Some rough notes on GravitoElectroMagnetism. Peeter Joot October 26, 2008. Last Revision: Date : 2008/10/2904 : 01 : 33

Contents 1

Motivation.

1

2

Definitions.

1

3

STA form.

2

4

Lagrangians. 4.1 Field Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Interaction Lagrangian. . . . . . . . . . . . . . . . . . . . . . . .

4 4 4

5

Conclusion.

5

1

Motivation.

I found the GEM equations interesting, and explored the surface of them slightly. Here are some notes, mostly as a reference for myself ... looking at the GEM equations mostly generates questions, especially since I don’t have the GR background to understand where the potentials (ie: what is that stress energy tensor Tµν ) nor the specifics of where the metric tensor (pertubation of the Minkowski metric) came from.

2

Definitions.

The article [Mashhoon(2003)] outlines the GEM equations, which in short are Scalar and potential fields Φ≈

GM , r

A≈

1

GJ×x c r3

(1)

Guage condition 1 ∂Φ +∇· c ∂t



1 A 2



= 0.

(2)

GEM fields E = −∇Φ −

1 ∂ c ∂t



 1 B , 2

B = ∇×A

(3)

and finally the Maxwell-like equations are 1 ∂ ∇×E = − c ∂t   1 B =0 ∇· 2



1 B 2



(5)

∇ · E = 4πGρ   1 1 ∂E 4πG ∇× B = + J 2 c ∂t c

3

(4)

(6) (7)

STA form.

As with Maxwell’s equations a clifford algebra representation should be possible to put this into a more symmetric form. Combining the spatial div and grads, following conventions from [Doran and Lasenby(2003)] we have   1 ∂ 1 IB ∇E = 4πGρ + c ∂t 2   1 1 ∂E 4πG ∇ IB = + J 2 c ∂t c

(8) (9)

Or 

1 ∂ ∇− c ∂t



1 E + IB 2



=

4πG (cρ + J) c

(10)

Left multiplication with γ0 , using a time positive metric signature ((γ0 )2 = 1), 

∇−  But ∇ −

1 ∂ c ∂t



1 ∂ c ∂t



   1 4πG  γ0 −E + IB = cργ0 + J i γi 2 c

(11)

γ0 = γi ∂i − γ0 ∂0 = −γµ ∂µ = −∇. Introduction of a four

vector mass density J = cργ0 + J i γi = J µ γµ , and a bivector field F = E − 21 IB this is 2

∇F = −

4πG J c

(12)

The guage condition suggests a four potential V = Φγ0 + Aγ0 = V µ γµ , where V 0 = Φ, and V i = Ai /2. This merges the space and time parts of the guage condition

∇ · V = γ µ ∂ µ · γν V ν = ∂ µ V µ =

1 ∂Φ 1 + ∂i Ai . c ∂t 2

It is reasonable to assume that F = ∇ ∧ V as in electromagnetism. Let’s see if this is the case 1 ∂ E − IB/2 = −∇Φ − c ∂t



 1 B − I ∇ × A/2 2

1 = −γi ∂i γ0 V 0 − ∂0 Ai γi γ0 + ∇ ∧ A/2 2 i 0 0 = γ ∂i γ0 V + γ ∂0 γi Ai /2 − γi ∂i ∧ γ j V j

= γi ∂i γ0 V 0 + γ0 ∂0 γi V i + γi ∂i ∧ γ j V j = γ µ ∂ µ ∧ γν V ν = ∇∧V Okay, so in terms of potential we have the form as Maxwell’s equation

∇(∇ ∧ V ) = −

4πG J. c

(13)

With the guage condition ∇ · V = 0, this produces the wave equation

∇2 V = −

4πG J. c

(14)

In terms of the author’s original equation 1.2 it appears that roughly V µ = ¯h0µ , and J µ ∝ T0µ . This is logically how he is able to go from that equation to the maxwell form since both have the same four-vector wave equation form (when Tij ≈ 0). To give the potentials specific values in terms of mass and current distribution appears to be where the retarded integrals are used. The author expresses T µν in terms of ρ, and mass current j, but the field equations are in terms of Tµν . What metric tensor is used to translate from upper to lower indexes in this case. ie: is it gµν , or ηµν ?

