Electrodynamics on cosmological scales

Abstract

Maxwell’s equations cannot describe a homogeneous and isotropic universe with a uniformly distributed net charge, because the electromagnetic field tensor in such a universe must be vanishing everywhere. For a closed universe with a nonzero net charge, Maxwell’s equations always fail regardless of the spacetime symmetry and the charge distribution. The two paradoxes indicate that Maxwell’s equations need be modified to be applicable to the universe as a whole. We consider two types of modified Maxwell equations, both can address the paradoxes. One is the Proca-type equation which contains a photon mass term. This type of electromagnetic field equations can naturally arise from spontaneous symmetry breaking and the Higgs mechanism in quantum field theory, where photons acquire a mass by eating massless Goldstone bosons. However, photons loose their mass again when the symmetry is restored, and the paradoxes reappear. The other type of modified Maxwell equations, which are more attractive in our opinions, contain a term with the electromagnetic field potential vector coupled to the spacetime curvature tensor. This type of electromagnetic field equations do not introduce a new dimensional parameter and return to Maxwell’s equations in a flat or Ricci-flat spacetime. We show that the curvature-coupled term can naturally arise from the ambiguity in extending Maxwell’s equations from a flat spacetime to a curved spacetime through the “minimal substitution rule”. Some consequences of the modified Maxwell equations are investigated. The results show that for reasonable parameters the modification does not affect existing experiments and observations. However, we argue that, the field equations with a curvature-coupled term can be testable in astrophysical environments where the mass density is high or the gravity of electromagnetic radiations plays a dominant role in dynamics, e.g., the interior of neutron stars and the early universe.

Appendices

Appendix 1: Derivation of Eq. (40)

By the equation 7.5.14 of [27], we have

$$\begin{aligned} 2\delta R_{ac} = -g^{bd}\nabla _a\nabla _c\delta g_{bd}-g^{bd}\nabla _b\nabla _d\delta g_{ac} +2g^{bd}\nabla _b\nabla _{(a}\delta g_{c)d} , \end{aligned}$$

(90)

and

$$\begin{aligned} 2A^aA^c\delta R_{ac} = -g^{bd}A^aA^c\nabla _a\nabla _c\delta g_{bd}-g^{bd}A^aA^c\nabla _b\nabla _d\delta g_{ac} +2g^{bd}A^aA^c\nabla _b\nabla _a\delta g_{cd} . \nonumber \\ \end{aligned}$$

(91)

Let us evaluate each term on the right-hand side of Eq. (91) separately. For the first term, we have

$$\begin{aligned} -g^{bd}A^aA^c\nabla _a\nabla _c\delta g_{bd}= & {} -g^{bd}\nabla _a\left( A^aA^c\nabla _c\delta g_{bd}\right) +g^{bd}\nabla _a(A^aA^c)\nabla _c\delta g_{bd} \nonumber \\= & {} -g^{bd}\nabla _a\left( A^aA^c\nabla _c\delta g_{bd}\right) +g^{bd}\nabla _c\left[ \nabla _a(A^aA^c)\delta g_{bd}\right] \nonumber \\&\quad -g^{bd}\nabla _c\nabla _a(A^aA^c)\delta g_{bd} \nonumber \\= & {} \nabla ^a[\ldots ]_a-g^{bd}\nabla _c\nabla _a(A^aA^c)\delta g_{bd} , \end{aligned}$$

(92)

where $\nabla ^a[\ldots ]_a$ is not written out explicitly since it does not contribute to the integral of action so does not affect the derivation of stress-energy tensor.

