Influence of Riemann Space-time Curvature on the Laws of Electromagnetism

1. Electromagnetism in Curved Space-Time

The background space-time curvature of the geometry influences the electromagnetic fields. Here, we consider the electromagnetic fields in a curved space-time geometry; however, we will ignore the influence of the electromagnetic field on the space-time geometry, which is discussed in the next section. For that, we describe the electromagnetic field equations, derived from the action principle.

The covariant derivative of the electromagnetic field covariant vector potential $A_{\mu }$ (or contravariant vector $A^{\mu }$) in the torsionless space-time manifold is

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\nu }{A}_{\mu }& ={\nabla }^{\nu }{A}_{\mu }\hfill \\ \hfill {\mathbb{D}}_{\nu }{A}^{\mu }& ={\nabla }_{\nu }{A}^{\mu }\hfill \end{array}$$

(1)

and hence, the anti-symmetric part of the connection, ${\Gamma }_{\sigma \nu }^{\mu }{A}^{\sigma }=0$. Thus, the field strength tensor in covariant form is:

$$\begin{array}{cc}\hfill F_{\mu \nu }& ={\mathbb{D}}_{\mu }{A}_{\nu }-{\mathbb{D}}_{\nu }{A}_{\mu }={\nabla }_{\mu }{A}_{\nu }-{\nabla }_{\nu }{A}_{\mu }\hfill \end{array}$$

(2)

and its contravariant form is

$$\begin{array}{cc}\hfill F^{\mu \nu }& ={\mathbb{D}}^{\mu }{A}^{\nu }-{\mathbb{D}}^{\nu }{A}^{\mu }={\nabla }^{\mu }{A}^{\nu }-{\nabla }^{\nu }{A}^{\mu }\hfill \end{array}$$

(3)

Furthermore, the electric and magnetic fields of the electromagnetic field components are given as follows:

$$\begin{array}{cc}\hfill F_{0i}& ={\nabla }_0{A}_i-{\nabla }_i{A}_0={\displaystyle \frac{E_i}{c}}=-{\displaystyle \frac{E^i}{c}}\hfill \\ \hfill F_{ij}& ={\nabla }_i{A}_j-{\nabla }_j{A}_i=-B_{ij}=-{\epsilon }_{kij}{B}^k\hfill \end{array}$$

(4)

Thus, $F_{\mu \nu }$ are the components of the electromagnetic field strength 2-form $\mathbf{F}$ and $A_{\mu }$ are the components of the electromagnetic field potential 1-form $\mathbf{A}$. On the other hand [21], the electric field vector $\mathbf{E}$ is related to the line integrand, and hence it is represented by a covector or a 1-form in three-dimensional space; the magnetic field vector $\mathbf{B}$ is related to surface and it is represented by a 2-form with components $B_{ij}$ that are expressed in terms of the contravariant components of the density vector $B^k$ being surface integrands:

$$\begin{array}{c}\hfill B_{ij}={\epsilon }_{kij}{B}^k\end{array}$$

(5)

The Maxwell’s equations are written as

$$\begin{array}{cc}\hfill \nabla ·\mathbf{D}& =\rho \hfill \\ \hfill \nabla \times \mathbf{E}& =-{\displaystyle \frac{\partial \mathbf{B}}{\partial t}}\hfill \\ \hfill \nabla ·\mathbf{B}& =0\hfill \\ \hfill \nabla \times \mathbf{H}& =\mathbf{J}+{\displaystyle \frac{\partial \mathbf{D}}{\partial t}}\hfill \end{array}$$

(6)

which are equivalent to conservation laws of charge and magnetic flux. These equations, under the influence of the curved space-time geometry, are written in terms of covariant derivative as

$$\begin{array}{cc}\hfill & {\mathbb{D}}_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu }\hfill \\ \hfill & {\mathbb{D}}_{\mu }{F}_{\nu \alpha }+{\mathbb{D}}_{\nu }{F}_{\alpha \mu }+{\mathbb{D}}_{\alpha }{F}_{\mu \nu }=0\hfill \end{array}$$

(7)

Besides, it is required that

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\sigma }{g}_{\mu \nu }& =0\hfill \end{array}$$

(8)

Note that Eq. (8) indicates that during the covariant differentiation the metric tensor components $g_{\mu \nu }$ can be considered a constant.

The actions of the covariant and contravariant derivatives on a vector can be written as

$$\begin{array}{cc}\hfill {\mathbb{D}}^{\nu }{A}_{\mu }& ={\nabla }^{\nu }{A}_{\mu }+A_{\sigma }{\Gamma }_{\mu }^{\sigma \nu }\hfill \\ \hfill {\mathbb{D}}_{\nu }{A}^{\mu }& ={\nabla }_{\nu }{A}^{\mu }+A^{\sigma }{\Gamma }_{\sigma \nu }^{\mu }\hfill \\ \hfill {\mathbb{D}}^{\mu }& =g^{\mu \nu }\left(\mathbf{x}\right){\mathbb{D}}_{\nu }\hfill \end{array}$$

(9)

where ${\Gamma }_{\sigma \nu }^{\mu }$ is the space-time connection, the so-called Christoffel’s symbols, and $\mathbf{x}$ is a vector characterising $x^{\mu }$ point in the space-time. Besides, the actions of the covariant derivative on the second-rank tensors are determined as

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mu }{F}^{\sigma \nu }& ={\nabla }_{\mu }{F}^{\sigma \nu }+{\Gamma }_{\lambda \mu }^{\sigma }{F}^{\lambda \nu }+{\Gamma }_{\lambda \mu }^{\nu }{F}^{\sigma \lambda }\hfill \\ \hfill {\mathbb{D}}_{\mu }{F}_{\sigma \nu }& ={\nabla }_{\mu }{F}_{\sigma \nu }+{\Gamma }_{\sigma \mu }^{\lambda }{F}_{\lambda \nu }+{\Gamma }_{\nu \mu }^{\lambda }{F}_{\sigma \lambda }\hfill \end{array}$$

(10)

For that, the general form of the covariant derivative of the anti-symmetric electromagnetic field strength tensor $F^{\mu \nu }$ in Riemann curved space-time is used, given as