3

4

Lagrangians.

4.1

Field Lagrangian.

Since the electrodynamic equation and corresponding field Lagrangian is J e0 c e c L = − 0 (∇ ∧ A)2 + A · J 2

∇(∇ ∧ A) =

Then, from 13, the GEM field Lagrangian in covariant form is

L=

c (∇ ∧ V )2 + V · J 8πG

Writing F µν = ∂µ V ν − ∂ν V µ , the scalar part of this Lagrangian is:

L=−

c F µν Fµν + V σ Jσ 16πG

Is this expression hiding in the Einstein field equations? What is the Lagrangian for newtonian gravity, and how do they compare?

4.2

Interaction Lagrangian.

The metric (equation 1.4) in the article is given to be   Φ 4 ds = −c dt + (A · dx) dt + 1 + 2 2 δij dxi dx j c c V 8 V 2 0 =⇒ ds = c2 (dτ )2 = (dx0 )2 − ∑(dxi )2 − 2 2 (dx0 )2 − 2 Vi dxi dx0 − 2 20 δij dxi dx j c c c i 2

2



Φ 1−2 2 c



2

With v = γµ dx µ /dτ, the Lagrangian for interaction is 1 ds 2 L = m 2 dτ 1 = mc2 2 1 mV = mv2 − 2 2 0 2 c

∑(x˙ µ )2 − µ

4

8m Vi x˙ 0 x˙ i c2

1 L = mv2 − 2m V0 ∑( x˙ µ /c)2 + 4Vi ( x˙ 0 /c)( x˙ i /c) 2 µ

! (15)

Now, unlike the Lorentz force Lagrangian

L=

1 2 mv + qA · v/c, 2

the Lagragian of 15 is quadradic in powers of x˙ µ . There are remarks in the article saying that the non-covariant Lagrangian used to arrive at the Lorentz force equivalent was a first order approximation. Evaluation of this interaction Lagrangian does not produce anything like the p˙ µ = κFµν x˙ ν that we see in electrodynamics. The calculation isn’t interesting but the end result for reference is  4m  (v · ∇V0 )γµ vµ + 2(v · ∇Vi )(vi γ0 + v0 γi ) 2 c  4m  + 2 V0 γµ aµ + 2Vi ( ai γ0 + a0 γi ) c ! 2m − 2 ∑(vµ )2 ∇V0 + 4v0 vi ∇Vi c µ

p˙ =

This can be simplified somewhat, but no matter what it will be quadratic in the velocity coordinates. The article also says that the line element is approximate. Has some of what is required for a more symmetric covariant interaction proper force been discarded?

5

Conclusion.

The ideas here are interesting. At a high level, roughly, as I see it, the equation

∇2 h0µ = T0µ has exactly the same form as Maxwell’s equations in covariant form, so you can define an antisymmetric field tensor equation in the same way, treating these elements of h, and the corresponding elements of T as a four vector potential and mass current. That said, I don’t have the GR background to know understand the introduction. For example, how to actually arrive at 1.2 or how to calculated your metric tensor in equation 1.4. I would have expected 1.4 to have a more symmetric form like the covariant Lorentz force Lagrangian (v2 + kA.v), since you 5

can get a Lorentz force like equation out of it. Because of the quadratic velocity terms, no matter how one varies that metric with respect to s as a parameter, one cannot get anything at all close to the electrodynamics Lorentz force equation m x¨ µ = qFµ ν x˙ ν , so the coorrespondance between electromagnetism and GR breaks down once one considers the interaction.

References [Doran and Lasenby(2003)] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, 2003. [Mashhoon(2003)] B. Mashhoon. Gravitoelectromagnetism: A brief review. Arxiv preprint gr-qc/0311030, 2003.

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