Similarly, for the second term on the right-hand side of Eq. (91), we have

$$\begin{aligned} -g^{bd}A^aA^c\nabla _b\nabla _d\delta g_{ac} = \nabla ^a[\ldots ]_a-g^{bd}\nabla _d\nabla _b(A^aA^c)\delta g_{ac} . \end{aligned}$$

(93)

For the third term, we have

$$\begin{aligned} +2g^{bd}A^aA^c\nabla _b\nabla _a\delta g_{cd} =\nabla ^a[\ldots ]_a+2g^{bd}\nabla _a\nabla _b(A^aA^c)\delta g_{cd} . \end{aligned}$$

(94)

Hence we get

$$\begin{aligned} 2A^aA^c\delta R_{ac}\doteq & {} -g^{bd}\nabla _c\nabla _a(A^aA^c)\delta g_{bd} -g^{bd}\nabla _d\nabla _b(A^aA^c)\delta g_{ac}\nonumber \\&\quad +\,2g^{bd}\nabla _a\nabla _b(A^aA^c)\delta g_{cd} , \end{aligned}$$

(95)

where $\doteq$ means “equal up to a term like $\nabla ^a[\ldots ]_a$”. Making use of the identity $\delta g_{ab}=-g_{ac}g_{bd}\delta g^{cd}$, we get

$$\begin{aligned} 2A^aA^c\delta R_{ac} \doteq \left[ \nabla _c\nabla _d(A^cA^d)g_{ab} +\nabla ^d\nabla _d(A_aA_b)-2\nabla ^c\nabla _b(A_aA_c)\right] \delta g^{ab} . \end{aligned}$$

(96)

On the other hand, we have

$$\begin{aligned} R_{ac}\delta (A^aA^c) = 2A^cR_{ca}A_b\delta g^{ab} , \end{aligned}$$

(97)

and

$$\begin{aligned} R_{ac}A^aA^c\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}R_{cd}A^cA^dg_{ab}\delta g^{ab} . \end{aligned}$$

(98)

Therefore, we have

$$\begin{aligned}&\delta \left( \sqrt{-g}R_{ac}A^aA^c\right) \doteq \sqrt{-g}\left\{ \frac{1}{2}\nabla ^c\nabla _c(A_aA_b) -\nabla ^c\nabla _b(A_cA_a)+2A^cR_{ca}A_b\right. \nonumber \\&\quad \left. +\,\frac{1}{2}g_{ab}\left[ \nabla _c\nabla _d(A^cA^d) -R_{cd}A^cA^d\right] \right\} \delta g^{ab} . \nonumber \\ \end{aligned}$$

(99)

Then, by Eq. (17), the stress-energy tensor in Eq. (40) is derived after symmetrization of the tensor index as required by the definition of $T_{ab}$.

Appendix 2: Derivation of Eq. (41)

For the simplicity of calculations, let us define

$$\begin{aligned} \Pi _{ab}^{(1)} = \nabla _c\nabla _d(A^cA^d)g_{ab} +\nabla ^c\nabla _c(A_aA_b) -2\nabla ^c\nabla _{(a}(A_{b)}A_c) , \end{aligned}$$

(100)

and

$$\begin{aligned} \Pi _{ab}^{(2)} = 4A^cR_{c(a}A_{b)}-R_{cd}A^cA^dg_{ab} . \end{aligned}$$

(101)

We have then

$$\begin{aligned} T_{\mathrm{EM},ab} = ~^{(0)}T_{\mathrm{EM},ab}+\frac{\kappa }{8\pi }\left( \Pi _{ab}^{(1)}+\Pi _{ab}^{(2)}\right) , \end{aligned}$$

(102)

and

$$\begin{aligned} \nabla ^{aT_{\mathrm{EM},ab}} = \nabla ^{a(0)} {T_{\mathrm{{EM}},ab}}+\frac{\kappa }{8\pi }\nabla ^a\left( \Pi _{ab}^{(1)}+\Pi _{ab}^{(2)}\right) , \end{aligned}$$

(103)

where

$$\begin{aligned} \nabla ^{a(0)} T_{\mathrm{EM},ab} = \frac{1}{4\pi }F_{ba}\nabla _cF^{ca} . \end{aligned}$$

(104)

