A Scalar-Tensor Theory of Electromagnetism Fredrick W. Cotton

http://sites.google.com/site/fwcotton/em-26.pdf [email protected]

Abstract

The addition of a scalar term to the Maxwell stress-energy tensor is sucient to describe force-free, spherically symmetric charge distributions in both at space and curved space. The solutions can be constructed so that they are mathematically well-behaved. The electromagnetic rest mass agrees with the gravitational rest mass.

© 2014-2015 Fredrick W. Cotton, rev. 31 January

PACS numbers: 03.50.De, 04.40.Nr

1. Introduction

There is a rich history of adding scalar elds to General Relativity beginning with Einstein's cosmological constant as the simplest case. In this paper, we will consider a dierent approach that turns out to be much simpler. We will add a scalar term to the Maxwell electromagnetic stress-energy tensor. The result will be that for any given spherically symmetric charge distribution, the scalar term is uniquely dened by the condition that the electric eld be force-free everywhere. We will derive the results in at space, then show that the curved space metric can be obtained by solving the Einstein-Maxwell equations. The electromagnetic rest mass agrees with the gravitational rest mass. More complicated examples, including examples with nonlinear dielectric functions as in Born and Infeld [1], can be found in an earlier paper by the author [2]. We will use SI units. In a sense, adding a scalar term to the Maxwell stress-energy tensor can be regarded as simply moving a generalized cosmological term from the gravitational side of the Einstein-Maxwell equations to the electromagnetic side. However the physical interpretation is dierent. Adding a scalar term to the Maxwell stress-energy tensor is the simplest way of obtaining force-free solutions that have a mathematically well-behaved self-energy. Both classically and quantum mechanically, the problem of innite self-energies was considered by many physicists to be a signicant problem until it was eventually swept under the rug by renormalization procedures in quantum eld theory. The Born-Infeld [1] approach was one way of attempting to look at nite self-energies in the context of both classical electromagnetic theory and General Relativity. The approach here is simpler and opens up a broader realm of mathematically well-behaved solutions which we hope can be used to gain new insights in aspects of black hole theory and quantum eld theory that are presently obscured by singularities. 2. Maxwell's Equations

In 4-dimensional notation, the electromagnetic elds and the current density are dened by fµν = Aν ,µ − Aµ,ν = Aν ;µ − Aµ;ν µ

j = 0 f

µν ;ν

(2.1a) (2.1b)

In terms of the 3-dimensional potentials, Aµ = c(A, − φ) . We will dene the stress-energy tensor and the force density T µν = 0 f µτ f ντ − g µν ( fµ = −Tµ

ν ;ν

1

1 4

0 fκτ f κτ − Q)

(2.2a) (2.2b)

For a spherically symmetric charge distribution with metric ds 2 = gµν dx µ dx ν = dr 2 + r2 dθ2 + r2 sin2 (θ)dϕ2 − c2 dt 2

(2.3)

let Z φ(r) = −

dr fe (r)

A=0

(2.4)

Then in vector notation, where er is the unit vector in the radial direction, (2.5a) (2.5b)

E = fe (r)er ρ(r) = 0 r

−2

2

{∂r [r fe (r)]}

The force-free condition fµ = 0 can be expressed as Q0 (r) = 0 fe (r)[2fe (r) + rfe0 (r)]r−1

which gives Q(r) = 21 0 fe2 (r) − 20

Z

∞

dr 0 (r0 )−1 fe2 (r0 )

(2.6) (2.7)

r

Note that the expression for Q(r) is an integral expression in the electromagnetic eld rather than a local expression. The limits of the integral have been chosen to insure the correct asymptotic behavior as r → ∞. We will show later that Q(r) is local in terms of the electromagnetic eld and the curved metric. It will then be the relation between the electromagnetic eld and the metric that will be an integral expression. Whether this makes a philosophical dierence, we leave to the reader. Perhaps it will give some insight into the question of how to quantize the theory. Looking ahead to curved space, we dene the energy density and the rest mass energy as En(r) = −T 44 = 12 0 fe2 (r) − Q(r) = 20 m0 c2 =

Z

∞

0

dr r2 Z ∞

= 8π0

Z

∞

dr 0 (r0 )−1 fe2 (r0 )

Z 2π dθ sin(θ) dϕ En(r) 0 0 Z ∞ dr r2 dr 0 (r0 )−1 fe2 (r0 ) Z

(2.8a)

r

π

0

(2.8b)

r

At r = 0 , we must have fe (0) = 0 . For a non-zero charge, we must also have lim fe (r) = q(4π0 r2 )−1

r→∞

(2.9)

In order to minimize any disagreement with experimental results in the far eld, we will require that the limit in (2.9) be approached exponentially rather than polynomially. If there is an external constant electric eld E 0 and if we assume that the external eld does not, to a rst approximation, modify Q(r); then the self force is still zero and the total external force is the usual qE 0 . This assumption is necessary in order to preserve the standard results of classical theory. Suciently strong external electric elds will undoubtedly modify Q(r). Perhaps quantizing the theory and looking at the interaction with the quantum vacuum will give some insight. Perhaps the theory of chaotic attractors will give some insight. Perhaps we need the results of appropriate experiments. For any given total charge q , it is obviously possible to construct to construct mathematically well-behaved charge distributions with dierent rest masses. An example of a charged particle is fe (r) = q(4π0 )−1 {1 − [1 − λ(r/r0 )3 ] exp[−(r/r0 )3 ]}r−2 m0 c2 = q 2 (108π0 r0 )−1 [(18 + 6λ)(2 − 21/3 ) + 21/3 λ2 ]Γ(2/3)

