Maxwell Equations and Riemann Geometry

Abstract

If we adopt a model of space–time based on the Riemann geometry, we have to face the problem of how to transfer to such a generalized context the old, standard results of relativistic physics obtained in the context of the Minkowski geometry.

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If we adopt a model of space–time based on the Riemann geometry, we have to face the problem of how to transfer to such a generalized context the old, standard results of relativistic physics obtained in the context of the Minkowski geometry. The equivalence principle tell us that the equations of special relativity still hold in a suitable inertial chart, but only locally, over a space–time region of small enough (infinitesimal) extension (see Sect. 2.2). In order to be globally extended on a general Riemann manifold, such equations are be suitably generalized.

The correct generalization procedure is provided by the so-called minimal coupling principle, which will be introduced in the next section and which will be applied, in this chapter, to the theory of the electromagnetic interactions. The validity of such a procedure is not restricted to the electromagnetic phenomena, however, and can be generally extended to all known physical systems and interactions. In the following chapters the minimal coupling principle will be indeed applied to many (and largely different) physical situations.

4.1 The Minimal Coupling Principle

The generalized relativity principle introduced in Chap. 2 imposes on the physical laws to respect an exact covariance property under the action of general coordinate transformations (more precisely, under the action of the diffeomorphism group).

If the physical system we are considering is described by equations which are already covariant (with respect to the Lorentz group) in the context of a Minkowski geometric structure, then such a system can be easily embedded into a more general Riemann structure—namely, its equations can be lifted to a general-covariant form—by applying a standard procedure called “minimal coupling principle”. Such a procedure, in practice, amounts to the following set of operations:

• replace the Minkowski metric with the Riemann metric, $\eta _{\mu \nu } \rightarrow g_{\mu \nu }$, for all scalar products and all raising (and lowering) indices operations;

• replace all partial (and total) derivatives with the corresponding covariant derivatives, $\partial _\mu \rightarrow \nabla _\mu$;

• use the appropriate powers of $\sqrt{-g}$ to saturate to zero the weights of all tensor densities. In the action integral, in particular, the covariant measure is given by the prescription $d^4x \rightarrow d^4 x \sqrt{-g}$.

By applying the above procedure to the equations of motion or—better—to the action of the physical system, the system turns out to be “coupled” to the geometry of the given Riemann manifold. The coupling is “minimal”, in the sense that it depends only on the metric and on its first derivatives (the connection), and thus it can be locally neutralized in the limit in which $g \rightarrow \eta$ and $\varGamma \rightarrow 0$ (in agreement with the principle of equivalence). A coupling procedure introducing higher-order derivatives of the metric would involve the space–time curvature (see Chap. 6), and it would be impossible to eliminate the associated (non-local) “tidal” forces.

Also, such a coupling is universal, in the sense that it necessarily affects (in the same way) all physical systems, with no exceptions. Obviously, the coupling is implemented differently for different geometric objects (the explicit form of the covariant derivative, for instance, depends on the type of object we are considering). To the best of our present knowledge, however, there is no physical system which is “geometrically neutral”, i.e. insensitive to the geometric properties (and in particular to the curvature) of the given Riemann manifold.

Let us finally notice that the minimal coupling principle is not a peculiar aspect of the Riemann model of space–time, but is also a typical ingredient of the so-called gauge theories, where such a principle is needed to restore the invariance of the theory under a group of local symmetry transformations. Indeed, also in the Riemann case the minimal coupling is introduced to make the model covariant with respect to the group of diffeomorphism transformations, thus lifting to a local level the symmetry associated to the “rigid” (i.e. global) Lorentz transformations of the Minkowski space–time. From this point of view, as already stressed in Sect. 3.4, the connection $\varGamma$ represents the “gauge field” associated to a local symmetry.

This last point will be further discussed and better clarified in Chap. 12. Here we will limit ourselves to note that the “gauge paradigm” seems to be, at present, a crucial ingredient of all successful models of the fundamental interactions. Hence, the above link between gauge theories and Riemann space–time strongly suggests that the Riemann geometry could provide a suitable tool to describe a fundamental interaction like (in particular) the gravitational one.