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mu }{F}^{\mu \nu }& ={\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}{F}^{\mu \nu }\right)\hfill \\ \hfill {\Gamma }_{\mu \nu }^{\sigma }& ={\nabla }_{\mu }\left(ln\left(\sqrt{-g}\right)\right)\hfill \end{array}$$

(11)

Combining Eq. (7) and Eq. (11), we obtain

$$\begin{array}{c}\hfill {\nabla }_{\mu }{F}^{\mu \nu }+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)F^{\mu \nu }={\mu }_0{J}^{\nu }\end{array}$$

(12)

Therefore,

$$\begin{array}{cc}\hfill & g^{\sigma \mu }{g}^{\lambda \nu }{\nabla }_{\mu }{F}_{\sigma \lambda }\hfill \\ \hfill & +F_{\sigma \lambda }\left({\nabla }_{\mu }\left(g^{\lambda \nu }{g}^{\sigma \mu }\right)+g^{\sigma \mu }{g}^{\lambda \nu }{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\right)={\mu }_0{J}^{\nu }\hfill \end{array}$$

(13)

where the following relations were used:

$$\begin{array}{cc}\hfill F^{\mu \nu }& =g^{\sigma \mu }{g}^{\lambda \nu }{F}_{\sigma \lambda }\hfill \\ \hfill {\nabla }_{\mu }{F}^{\mu \nu }& =\left({\nabla }_{\mu }\left(g^{\sigma \mu }{g}^{\lambda \nu }\right)\right)F_{\sigma \lambda }+g^{\lambda \nu }{g}^{\sigma \mu }\left({\nabla }_{\mu }{F}_{\sigma \lambda }\right)\hfill \end{array}$$

(14)

In terms of the components of the electric and magnetic fields, Eq. (13) can be written as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c}}\left({\nabla }_{\mu }{E}_i\right)\left(g^{0\mu }{g}^{i\nu }-g^{i\mu }{g}^{0\nu }\right)+{\displaystyle \frac{1}{c}}{E}_i\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{0\mu }{g}^{i\nu }-g^{i\mu }{g}^{0\nu }\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{0\mu }{g}^{i\nu }-g^{i\mu }{g}^{0\nu }\right)\right)\hfill \\ \hfill & -{\epsilon }_{klm}\left({\nabla }_{\mu }{B}^k\right)g^{l\mu }{g}^{m\nu }-{\epsilon }_{klm}{B}^k\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{l\mu }{g}^{m\nu }\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{l\mu }{g}^{m\nu }\right)\right)={\mu }_0{J}^{\nu }\hfill \end{array}$$

(15)

where $J^{\nu }=(\rho c,\mathbf{J})$ (with $\rho$ being the charge density and $\mathbf{J}=\rho \mathbf{v}$ the current density, where c is the speed of light in vacuum and $\mathbf{v}$ velocity vector in three-dimensional space.)

For $\nu =0$, Eq. (15) can be written as

$$\begin{array}{cc}\hfill & \left({\nabla }_{\mu }{E}_i\right)\left(g^{0\mu }{g}^{i0}-g^{i\mu }{g}^{00}\right)\hfill \\ \hfill & +E_i\left({\nabla }_{\mu }\left(g^{0\mu }{g}^{i0}-g^{i\mu }{g}^{00}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{0\mu }{g}^{i0}-g^{i\mu }{g}^{00}\right)\right)\hfill \\ \hfill & -c{\epsilon }_{klm}\left({\nabla }_{\mu }{B}^k\right)g^{l\mu }{g}^{m0}-c{\epsilon }_{klm}{B}^k\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{l\mu }{g}^{m0}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{l\mu }{g}^{m0}\right)\right)={\mu }_0\rho c^2={\displaystyle \frac{\rho }{{ϵ}_0}}\hfill \end{array}$$

(16)

where the relation $c^2=1/\left({\mu }_0{ϵ}_0\right)$ is used. Note that Eq. (16) is the generalisation of Gauss’s law of Maxwell’s equations in Minkowski (flat) space-time for the case of Riemann (curved) space-time. Interestingly, this equation is showing some new phenomena of the electromagnetism in the presence of the gravitational fields modifying the curvature of the space-time. For instance, in the absence of charges (that is, $\rho =0$), a magnetic field (including a static magnetic field) creates an electric field. Furthermore, in the case of the weak gravitational fields, when the off-diagonal elements of metric tensor are zero, the magnetic field cancels out of Eq. (16). In particular, for $g^{0i}=g^{i0}=g^{ij}=0$ (for $i\ne j$), we get

$$\begin{array}{cc}\hfill & -g^{jj}{g}^{00}{\nabla }_j{E}_j-E_j\left(g^{jj}{g}^{00}{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_j\left(\sqrt{-g}\right)+{\nabla }_j\left(g^{jj}{g}^{00}\right)\right)={\displaystyle \frac{\rho }{{ϵ}_0}}\hfill \end{array}$$

(17)

From Eq. (17), for the Riemann space-time geometries for which the second term is non-zero, then the electric field in vacuum (that is, $\rho =0$) is non-vanishing and it is necessary non-uniform. Therefore, the curvature of the space-time introduces a spatial change on the electric field in vacuum, even for weak gravitational fields. Furthermore, in the case of Minkowski space-time (that is, $g^{00}=1$ and $g^{jj}=-1$), Eq. (17) reduces to the first Maxwell’s equation, $\nabla ·\mathbf{E}=\rho /{ϵ}_0$.

