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7.1 Exact Gravitation of Extremely Charged Matter and the Electric Field

Lecture

The exact solution to Einstein's field equations for nonmagnetic extremely charged matter of arbitrary static configuration is

(7.1.1a,b)

And the corresponding rank two electric field tensor is

(7.1.2a,b)

The stress energy of the electromagnetic field tensor and the stress energy of the sources can be expressed in the stress energy tensor

(7.1.3)

(7.1.4)

F is the gravitational potential, equal in magnitude to the electric potential converted to the same units.

The exact solution to Einstein's field equations that was the first step and key to finding the above solution was the solution below for a uniform electric field in nonsingular coordinates along the direction of a coordinate axis.

ds² = {exp[2(a/c²)z]}dct² -{exp[-2(a/c²)z]}dx² -{exp[-2(a/c²)z]}dy² -{exp[-4(a/c²)z]}dz²

(7.1.5)

|a| = |E |sqrt(4pGe₀)

(7.1.6)

This uniform field solution also works to describe a uniform magnetic field as long as the magnetic field is in the same direction as the electric field.

(7.1.7a-j)

and the electromagnetic field tensors for either solution obey the general relativistic version of Maxwell's equations just above.

Upon finding the uniform field solution it immediately it became apparent that there is a gravitational acceleration of a whose strength was proportional to the electric field, which lay in the same direction.

The next generalizing step would be to allow an arbitrary orientation to the field. An appropriate coordinate choice allows such a uniform field solution to be expressed as

ds² = {exp[2(a_x/c²)x + 2(a_y/c²)y + 2(a_z/c²)z]}dct²

-{exp[-4(a_x/c²)x - 2(a_y/c²)y - 2(a_z/c²)z]}dx²

-{exp[-2(a_x/c²)x - 4(a_y/c²)y - 2(a_z/c²)z]}dy²

-{exp[-2(a_x/c²)x - 2(a_y/c²)y - 4(a_z/c²)z]}dz²

|a_i | = |E_i |sqrt(4pGe₀)

(7.1.8a,b)

This represents then the exact solution to Einstein's field equations for general relativity for a uniform electric field of arbitrary orientation in nonsingular coordinates.

The next step of generalization for general relativity was to find the exact solution that allowed the electric field to vary through space. The hint suggesting how to find this came from expressing the uniform solution along one axis in isometric coordinates.

Doing the following coordinate transformation

exp[(a/c²)z] ® (1-az/c²) ^-1

(7.1.9)

The electric field solution along the z axis can then be expressed as

ds² = (1-az/c²) ^-2 dct² - (1-az/c²)² [dx² + dy² + dz²]

(7.1.10)

From that step it was merely a matter of calculating the Einstein tensor and Ricciscalar to verify that the exact solution to Einstein's field equations for arbitrarily configured matter of extreme charge is

(7.1.1a,b)

The exact calculation of geodesic motion for this spacetime is

(7.1.11a,b)

where g is a constant of the motion and the gravitational acceleration's relation to the gradient of this potential whose gradient also yields the electric field implied an equivalence of the gravitational and electric potentials when it is for extreme charge.

There are a couple of other isometric static spacetime solutions that exist for charged matter with extra mass perturbing the solution. For example, the Reissner-Nordstrom solution for a charged black hole can be expressed as

(7.1.12a,b,c)

(7.1.13)

In the case of extreme charge |e| = GM/c² , and the isometric coordinate expression for the Reissner-Nordstrom solution then becomes a case of my solution.

One other exact solution to Einstein's field equations I found which allows an isometric static spacetime with extra mass perturbing the solution is

(7.1.14)

This is the exact solution for a uniformly charged thin sheet of matter of mass per area s_m and charge per area s_q converted to the same units. In the case that the charge per area is zero this reduces of course to a vacuum solution for

z ¹ 0, and in the case of extreme charge s_m = |s_q| this, like the Reissner-Nordstrom solution, also reduces to a case of my solution for extreme charge distributions and the electric field.

The exact solution for an infinite line charge of arbitrary mass per length l_m perturbing the solution and charge per length converted to units of mass per length of l_q is

(7.1.15)

Interestingly this solution infers that two hypothetical line charges would yield the exact same spacetime geometry if

l²_q1/l_m1 = l²_q2/l_m2. And therefor one can't tell from the spacetime geometry itself if the line charge actually has a mass different from one of extremal charge.

Due to that fact especially as well as the differences in the above three solutions which allow extra mass above that of extremal charge, it is apparent that mass does not exist in nature independent of charge, but exists as a perturbance to the spacetime solution that keeps the Ricci-scalar zero for the space surrounding what is really just a charge source. That being the case, all of our weak field approximations for vacuum solutions are cast into doubt because the space around matter of various odd shapes must be solutions for the electric and magnetic fields extending from the matter, solutions for zero Ricci-scalar electromagnetic fields, but not actually vacuum solutions at all. It also brings into question the validity of any unification schemes such as strings which can describe gravitation as a mass-mass force exchange via gravitons. The real sources being charges should exchange real force through photon exchange. Gravitation is merely the large scale description of the geometry of spacetime corresponding to a given charge and current distribution.

The exact solution to Einstein's field equations for a static uniform electric field in the x direction at right angle to a static uniform magnetic field in the y direction is

(7.1.16)

I suspect that these are merely a coordinate transformation of the uniform electric field case and uniform magnetic field case equation 7.1.5, which results in the rise of the perpendicular field. What is interesting that was not obvious a-priori is that the theory here predicts a theoretical limit on how close the induced fields magnitude of the static uniform field can be to the magnitude of the other field. One can only come up to about 41.4% of the other in magnitude.