4.2 Coupling Geometry to Electromagnetic Fields

By applying the minimal coupling principle to the definition of the electromagnetic field tensor $F_{\mu \nu }$ we may note, first of all, that the usual relation between fields and potentials is left unchanged:

$$\begin{aligned} F_{\mu \nu } \rightarrow&\nabla _\mu A_\nu - \nabla _\nu A_\mu \nonumber \\&= \partial _{\mu } A_{\nu } - \partial _{\nu } A_{\mu } - \left( \varGamma _{\mu \nu }\,^{\alpha } - \varGamma _{\nu \mu }\,^{\alpha } \right) A_{\alpha } \nonumber \\&= \partial _\mu A_\nu - \partial _\nu A_\mu \equiv F_{\mu \nu }. \end{aligned}$$

(4.1)

The connection contribution automatically disappears, because of the symmetry property $\varGamma _{[\mu \nu ]}\,^\alpha =0$. However, the geometric independence of the relation between fields and potentials is not an accidental consequence of using a model with a symmetric connection—as it might appear from the above equations—being a very general result, which holds even in the presence of torsion.

In fact, the correct geometric description of the electromagnetic potential (as well as of all vector potentials associated to gauge fields, of Abelian or not-Abelian type) has to be referred not the vector representation of the diffeomorphism group, but to the so-called “exterior forms” (or differential forms), to be introduced in Appendix A. Without going for the moment into further details, it will be enough to recall here that the vector potential of a gauge field is geometrically represented, according to the minimal coupling principle, by the differential 1-form $A= A_\mu dx^\mu$ (see Appendix A). Such an object transforms as a scalar in the Minkowski space–time locally tangent to the given Riemann manifold (see also Chap. 12): hence, the exterior covariant derivatives of this scalar object always reduce to ordinary derivatives, quite independently of the type of connection. Of course, if the model is characterized by a symmetric connection (as in the case we are considering), the distinction between vector representation and 1-form representations becomes irrelevant.

In any case, the fact that the relation between $F_{\mu \nu }$ and $A_\nu$ is left unchanged has two important consequences.

The first consequence is that the minimal coupling principle does not modify the Maxwell equations concerning the divergence of the magnetic field and the curl of the electric field, expressed by the condition $\partial _{[\alpha } F_{\mu \nu ]}=0$. In order to derive this result we can use, in our case, the covariant language of the Riemann geometry, where

$$\begin{aligned} \nabla _{\alpha } F_{\mu \nu }= {\partial }_{\alpha } F_{\mu \nu }- {\varGamma }_{\alpha \mu }^{\beta } F_{\beta \nu } - {\varGamma }_{\alpha \nu }^{\beta } F_{\mu \beta }. \end{aligned}$$

(4.2)

In fact, by taking the totally antisymmetric part of this equations we immediately recover the result

$$\begin{aligned} \nabla _{[\alpha } F_{\mu \nu ]}= \partial _{[\alpha } F_{\mu \nu ]}=0, \end{aligned}$$

(4.3)

where the connection contributions are absent, again, due to the index symmetry $\varGamma _{[\mu \nu ]}\,^\alpha =0$. Hence, the minimal coupling to the geometry does not modify this sector of the Maxwell equations.

The second, important consequence concerns the invariance of $F_{\mu \nu }$ under the (electromagnetic) gauge transformation

$$\begin{aligned} A_\mu \rightarrow A_\mu + \partial _\mu f , \end{aligned}$$

(4.4)

generated by an arbitrary scalar function f(x). Such an invariance still holds, and still implies the exact conservation of the electric charge, quite independently of the space–time geometry in which the fields (and the sources) are embedded.

