Gravitational analog of electromagnetic force

[synch]

Is there a gravitational analog of electromagnetic force?
[Apart from the obvious "static" forces, ie electrostatic between fixed charges and gravitational between fixed masses.]
I am thinking of the classic situation of a moving charge (or current) creating a magnetic field which then affects other moving charges. So a question by analogy, does a moving mass have an additional field which is only detected by other moving mass ? It would be very very small I guess.

 

[sophiecentaur]

Are you trying to find things that the two types of force have in common? Google is your friend here. I tried "compare and contrast gravitational and EM forces" and there were loads of hits.
I don't think you can expect to get much out of PF responses to your question because any answers are going to be divergent (most likely correct but divergent). I think you will need to read around by yourself and come to your own conclusions. I'm not trying to dodge the question but from the way it's presented , I don't think you are sure what you actually want to know. Reading around can help you resolve this.

 

[malawi_glenn]

https://en.wikipedia.org/wiki/Gravitoelectromagnetism

 

[Nik_2213]

There's 'teleparallel gravity', which is still hypothetical. IIRC, it reckons space/time twisted up one way gives our familiar EM, in another gives gravity. As yet, the different 'flavours' are mutually exclusive...
Sadly, beyond a fascinating article in New Scientist a couple of years ago, I've seen nothing about it that was comprehensible without serious math...

 

[JD23]

Gravitational analog of electromagnetic force was proposed by Oliver Heaviside in 1893 as gravitoelectromagnetism: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
Now it is seen as the first correction to Newton toward general relativity, as magnetism in EM necessary for Lorentz invariance, directly confirmed by Gravity Probe B.

 

[synch]

Thanks everyone ! Well appreciated.
I (now) wonder if anything with a non-zero spin can be regarded as "fixed" apart from approximation. But that is a whole different question and topic. Thanks again

 

[tech99]

I think a traditional view is that gravity does not have "positive" and "negative" masses. These two are required for an electric current to create a magnetic field via Relativity (see Electricity and Magnetism by Purcell).

 

[JD23]

But the switch of sign between Coulomb and Newton seems a very difficult problem (I am recently thinking about) - having two masses as sources of curvature, taking them closer total energy should grow, what means repulsion.

E.g. in liquid crystals they get Coulomb-like interaction between topological charges:
https://pubs.rsc.org/en/content/articlelanding/2019/sm/c9sm01710k#!divAbstract
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.76.011707
https://www.nature.com/articles/s41598-017-16200-z
Below is example calculation of effective Coulomb potential for such two topological charges in various distances ... but how to switch sign for Newton?
Is there such calculation of approximately Newton effective potential from general relativity Einstein-Hilbert action (->Hamiltonian)?

 

[Vanadium 50]

Can one express some GR scenarios as having a "gravitoelectric" and "gravitomagnetic" part? Yes, Is it helpful in making calculations? Almost never.

 

[JD23]

"Gravitoelectric" is Newton force approximation, "gravitomagnetic" are the first order corrections (necessary for Lorentz invariance): frame dragging, Lense-Thirring, e.g. tested by Gravity Probe B: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
The big question are higher order corrections, their experimental verification.

Simple simulator for Kepler problem e.g. around rotating black hole with GEM approximation: https://demonstrations.wolfram.com/KeplerProblemWithClassicalSpinOrbitInteraction/

I have just found calculated Hamiltonian for Einstein-Hilbert (far nontrivial): page 24-27 of https://arxiv.org/pdf/2204.03537
But I cannot find further derivation of approximately Newton effective potential like above: place two masses in various distances, calculate energy-distance dependence from integration of Hamiltonian (?)

 

[olgerm]

Gravitational analog of electromagnetic force was proposed by Oliver Heaviside in 1893 as gravitoelectromagnetism: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
Now it is seen as the first correction to Newton toward general relativity, as magnetism in EM necessary for Lorentz invariance, directly confirmed by Gravity Probe B.

View attachment 328536

It is interesting to note, that these equations do not predict gravitational time dilation nor gravity bending light. But modifying these equations a little bit would give a model, that also predicts time dilation and gravity bending light. I wonder how many possible models of physics there are between Newtonian gravity and GR.