For $\nu =1,2,3$, we obtain the generalisation of Maxwell-Ampére’s law in the Riemann’s curved space-time. In particular, from Eq. (15), for $\nu \equiv j=1,2,3$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c}}\left({\nabla }_{\mu }{E}_i\right)\left(g^{0\mu }{g}^{ij}-g^{i\mu }{g}^{0j}\right)+{\displaystyle \frac{1}{c}}{E}_i\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{0\mu }{g}^{ij}-g^{i\mu }{g}^{0j}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{0\mu }{g}^{ij}-g^{i\mu }{g}^{0j}\right)\right)\hfill \\ \hfill & -{\epsilon }_{klm}\left({\nabla }_{\mu }{B}^k\right)g^{l\mu }{g}^{mj}-{\epsilon }_{klm}{B}^k\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{l\mu }{g}^{mj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{l\mu }{g}^{mj}\right)\right)={\mu }_0{J}^j\hfill \end{array}$$

(18)

For the case of the weak gravitational fields (i.e., $g^{0i}=g^{i0}=0$), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c}}\left({\nabla }_0{E}_i\right)\left(g^{00}{g}^{ij}\right)+{\displaystyle \frac{1}{c}}{E}_i\left({\nabla }_0\left(g^{00}{g}^{ij}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_0\left(\sqrt{-g}\right)\left(g^{00}{g}^{ij}\right)\right)\hfill \\ \hfill & -{\epsilon }_{klm}\left({\nabla }_n{B}^k\right)g^{ln}{g}^{mj}-{\epsilon }_{klm}{B}^k\hfill \\ \hfill & \times \left({\nabla }_n\left(g^{ln}{g}^{mj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_n\left(\sqrt{-g}\right)\left(g^{ln}{g}^{mj}\right)\right)={\mu }_0{J}^j\hfill \end{array}$$

(19)

Or,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c}}\left({\nabla }_0{E}_i\right)\left(g^{00}{g}^{ij}\right)+{\displaystyle \frac{1}{c}}{E}_i\left({\nabla }_0\left(g^{00}{g}^{ij}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_0\left(\sqrt{-g}\right)\left(g^{00}{g}^{ij}\right)\right)\hfill \\ \hfill & -\left({\nabla }_n{B}_{lm}\right)g^{ln}{g}^{mj}-B_{lm}\hfill \\ \hfill & \times \left({\nabla }_n\left(g^{ln}{g}^{mj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_n\left(\sqrt{-g}\right)\left(g^{ln}{g}^{mj}\right)\right)={\mu }_0{J}^j\hfill \end{array}$$

(20)

where the following relation is used

$$\begin{array}{cc}\hfill B_{lm}& ={\epsilon }_{lmk}{B}^k\hfill \end{array}$$

(21)

which transforms the electromagnetic vector field to the corresponding 2-form. Thus, Eq. (20) becomes

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^2}}\left(g^{00}{g}^{ij}\right)\left({\nabla }_t{E}_i\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}{E}_i\left({\nabla }_t\left(g^{00}{g}^{ij}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_t\left(\sqrt{-g}\right)\left(g^{00}{g}^{ij}\right)\right)\hfill \\ \hfill & -g^{ln}{g}^{mj}\left({\nabla }_n{B}_{lm}\right)\hfill \\ \hfill & -B_{lm}\left({\nabla }_n\left(g^{ln}{g}^{mj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_n\left(\sqrt{-g}\right)\left(g^{ln}{g}^{mj}\right)\right)={\mu }_0{J}^j\hfill \end{array}$$

(22)

where the following relation is used

$$\begin{array}{c}\hfill {\nabla }_0={\displaystyle \frac{\partial }{\partial x^0}}={\displaystyle \frac{\partial }{\partial \left(ct\right)}}={\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}}={\displaystyle \frac{1}{c}}{\nabla }_t\end{array}$$

(23)

Next, we consider a metric with vanishing the off-diagonal elements (i.e., $g^{ij}=0$, for $i\ne j$), and Eq. (22) reduces to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^2}}\left(g^{00}{g}^{jj}\right)\left({\nabla }_t{E}_j\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}{E}_j\left({\nabla }_t\left(g^{00}{g}^{jj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_t\left(\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & -g^{ll}{g}^{jj}\left({\nabla }_l{B}_{lj}\right)\hfill \\ \hfill & -B_{lj}\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_l\left(\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)={\mu }_0{J}^j\hfill \end{array}$$

(24)

Or,

$$\begin{array}{cc}\hfill & -{\epsilon }_{ljk}{g}^{ll}{g}^{jj}\left({\nabla }_l{B}^k\right)\hfill \\ \hfill & -{\epsilon }_{ljk}{B}^k\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_l\left(\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\mu }_0{J}^j+{\mu }_0{ϵ}_0\left(-g^{00}{g}^{jj}\right){\nabla }_t{E}_j\hfill \\ \hfill & +{\mu }_0{ϵ}_0{E}_j\left({\nabla }_t\left(-g^{00}{g}^{jj}\right)-{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_t\left(\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \end{array}$$

(25)

Interestingly, Eq. (25) indicates that the coupling between the magnetic and electric fields does not vanish from Maxwell-Ampére’s law, even for weak gravitational fields and diagonal Riemann metric, in contrast to Gauss’s law. Furthermore, if the geometry is time-independent, then Eq. (25) reduces to

$$\begin{array}{cc}\hfill & -{\epsilon }_{ljk}{g}^{ll}{g}^{jj}\left({\nabla }_l{B}^k\right)\hfill \\ \hfill & -{\epsilon }_{ljk}{B}^k\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_l\left(\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\mu }_0{J}^j+{\mu }_0{ϵ}_0\left(-g^{00}{g}^{jj}\right){\nabla }_t{E}_j\hfill \end{array}$$

(26)

because ${\nabla }_t\left(-g^{00}{g}^{jj}\right)-{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_t\left(\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)=0$. However, the time derivative term of electric field does not cancel out (to cancel this term out, a time-varying charge density is required based on Gauss’s law). For time-varying space-time geometry, both terms on the right-hand side of Eq. (26), because a non-stationary space-time geometry, will certainly induce a time-varying electric field (based on Gauss’s law). Furthermore, the gravitational waves will directly influence the magnetic fields. If we introduce a generalised charge current density as

$$\begin{array}{cc}\hfill {\mathbb{J}}^j& =\sqrt{-g}{J}^j\hfill \end{array}$$

(27)

a generalised displacement current as

$$\begin{array}{cc}\hfill {\mathbb{J}}_D^j& ={ϵ}_0\sqrt{-g}\left(-g^{00}{g}^{jj}{\nabla }_t{E}_j\right.\hfill \\ \hfill & -\left.E_j\left({\nabla }_t\left(g^{00}{g}^{jj}\right)-{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_t\left(\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\right)\hfill \end{array}$$

(28)

and by introducing the following five-dimensional object

$$\begin{array}{c}\hfill {\mathbb{B}}^{lljjk}=\sqrt{-g}{g}^{ll}{g}^{jj}{B}^k\end{array}$$

(29)

then Eq. (26) takes the form of the generalised Maxwell-Ampére’s law as

$$\begin{array}{c}\hfill {\epsilon }_{ljk}{\nabla }_l{\mathbb{B}}^{lljjk}={\mu }_0({\mathbb{J}}^j+{\mathbb{J}}_D^j)\end{array}$$

(30)

Therefore, in the absence of the currents (that is, $\mathbb{J}=0$), the electric field can be the source of the magnetic field, where an extra contribution is added from the displacement current that is induced by the time-varying space-time geometry. For stationary space-time geometry that contribution is zero; however, for high frequency gravitational waves (strongly varying waves), that contribution might be significant.