In fact, let us consider the action describing the dynamics of the electromagnetic field and its interactions with the charged currents. By applying the minimal coupling principle to the corresponding Minkowskian action we have:

$$\begin{aligned} S=-\int _\varOmega d^4x \, \sqrt{-g} \left( {1\over 16 \pi } F_{\mu \nu } F^{\mu \nu }+{1\over c}\widetilde{J}^\mu A_\mu \right) . \end{aligned}$$

(4.5)

Here all scalar products are performed with the Riemann metric g, and $\widetilde{J}$ is the electromagnetic current obtained by generalizing (according to the minimal coupling procedure) the corresponding current $J^\mu$ of the Minkowski space–time.

Performing the gauge transformation (4.4)—i.e., by varying the potential as $A_\mu \rightarrow A_\mu + \delta A_\mu$, with $\delta A_\mu = \partial _\mu f$— we find that the corresponding variation of the above action is given by:

$$\begin{aligned} \delta S&= -{1\over c}\int _\varOmega d^4x \, \sqrt{-g} \,\widetilde{J}^\mu \partial _\mu f \nonumber \\&=-{1\over c}\int _\varOmega d^4x \, \partial _\mu \left( \sqrt{-g} \widetilde{J}^\mu f\right) +{1\over c}\int _\varOmega d^4x \,f\partial _\mu \left( \sqrt{-g} \widetilde{J}^\mu \right) \end{aligned}$$

(4.6)

(since $\delta F_{\mu \nu }=0$). By applying the Gauss theorem we find that the first contribution to $\delta S$ is vanishing, provided the current $\widetilde{J}$ approaches zero rapidly enough on the boundary $\partial \varOmega$ of the four-dimensional integration volume (which we expect to be true if our physical sources are localized on a finite portion of space). In that case, given the arbitrary value of the gauge function f, we can conclude that the action is gauge invariant provided that:

$$\begin{aligned} \partial _\mu \left( \sqrt{-g} \widetilde{J}^\mu \right) =0. \end{aligned}$$

(4.7)

The above condition can also be rewritten as

$$\begin{aligned} \nabla _\mu \widetilde{J}^\mu =0, \end{aligned}$$

(4.8)

where we have used the result (3.103) for the covariant divergence of a vector.

We can say, therefore, that the gauge symmetry of the action is still associated to the presence of a conserved current (according to the Nöther theorem), but such a current ($\widetilde{J}$) seems to be different from its Minkowski analogue (J). Different currents, on the other hand, may correspond to different conserved charges (recall Eqs. (1.33)–(1.35), which show how to compute the conserved quantity associated to a given current). Hence, it would seem that the conserved charge depends not only on the intrinsic properties of the electromagnetic sources, but also on the metric, and thus on the geometric properties of the space–time in which the sources are embedded.

Actually, such a geometric influence on the electric charge is a fictitious effect, as can be easily checked by recalling the explicit relation existing between J and $\widetilde{J}$ according to the minimal coupling principle.

Let us first notice, in fact, that the current density of the Minkowski space–time is given by $J^\mu = \rho dx^\mu /dt$, where $\rho$ is the electric charge density. Multiplying by $d^4x$ (which is a scalar measure for the proper Lorentz group) one obtains the (Lorentz) four-vector

$$\begin{aligned} J^\mu d^4x= c dq dx^\mu , \end{aligned}$$

(4.9)

where the infinitesimal scalar $dq= \rho d^3x$ gives the total charge in the volume element $d^3x$, while $dx^\mu$ is the infinitesimal displacement along the “world line” spanned by the evolution of the charged sources. In order to generalize this expression to a Riemann space–time context we have to apply the minimal coupling procedure, which leads the corresponding (generally covariant) equation

$$\begin{aligned} \widetilde{J}^\mu d^4x \,\sqrt{-g}= c dq dx^\mu . \end{aligned}$$

(4.10)

A comparison with Eq. (4.9) then gives $\widetilde{J}^\mu = J^\mu / \sqrt{-g}$. It follows that Eqs. (4.7), (4.8) are nothing more than a transcription, in explicitly covariant form, of the conservation equation $\partial _\mu J^\mu =0$ valid for the electromagnetic current $J^\mu$ of the Minkowski space–time.