The divergence of $\Pi _{ab}^{(1)}$ is

$$\begin{aligned} \nabla ^a\Pi _{ab}^{(1)}= & {} \nabla _b\nabla _c\nabla _d(A^cA^d) +\nabla ^a\nabla ^d\nabla _d(A_aA_b)-2\nabla ^a\nabla ^c\nabla _{(a}(A_{b)}A_c) \nonumber \\= & {} \nabla _b\left( \nabla _c A^c\nabla _dA^d+A^c\nabla _c\nabla _dA^d +\,\nabla _cA^d\nabla _dA^c+A^d\nabla _c\nabla _dA^c\right) \nonumber \\&+\,\nabla ^a\left( 2\nabla ^dA_a\nabla _dA_b+A_a\nabla ^d\nabla _dA_b +A_b\nabla ^d\nabla _dA_a\right) \nonumber \\&-\,\nabla ^a\left( \nabla ^cA_b\nabla _aA_c+A_b\nabla ^c\nabla _aA_c +\nabla ^cA_c\nabla _aA_b+A_c\nabla ^c\nabla _aA_b\right) \nonumber \\&-\,\nabla ^a\left( \nabla ^cA_a\nabla _bA_c+A_a\nabla ^c\nabla _bA_c +\nabla ^cA_c\nabla _bA_a+A_c\nabla ^c\nabla _bA_a\right) \nonumber \\= & {} \left( 2\nabla _b\nabla _c A^c\nabla _dA^d +\nabla _bA^c\nabla _c\nabla _dA^d+A^c\nabla _b\nabla _c\nabla _dA^d+2\nabla _b\nabla _cA^d\nabla _dA^c\right. \nonumber \\&\left. +\,\nabla _bA^d\nabla _c\nabla _dA^c+A^d\nabla _b\nabla _c\nabla _dA^c\right) \nonumber \\&+\,\left( 2\nabla ^a\nabla ^dA_a\nabla _dA_b+2\nabla ^dA_a\nabla ^a\nabla _dA_b\right. \nonumber \\&\left. +\,\nabla ^aA_a\nabla ^d\nabla _dA_b\!+\!A_a\nabla ^a\nabla ^d\nabla _dA_b+\nabla ^aA_b\nabla ^d\nabla _dA_a+A_b\nabla ^a\nabla ^d\nabla _dA_a\right) \nonumber \\&-\,\left( \nabla ^a\nabla ^cA_b\nabla _aA_c+\nabla ^cA_b\nabla ^a\nabla _aA_c +\nabla ^aA_b\nabla ^c\nabla _aA_c+A_b\nabla ^a\nabla ^c\nabla _aA_c\right. \nonumber \\&+\,\nabla ^a\nabla ^cA_c\nabla _aA_b+\nabla ^cA_c\nabla ^a\nabla _aA_b \nonumber \\&\left. +\,\nabla ^aA_c\nabla ^c\nabla _aA_b+A_c\nabla ^a\nabla ^c\nabla _aA_b\right) -\left( \nabla ^a\nabla ^cA_a\nabla _bA_c+\nabla ^cA_a\nabla ^a\nabla _bA_c\right. \nonumber \\&+\,\nabla ^aA_a\nabla ^c\nabla _bA_c+A_a\nabla ^a\nabla ^c\nabla _bA_c\nonumber \\&\left. +\,\nabla ^a\nabla ^cA_c\nabla _bA_a+\nabla ^cA_c\nabla ^a\nabla _bA_a +\nabla ^aA_c\nabla ^c\nabla _bA_a+A_c\nabla ^a\nabla ^c\nabla _bA_a\right) \nonumber \\= & {} \nabla _aA^a\times \textcircled {1}+\nabla _bA^c\times \textcircled {2}+\nabla ^cA_b\times \textcircled {3}+\nabla _aA^c\times \textcircled {4}+[\text{ Rest }], \end{aligned}$$

(105)

where

$$\begin{aligned} \textcircled {1}= & {} 2\left( \nabla _b\nabla _c-\nabla _c\nabla _b\right) A^c =-2R_{bd}A^d, \quad \textcircled {2} =0 ,\nonumber \\ \textcircled {3}= & {} (\nabla _a\nabla _c-\nabla _c\nabla _a)A^a =R_{cd}A^d ,\nonumber \\ \textcircled {4}= & {} 2(\nabla _b\nabla _c-\nabla _c\nabla _b)A^a+\left( \nabla _c\nabla ^a-\nabla ^a\nabla _c\right) A_b=-2R_{bcd}^{\;\;\;\;\;\;a}A^d+R_{bdc}^{\;\;\;\;\;\;a}A^d ,\nonumber \\ \end{aligned}$$

