Electromagnetism : Electromagnetism in General Relativity

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7

7.1 Exact Gravitation of Extremely Charged Matter and the Electric Field

Lets start this section with the well known exact solution to Einstein's field equations, for a nonrotating point charge and the stress-energy tensor for its surrounding electric field, called the Reissner-Nordstrom solution. It is commonly expressed in the coordinates expressed in the first line here but the choice of coordinates for which to express it are not unique. For example one may express it in isotropic coordinates through the following transformation:

(7.1.1a,b,c,d,e)

For the purpose of comparison to other exact charge solution to Einstein's field equations introduced in this section, let us instead do the transformation:

(7.1.2a,b)

This results in the Reissner-Nordstrom solution being expressed as

(7.1.3)

The stress-energy tensor for for which this is the exact solution to Einstein's field equations is

(7.1.4)

Note that this looks like the stress-energy tensor of special relativity for the electric field of a point charge except for the

(1+GM/rc2)4 denominators which are an artifact of the coordinates we have chosen to express it with.

Akin to the isotropic coodinate expression of the Reissner-Nordstrom solution, there is an isotropic coordinate solution for an infinite charged sheet for a particular pressure state in the sheet which is given by

(7.1.5)

having a mass per area and charge per area . In the case that the charge per area is zero this reduces of course to a vacuum solution for .

For different pressure in the sheet, the exact solution to Einstein's field equations for the stress-energy tensor of an electric field off of z = 0 given by

(7.1.6)

is

(7.1.7)

where is a constant, a coefficient of the pressure terms of the sheet itself and is 1 in the case that transforms to 7.1.5. This 7.1.7 is the correct solution I have found for what is called an infinite charged "domain wall" with uniform pressure the same along x as the y direction.

Consider next the stress-energy tensor for an electric field from an infinite line charge off of r = 0 given by

(7.1.8)

The exact solution to Einstein's field equations for this is

(7.1.9)

This is the correct solution I have found for what is called an infinite charged "cosmic string".

Note that this becomes an exact vacuum solution for , the charge per length, being 0 as expected, however what shocked me was that even given that fact, there exists a frame transformation that transforms 7.1.9 into

(7.1.10)

which is the extreme charge case of 7.1.9. This means that with the presence of any charge at all, one can not actually distinguish the charge per length from the mass per length of the infinite line charge. This indicates that general relativity's static solutions are actually charge solutions, that in the spacetime structure of Einstein's field equations there is a fundamentally built in classical unification of mass and charge. It is an already classically unified theory of gravity and electromagnetism. What you have is a core extreme charge solution of

 

(7.1.11a,b)

And the type of perturbance to the line elements 7.1.3, 7.1.7, and 7.1.9

are an allowed disturbance of the spacetime geometry from that of the extreme charge solution that maintains the line element as an electric field solution, yet allows an observer to find a difference in the "observed charge" q, from the real source gravitational charge M.

The rank two electric field tensor which generates the stress-energy tensor for which 7.1.11 is the exact solution is

(7.1.12a,b)

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

(7.1.13)

(7.1.14)

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

The exact solutions described above are not in the chronological order in which I discovered them. The process started with the first step and key to finding the above solution, which was working out the solution below for a uniform electric field in nonsingular coordinates along the direction of a coordinate axis:

(7.1.15)

where I am defining the constant in accordance with

(7.1.16)

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.

(7.1.17a-j)

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

Upon finding the uniform field solution it became apparent in the calculation of geodesic motion that the constant could be interpreted as a gravitational acceleration field whose strength was proportional to the electric field, and which lay in the same direction.

After this I went looking for an exact solution that allowed the electric field to vary through space. The hint suggesting to me how to find this came from expressing this uniform solution along the one axis in isotropic coordinates.

Doing the following coordinate transformation

(7.1.18)

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

(7.1.19)

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 (the Majumdar-Papapetrau solution)

(7.1.11a,b)

That was the actual process leading me to the exact solution for static extreme charge of arbitrary distribution 7.1.11, and I came to the other solutions above later. I later found that the extreme charge case was first discovered by Majumdar and Papapetrau.

The exact calculation of geodesic motion for this 7.1.11 spacetime is

(7.1.20a,b,c)

where 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.

As discussed above where it is noted that there is a transformation that takes 7.1.9 to 7.1.10, due to this especially, and the differences in the three solutions 7.1.3, 7.1.7, and 7.1.9 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, an emergent property of electric fields effects on macroscopic spacetime.