The homogeneous equations of the electromagnetic fields are expressed as follows:

$$\begin{array}{cc}\hfill 0& ={\mathbb{D}}_{\mu }{F}_{\nu \alpha }+{\mathbb{D}}_{\nu }{F}_{\alpha \mu }+{\mathbb{D}}_{\alpha }{F}_{\mu \nu }\hfill \\ \hfill & ={\nabla }_{\mu }{F}_{\nu \alpha }+{\Gamma }_{\nu \mu }^{\sigma }{F}_{\sigma \alpha }+{\Gamma }_{\alpha \mu }^{\sigma }{F}_{\nu \sigma }\hfill \\ \hfill & +{\nabla }_{\nu }{F}_{\alpha \mu }+{\Gamma }_{\alpha \nu }^{\sigma }{F}_{\sigma \mu }+{\Gamma }_{\mu \nu }^{\sigma }{F}_{\alpha \sigma }\hfill \\ \hfill & +{\nabla }_{\alpha }{F}_{\mu \nu }+{\Gamma }_{\mu \alpha }^{\sigma }{F}_{\sigma \nu }+{\Gamma }_{\nu \alpha }^{\sigma }{F}_{\mu \sigma }\hfill \end{array}$$

(31)

Thus, for a torsionless manifold, as it is the case of Riemann space-time geometry, we get

$$\begin{array}{cc}\hfill {\nabla }_{\mu }{F}_{\nu \alpha }+{\nabla }_{\nu }{F}_{\alpha \mu }+{\nabla }_{\alpha }{F}_{\mu \nu }& =0\hfill \end{array}$$

(32)

Eq. (31) indicates that the homogeneous equations of the electromagnetic fields are not influenced by the curvature of Riemann’s space-time, and hence they are equivalent to Faraday’s law (for $\mu =1,2,3$) and Gauss’s law for magnetic field (for $\mu =0$):

$$\begin{array}{cc}\hfill {\nabla }_t{B}^i& =-{\epsilon }^{ijk}{\nabla }_j{E}_k\hfill \\ \hfill {\nabla }_j{B}^j& =0\hfill \end{array}$$

(33)

where the summation of repeating indices is assumed.

1.1. Equation of Electromagnetic Field Potential Wave

In the following, the inhomogeneous equations will be expressed in terms of the electromagnetic field 4-potential, namely $A^{\mu }$. For that, the covariant derivative of field strength tensor is

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mu }{F}^{\mu \nu }& ={\mathbb{D}}_{\mu }{\mathbb{D}}^{\mu }{A}^{\nu }-g^{\sigma \nu }\left(\left[{\mathbb{D}}_{\mu },{\mathbb{D}}_{\sigma }\right]A^{\mu }+{\mathbb{D}}_{\sigma }{\mathbb{D}}_{\mu }{A}^{\mu }\right)\hfill \end{array}$$

(34)

where

$$\begin{array}{cc}\hfill \left[{\mathbb{D}}_{\mu },{\mathbb{D}}_{\sigma }\right]& ={\mathbb{D}}_{\mu }{\mathbb{D}}_{\sigma }-{\mathbb{D}}_{\sigma }{\mathbb{D}}_{\mu }\hfill \end{array}$$

(35)

Furthermore,

$$\begin{array}{cc}\hfill \left[{\mathbb{D}}_{\mu },{\mathbb{D}}_{\sigma }\right]A^{\nu }& =R_{\lambda \mu \sigma }^{\nu }{A}^{\lambda }\hfill \end{array}$$

(36)

where $R_{\lambda \mu \sigma }^{\nu }$ are the components of Riemann tensor. Therefore, we obtain Maxwell’s equation in the following form:

$$\begin{array}{c}\hfill {\mathbb{D}}_{\mu }{\mathbb{D}}^{\mu }{A}^{\nu }-g^{\sigma \nu }{R}_{\lambda \sigma }{A}^{\lambda }-{\mathbb{D}}^{\nu }\left({\mathbb{D}}_{\mu }{A}^{\mu }\right)={\mu }_0{J}^{\nu }\end{array}$$

(37)

where $R_{\lambda \sigma }$ is Ricci tensor

$$\begin{array}{c}\hfill R_{\lambda \sigma }=R_{\lambda \mu \sigma }^{\mu }\end{array}$$

(38)

Using Lorentz condition for the electromagnetic field potential in the curved space-time:

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mu }{A}^{\mu }& ={\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}{A}^{\mu }\right)=0\hfill \end{array}$$

(39)

we obtain the following

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mu }{\mathbb{D}}^{\mu }{A}^{\nu }-g^{\sigma \nu }{R}_{\lambda \sigma }{A}^{\lambda }& ={\mu }_0{J}^{\nu }\hfill \end{array}$$

(40)

The formula of generalised Laplacian in the curved space-time is

$$\begin{array}{c}\hfill {\mathbb{D}}_{\mu }{\mathbb{D}}^{\mu }\Phi ={\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}{g}^{\mu \sigma }{\nabla }_{\sigma }\Phi \right)\end{array}$$

(41)

Combining Eq. (40) and Eq. (41), we get

$$\begin{array}{cc}\hfill {\nabla }_{\mu }{\nabla }^{\mu }{A}^{\nu }+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}{g}^{\mu \sigma }\right){\nabla }_{\sigma }{A}^{\nu }-g^{\sigma \nu }{R}_{\lambda \sigma }{A}^{\lambda }& ={\mu }_0{J}^{\nu }\hfill \end{array}$$

(42)