The charge (of a given source) conserved in a Riemann space–time thus exactly coincides with the charge (of the same source) conserved in the Minkowski space–time.

4.3 The Generalized Maxwell Equations

As shown in the previous section, the generalization of the Minkowski space–time to the case of a Riemann space–time does not modify neither the relation between electromagnetic fields and potentials, nor the conservation law of the electric charge. Hence, one may wonder if there is or not some new physical effect associated with the presence of a generalized geometric structure. The answer is in the affirmative: there is indeed a modification of the dynamical equations describing the propagation of the electromagnetic fields in a general Riemann context. Such equations turn out to be crucially dependent on the geometric properties of the space–time itself.

In order to elucidate this effect let us recall the action (4.5), which we rewrite as

$$\begin{aligned} S= \int _\varOmega d^4x \,\sqrt{-g} \,\mathcal{L}(A, \partial A), \end{aligned}$$

(4.11)

where $\mathcal{L}$ is the term in round brackets of Eq. (4.5). By varying with respect to $A_\nu$, and imposing the condition of stationary action, we get the corresponding Euler–Lagrange equations

$$\begin{aligned} \partial _\mu {\partial \left( \sqrt{-g} \mathcal{L}\right) \over \partial \left( \partial _\mu A_\nu \right) }= {\partial \left( \sqrt{-g} \mathcal{L}\right) \over \partial A_\nu }, \end{aligned}$$

(4.12)

written for the “effective” Lagrangian $\sqrt{-g} \mathcal{L}$ (which is not a scalar, but a scalar density of weight $w=-1$). By computing the partial derivatives, and dividing by $\sqrt{-g}$, we then arrive at the equations of motion:

$$\begin{aligned} {1\over \sqrt{-g}}\partial _\mu \left( \sqrt{-g} F^{\mu \nu }\right) ={4 \pi \over c} \widetilde{J}^\nu . \end{aligned}$$

(4.13)

Let us now recall that

$$\begin{aligned} \nabla _\mu F^{\mu \nu }&= \partial _\mu F^{\mu \nu }+\varGamma _{\mu \alpha }\,^\mu F^{\alpha \nu }+ \varGamma _{\mu \alpha }\,^\nu F^{\mu \alpha } \nonumber \\&=\partial _\mu F^{\mu \nu }+{1\over \sqrt{-g}}\partial _\alpha \left( \sqrt{-g}\right) F^{\alpha \nu } \nonumber \\&={1\over \sqrt{-g}}\partial _\mu \left( \sqrt{-g}F^{\mu \nu }\right) \end{aligned}$$

(4.14)

(the last term of the first line is vanishing because $F^{\mu \alpha }= F^{[\mu \alpha ]}$, and we have used Eq. (3.97) for the trace of the Christoffel connection). Equation (4.13) can thus be rewritten as:

$$\begin{aligned} \nabla _\mu F^{\mu \nu } ={4 \pi \over c} \widetilde{J}^\nu . \end{aligned}$$

(4.15)

This is exactly the equation we would obtain by applying the minimal coupling principle directly to the Maxwell equations in Minkowski space (see Eq. (1.78)), and is clearly different from the original electromagnetic equations.

To summarize our results, and emphasize the possible physical effects of the minimal electromagnetic coupling to the Riemann geometry, it is now convenient to write down the full set of generalized Maxwell equations in terms of the field variables that are unchanged with respect to the Minkowski space–time. Those variables are the covariant tensor $F_{\mu \nu }$ (see Eq. (4.1)) and the vector current $J^\mu = \sqrt{-g} \widetilde{J}^\mu$ (see Eqs. (4.9) and (4.10)). From Eqs. (4.3), (4.13) we then obtain:

$$\begin{aligned}&\partial _\mu \left( \sqrt{-g} g^{\mu \alpha } g^{\nu \beta } F_{\alpha \beta } \right) = {4 \pi \over c} J^\nu , ~~~~~~~~~~\partial _{[\mu } F_{\alpha \beta ]}=0, \nonumber \\&F_{\alpha \beta }= \partial _\alpha A_\beta -\partial _\beta A_\alpha . \end{aligned}$$

(4.16)

All geometric contributions, described by the Riemann metric g, here appear explicitly, and their form suggests the existence of a close analogy between the electromagnetic equations written in a Riemann space–time and the same equations written in the Minkowski space–time but in the presence of a continuous optical medium.