 

It turns out that for extremely charged static matter distributions the rank 2 electric field tensor equation 7.1.12a can be related to the Christoffel symbols which are the gravitational acceleration field for general relativity by

(7.1.21)

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 7.1.12a. Consider the time like Killing vector T = (1,0,0,0). The covariant derivative of the timelike Killing vector is equation 4.4.8

For these coordinates this reduces to

We now can write 7.1.7 as

(7.1.22)

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

(7.1.23)

Here I've absorbed 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.

(7.1.24)

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

(7.1.25)

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 the matter of unification will require some postulation so from here that discussion will be continued in the fringe physics unit at the end of the chapter on Unification theories.

The next step in solving for the metric for a more general case of electromagnetism may be to add time dependence. Though the complete generalization to a time dependent electromagnetic solution, that would reduce to equation 7.1.1 for a static potential, is as yet unknown, any satisfying the ordinary wave equation yields a zero Ricci-scalar for a metric given by

(7.1.26)

Which suggests that transformations yielding different exponents on the coefficients of the line elements differentials may lead to solutions that can incorporate time dependence.

I have found that

(7.1.27)

is an exact vacuum solution and since

n(z+ct)+m(z-ct) = az+bct

(7.1.28)

describes colliding right and left moving wave forms, and the Riemann tensor is not zero for , 7.1.27 is an exact vacuum solution for head on colliding gravitational plane wave forms.

So, consider the line element

(7.1.29)

The Ricci-scalar for the line element is

(7.1.30)

which gives zero Ricci-scalar solutions for

(7.1.31)

So for example let

f = az+bct - 2gzct + kz2 + kct2

(7.1.32)

The Ricciscalar is zero and the resulting stress-energy tensor is

(7.1.33)

Trying to model a static electric field at right angle to a static magnetic field with a "static" metric yield's questionably valid metrics as the results. Here we see an electric field can be modeled at right angle to a magnetic field of equal magnitude with equation 7.1.29, but the metric itself becomes time dependent.

The full stress-energy tensor for which equation 7.1.29 is the exact solution is

(7.1.34)

where f is a function of ct and z.

In general as mentioned in chapter 6 it would be appropriate to repeat in this section that the stress-energy of an electromagnetic field is

(7.1.35)

which in terms of E&B in Cartesian inertial frame coordinates for flat spacetime is

(7.1.36-38)

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 7.1.11 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?

______________________________________________________________________________

7.2 Maxwell's Equations

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

(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 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 does.

Consider a a four-vector current and four-vector potential for electromagnetism where in special relativity and Lorentz guage we have

(7.2.2a)

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

(7.2.2b)

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

(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

and if one inserts the unprimed frame observers four-vector velocity then it can be written as a product of tensors

, but even so is still not a tensor. As is the four-vector velocity of whoever is the observer, everyone would use for his own 4-velocity , as a result and then the expression does not transform as a four-vector.

. If 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 is the electromagnetic field already expressed according to his own frame. Likewise the magnetic field given by

which can then be written in terms of the unprimed frame observers own four-vector velocity as

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

(7.2.4)

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

(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

(7.2.7a)

Where the fourth rank Levi-Chivita tensor is given by 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

(7.2.8)

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

.

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

(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 , , and . 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

______________________________________________________________________________

7.3 Larmor Radiation and the Abraham-Lorentz Formulae

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

(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.

(7.3.2)

This is called the Larmor radiation formulae.

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

.

86 Chapter 7 Electromagnetism

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

(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.

.

(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.

(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

(7.3.6)

The should be intuitive. The term is inserted so that the four-vector obeys an energy conservation property of force four-vectors . 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

(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.

 

Citations References

Bertotti, B., 1959, "Uniform Electric Field in the Theory of General Relativity," Phys. Rev. 116, 1331-1333

Gamow, G., 1967, "Electricity, Gravity, and Cosmology," Phys. Rev. Lett. 19, 759-761

Gupta, S. N., 1954, "Gravitation and Electromagnetism," Phys. Rev. 96, 1683-1685

Israel, W., 1968, "Event Horizons in Static Electrovac Space-Times," Commun. Math. Phys. 8, 245-260

Kuchar, K., 1968, "Charged Shells in General R. and Their Gravitational Collapse," Czech J. Phys. B 18, 435-463

Misner, C. W., and J. A. Wheeler, 1957, "Classical Physics as Geometry: Gravitation, Electromagnetism, Unquantized Charge, and Mass as Properties of Curved Empty Space," Ann. Phys. 2, 525-603

Robinson, I., 1961, "Null Electromagnetic Fields," J. Math. Phys. 2, 290-291

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