(2.10a) (2.10b)

An example of a neutral particle is fe (r) = βr exp(−r/r0 ) 2

m0 c = 2β

© 2014-2015 Fredrick W. Cotton

2

2

π0 r05

(2.11a) (2.11b)

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These are classical eld structures. We have not attempted to quantize them. The details of the construction are arbitrary so long as they obey the boundary conditions. In the next section we derive the metric that solves the Einstein-Maxwell equations. This should have implications for various stellar structures including some types of black holes. We have not attempted to look at those implications. 3. Einstein-Maxwell Equations

The Einstein-Maxwell equations are gµν R = 8πGc−4 Tµν

1 2

Gµν = Rµν −

(3.1)

where G is Newton's gravitational constant. In the rest frame of the particle, let the metric be given by ds 2 = gµν dx µ dx ν = fg−1 (r)dr 2 + r2 dθ2 + r2 sin2 (θ)dϕ2 − c2 fg (r)dt 2

(3.2)

Then the non-zero components of Tµν and Gµν are T44 = −c

2

fg2 (r)T11

∞

Z

2

dr 0 (r0 )−1 fe2 (r0 )

= 2c 0 fg (r)

(3.3a)

r

T33 = sin2 (θ)T22 = 0 r2 [fe2 (r) − 2

∞

Z

dr 0 (r0 )−1 fe2 (r0 )] sin2 (θ)

(3.3b)

r

G44 = −c2 fg2 (r)G11 = −c2 r−2 fg (r)[−1 + fg (r) + rfg0 (r)] 2

G33 = sin (θ)G22 =

[rfg0 (r)

+

2 1 2 00 2 r fg (r)] sin (θ)

(3.3c) (3.3d)

Equations (3.3) reduce to ∞

Z

dr 0 (r0 )−1 fe2 (r0 ) Z ∞ r2 fg00 (r) = 8πGc−4 0 r2 [fe2 (r) − 2 dr 0 (r0 )−1 fe2 (r0 )]

−1 + fg (r) + rfg0 (r) = −16πGc−4 0 r2

(3.4a)

r

rfg0 (r) +

1 2

(3.4b)

r

Integrating (3.4b) and substituting into (3.4a) gives −4 −1

fg (r) = 1 − 16πG0 c

Z

r

r 0

0 2

Z

∞

dr (r )

r0

0

dr 00 (r00 )−1 fe2 (r00 )

(3.5)

Comparison with the Schwarzschild metric, for which fg (r) = 1 − 2Gm0 c−2 r−1 , shows that m0 = 8π0 c

−2

Z

∞

dr r 0

2

Z

∞

dr 0 (r0 )−1 fe2 (r0 )

(3.6)

r

which agrees with (2.8b). The entire rest mass is electromagnetic. Furthermore, for the example in (2.10), we can show that in the far eld the metric approaches the Reissner-Nordström metric exponentially. lim fg (r) = 1 − 2Gm0 c−2 r−1 + q 2 G(4π0 c4 r2 )−1 + exp[−(r/r0 )3 ]O[r−2 ]

r→∞

(3.7)

From (2.7) and (3.4a). we can show that Q(r) = 12 0 fe2 (r) + c4 (8πGr2 )−1 [fg (r) + rfg0 (r) − 1]

(3.8)

Thus Q(r) can be expressed as a local function in terms of the electromagnetic eld and the curved metric. Perhaps a quantized theory will show that setting the self-force equal to zero is valid only on average and is constrained by some form of the Heisenberg uncertainty relation. Since the standard form of Maxwell's equations and Einstein's equations are both derivable from Lagrangians, it is trivial to say that the simplest Lagrangian for Q(r) is the dierence. However, the actual Lagrangian will depend on the physical content of Q(r), including the behavior in the presence of strong external elds and high

© 2014-2015 Fredrick W. Cotton

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accelerations. One open question is whether the physical content is the same or dierent for black holes and particle wave functions in quantum eld theory. 4. Conclusions

We have shown that by adding a scalar term to the Maxwell stress-energy tensor, we can construct force-free, spherically symmetric charge distributions in both at space and curved space. The details of the construction are arbitrary so long as they obey the boundary conditions. The electromagnetic rest mass agrees with the gravitational rest mass. Acknowledgments

Many of the calculations were done using Package.

Mathematica ® 8.01 [3] with the MathTensor — 2.2.1 [4] Application References

1. M. Born and L. Infeld, Proc. Roy. Soc. A144, 425-451 (1934). 2. F.W. Cotton, BAPS.2013.APR.S2.10 (http://absimage.aps.org/image/APR13/MWS_APR13-2012-000003.pdf) (http://sites.google.com/site/fwcotton/em-25.pdf). 3. Wolfram Research, Mathematica ® 8.01 (http://www.wolfram.com/).

4. L. Parker and S.M. Christensen, MathTensor — 2.2.1 (http://smc.vnet.net/MathTensor.html).

© 2014-2015 Fredrick W. Cotton

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