 

[Vanadium 50]

I wonder how many possible models of physics there are between Newtonian gravity and GR.

An infinite number.

Force = x(Predicted by Theory A) + (1-x)(Predicted by Theory B).

How many values of x can there be.

 

[olgerm]

We can also write down gravitomagnetic equations using gravitomagnetic-4-potential.


EM equations:
$dA_{E0}/dx0+dA_E1/dx1+dA_E2/dx2+dA_E3/dx3=0$ (lorentz gauge condition)
$-dA_{E0}^2/dx0^2+dA_{E0}^2/dx1^2+dA_{E0}^2/dx2^2+dA_{E0}^2/dx3^2+J_{E0}=0$
$-dA_{E1}^2/dx0^2+dA_{E1}^2/dx1^2+dA_{E1}^2/dx2^2+dA_{E1}^2/dx3^2+J_{E1}=0$
$-dA_{E2}^2/dx0^2+dA_{E2}^2/dx1^2+dA_{E2}^2/dx2^2+dA_{E2}^2/dx3^2+J_{E2}=0$
$-dA_{E3}^2/dx0^2+dA_{E3}^2/dx1^2+dA_{E3}^2/dx2^2+dA_{E3}^2/dx3^2+J_{E3}=0$

GEM analog equations:
for gravity
$dA_{G0}/dx0+dA_E1/dx1+dA_E2/dx2+dA_E3/dx3=0$ (lorentz gauge condition)
$-dA_{G0}^2/dx0^2+dA_{G0}^2/dx1^2+dA_{G0}^2/dx2^2+dA_{G0}^2/dx3^2+J_{G0}=0$
$-dA_{G1}^2/dx0^2+dA_{G1}^2/dx1^2+dA_{G1}^2/dx2^2+dA_{G1}^2/dx3^2+J_{G1}=0$
$-dA_{G2}^2/dx0^2+dA_{G2}^2/dx1^2+dA_{G2}^2/dx2^2+dA_{G2}^2/dx3^2+J_{G2}=0$
$-dA_{G3}^2/dx0^2+dA_{G3}^2/dx1^2+dA_{G3}^2/dx2^2+dA_{G3}^2/dx3^2+J_{G3}=0$
for electromagnetism:
$dA_{E0}/dx0+dA_E1/dx1+dA_E2/dx2+dA_E3/dx3=0$ (lorentz gauge condition)
$-dA_{E0}^2/dx0^2+dA_{E0}^2/dx1^2+dA_{E0}^2/dx2^2+dA_{E0}^2/dx3^2+J_{E0}=0$
$-dA_{E1}^2/dx0^2+dA_{E1}^2/dx1^2+dA_{E1}^2/dx2^2+dA_{E1}^2/dx3^2+J_{E1}=0$
$-dA_{E2}^2/dx0^2+dA_{E2}^2/dx1^2+dA_{E2}^2/dx2^2+dA_{E2}^2/dx3^2+J_{E2}=0$
$-dA_{E3}^2/dx0^2+dA_{E3}^2/dx1^2+dA_{E3}^2/dx2^2+dA_{E3}^2/dx3^2+J_{E3}=0$

Or to make also gravity bend light and dilate time:
for gravity:
$dA_{G0}/dx0/A_{G0}+dA_{G1}/dx1/A_{G1}+dA_{G2}/dx2/A_{G2}+dA_{G3}/dx3/A_{G3}=0$
$-dA_{G0}^2/dx0^2/A_{G0}+dA_{G0}^2/dx1^2/A_{G1}+dA_{G0}^2/dx2^2/A_{G2}+dA_{G0}^2/dx3^2/A_{G3}+J_{G0}=0$
$-dA_{G1}^2/dx0^2/A_{G0}+dA_{G1}^2/dx1^2/A_{G1}+dA_{G1}^2/dx2^2/A_{G2}+dA_{G1}^2/dx3^2/A_{G3}+J_{G1}=0$
$-dA_{G2}^2/dx0^2/A_{G0}+dA_{G2}^2/dx1^2/A_{G1}+dA_{G2}^2/dx2^2/A_{G2}+dA_{G2}^2/dx3^2/A_{G3}+J_{G2}=0$
$-dA_{G3}^2/dx0^2/A_{G0}+dA_{G3}^2/dx1^2/A_{G1}+dA_{G3}^2/dx2^2/A_{G2}+dA_{G3}^2/dx3^2/A_{G3}+J_{G3}=0$