(106)

and

$$\begin{aligned} {[}\text{ Rest }]= & {} A^c\nabla _b\nabla _c\nabla _dA^d +A^d\nabla _b\nabla _c\nabla _dA^c +A_a\nabla ^a\nabla ^d\nabla _dA_b +A_b\nabla ^a\nabla ^d\nabla _dA_a\nonumber \\&-A_b\nabla ^a\nabla ^c\nabla _aA_c -A_c\nabla ^a\nabla ^c\nabla _aA_b \nonumber \\&-A_a\nabla ^a\nabla ^c\nabla _bA_c -A_c\nabla ^a\nabla ^c\nabla _bA_a \nonumber \\= & {} A^c\left[ \nabla _b\left( \nabla _c\nabla _aA^a+\nabla _a\nabla _cA^a\right) -\left( \nabla _c\nabla _a+\nabla _a\nabla _c\right) \nabla _bA^a\right] \nonumber \\&+A_b\left( \nabla _a\nabla _c-\nabla _c\nabla _a\right) \nabla ^cA^a \nonumber \\&+A^a\left( \nabla _a\nabla _c-\nabla _c\nabla _a\right) \nabla ^cA_b . \end{aligned}$$

(107)

By the equation 3.2.12 of [27], we have $\nabla _a\nabla _b\omega _{cd}-\nabla _b\nabla _a\omega _{cd}=R_{abc}^{\;\;\;\;\;\;e}\omega _{ed}+R_{abd}^{\;\;\;\;\;\;e}\omega _{ce}$ for any tensor $\omega _{ab}$, from which we get $\nabla _a\nabla _b\omega ^c_{\;\,d}-\nabla _b\nabla _a\omega ^c_{\;\,d}=-R_{abe}^{\;\;\;\;\;\;c}\omega ^e_{\;\,d}+R_{abd}^{\;\;\;\;\;\;e}\omega ^c_{\;\,e}$. Hence, we have

$$\begin{aligned} \left( \nabla _c\nabla _a+\nabla _a\nabla _c\right) \nabla _bA^a = 2\nabla _c\nabla _a\nabla _bA^a+R_{acb}^{\;\;\;\;\;\;e}\nabla _eA^a+R_{ce}\nabla _bA^e , \end{aligned}$$

(108)

where we have used the identities $R_{ace}^{\;\;\;\;\;\;a}=-R_{cae}^{\;\;\;\;\;\;a}=-R_{ce}$.

We also have $\nabla _c\nabla _aA^a+\nabla _a\nabla _cA^a=2\nabla _c\nabla _aA^a+R_{cd}A^d$. Then, in the expression for [Rest],

$$\begin{aligned}&A^c\left[ \nabla _b\left( \nabla _c\nabla _aA^a+\nabla _a\nabla _cA^a\right) -\left( \nabla _c\nabla _a+\nabla _a\nabla _c\right) \nabla _bA^a\right] \nonumber \\&\quad =A^c\left[ 2\nabla _c\left( \nabla _b\nabla _aA^a-\nabla _a\nabla _bA^a\right) +\nabla _b\left( R_{cd}A^d\right) \right. \nonumber \\&\quad \left. -R_{acb}^{\;\;\;\;\;\;e}\nabla _eA^a-R_{ce}\nabla _bA^e\right] =A^c\left[ -2\nabla _c\left( R_{bd}A^d\right) \right. \nonumber \\&\quad \left. +\nabla _b\left( R_{cd}A^d\right) -R_{acb}^{\;\;\;\;\;\;e}\nabla _eA^a-R_{ce}\nabla _bA^e\right] , ~ \end{aligned}$$

(109)

where we have used the identity $\nabla _b\nabla _c\nabla _aA^a=\nabla _c\nabla _b\nabla _aA^a$.

