On gravity as a medium property in Maxwell equations

Abstract

The effect of gravity in Maxwell’s equations is often treated as a medium property. The commonly used formulation is based on managing Maxwell’s equations in exactly the same form as in Minkowski spacetime and expressing the effect of gravity as a set of constitutive relations. We show that such a set of Maxwell’s equations is, in fact, a combination of the electric and magnetic fields defined in two different non-covariant ways, both of which fail to identify the associated observer’s four-vectors. The suggested constitutive relations are also ad hoc and unjustified. To an observer with a proper four-vector, the effect of gravity can be arranged as effective polarizations and magnetizations appearing in both the homogeneous and inhomogeneous parts. Modifying the homogeneous part by gravity is inevitable to any observer, and the result cannot be interpreted as the medium property. For optical properties one should directly handle Maxwell’s equations in curved spacetime.

Appendices

Appendix A: Covariant formulation

Maxwell’s equations in a medium using the electromagnetic field strength tensors are [29]

$$\begin{aligned} H^{ab}_{\;\;\;;b} = {1 \over c} J^a, \quad \eta ^{abcd} F_{bc,d} = 0, \end{aligned}$$

(A1)

where $H_{ab}$ is associated with ($D_a$, $H_a$) as $F_{ab}$ is associated with ($E_a$, $B_a$). Using dual tensors, $F^*_{ab} \equiv {1 \over 2} \eta _{abcd} F^{cd}$ and similarly for $H^*_{ab}$, we have

$$\begin{aligned} - {1 \over 2} \eta ^{abcd} H^*_{bc,d} = {1 \over c} J^a, \quad F^{*ab}_{\;\;\;\;\;;b} = 0. \end{aligned}$$

(A2)

Notice the ordinary derivatives appearing in the second of Eq. (A1) and the first of Eq. (A2). We have

$$\begin{aligned} \eta _{0ijk} = - \sqrt{-g} \eta _{ijk}, \quad \eta ^{0ijk} = {1 \over \sqrt{-g}} \eta ^{ijk}, \end{aligned}$$

(A3)

where $g \equiv \textrm{det}(g_{ij})$ and indices of $\eta _{ijk}$ are raised and lowered using $\delta _{ij}$ and its inverse. In this work, we consider a vacuum, thus $H_{ab} = F_{ab}$.

Using a generic time-like four-vector $u_a$ of an observer, with $u_a u^a \equiv -1$, we define the EM fields, $E_a$ and $B_a$, associated with the four-vector as [7, 32, 33]

$$\begin{aligned} F_{ab}\equiv & {} u_a E_b - u_b E_a - \eta _{abcd} u^c B^d, \nonumber \\ F^*_{ab}= & {} u_a B_b - u_b B_a + \eta _{abcd} u^c E^d, \end{aligned}$$

(A4)

with $E_a u^a \equiv 0 \equiv B_a u^a$. The EM fields are

$$\begin{aligned} E_a = F_{ab} u^b, \quad B_a = F_{ab}^* u^b. \end{aligned}$$

(A5)

The current four-vector is decomposed using the same four-vector as

$$\begin{aligned} J^a \equiv \varrho c u^a + j^a, \quad j_a u^a \equiv 0, \end{aligned}$$

(A6)

where $\varrho$ and $j_a$ are charge and current densities, respectively, measured by the same observer. The covariant form of Maxwell’s equations in terms of $E_a$ and $B_a$ are derived in [33], see also [34]. In other sections we often use tildes to indicate the spacetime covariant quantities.

Appendix B: Two ($3+1$) decompositions

We introduce two different decompositions of the spacetime metric. The ADM metric and its inverse are [35]

$$\begin{aligned} g_{00}\equiv & {} - N^2 + N^i N_i, \quad g_{0i} \equiv N_i, \quad g_{ij} \equiv {{\overline{h}}}_{ij}, \nonumber \\ g^{00}= & {} - {1 \over N^2}, \quad g^{0i} = {N^i \over N^2}, \quad g^{ij} = {{\overline{h}}}{}^{ij} - {N^i N^j \over N^2}, \end{aligned}$$

(B1)

where ${{\overline{h}}}{}^{ij}$ is an inverse of the three-space intrinsic metric ${{\overline{h}}}_{ij}$, thus ${{\overline{h}}}{}^{ik} {{\overline{h}}}_{jk} \equiv \delta ^i_j$, and the index of $N_i$ is raised and lowered by ${{\overline{h}}}_{ij}$ and its inverse. We have

$$\begin{aligned} \eta _{0ijk} = - N {\overline{\eta }}_{ijk}, \quad \eta {}^{0ijk} = {1 \over N} {\overline{\eta }}{}^{ijk}, \end{aligned}$$

(B2)

where indices of ${\overline{\eta }}_{ijk}$ is raised and lowered using ${{\overline{h}}}_{ij}$ and its inverses.