Using the relationship with Christoffel’s symbol for Riemann’s space-time geometry:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}{g}^{\mu \sigma }\right)& =-g^{\mu \nu }{\Gamma }_{\mu \nu }^{\sigma }\hfill \end{array}$$

(43)

Eq. (42) can also be written as follows:

$$\begin{array}{cc}\hfill {\nabla }_{\mu }{\nabla }^{\mu }{A}^{\nu }-g^{\mu \nu }{\Gamma }_{\mu \nu }^{\sigma }{\nabla }_{\sigma }{A}^{\nu }-g^{\sigma \nu }{R}_{\lambda \sigma }{A}^{\lambda }& ={\mu }_0{J}^{\nu }\hfill \end{array}$$

(44)

Consider a diagonal metric in vacuum (that is, $J^{\nu }=0$), Eq. (42) reduces to

$$\begin{array}{cc}\hfill {\nabla }_{\mu }{\nabla }^{\mu }{A}^{\nu }+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}{g}^{\mu \mu }\right){\nabla }_{\mu }{A}^{\nu }-g^{\nu \nu }{R}_{\lambda \nu }{A}^{\lambda }& =0\hfill \end{array}$$

(45)

Thus, the 4-potential components of the electromagnetic field are coupled in either Eq. (42) or Eq. (45), which is not the case of Minkowski flat space-time geometry. Furthermore, Eq. (45) indicates that even when Ricci’s tensor components are zero, there will be a coupling between the components of the 4-potential of the electromagnetic field. Therefore, these results indicate that presence of the geometry dependent terms for the gravitational fields suggests new phenomena may arise for electromagnetism under the influence of space-time curvature.

1.2. Equations of Electromagnetic Waves

Using the electromagnetic field equations in vacuum ($J^{\mu }=0$), we write

$$\begin{array}{cc}\hfill & {\mathbb{D}}_{\mu }{F}^{\mu \nu }={\nabla }_{\mu }{F}^{\mu \nu }+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)F^{\mu \nu }=0\hfill \\ \hfill & {\mathbb{D}}_{\mu }{F}_{\alpha \beta }+{\mathbb{D}}_{\alpha }{F}_{\beta \mu }+{\mathbb{D}}_{\beta }{F}_{\mu \alpha }=0\hfill \end{array}$$

(46)

These two equations are gauge invariant.

First, consider the first expression in Eq. (46) for $\nu \equiv j=1,2,3$ (which is Maxwell-Ampére’s law). Then using Eq. (19) for $J^j=0$ (that is, vacuum), we obtain:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c}}\left({\nabla }_{\mu }{E}_i\right)\left(g^{0\mu }{g}^{ij}-g^{i\mu }{g}^{0j}\right)+{\displaystyle \frac{1}{c}}{E}_i\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{0\mu }{g}^{ij}-g^{i\mu }{g}^{0j}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{0\mu }{g}^{ij}-g^{i\mu }{g}^{0j}\right)\right)\hfill \\ \hfill & -{\epsilon }_{klm}\left({\nabla }_{\mu }{B}^k\right)g^{l\mu }{g}^{mj}-{\epsilon }_{klm}{B}^k\hfill \\ \hfill & \times \left({\nabla }_{\mu }\left(g^{l\mu }{g}^{mj}\right)+{\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_{\mu }\left(\sqrt{-g}\right)\left(g^{l\mu }{g}^{mj}\right)\right)=0\hfill \end{array}$$

(47)

For weak gravitational fields (i.e., $g^{0i}=g^{i0}=0$) with a diagonal metric (i.e., $g^{ij}=0$ for $i\ne j$), from Eq. (47), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^2}}{g}^{00}{g}^{jj}{\nabla }_t{E}_j+{\displaystyle \frac{1}{c^2}}{E}_j\left({\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\epsilon }_{ljk}{g}^{ll}{g}^{jj}{\nabla }_l{B}^k+{\epsilon }_{ljk}{B}^k\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \end{array}$$

(48)

where the following identity is used

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\sqrt{-g}}}{\nabla }_t\left(\sqrt{-g}\right)={\nabla }_t\left(ln\sqrt{-g}\right)\end{array}$$

(49)

Taking the time derivative of both sides of Eq. (48), we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^2}}{g}^{00}{g}^{jj}{\nabla }_{tt}^2{E}_j\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}\left({\nabla }_t{E}_j\right)\left(2{\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}{E}_j\left({\nabla }_{tt}^2\left(g^{00}{g}^{jj}\right)+{\nabla }_{tt}^2\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_t\left(ln\sqrt{-g}\right){\nabla }_t\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\epsilon }_{ljk}{\nabla }_t\left(g^{ll}{g}^{jj}\right)\left({\nabla }_l{B}^k\right)\hfill \\ \hfill & +{\epsilon }_{ljk}{g}^{ll}{g}^{jj}\left({\nabla }_l{\nabla }_t{B}^k\right)\hfill \\ \hfill & +{\epsilon }_{ljk}\left({\nabla }_t{B}^k\right)\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & +{\epsilon }_{ljk}{B}^k\left({\nabla }_{lt}^2\left(g^{ll}{g}^{jj}\right)+{\nabla }_{lt}^2\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_l\left(ln\sqrt{-g}\right){\nabla }_t\left(g^{ll}{g}^{jj}\right)\right)\hfill \end{array}$$

(50)

Using Faraday’s law in the following form

$$\begin{array}{c}\hfill -{\epsilon }_{mn}^k{\nabla }^m{E}^n={\nabla }_t{B}^k\end{array}$$

(51)

and the mathematical identity

$$\begin{array}{c}\hfill {\epsilon }_{ljk}{\epsilon }_{mn}^k={\delta }_{lm}{\delta }_{jn}-{\delta }_{ln}{\delta }_{jm}\end{array}$$

(52)

then, we have

$$\begin{array}{cc}\hfill {\epsilon }_{ljk}{g}^{ll}{g}^{jj}\left({\nabla }_l{\nabla }_t{B}^k\right)& =-{\epsilon }_{ljk}{g}^{ll}{g}^{jj}\left({\epsilon }_{mn}^k{\nabla }_l{\nabla }^m{E}^n\right)\hfill \\ \hfill & =-{\delta }_{lm}{\delta }_{jn}{g}^{ll}{g}^{jj}{\nabla }_l{\nabla }^m{E}^n\hfill \\ \hfill & +{\delta }_{ln}{\delta }_{jm}{g}^{ll}{g}^{jj}{\nabla }_l{\nabla }^m{E}^n\hfill \\ \hfill & =-g^{mm}{g}^{nn}{\nabla }_m{\nabla }^m{E}^n+g^{nn}{g}^{mj}{\nabla }_n{\nabla }^m{E}^n\hfill \\ \hfill & =-g^{nn}{\nabla }^m{\nabla }_m{E}_n\hfill \\ \hfill & =-g^{jj}{\nabla }^k{\nabla }_k{E}_j\hfill \end{array}$$

(53)

where $g^{mj}=0$, $g^{mm}{\nabla }_m={\nabla }^m$, and $E_j=g_{kj}{E}^k$ are used.