for light:
$dA_{E0}/dx0/A_{G0}+dA_{E1}/dx1/A_{G1}+dA_{E2}/dx2/A_{G2}+dA_{E3}/dx3/A_{G3}=0$
$-dA_{E0}^2/dx0^2/A_{G0}+dA_{E0}^2/dx1^2/A_{G1}+dA_{E0}^2/dx2^2/A_{G2}+dA_{E0}^2/dx3^2/A_{G3}+J_{E0}=0$
$-dA_{E1}^2/dx0^2/A_{G0}+dA_{E1}^2/dx1^2/A_{G1}+dA_{E1}^2/dx2^2/A_{G2}+dA_{E1}^2/dx3^2/A_{G3}+J_{E1}=0$
$-dA_{E2}^2/dx0^2/A_{G0}+dA_{E2}^2/dx1^2/A_{G1}+dA_{E2}^2/dx2^2/A_{G2}+dA_{E2}^2/dx3^2/A_{G3}+J_{E2}=0$
$-dA_{E3}^2/dx0^2/A_{G0}+dA_{E3}^2/dx1^2/A_{G1}+dA_{E3}^2/dx2^2/A_{G2}+dA_{E3}^2/dx3^2/A_{G3}+J_{E3}=0$


$J_G$ is energy-flow density. It's 0th component $J_{G0}$ is energy-density.

 

[olgerm]

An infinite number.

Force = x(Predicted by Theory A) + (1-x)(Predicted by Theory B).

How many values of x can there be.

Newtonian gravity and GEM can not be mixed that way because Newtonian gravity is Galilei-invariant, but GEM is lorenz-invariant.

1 interesting thing about GEM is that GM field itself has energy density ($J_{G0}=E_{G0}^2/2+B_{G12}^2/2+B_{G13}^2/2+E_{B23}^2/2$). Would it not cause some positive feedback runaway loop that causes energy density to increase unlimitlessly? where GM-field creates more energy density and more energy-density creates more GM field?

 

[pervect]

Is there a gravitational analog of electromagnetic force?
[Apart from the obvious "static" forces, ie electrostatic between fixed charges and gravitational between fixed masses.]
I am thinking of the classic situation of a moving charge (or current) creating a magnetic field which then affects other moving charges. So a question by analogy, does a moving mass have an additional field which is only detected by other moving mass ? It would be very very small I guess.

There is a way to break down, or decompose, the curvature tensor in General relativity (called the Riemann tensor) into parts, which can be loosely interpreted as an "electric part", a "magnetic part", and a "topological (curvature) part". This is a bit different than the GEM formalism, also known as Gravitoelectromagnetism, which has also been mentioned in this thread, in that it works even in the strong field case, rather than being a linear approximation.

The first two parts would probably be described in popularizations as a 'force'. (That's not 100% accurate). The last part would be described in popularizations as "the curvature of space" rather than as a force. (That description is similarly not 100% accurate). I personally think it's close enough to not be horribly misleading, which is pretty good for a popularization about General Relativity.

The original paper is in French and I haven't been able to get a hold of it or a translation (or even the original). See the wiki article https://en.wikipedia.org/wiki/Bel_decomposition for the Wiki reference. Unlike the GEM formalism, the decomposition works even in strong fields. There are some discussions in "Gravitation" which break the Riemann tensor into different parts, but they don't mention Bel by name.

To perform the decomposition, one needs to define what might be called "the observers flow of time", more formally a timelike congruence, or a unit timelike vector field. This additional information given by the flow of time gives the necessary information to sensibly decompose spacetime as a unified entiity into a spatial part and a time part, and similarly how to decompose the electromagnetic field into an "electric" part and a "magnetic" part.