The normal frame is introduced as

$$\begin{aligned} n_i \equiv 0, \quad n_0 = - N, \quad n{}^i = - {N^i \over N}, \quad n{}^0 = {1 \over N}. \end{aligned}$$

(B3)

For the EM fields and current density in the normal frame, we set

$$\begin{aligned} {{\widetilde{B}}}_i \equiv {{\overline{B}}}_i, \quad {{\widetilde{B}}}_0 = {{\overline{B}}}_i N^i, \quad {{\widetilde{B}}}^i = {{\overline{B}}}{}^i, \quad {{\widetilde{B}}}^0 = 0, \end{aligned}$$

(B4)

and similarly for ${{\overline{E}}}_a$, and ${{\overline{j}}}_a$ with ${{\widetilde{E}}}_i \equiv {{\overline{E}}}_i$ etc.; indices of ${{\overline{B}}}_i$ etc. are raised and lowered by ${{\overline{h}}}_{ij}$ and its inverse. Based on the normal frame, Eqs. (A4) and (A6) give

$$\begin{aligned} F_{0i}= & {} - N {{\overline{E}}}_i - {\overline{\eta }}_{ijk} N^j {{\overline{B}}}{}^k, \quad F_{ij} = {\overline{\eta }}_{ijk} {{\overline{B}}}{}^k, \nonumber \\ F{}^*_{0i}= & {} - N {{\overline{B}}}_i + {\overline{\eta }}_{ijk} N^j {{\overline{E}}}{}^k, \quad F{}^*_{ij} = - {\overline{\eta }}_{ijk} {{\overline{E}}}{}^k, \nonumber \\ J^0= & {} \varrho c {1 \over N}, \quad J^i = - \varrho c {N^i \over N} + {{\overline{j}}}^i. \end{aligned}$$

(B5)

By introducing ${{\overline{B}}}_i \equiv B_i$ with the index of $B_i$ raised and lowered using $\delta _{ij}$ and its inverse, and similarly for $E_i$ and $j_i$, from Eq. (A1) we can derive Eqs. (1)–(4), see also [13].

In [7, 8] an alternative ($3+1$) decomposition is used with (in our notation)

$$\begin{aligned} g_{00}\equiv & {} - {{\overline{N}}}{}^2, \quad g_{0i} \equiv {{\overline{N}}}_i, \quad g_{ij} = {\overline{\gamma }}_{ij} - {{{\overline{N}}}_i {{\overline{N}}}_j \over {{\overline{N}}}{}^2}, \nonumber \\ g^{00}= & {} - {1 \over {{\overline{N}}}{}^2} \left( 1 - {{{\overline{N}}}{}^k {{\overline{N}}}_k \over {{\overline{N}}}{}^2} \right) , \quad g^{0i} = {{{\overline{N}}}{}^i \over {{\overline{N}}}{}^2}, \quad g^{ij} \equiv {\overline{\gamma }}{}^{ij}, \end{aligned}$$

(B6)

where the index of ${{\overline{N}}}_i$ is associated with ${\overline{\gamma }}_{ij}$ and its inverse metric ${\overline{\gamma }}^{ij}$; in [8], ${{\overline{N}}}{}^2 \equiv h$ and ${{\overline{N}}}_i \equiv h g_i$. We have

$$\begin{aligned} \eta _{0ijk}= & {} - \sqrt{-g} \eta _{ijk} = - {{\overline{N}}} {\overline{\eta }}_{ijk} = - \sqrt{-g_{00}} {\overline{\eta }}_{ijk}, \nonumber \\ \eta ^{0ijk}= & {} {1 \over \sqrt{-g}} \eta ^{ijk} = {1 \over {{\overline{N}}}} {\overline{\eta }}^{ijk} = {1 \over \sqrt{-g_{00}}} {\overline{\eta }}^{ijk}, \end{aligned}$$

(B7)

where $\sqrt{-g} = {{\overline{N}}} \sqrt{{\overline{\gamma }}} = N \sqrt{{{\overline{h}}}}$ with ${\overline{\gamma }} \equiv \textrm{det}({\overline{\gamma }}_{ij})$; here, indices of ${\overline{\eta }}_{ijk}$ is associated with ${\overline{\gamma }}_{ij}$ as the metric. Compared with the ADM decomposition, we have

$$\begin{aligned} ds^2= & {} - {{\overline{N}}}{}^2 \left( d x^0 - {{{\overline{N}}}_i \over {{\overline{N}}}{}^2} d x^i \right) \left( d x^0 - {{{\overline{N}}}_j \over {{\overline{N}}}{}^2} d x^j \right) \nonumber \\{} & {} + {\overline{\gamma }}_{ij} d x^i d x^j, \end{aligned}$$

(B8)

$$\begin{aligned} d s^2= & {} - N^2 d x^0 d x^0 \nonumber \\{} & {} + {{\overline{h}}}_{ij} \left( d x^i + N^i d x^0 \right) \left( d x^j + N^j d x^0 \right) . \end{aligned}$$

(B9)

We can express the normal four-vector, the EM fields, field strength tensor and four-current in the normal frame using this decomposition. But these are more complicated compared with Eqs. (B3)–(B5) for the ADM decomposition. For example, we have

$$\begin{aligned} n_i\equiv & {} 0, \quad n_0 = - {{{\overline{N}}}^2 \over \sqrt{ {{\overline{N}}}^2 - {{\overline{N}}}^k {{\overline{N}}}_k}}, \nonumber \\ n{}^i= & {} - {{{\overline{N}}}^i \over \sqrt{ {{\overline{N}}}^2 - {{\overline{N}}}^k {{\overline{N}}}_k}}, \quad n{}^0 = {\sqrt{ {{\overline{N}}}^2 - {{\overline{N}}}^k {{\overline{N}}}_k} \over {{\overline{N}}}^2}. \end{aligned}$$

(B10)

This decomposition is more adapted to the coordinate frame with

$$\begin{aligned} {{\bar{n}}}_i = {{{\overline{N}}}_i \over {{\overline{N}}}}, \quad {{\bar{n}}}_0 = - {{\overline{N}}}, \quad {{\bar{n}}}{}^i \equiv 0, \quad {{\bar{n}}}{}^0 = {1 \over {{\overline{N}}}}, \end{aligned}$$

(B11)

etc., but Maxwell’s equations are still more complicated compared with the compact forms available in the normal frame in the ADM decomposition, see Eqs. (20)–(23) in [13].