Furthermore,

$$\begin{array}{cc}\hfill & {\epsilon }_{ljk}\left({\nabla }_t{B}^k\right)\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\epsilon }_{ljk}(-{\epsilon }_{mn}^k{\nabla }^m{E}^n)\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & =(-{\delta }_{lm}{\delta }_{jn}+{\delta }_{ln}{\delta }_{jm}){\nabla }^m{E}^n\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & =-{\nabla }^m{E}^n\left({\nabla }_l\left(g^{ml}{g}^{nj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ml}{g}^{nj}\right)\right)\hfill \\ \hfill & +{\nabla }^m{E}^n\left({\nabla }_l\left(g^{nl}{g}^{mj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{nl}{g}^{mj}\right)\right)\hfill \\ \hfill & =0\hfill \end{array}$$

(54)

where $g^{ml}=g^{nj}=g^{nl}=g^{mj}=0$ is used.

Using Gauss’s law in vacuum ($\rho =0$), from Eq. (18), we write

$$\begin{array}{cc}\hfill & g^{jj}{g}^{kk}{\nabla }_k{E}_k-g^{jj}{g}_{00}{E}_k{a}^k\left(\mathbf{x}\right)=0\hfill \end{array}$$

(55)

where

$$\begin{array}{cc}\hfill a^k\left(\mathbf{x}\right)& =-\left(g^{kk}{g}^{00}{\nabla }_k\left(ln\sqrt{-g}\right)+{\nabla }_k\left(g^{kk}{g}^{00}\right)\right)\hfill \end{array}$$

(56)

which is a function of the space-time point.

Taking the derivative ${\nabla }_j$ of both sides and summing for $j=1,2,3$, we obtain

$$\begin{array}{cc}\hfill & {\nabla }_j\left(g^{jj}{g}^{kk}\right){\nabla }_k{E}_k+g^{jj}{g}^{kk}{\nabla }_j{\nabla }_k{E}_k\hfill \\ \hfill & +\left({\nabla }_j{g}^{jj}{g}_{00}\right)a^k{E}_k+g^{jj}{g}_{00}{a}^k{\nabla }_j{E}_k+g^{jj}{g}_{00}\left({\nabla }_j{a}^k\right)E_k=0\hfill \end{array}$$

(57)

Or,

$$\begin{array}{cc}\hfill & \left({\nabla }_j\left(g^{jj}{g}^{kk}\right)+g^{jj}{g}^{kk}{\nabla }_j\right){\nabla }_k{E}_k\hfill \\ \hfill & +\left(\left({\nabla }_j{g}^{jj}{g}_{00}\right)a^k+g^{jj}{g}_{00}\left({\nabla }_j{a}^k\right)\right)E_k+g^{jj}{g}_{00}{a}^k{\nabla }_j{E}_k=0\hfill \end{array}$$

(58)

Combining Eq. (50), Eq. (53), Eq. (54), and Eq. (58) we obtain

$$\begin{array}{cc}\hfill g^{jj}\left({\displaystyle \frac{g^{00}}{c^2}}{\nabla }_{tt}^2{E}_j+{\nabla }^k{\nabla }_k{E}_j\right)& ={\alpha }^{jj}{\nabla }_t{E}_j\hfill \\ \hfill & +{\eta }^{jkk}{\nabla }_k{E}_k+{\theta }^{jjk}{\nabla }_j{E}_k\hfill \\ \hfill & +{\beta }^{jj}{E}_j+{\psi }^{jk}{E}_k\hfill \\ \hfill & +{\gamma }_k^{jl}\left({\nabla }_l{B}^k\right)+{\sigma }_k^j{B}^k\hfill \end{array}$$

(59)

where

$$\begin{array}{cc}\hfill & {\alpha }^{jj}=-{\displaystyle \frac{1}{c^2}}\left({\nabla }_t{E}_j\right)\left(2{\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & {\beta }^{jj}=-{\displaystyle \frac{1}{c^2}}\left({\nabla }_{tt}^2\left(g^{00}{g}^{jj}\right)+{\nabla }_{tt}^2\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_t\left(ln\sqrt{-g}\right){\nabla }_t\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & {\gamma }_k^{jl}=-{\epsilon }_{ljk}{\nabla }_t\left(g^{ll}{g}^{jj}\right)\hfill \\ \hfill & {\sigma }_k^j=-{\epsilon }_{ljk}\left({\nabla }_{lt}^2\left(g^{ll}{g}^{jj}\right)+{\nabla }_{lt}^2\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_l\left(ln\sqrt{-g}\right){\nabla }_t\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & {\eta }^{jkk}={\nabla }_j\left(g^{jj}{g}^{kk}\right)+g^{jj}{g}^{kk}{\nabla }_j\hfill \\ \hfill & {\theta }^{jjk}=g^{jj}{g}_{00}{a}^k\hfill \\ \hfill & {\psi }^{jk}=\left({\nabla }_j{g}^{jj}{g}_{00}\right)a^k+g^{jj}{g}_{00}{\nabla }_j{a}^k\hfill \end{array}$$

(60)

Eq. (53) represents the wave equation of the electric field in a curved space-time geometry. It can be seen that is not a homogeneous second order differential equation, as derived for the flat space-time geometry. The presence of non-vanishing terms on the right-hand side indicates the influence of the space-time curvature on the electric field polarisation. Furthermore, the wave equation of the electric field is not independent on the magnetic field, even in the case of stationary space-time curvature indicating that the electromagnetic wave is a mixture of transverse and longitudinal waves, which is in contrast to Lorentz flat space-time geometry.