4.3.1 Analogy with the Equations in an Optical Medium

In the presence of a continuous dielectric medium, and in the context of the Minkowski space–time, it is well known that the Maxwell equations can be conveniently expressed in terms of two different field tensors. The usual tensor $F_{\mu \nu }$, whose components $F_{0i}=E_i$ and $F_{ij}=-\epsilon _{ijk} B^k$ describe the electromagnetic field in vacuum, related to the total charge density and current density; and another tensor $G^{\mu \nu }$, whose components $G^{i0}=D^i$ and $G^{ij}=-\epsilon ^{ijk} H_k$ describe the electromagnetic field inside the medium (also called the “induction field”), related to the free charge density and current density. The two fields F and G satisfy the following equations:

$$\begin{aligned}&\partial _\mu G^{\mu \nu }= {4 \pi \over c} J^\nu , ~~~~~~~~~~~~~~~~\partial _{[\mu } F_{\alpha \beta ]}=0, \nonumber \\&F_{\alpha \beta }= \partial _\alpha A_\beta -\partial _\beta A_\alpha . \end{aligned}$$

(4.17)

and are related by the so-called “constitutive equation”,

$$\begin{aligned} G^{\mu \nu }= \chi ^{\mu \nu \alpha \beta } F_{\alpha \beta }, \end{aligned}$$

(4.18)

which describes the electromagnetic properties of the considered medium.

The tensor $\chi$ satisfies the general properties:

$$\begin{aligned} \chi ^{\mu \nu \alpha \beta } = \chi ^{[\mu \nu ][\alpha \beta ]}=\chi ^{\alpha \beta \mu \nu }, ~~~~~~~~~~\chi ^{[\mu \nu \alpha \beta ]}=0. \end{aligned}$$

(4.19)

In order to give a simple example we can take, in particular, an isotropic and non-conducting medium, characterized by a dielectric constant $\epsilon$ and a magnetic permeability $\mu$. In that case, and in the reference system at rest with the medium, we have

$$\begin{aligned} \chi ^{i0j0}= -\epsilon \delta ^{ij}, ~~~~~~~~~~~ \chi ^{ijkl}={1\over 2 \mu } \left( \delta ^{ik} \delta ^{jl} - \delta ^{il} \delta ^{jk} \right) , \end{aligned}$$

(4.20)

so that Eq. (4.18) provides us with the well known constitutive relations:

$$\begin{aligned} \varvec{D}= \epsilon \varvec{E}, ~~~~~~~~~~~~~~~~ \varvec{B}= \mu \varvec{H}. \end{aligned}$$

(4.21)

A comparison of Eqs. (4.16) and (4.17) clearly shows that a Riemann space–time manifold formally behaves, from an electrodynamic point of view, as a continuous optical medium whose dielectric properties are controlled by the metric, through an “effective” constitutive tensor given by

$$\begin{aligned} \chi ^{\mu \nu \alpha \beta }= {1\over 2} \sqrt{-g} \left( g^{\mu \alpha } g^{\nu \beta } -g^{\mu \beta } g^{\nu \alpha } \right) . \end{aligned}$$

(4.22)

This analogy is not only a formal result. Indeed, as we shall see later in Chap. 8, a space–time geometry described by a suitable Riemann metric is able to deflect and slow down light rays—and, more generally, any other electromagnetic signal—just like a transparent, inhomogeneous dielectric. Also: a geometry described by a non-static metric, with off-diagonal components $g^{0i} \not =0$, my act as an optically active medium, able to rotate the polarization plane of a propagating electromagnetic wave. Further effects of the Riemann geometry on the propagation of light rays and electromagnetic signal will be illustrated in the following chapter.