For the other two terms in [Rest], we have

$$\begin{aligned}&A_b\left( \nabla _a\nabla _c-\nabla _c\nabla _a\right) \nabla ^cA^a = A_b\left( -R_{ace}^{\;\;\;\;\;\;c}\nabla ^eA^a-R_{ace}^{\;\;\;\;\;\;a}\nabla ^cA^e\right) \nonumber \\&\quad = -A_bR_{ae}\left( \nabla ^eA^a-\nabla ^aA^e\right) = 0 , \end{aligned}$$

(110)

and

$$\begin{aligned} A^a\left( \nabla _a\nabla _c-\nabla _c\nabla _a\right) \nabla ^cA_b = A^a\left( -R_{ae}\nabla ^eA_b +R_{acb}^{\;\;\;\;\;\;e}\nabla ^cA_e\right) . \end{aligned}$$

(111)

Therefore, we have

$$\begin{aligned}{}[\text{ Rest }]= & {} A^c\left[ -2\nabla _c\left( R_{bd}A^d\right) +\nabla _b\left( R_{cd}A^d\right) -R_{acb}^{\;\;\;\;\;\;e}\nabla _eA^a-R_{ce}\nabla _bA^e\right] \nonumber \\&\quad +A^a\left( -R_{ae}\nabla ^eA_b+R_{acb}^{\;\;\;\;\;\;e}\nabla ^cA_e\right) \nonumber \\= & {} A^c\left[ -2\nabla _c\left( R_{bd}A^d\right) +\nabla _b\left( R_{cd}A^d\right) \right. \nonumber \\&\left. \quad +R_{cabe}(\nabla ^eA^a+\nabla ^aA^e)-R_{ce}\left( \nabla _bA^e+\nabla ^eA_b\right) \right] . \end{aligned}$$

(112)

Putting the above results together, we get

$$\begin{aligned} \nabla ^a\Pi _{ab}^{(1)} = -2\nabla _c\left( A^cR_{bd}A^d\right) +A^c\nabla _b\left( R_{cd}A^d\right) -A^dR_{dc}\nabla _bA^c , \end{aligned}$$

(113)

where we have used the identity $-R_{bcda}+R_{bdca}+R_{dcba}=0$ (since $R_{[abc]}^{\;\;\;\;\;\;\;d}=0$).

The divergence of $\Pi _{ab}^{(2)}$ is

$$\begin{aligned} \nabla ^a\Pi _{ab}^{(2)} = 2\nabla _c\left( A^dR_{db}A^{c}\right) +2\nabla ^c\left( A^dR_{dc}A_{b}\right) -\nabla _b\left( R_{cd}A^cA^d\right) . \end{aligned}$$

(114)

Therefore, we have

$$\begin{aligned}&\nabla ^a\left( \Pi _{ab}^{(1)}+\Pi _{ab}^{(2)}\right) = 2\nabla ^c\left( A^dR_{dc}A_{b}\right) -2A^dR_{dc}\nabla _bA^c\nonumber \\&\quad = -2F_b^{\;\,c}R_{cd}A^d+2A_b\nabla ^c\left( R_{cd}A^d\right) . \end{aligned}$$

(115)

Substituting Eqs. (104) and (115) into Eq. (103), we get Eq. (41).

Note, when we derived the Eq. (115) we did not make use of the electromagnetic field equation. When $R_{ab}=0$, we have $\Pi _{ab}^{(2)}=0$ and $\nabla ^a\Pi _{ab}^{(1)}=0$.

Appendix 3: Derivation of the Poynting flux vector

In this appendix we derive the Poynting flux vector corresponding to the electromagnetic field Eq. (10).

By Eqs. (5) and (6), the $F_{ab}$ can be expressed in terms of $E_a$ and $B_a$ as

$$\begin{aligned} F_{ab}=-2E_{[a}u_{b]}-\epsilon _{abcd}B^cu^d , \end{aligned}$$

(116)

where $u^au_a=-1$ and $E_au^a=B_au^a=0$. We can derive

$$\begin{aligned} F_{ab}F^{ab}=-2\left( E_aE^a-B_aB^a\right) , \end{aligned}$$

(117)

and

$$\begin{aligned} F_{ac}F_b^{\;\,c} = -E_aE_b-B_aB_b+E_cE^cu_au_b +(g_{ab}+u_au_b)B_cB^c+2u_{(a}\tilde{\epsilon }_{b)cd}E^cB^d , \nonumber \\ \end{aligned}$$

(118)

where $\tilde{\epsilon }_{abc}\equiv -\epsilon _{abcd}u^d$.