The following steps can be taken to derive the wave equation for magnetic field. From homogeneous electromagnetic equation in Eq. (46), for $\mu =0$, we obtain

$$\begin{array}{cc}\hfill {\epsilon }^{klj}{\nabla }_l{E}_j& =-{\nabla }_t{B}^k\hfill \end{array}$$

(61)

Taking the derivative for time t of both sides in Eq. (61), we will get

$$\begin{array}{cc}\hfill {\epsilon }^{klj}{\nabla }_t{\nabla }_l{E}_j& =-{\nabla }_{tt}^2{B}^k\hfill \end{array}$$

(62)

Furthermore,

$$\begin{array}{cc}\hfill {\nabla }_t{\nabla }_l{E}_j& ={\epsilon }_{ljk}{\nabla }_{tt}^2{B}^k\hfill \end{array}$$

(63)

Taking the derivative ${\nabla }_l$ in both sides of Eq. (48) and summing for $l=1,2,3$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^2}}{\nabla }_l\left(g^{00}{g}^{jj}\right)\left({\nabla }_t{E}_j\right)+{\displaystyle \frac{1}{c^2}}{g}^{00}{g}^{jj}{\nabla }_t{\nabla }_l{E}_j\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}\left({\nabla }_l{E}_j\right)\left({\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}{E}_j\left({\nabla }_l{\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_l{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_t\left(ln\sqrt{-g}\right){\nabla }_l\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\epsilon }_{ljk}\left({\nabla }_l{g}^{ll}{g}^{jj}\right)\left({\nabla }_l{B}^k\right)+{\epsilon }_{ljk}{g}^{ll}{g}^{jj}\left({\nabla }_l{\nabla }_l{B}^k\right)\hfill \\ \hfill & +{\epsilon }_{ljk}{\nabla }_l{B}^k\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & +{\epsilon }_{ljk}{B}^k\left({\nabla }_l{\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_l\left(ln\sqrt{-g}\right){\nabla }_l\left(g^{ll}{g}^{jj}\right)\right)\hfill \end{array}$$

(64)

Substituting Eq. (63) into Eq. (64), we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^2}}{\nabla }_l\left(g^{00}{g}^{jj}\right)\left({\nabla }_t{E}_j\right)+{\epsilon }_{ljk}{\displaystyle \frac{g^{00}}{c^2}}{g}^{jj}{\nabla }_{tt}^2{B}^k\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}\left({\nabla }_l{E}_j\right)\left({\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{c^2}}{E}_j\left({\nabla }_l{\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_l{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_t\left(ln\sqrt{-g}\right){\nabla }_l\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill & ={\epsilon }_{ljk}\left({\nabla }_l{g}^{ll}{g}^{jj}\right)\left({\nabla }_l{B}^k\right)+{\epsilon }_{ljk}{g}^{jj}\left({\nabla }^l{\nabla }_l{B}^k\right)\hfill \\ \hfill & +{\epsilon }_{ljk}{\nabla }_l{B}^k\left({\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill & +{\epsilon }_{ljk}{B}^k\left({\nabla }_l{\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_l\left(ln\sqrt{-g}\right){\nabla }_l\left(g^{ll}{g}^{jj}\right)\right)\hfill \end{array}$$

(65)

Or,

$$\begin{array}{cc}\hfill & g^{jj}\left({\displaystyle \frac{g^{00}}{c^2}}{\nabla }_{tt}^2{B}^k+{\nabla }^l{\nabla }_l{B}^k\right)=A^{jjl}{\nabla }_l{B}^k+C^{jj}{B}^k\hfill \\ \hfill & +D^{jjjk}{\nabla }_t{E}_j+F_l^{jjjk}{\nabla }^l{E}_j+M^{jjjk}{E}_j\hfill \end{array}$$

(66)

where

$$\begin{array}{cc}\hfill A^{jjl}& =-\left(2{\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill C^{jj}& =-\left({\nabla }_l{\nabla }_l\left(g^{ll}{g}^{jj}\right)+{\nabla }_l{\nabla }_l\left(ln\sqrt{-g}\right)\left(g^{ll}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_l\left(ln\sqrt{-g}\right){\nabla }_l\left(g^{ll}{g}^{jj}\right)\right)\hfill \\ \hfill D^{jjjk}& =-{\epsilon }_{ljk}{\displaystyle \frac{1}{c^2}}{\nabla }_l\left(g^{00}{g}^{jj}\right)\hfill \\ \hfill F_l^{jjjk}& =-{\epsilon }_{ljk}{\displaystyle \frac{1}{c^2}}\left({\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right)\hfill \\ \hfill M^{jjjk}& =-{\epsilon }_{ljk}{\displaystyle \frac{1}{c^2}}\left({\nabla }_l{\nabla }_t\left(g^{00}{g}^{jj}\right)+{\nabla }_l{\nabla }_t\left(ln\sqrt{-g}\right)\left(g^{00}{g}^{jj}\right)\right.\hfill \\ \hfill & +\left.{\nabla }_t\left(ln\sqrt{-g}\right){\nabla }_l\left(g^{00}{g}^{jj}\right)\right)\hfill \end{array}$$

(67)

Eq. (66) represents the wave equation of the magnetic field in a curved space-time geometry. Again, it is not a homogeneous second order differential equation, in contrast to the wave equation derived for the flat space-time geometry. The presence of non-vanishing terms on the right-hand side indicates the influence of the space-time curvature on the magnetic field. Furthermore, the wave equation of the magnetic field is coupled to the electric field, even in the case of stationary space-time curvature indicating that the electromagnetic wave is a mixture of transverse and longitudinal waves, which is in contrast to Lorentz flat space-time geometry.