Let us conclude this chapter by noticing, however, that it would be wrong to take too seriously such an analogy between optical media and geometry. There are indeed various crucial differences between the set of equations (4.16)—describing electromagnetic fields in vacuum, embedded in a Riemann space–time—and the set of equations (4.17)—describing electromagnetic fields in a medium, embedded in the Minkowski space–time—which prevent a total physical equivalence of the two above configurations. Unlike a real dielectric medium, in fact, the “geometric medium” satisfies the equivalence principle, and thus affects in a universal way all physical systems.

In this respect, an important example concerns the Cherenkov effect. In a typical dielectric medium, where the photon speed is slowed down, it may become possible for a charged particle to propagate at a speed faster than light in that medium. It that case, as is well known, there is emission of Cherenkov radiation.

In the geometric analogue of the dielectric medium, on the contrary, the Cherenkov effect cannot occur.¹ A curved geometry, indeed, may slow down the photon propagation but, at the same time, it also slows down—exactly at the same rate—the propagation of any other signal and test particle. If a particle is slower than light in the empty Minkowski space, then it will keep slower than light also in empty Riemann space, quite independently of the given geometry. Only a dielectric medium can act in a non-universal way, slowing down photons more than other types of particle, and triggering the emission of Cherenkov radiation.

Appendices

Exercises Chap. 4

4.1

Electrostatic Field in a Spherically Symmetric Space–Time Compute the electrostatic field of a point-like charge e embedded in a space–time manifold described by the following Riemann metric:

$$\begin{aligned} g_{00}= f(r), ~~~~~~~~~ g_{ij}= - \delta _{ij}, ~~~~~~~~~ g_{i0}=0, \end{aligned}$$

(4.23)

where $r= (x_ix^i)^{1/2}$.

4.2

Conformal Invariance of the Maxwell Equations Derive the propagation equation for the vector potential $\varvec{A}$ in the absence of charged sources, in the radiation gauge ($\varvec{\nabla }\cdot \varvec{A}=0$, $A_0=0$), and in a space–time geometry described by the Riemann metric

$$\begin{aligned} g_{00}= 1, ~~~~~~~~~ g_{ij}= - a^2(t) \delta _{ij}, ~~~~~~~~~ g_{i0}=0 \end{aligned}$$

(4.24)

(using for simplicity natural units in which $c=1$). Show that such equation can always be recast in the standard form of the d’Alembert wave equation through a suitable redefinition of the time variable. Finally, compute the explicit form of the metric tensor in the transformed system of coordinates.

Solutions

4.1

Solution Let us consider the electromagnetic equations (4.16), and set

$$\begin{aligned} J^i=0, ~~~~~~ J^0= e c \delta ^{(3)}(x), ~~~~~~~ F_{ij}=0, ~~~~~~~~ F_{0i}=E_i. \end{aligned}$$

(4.25)

Noticing that $\sqrt{-g}=f^{1/2}$ and $g^{00}=f^{-1}$ we get

$$\begin{aligned} \partial _i\left( \sqrt{-g} g^{ij} g^{00} F_{j0}\right) = \partial _i \left( f^{-1/2} E^i\right) = 4 \pi e \delta ^{(3)}(x). \end{aligned}$$

(4.26)

Let us now introduce a scalar function $\chi (r)$ such that

$$\begin{aligned} f^{-1/2} E^i=- \partial ^i \chi , \end{aligned}$$

(4.27)

and insert this definition into Eq. (4.26). Solving the Poisson equation for the scalar variable $\chi$ we easily obtain $\chi = e/r$, and the components of the electric field are thus given by

$$\begin{aligned} E^i= - f^{1/2} \partial ^i \chi = f^{1/2} {e x^i \over r^3}. \end{aligned}$$

