where the underlying analogies take us, so maybe we should be thinking about whether the underlying analogies are a useful way to introduce GR?

I think we need to hear from OP before any further headway can be made.
I don't really know much, I'm actually reliant on you guys.

No, I don't know of anyone.

Simon Bridge
Homework Helper
You asked the question: were you asking as a student or a teacher?
i.e. were you hoping to use GEM for your own studies as someone who wants to know more about GR?

You asked the question: were you asking as a student or a teacher?
i.e. were you hoping to use GEM for your own studies as someone who wants to know more about GR?
for both purposes

Simon Bridge
Homework Helper
OK so lets recap: you don't know anything... you are at introductory level with GR yourself... you want to learn more, so you can teach it to others?
Is that correct?

Focus on you: you should become competent with GR before attempting to teach it.
Probably not a good idea to self-study GR from scratch... you need a guide.
Do you have a solid background in EM? (i.e please describe your education to date.)

Just checked a bunch of previous posts ... I don't see reason to be confident that GEM will help you understand GR.

OK so lets recap: you don't know anything... you are at introductory level with GR yourself... you want to learn more, so you can teach it to others?
Is that correct?
I don't want to teach others (not anytime soon at least), but I'm wondering if it would make a good educational tool in general.

Jonathan Scott
Gold Member
I thought "c" was just the (invariant) scale factor to get the units to come out how you want which is why we usually pick units so that c=1.
As I said earlier in my post, I'm using $c$ here to temporarily mean the coordinate speed of light, to make the notation more familiar and easier to compare with electromagnetism. It would be natural to use units such that the value is 1 in flat space. The concept of a single variable designating the coordinate speed of light only exists for an isotropic coordinate system when the spatial factor in the metric is the same in all directions, otherwise the coordinate speed of light is different in different directions, requiring a tensor representation.
Wouldn't the coordinate E usually vary with relative speed and mass?
In a static gravitational field, the total energy of a test particle is constant. This is equivalent to the Newtonian way in which potential plus kinetic energy is constant. The Newtonian potential energy corresponds to the effect of time-dilation on the energy in GR.

Jonathan Scott
Gold Member
I don't want to teach others (not anytime soon at least), but I'm wondering if it would make a good educational tool in general.
I don't think it's much more than a useful analogy but a very weak approximation. Gravity is somewhat like electromagnetism. The gravitational field acting on masses is somewhat like electric fields operating on charges. In gravity, moving masses induce a rotational effect which is analogous to the way in which moving charges induce a magnetic field in electromagnetism.

Note that GEM goes wrong immediately for fast-moving test particles (including light beams), but the approximate gravitational equation of motion which I mentioned above (with the square of the four-velocity) is very accurate even for relativistic motion and for sensitive solar system tests such as the perihelion precession of Mercury. It only becomes inaccurate near to extremely dense sources such as neutron stars or black holes.

Simon Bridge
Homework Helper
@Jonathan Scott Oh I see - reference for people who don't know what this means:

greswd said:
I don't want to teach others (not anytime soon at least), but I'm wondering if it would make a good educational tool in general.
Well I asked if you were asking as a teacher or a student and you replied "both" ... it goes to the purpose of acquiring the knowledge.
To answer this question: in general no. In very specific cases maybe - depends on what conceptual problem the teacher hopes to overcome, and how well the teacher can impress the student with the limits of any analogies and approximations made.

Simon Bridge
Homework Helper
Aside: People are often surprised at my response when I'm asked about analogies.
Donald Simanek (for eg.) has a similar perspective, and discusses it here:
https://www.lhup.edu/~dsimanek/scenario/analogy.htm
He lists the properties of a useful analogy as follows:
1. The analogy should compare the unfamiliar to the already-familiar. The analogy of currents to the flow of water in pipes is useless unless the student already has a good grasp of hydraulics.
2. The analogy should be simple and easy to present. If the comparison requires elaborate justification or explanations, forget it. If the analogy requires a long list of "exceptions" and qualifications, the time would be better spent on a direct and fuller physical and mathematical treatment.
3. The analogy should be reasonably complete in all important details. The non-analogous details shouldn't require elaborate explanation.
4. The analogy should be mathematically analogous. The two cases being related ought to obey the very same mathematical equations without exception.
5. The analogy should be physically analogous. The physical principles in the cases being compared ought to also be meaningfully analogous.
6. Analogies ought never be represented as a demonstrations, arguments, or proofs. Analogies must never replace rigorous mathematical and physical development.
7. The analogy should not be restricted to a single case or a special case.
8. All obvious extrapolations of the analogy should be valid. The analogy should continue to give correct predictions for other cases that will occur later in the course, and for other cases that a thoughtful student might apply it to.
9. There should be no hidden or unstated assumptions required to make the analogy "work."
... how does EM as an analogy for GR work in above? (Remember GEM is not an analogy, but the result of an analogy.)