It turns out that for extremely charged static matter distributions the rank 2 electric field tensor equation 7.1.2a can be related to the Christoffel symbols which are the gravitational acceleration field for general relativity by

(7.1.17)

Where the overall sign is not determined by theory. This result for extremely charged static matter distributions is a unified gravitational-electric field giving a clue how gravity and electromagnetism are already unified more generally in general relativity. It demonstrates how the electric field tensor is equivalent to a gravitational field at least for this case of matter.

To start to generalize we must find a tensor that yields the Christoffel symbol on the right hand side of this equation for the coordinates used in equation 7.1.2a. Consider the time like Killing vector T = (1,0,0,0). The covariant derivative of the timelike Killing vector is equation 4.4.8

T^k;_n = T^k,_n + G^k_rnT^r

For these coordinates this reduces to

T^k;_n = G^k_0n

We now can write 7.1.7 as

F_mn = (c²/sqrt(4pGe₀g₀₀))g_mkT^k;_n

(7.1.18)

Next to write the right hand side as a tensor so that the relation is retained for all frames we make the replacement

g₀₀ = g_mnT^mTⁿ = T_mT^m.

F_mn = (c²/sqrt(4pGe₀T_lT^l))g_mkT^k;_n

(7.1.19)

Here I've absoarbed the overall sign into the definition of the timelike Killing vector so that changing the direction of the electric field or the standard sign of the charges is the same thing as doing a time reversal. Using this with the force equation 7.2.9 and using that in the nongeodesic equation 5.2.1b yields an equation of motion for a test charge that can be interpreted as 100% due to the spacetime geometry.

dU^l/dt = (q/m)(c²/sqrt(4pGe₀T_sT^s))(U^m/c)g^lrg_mkT^k;_r - G^l_mnU^mUⁿ

(7.1.20)

In principle this means that when the exact solutions for more general time dependent matter distributions will be found, one can find the equivalence between the electromagnetic field tensor and the Christoffel symbols and express general relativistic motion of charges in such a way that gravitation and electromagnetism are already a unified field in general relativity.

Though they are not as simple for this spacetime there one can generally find three more rank 1 Killing vectors. The covariant derivative of any Killing four-vector is by definition anti-symmetric

K_m ;_n + K_n ;_m = 0

(7.1.21)

which the electromagnetic field tensor is. So it makes sense that what acts like a vector potential whose covariant derivative yields the electromagnetic field here should be proportional to a killing vector.

Everything up to this point has been concrete results of mainstream general relativity, but to continue investigating this matter will require some postulation so from here the discussion will be continued in the fringe physics unit at the end of the chapter on Unification theories.

Though the generalization to a time dependent electromagnetic solution that would reduce to equation 7.1.1 for a static potential is as yet unknown, any F(ct,x,y,z) satisfying the ordinary wave equation and representing a standing wave yields a zero Ricci-scalar solution that is similar

ds² = (1-F/c²)²dct² - (1-F/c²)²(dx² + dy² + dz²)

(7.1.22)

Exercises

Problem 7.1.1

In the isometric coordinate r where are the locations that correspond to r' = 0 and the event horizons of the charged black hole, not extremely charged as well as extremely charged? Hint r' = 0 is not at r = 0, note the terms in the transformation equation that diverge there.

Problem 7.1.2

Does the extreme charge solution for arbitrary static matter distribution equation 7.1.1 allow extreme charged matter to have structure or extent when compacted under what would be the inner-outer horizon which have met up at the same location of an extreme charged black hole? Why or why not?

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7.2 Maxwell's Equations

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

(7.2.1)

These are correct for Einstein's special relativity theory, 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

(7.2.2a)

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

f^m ;ⁿ_n = cm₀J^m

(7.2.2b)

The vector potential gives rise to an electromagnetic field tensor for general relativity according to

F_mn = (f_m ;_n - f_n ;_m)

(7.2.3)

(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 7.2.8.

[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^m0lrF_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^m0lrF_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.]

83

84 Chapter 7 Electromagnetism

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

J^m = r₀U^m

(7.2.4)

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

(7.2.5)

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

(7.2.6)

The duel being related to the original by

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

(7.2.7a)

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 for general relativity as

F^mn;_n = cm₀J^m

D^mn;_n = 0

(7.2.8)

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 for general relativity is

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

(7.2.9)

7.2 Maxwell's Equations 85

Exercises

Problem 7.2.1

Show that Eqn.7.2.5 is equivalent to Eqn.7.2.8

Problem 7.2.2

Show that Eqn.7.2.8 generates Eqn.7.2.1

Problem 7.2.3

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 7.2.4

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

Problem 7.2.5

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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7.3 Larmor Radiation and the Abraham-Lorentz Formulae

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

(7.3.1)

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

(7.3.2)

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

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

(7.3.3)

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.

(Only for a zero Reimann tensor) (7.3.4)

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)

(7.3.5)

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.

7.3 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)

(7.3.6)

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)

(Only for a zero Reimann tensor) (7.3.7)

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

Exercises

Problem 7.3.1

Verify that Eqn.7.3.4 becomes Eqn.7.3.2 with the appropriate choice of transformation.

Problem 7.3.2

Verify that Eqn.7.3.7 becomes 7.3.5 with the appropriate choice of transformation.

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General Relativity