By Eq. (19), we get

$$\begin{aligned} 4\pi ^{(0)}T_{\mathrm{EM},ab}= & {} \left( E_cE^c+B_cB^c\right) \left( u_au_b+\frac{1}{2}g_{ab}\right) \nonumber \\&-\,E_aE_b-B_aB_b+2u_{(a}\tilde{\epsilon }_{b)cd}E^cB^d . \end{aligned}$$

(119)

To express the law of energy conservation in terms of $E_a$ and $B_a$, we assume that the spacetime has a timelike Killing vector $t^a$ parallel to the $u^a$, i.e., $t^a=\alpha u^a$, where $\alpha$ is the lapse function. By the Killing equation [27], we get

$$\begin{aligned} 0=\nabla _{(a}t_{b)}=\alpha \nabla _{(a}u_{b)}+u_{(a}\nabla _{b)}\alpha , \end{aligned}$$

(120)

from which we can derive

$$\begin{aligned} \alpha \nabla _au^a+u^a\nabla _a\alpha =0 \end{aligned}$$

(121)

and

$$\begin{aligned} u^a\nabla _au_b=\nabla _b\ln \alpha -u_bu^a\nabla _a\ln \alpha . \end{aligned}$$

(122)

Since $u^bu_b=-1$ and $u^b\nabla _au_b=0$, Eq. (122) leads to

$$\begin{aligned} u^a\nabla _a\alpha =0 , u^a\nabla _au_b=\nabla _b\ln \alpha . \end{aligned}$$

(123)

Then, by Eq. (121) we have

$$\begin{aligned} \nabla _au^a=0 . \end{aligned}$$

(124)

An energy-momentum flux vector $\mathcal{J}^a$ is defined by

$$\begin{aligned} \mathcal{J}_a \equiv -t^b T_{\mathrm{EM},ab} = -\alpha u^b T_{\mathrm{EM},ab}. \end{aligned}$$

(125)

By Eqs. (18), (119), and (20), we get

$$\begin{aligned} \mathcal{J}^a = \frac{\alpha }{4\pi }\left\{ \frac{1}{2} \left[ E_cE^c+B_cB^c+\xi \left( \Phi ^2+\tilde{A}_c\tilde{A}^c\right) \right] u^a +\left( \tilde{\epsilon }^{acd}E_cB_d+\xi \Phi \tilde{A}^a\right) \right\} , \nonumber \\ \end{aligned}$$

(126)

where we have written $A^a$ in terms of a scalar potential $\Phi$ and a spatial vector potential $\tilde{A}^a$ (i.e., $A^a=\Phi u^a+\tilde{A}^a$, $u_a\tilde{A}^a=0$).

With the expression in Eq. (126), the energy-momentum flux vector is decomposed into an energy density component parallel to $u^a$, and a spatial momentum component perpendicular to $u^a$:

$$\begin{aligned} \mathcal{J}^a=\rho _\mathrm{EM}u^a+\tilde{\mathcal{J}}^a , \end{aligned}$$

(127)

where

$$\begin{aligned} \rho _\mathrm{EM}=\frac{\alpha }{8\pi }\left[ E_cE^c+B_cB^c+\xi \left( \Phi ^2+\tilde{A}_c\tilde{A}^c\right) \right] , \end{aligned}$$

(128)

and

$$\begin{aligned} \tilde{\mathcal{J}}^a=\frac{\alpha }{4\pi }\left( \tilde{\epsilon }^{acd}E_cB_d+\xi \Phi \tilde{A}^a\right) . \end{aligned}$$

(129)

In $\tilde{\mathcal{J}}^a$, the first term is just the usual Poynting flux vector, and the second term is an additional momentum flux due to the $\xi$-term in the electromagnetic equation. We may call $\tilde{\mathcal{J}}^a$ the generalized Poynting flux vector.