(4.28)

4.2

Solution Let us insert the metric components (4.24) into Eqs. (4.16), and note that

$$\begin{aligned} g^{00}=1, ~~~~~~~~ g^{ij}=-a^{-2} \delta ^{ij}, ~~~~~~~\sqrt{-g}= a^3. \end{aligned}$$

(4.29)

By using the radiation gauge, $A_0=0$, $\partial ^i A_i=0$, we obtain

$$\begin{aligned} -\partial _0 \left( a \delta ^{ij} \partial _0 A_j\right) +{1\over a} \delta ^{kj}\delta ^{il} \partial _k \partial _j A_l=0, \end{aligned}$$

(4.30)

from which, dividing by a,

$$\begin{aligned} \left( {\partial ^2 \over \partial t^2} + {\dot{a} \over a} {\partial \over \partial t} -{\nabla ^2 \over a^2} \right) \varvec{A}=0, \end{aligned}$$

(4.31)

where $\dot{a}= da/dt$, and where $\nabla ^2= \delta ^{ij} \partial _i\partial _j$ is the usual Laplace operator of three-dimensional Euclidean space (we have set $c=1$).

The above equation can be recast in standard d’Alembertian form by introducing a new time parameter $\tau$, related to t by the differential condition $dt = a d\tau$. Using such a new coordinate we have, in fact,

$$\begin{aligned}&{\partial A \over \partial \tau }= a {\partial A \over \partial t},\nonumber \\&{\partial ^2 A \over \partial \tau ^2}= a {\partial \over \partial t} \left( a {\partial A \over \partial t}\right) = a^2 {\partial ^2 A \over \partial t^2}+ a \dot{a} {\partial A \over \partial t}, \end{aligned}$$

(4.32)

and Eq. (4.31) can be rewritten as

$$\begin{aligned} \left( {\partial ^2 \over \partial \tau ^2} - \nabla ^2\right) \varvec{A}=0. \end{aligned}$$

(4.33)

It is important to stress that this result is a consequence of the so-called conformal invariance of the Maxwell Lagrangian,

$$\begin{aligned} \sqrt{-g} g^{\mu \alpha } g^{\nu \beta } F_{\mu \nu }F_{\alpha \beta }, \end{aligned}$$

(4.34)

which is invariant under the following class of transformations

$$\begin{aligned} g_{\mu \nu } \rightarrow \widetilde{g}_{\mu \nu } = f(x) g_{\mu \nu }, ~~~~~~~~~~ g^{\mu \nu } \rightarrow \widetilde{g}^{\mu \nu } = f^{-1}(x) g^{\mu \nu } \end{aligned}$$

(4.35)

(called “local scale transformations”, or also “Weyl transformations”). As a consequence of this invariance, the form of the Maxwell equations is the same in the two (different) geometries described by the metrics g and $\widetilde{g}$ related by the transformation (4.35).

With the time redefinition $t \rightarrow \tau$, on the other hand, the line-element of the space–time (4.24) takes the form

$$\begin{aligned} ds^2= dt^2- a^2 dx_i dx^i= a^2 \left( d \tau ^2 - dx_i dx^i \right) , \end{aligned}$$

(4.36)

and the geometry is described by a new metric $\widetilde{g}_{\mu \nu }$,

$$\begin{aligned} \widetilde{g}_{\mu \nu }= a^2 (\tau ) \eta _{\mu \nu }. \end{aligned}$$

(4.37)

This metric is called “conformally flat”, as it is related to the Minkowski metric $\eta$ by a transformation of type (4.35), with $f=a^2$. Since the Maxwell equations must be identical in the two metrics $\widetilde{g}$ and $\eta$, we can immediately conclude that the wave equation for the vector potential, if expressed in terms of the coordinate $\tau$ of the metric $\widetilde{g}$, must coincide with the equation one would obtain in the space–time described by the Minkowski metric $\eta$ (namely, with the d’Alembert wave equation), as indeed obtained in Eq. (4.33).