Nasa's descritpion of Gravity Probe B disucssses the use of the GEM analogy in the context of the Gravity probe B experiment, though I find their presentation a bit murky. If you are not familiar, gravity probe B was designed to detect frame dragging effects predicted by GR, effects which are even smaller than the GEM effects, which are also present and also measured by the GPB experiment. To give an idea of the magnitude of the effects, it would take only about 200,000 years for the GPB gyroscopes to precess through a full 360 degree arc due to the geodetic effect.

 

[robphy]

The original paper is in French and I haven't been able to get a hold of it or a translation (or even the original). See the wiki article https://en.wikipedia.org/wiki/Bel_decomposition for the Wiki reference. Unlike the GEM formalism, the decomposition works even in strong fields. There are some discussions in "Gravitation" which break the Riemann tensor into different parts, but they don't mention Bel by name.

Is this the paper you seek? From the WIkipedia article....
Bel, L. (1958), "Définition d'une densité d'énergie et d'un état de radiation totale généralisée", Comptes rendus hebdomadaires des séances de l'Académie des sciences 246: 3015
It is hyperlinked to
https://gallica.bnf.fr/ark:/12148/bpt6k723q/f965.item.langEN

from poking around... this also may be of interest:
https://www.numdam.org/item/SJ_1958-1959__2__A12_0.pdf
LOUIS BEL
La radiation gravitationnelle
Séminaire Janet. Mécanique analytique et mécanique céleste, tome 2 (1958-1959),
exp.no12, p.1-16
http://www.numdam.org/item?id=SJ_1958-1959__2__A12_0

Here are some translations (by a classmate from grad school):

https://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/bel_-_energy_density.pdf
https://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/bel_-_grav._rad..pdf
https://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/bel_-_curvature_tensor.pdf

 

[robphy]

This reminds me of a very old post of mine (reconstructed here because the old-style LaTeX on PF appears broken):

robphy (Dec 1, 2004)
in What are the implications of GR that are not captured by gravitomagnetism?

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: $$F^{ab}{}_{;b}=J^a$$
whereF^{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?

It might be interesting to see if one could formulate a massless spin-3 analogue of these equations from the appropriate Bianchi identity. If not, what goes wrong?

---

related followup references by me to the 2004-post:

(2005) in There is no gravitational dipole

Note that the equations above are not approximations. They are algebraically related to the Einstein Equations. In addition, like the electromagnetic tensor F, the Weyl tensor C can be decomposed into "electric" and "magnetic" parts using an observer's 4-velocity.

(2016) in Is Maxwellian Gravitoelectromagnetism a useful analogy?

[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)

(2018) in Why Not Develop Relativistic Gravitational Theory Analogy to Electromagnetism?

 

[olgerm]

1 of famous examples where predictions of Newtonian-gravity and GR differ is perihelion precession of Mercury. Would predictions from GEM-model agree more with predictions of Newtonian-gravity or predictions of GR? GEM surely would predict Lense–Thirring precession. But would the overall prediction of GEM be more similar to that of Newtonian gravity or that of GR?

 

[pervect]

I was doing some more reading, and apparently the "modern" approach to the topic involves decomposing the Weyl tensor rather than the Riemann tensor. The process I was refering too, which I described colorfully as requiring "the flow of time of an observer" is more formally called the "3+1 decomposition of a tensor", in this case the Riemann or Weyl tensors. I don't use the Weyl tensor much - I've read about it but it doesn't play a large part in my intuition.

MTW only briefly mentions the topic in an exercise, $14.14 and $14.15 (at least, that's all I could find). They basically just re-organize the non-zero components of the Riemann in an orthonormal basis into four 3x3 matrices. Two of the four resulting parts (which are considered the magnetic parts) are the negative-transpose of each other, which leaves us with three parts. In a vacuum, the remaining two parts (which can be considered as the electric and topological parts) are equal, though they are not equal in general.

Two or three 3x3 matrices are a lot easier to get an intuition for than the 4x4x4x4 entity that's the Riemann. Each 3x3 matrix has at most 9 components - the 4x4x4x4 tensor has 256 components, most of which are zero or repeats of each other due to the high degree of symmetry the tensor has.

