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Any physicist should recognize the following equations as Maxwell's equations in vacuum.

Ñ×E = r/e₀

Ñ´E = -¶B/¶t

Ñ×B = 0

Ñ´B = m₀J + m₀e₀¶E/¶t

(7.1.1)

These are correct for special relativity, but they are not globally correct for general relativity. Beginners should be careful here. Here r is not a mass density as discussed before. In the context of this section it is a charge density. The general relativistic version of these are of course tensor equations. The electric field by itself does not transform as a tensor and the magnetic field by itself does not either. However, the electromagnetic tensor F_mn does.

Consider a a four-vector current J^m = r₀U^m and four-vector potential for electromagnetism [f^m] = [f, f] where in special relativity and Lorentz guage we have

Ñ ²f^m - ¶ ²f^m/¶ct² = - cm₀J^m

(7.1.2a)

And in general relativity, the Lorentz guage expression for this becomes

f^m ;ⁿ_n = cm₀J^m

(7.1.2b)

The vector potential gives rise to an electromagnetic field tensor generally according to

F_mn = (f_m ;_n - f_n ;_m)

(7.1.3)

(Sign Convention)

The electromagnetic tensor in local Cartesian coordinates is Eqn. 6.3.11

Be aware that there is a sign convention chosen here for the electromagnetic field tensor that not all authors choose the same. It effects the order of terms in 7.1.3 and which index is contracted over in the first line of 7.1.8.

[note - the electric field is given by

E^m = F^0m

and if one inserts the unprimed frame observers four-vector velocity Uⁿ = (c,0,0,0) then it can be written as a product of tensors

E^m = F^0m = F^nmU_n/c, but even so E^m is still not a tensor. As Uⁿ is the four-vector velocity of whoever is the observer everyone would use (c,0,0,0) as a result and then the expression does not transform as a four-vector. E'^m = F'_m0 ¹ (¶x^l/¶x'^m)F_l0. If Uⁿ were the four-vector velocity of one "particular" observer then the expression would transform as a tensor, but then it wouldn't represent the electric field to anyone except that observer and it would then only when F_mn is the electromagnetic field already expressed according to his own frame. Likewise the magnetic field given by

B^m = (1/2)e^m0^lrF_lr/c

which can then be written in terms of the unprimed frame observers four-vector velocity Uⁿ = (c,0,0,0) as

B^m = - (1/2)e^m0^lrF_lr/c = - (1/2)e^mnlrF_lrU_n/c² is also not a tensor. An easy way to conceptually prove that it is not is to imagine a proton beam's magnetic field and note that it vanishes according to the proton frame. A tensor can not be transformed away.]

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84 Chapter 7 Electromagnetism in GR

The current density was related to a four-vector velocity for the current according to

J^m = r₀U^m

r₀ is an invariant which is the charge density according to a local frame moving with the bit of charge there. Now we have the tensors necessary in order to write general relativity's version of Maxwell's equations in vacuum. These are

F^mn;_n = cm₀J^m

F_mn;_l + F_nl;_m + F_lm;_n = 0

(7.1.5)

The electromagnetic field can also be expressed in the electromagnetic duel tensor. The electromagnetic duel tensor can be written in local Cartesian coordinates as

(7.1.6)

The duel being related to the original by

D^mn = (1/2)e^mnlr F_lr

(7.1.7a)

Where the fourth rank Levi-Chivita tensor is given by e⁰¹²³ and every even permutation equaling 1, and for every odd permutation equaling -1, and is zero otherwise.(This is also how any second rank antisymmetric tensor transforms.)

In terms of the electromagnetic duel tensor, Maxwell's equations can be written

F^mn;_n = cm₀J^m

D^mn;_n = 0

(7.1.8)

In special relativity the ordinary electromagnetic force on a charged particle is Eqn.3.2.3

f = q(E + u´B).

In general a tensor equation version replaces this equation. The electromagnetic four-force on a charged particle is

F^l = qg_mn(U^m/c)F^nl

(7.1.9)

7.1 Maxwell's Equations 85

Problem 7.1.1

Show that Eqn.7.1.5 is equivalent to Eqn.7.1.8

Problem 7.1.2

Show that Eqn.7.1.8 generates Eqn.7.1.1

Problem 7.1.3

Determine the transformation equations for the electric and magnetic fields for transformation between inertial frames boosting along the x¹ direction, given that the electromagnetic field tensor would transform according to the definition of a rank two covariant tensor in special relativity.

