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.

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).