Using the modern approach, one first strips out "trace" components from the Riemann to get the Weyl, then the Weyl can be divided into "electric" and "magnetic" parts.

The organization of components approach is similar to how one splits the electromagnetic Faraday tensor into magnetic and electric parts - for instance, the electric field is just $E_{\hat{i}} = F_{0\hat{i})$ (with possible unit conversions, E being the electric field and F the faraday tensor). The more mathematically mature approach uses the idea of the Hodges dual, also known as the star operator, which doesn't need a specific basis choice.

 

[PeterDonis]

the 4x4x4x4 tensor has 256 components, most of which are zero or repeats of each other due to the high degree of symmetry the tensor has.

The number of independent components of the Riemann tensor in a general curved spacetime is 20. In spacetimes with symmetries, the number is smaller.

If we split the Riemann tensor into the Weyl tensor and the Ricci tensor, each of the latter two has 10 independent components (half of the total of 20). In vacuum, the Ricci tensor vanishes, so those 10 independent components are all zero, which of course simplifies things a lot.

General spacetimes can be classified according to the properties fo the Weyl tensor; this Wikipedia article gives a decent overview:

https://en.wikipedia.org/wiki/Petrov_classification

 

[pervect]

The organizational structure in MTW actually illustrates why there are 20 degrees of freedom and 21 unique (up to sign) nonzero components in the Riemann.

MTW start with the Riemann $R^{\hat{a}\hat{b}}_{\hat{c}\hat{d}}$ as per MTW's exercise $14.14, Note that this is presumed to be in an orthonormal basis, MTW uses the "hats" to convey this important information symbolically but I'll mention that as people not familiar with the books conventions wouldn't necessarily know that.

We start out with considering the "left half", $R^{\hat{a} \hat{b}}$ and the "right half" $R_{\hat{c}\hat{d}}$. Which are equivalent as far as numerical component values go, because the basis inner product $n_{\hat{a}\hat{b}}$ is diag(-1,1,1,1), and we multiply by it twice to raise and lower both indices which gives us an identity matrix for the numerical values of the components.

It's unclear to me why MTW uses $R^{\hat{a}\hat{b}}{}_{\hat{c}\hat{d}}$ rather than $R_{\hat{a}\hat{b}\hat{c}\hat{d}}$ though, to be honest.

Each half has two components belonging to the set (t,x,y,z) or (0,1,2,3) depending on your choice of notation (MTW uses the numeric indices). So a pair has 16 possible values.

So far, we've simply reorganized the 4x4x4x4 Rimenn as a 16x16 matrix, we haven't gained much yet.

But terms in each half with a repeated index are zero due to symmetry, so that leaves only 12 nonzero components. And (Anti)symmetry lowers this to six.

So, we've gone from a 4x4x4x4 display with 256 components to a 6x6 display with only 36 components. Which is a lot of progress, but we're not done yet.

In this 6x6 display, there are 6 diagonal elements and 30 off-diagonal elements. The off-diagonal elements are symmetrical (up to sign?). So that's 6+15=21 unique components.

The 21 components are not all independent, though - there's an identiy that lowers this from 21 to 20.

The 6x6 matrix form is further split up into 4 3x3 matrices. (That's back to the 36).

It looks like

$$\begin{bmatrix}E & H \\-H^T & F\\ \end{bmatrix}$$

The electrogravitic part is the "upper left" of the 4 3x3 matrices in MTW's scheme, each 3x3 matrix in the electrogravitic part has component pairs like tx, ty, and tz, pairs that include a time coordinate.

For a flow/vector field that points strictly along the coordinate time, the electrogravitic part is the only source of geodesic deviation, i.e. tidal forces. So a static frame for a static arrangment of masses only has the E part.

The split between E, H, and F is based on the number of "time" components. E has two, H has one, F has zero.

In Bel's original, F is named the topogravitic part, but the "modern" approach organizes things a bit differently as my recent reading mentioned. But I am still used to the "old" split.