Problem 7.1.4

Work out F_m^m, D_m^m, and F_mnD^mn. The results are invariants.

Problem 7.1.5

Show that the covariant duel of the contravariant duel of the a covarient antisymmetric tensor of rank two is the original tensor times -1. i.e. that

F_mn = - (-1/2)e_mnab (-1/2)e^ablr F_lr

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Take a global inertial frame in which a charged particle has an acceleration of a along the

q = 0 (z-axis) and is instantaneously at rest. According to this frame, the power radiated per area S will be given by

S = (1/4pe₀)(q²/4pc³)(a/r)²sin²q

(7.2.1)

Notice that this is symmetric with respect to z.

Integrating this around a closed surface area enclosing the charge results in the proper frame power radiated.

P = (1/4pe₀)(2/3)q²a²/c³

(7.2.2)

This is called the Larmor radiation formulae.

Now consider the invariant, h_mnA^mAⁿ . It is easy to show that this frame results in

a² = -h_mnA^mAⁿ .

86 Chapter 7 Electromagnetism in GR

But the contraction of any tensor is an invariant and so the proper frame power can be written more generally

P = -(1/4pe₀)(2/3)q²h^mnA^mAⁿ/c³

(7.2.3)

In this form it is evident that the proper power radiated is an invariant associated with the contraction of the acceleration four-vector. Now one might jump to the conclusion that the proper power radiated is also the invariant associated with the contraction in general relativity.

P = -(1/4pe₀)(2/3)q²g_mnA^mAⁿ/c³.

(not always right) (7.2.4)

Unfortunately, this conclusion is not justifiable because of the following. We assumed that we could find a global inertial frame in which the particle was instantaneously at rest. In such a global inertial frame, S was symmetric across the z axis taking a known equation form that was integrated to get the formulae for the proper frame power radiated. In general relativity it is not always the case that such a global inertial frame exists. There are cases where the local inertial frame in which the particle is instantaneously at rest even nearby still has a tidal gradient associated with a nonzero Riemann tensor. In this case, the local inertial frame observes a gravitational field that is not zero a finite extent away from the charged particle. But this would lead to a Doppler shift in the radiation field of the particle. In that case we no longer have the symmetry along the z axis nor a general expression for S and so the integral for the proper power radiated no longer results in a² which gave us the invariant resulting above.

Since this result breaks with the result of special relativity, this superficially seems to defy the semi-strong level of the equivalence principle, but this level of equivalence is actually a reference to the results of local experiments. In order to find the correct power radiated one would need to know the behavior of the electromagnetic fields far from the charge and do the integral of S far away. So in this experiment the effect is due to the remote behavior of the electromagnetic fields. Even if the measurements are taken locally, the experiment is intrinsically remote. Therefor, the semi-strong level of equivalence is not even a reference to this kind of experiment.

At speeds small compared to the speed of light there is a formulae that is intended to give the force on a charged particle due to the emission of its radiation. This formulae is called the Abraham-Lorentz formulae.

f_rad = (1/4pe₀)(2/3)(q²/c³)(da/dt)

(7.2.5)

In my opinion, this formulae and its implications are not well understood by any physicist to date. Even so, it is not too difficult to express this relativisticly as a tensor equation in the case of special relativity.

7.2 Larmor Radiation and the Abraham-Lorentz Formulae 87

The special relativistic tensor equation version is

F^l_rad = (1/4pe₀)(2/3)(q²/c³)(dA^l/dt + U^lh_mnA^mAⁿ/c²)

(7.2.6)

The dA^l/dt should be intuitive. The U^lh_mnA^mAⁿ/c² term is inserted so that the four-vector F^l_rad obeys an energy conservation property of force four-vectors (g_mlU^mF^l = 0) . At speeds much less than the speed of light, this does reduce to the ordinary force expression above it.

Again, one might suppose from this that the general relativistic version should be

F^l_rad = (1/4pe₀)(2/3)(q²/c³)(dA^l/dt + G^l_mnU^mAⁿ + U^lg_mnA^mAⁿ/c²)

(not always right) (7.2.7)

but again this would be unjustified for the same kind of reason in the Larmor radiation discussion above.

Problem 7.2.1

Verify that Eqn.7.2.4 becomes Eqn.7.2.2 with the appropriate choice of transformation.

Problem 7.2.2

Verify that Eqn.7.2.7 becomes 7.2.5 with the appropriate choice of transformation.