2. Unification of Gravitation and Electromagnetism in Framework of General Relativity

It is important to note that $T^{\mu \nu }$ gives the energy-momentum density tensor of the matter, which may includes the mass distribution and they current of charges in matter. In general, it may also include the electromagnetic field that exists in the space; in that general case, $T_{\mu \nu }$ can be written in the covariant form as

$$\begin{array}{c}\hfill T_{\mu \nu }={\mathcal{G}}_{\mu \nu }+{\mathcal{T}}_{\mu \nu }\end{array}$$

(68)

where ${\mathcal{T}}_{\mu \nu }$ is the electromagnetic field energy-momentum density tensor including the interaction of electromagnetic field with currents and charges present, and it is given by the expression:

$${\mathcal{T}}^{\mu \nu }={\displaystyle \frac{1}{{\mu }_0}}{g}^{\mu \alpha }{F}_{\alpha \beta }{F}^{\beta \nu }+{\displaystyle \frac{1}{4{\mu }_0}}{g}^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }+g^{\mu \nu }{J}_{\alpha }{A}^{\alpha }$$

(69)

${\kappa }^{-1}\Lambda g_{\mu \nu }$ is the total energy-momentum density tensor; and $-{\kappa }^{-1}{G}_{\mu \nu }$ gives the energy-momentum density tensor of the gravitation field only. That is the viewpoint of the un-unified theory of the gravitation and electromagnetic fields.

Furthermore, there is an essential difference between the $-{\kappa }^{-1}{G}_{\mu \nu }$, which represents the storage of gravitation field energy and momentum density surrounding the masses, and the gravitational potential energy, which expresses the interaction energy between the masses, and hence it represents the interaction forces between the masses and it depends on the mass distribution. Therefore, their physical origin is completely different. The gravitational field energy and momentum distribution (which is the reality) is represented by the curvature of the space-time described by the Riemann metric tensor $g_{\mu \nu }\left(x^{\sigma }\right)$ a function of the space-time point $x^{\sigma }$ (which is a picture of that reality). Besides, both can not be localised in space; thus, to obtain both, one has to integrate overall space. Moreover, the gravitational potential energy density is always negative and it vanishes in the empty space-time. On the other hand, the gravitation field energy density is always negative inside the matter, and outside it depends on the metric used; for instance, for some metrics, it is positive and thus $\Lambda >0$.

Similarly, one can think about the electromagnetic field energy (which is a reality) and the electromagnetic interactions between the currents and the electromagnetic field; in this view, the electromagnetic field energy represents the energy stored around the space of the sources that created it, and the electromagnetic potential energy represents the interaction between the charges and the currents with the electromagnetic field, and so it depends on the charge and the current distribution of the matter. Therefore, in analogy, the distribution of the stored electromagnetic field energy and momentum (which is the reality) can introduce a new fabrication of the geometry, which can be described by an effective metric tensor, namely ${\tilde{g}}_{\mu \nu }$:

$$\begin{array}{c}\hfill {\tilde{g}}_{\mu \nu }=g_{\mu \nu }+\alpha F_{\mu \nu }\end{array}$$

(70)

which is a projection of the reality influenced by both gravitational and electromagnetic fields. In Eq. (70), $F_{\mu \nu }$ denotes the electromagnetic field strength tensor, and $\alpha$ is a constant having the inverse units of $F_{\mu \nu }$ such that $\alpha F_{\mu \nu }$ is dimensionless.

Then, we have the following covariant form:

$${\kappa }^{-1}\Lambda g_{\mu \nu }={\mathcal{G}}_{\mu \nu }+{\mathcal{T}}_{\mu \nu }-{\kappa }^{-1}{G}_{\mu \nu }$$

(71)

where ${\mathcal{T}}_{\mu \nu }$ is given by

$$\begin{array}{cc}\hfill {\mathcal{T}}_q^{0i}& ={\kappa }_e{J}^0{u}^i=\gamma {\kappa }_e{\rho }_{e}c{v}^i\hfill \\ \hfill {\mathcal{T}}_q^{ij}& ={\kappa }_e{J}^i{u}^j=\gamma {\kappa }_e{\rho }_e{v}^i{v}^j\hfill \end{array}$$

(72)

where ${\kappa }_e$ is a scaling factor [11]

$$\begin{array}{c}\hfill {\kappa }_e={\displaystyle \frac{4\pi \sqrt{G}/s^2}{\kappa }}={\displaystyle \frac{c^4}{2\sqrt{G}{s}^2}}\end{array}$$

(73)

where $s=1$ and its SI unit is m²/s², and

$$\begin{array}{cc}\hfill {\mathcal{T}}_q^{00}& =\gamma {\kappa }_e{\rho }_e{c}^2\hfill \end{array}$$

(74)

and ${\mathcal{G}}_{\mu \nu }$ by

$${\mathcal{G}}^{\mu \nu }=\left(\rho +{\displaystyle \frac{P}{c^2}}\right)u^{\mu }{u}^{\nu }+g^{\mu \nu }P$$

(75)

where $\rho$ is the mass density of macroscopic mass distribution and P is the pressure. In Eq. (75), $g^{\mu \nu }$ is the metric tensor of Riemann space-time. The zeroth component is

$$\begin{array}{cc}\hfill {\mathcal{G}}^{00}& =\left(\rho +{\displaystyle \frac{P}{c^2}}\right){\gamma }^2{c}^2+g^{00}P\hfill \\ \hfill & ={\gamma }^2\rho c^2+{\gamma }^{2}P+g^{00}P\hfill \\ \hfill & =\gamma \epsilon +P\left({\gamma }^2+g^{00}\right)\hfill \end{array}$$

(76)

where $\epsilon$ is the relativistic energy density. Eq. (76) indicates that ${\mathcal{G}}^{00}>0$, as expected.

In Eq. (71), ${\kappa }^{-1}\Lambda g_{\mu \nu }$ is the total energy-momentum density tensor; ${\mathcal{G}}_{\mu \nu }+{\mathcal{T}}_{\mu \nu }$ gives the energy-momentum density tensor of the matter (which includes the mass distribution and the current of charges in matter only); and $-{\kappa }^{-1}{G}_{\mu \nu }$ gives the energy-momentum density tensor of the gravitation field and electromagnetic field. Therefore, the rays of the electromagnetic field bend when they travel as they would bend in a space with only gravitation field with the effective metric tensor ${\tilde{g}}_{\mu \nu }$.

3. Conclusions

We focused on the influence of Riemann space-time curvature on the laws of electromagnetism, such as Gauss’s law, Maxwell-Ampére’s law, and Faraday’s law. Furthermore, we also introduced some work done on the unification of the gravitation and electromagnetic fields.