Unfortunate Similarities Between Geometrical Optics and General Relativity

General relativity generalizes the line element

$$-c^2d\tau^2=-Bc^2dt^2+Adr^2+r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2),$$

to the case where the coefficients, $A$ and $B$, appearing the metric can be functions of $r$, and perhaps also $t$. However, if they are functions of $r$, the coefficients will necessarily be functions of $t$ on account of the geodesic equations. The time variables $\tau$ and $t$ distinguish between 'proper', or local, time, and coordinate time, respectively. In the former, the clock is attached to the object in motion, while in the latter, it measures 'laboratory' time. Potentials, whether they be electromagnetic or gravitational, appear in the coefficient of $dt^2. Neglecting the space part, one arrives at conclusions that the ticks of a clock become further apart in a gravitational field, or the red-shift in the frequency is due to the presence of a massive body. These are unfortunate conclusions.

Geometrical optics has a lagrangian of the form

$$\mathcal{L}=\frac{1}{2}\left(A\dot{r}^2+r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)-n^2(r)\right),$$

where $n(r)$ is a varying index of refraction and $A$, a coefficient. The dot denotes a differential with respect to an affine parameter $\tau$, and the index of refraction can be written in terms of the differential of one affine parameter $t$ with respect to it,

$$n^2(r)=B\dot{t}^2,$$

where $B$ is another coefficient. The similarity between coordinate $t$ and 'proper' times $\tau$ is striking---but extremely misleading. Explicit time dependencies does not enter into geometric optics, and any difference in the affine parameters will show up in wavelength changes, and not frequency changes. Time does appear however in the rate of change of $r,\vartheta,\varphi,$ and $t$.

Geometrical optics has been likened to a zero-energy mechanics, by requiring the difference $\dot{r}^2-n^2$, in a spherically symmetric system to vanish [Evans and Rosenquist, AJP 54 (1986) 876]. This is incorrect since it does not introduce any limiting speed. Rather, we have shown in a previous blog that the index of refraction can be determined by the Legendre transform of the kinetic energy, $T=(1/2)A\dot{r}^2$, viz.,

$$\Psi(\dot{t})=\dot{r}\frac{\partial T}{\partial\dot{r}}-T,$$

where a new variable $\dot{t}$ is defined by

$$\dot{t}=\frac{\partial T}{\partial\dot{r}}=A\dot{r},$$

so that the new potential $\Psi$ is

$$\Psi(\dot{t})=\frac{\dot{t}^2}{2A}.$$

Subtracting this off the total kinetic energy gives the lagrangian

$$\mathcal{L}=\frac{1}{2}\left(A\dot{r}^2+r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)-\frac{\dot{t}^2}{A}\right).$$

Comparing this with the expression for $n$, we find $B=1/A$.

The Euler-Lagrange equations are given by

$$\ddot{r}+\frac{1}{2}A^{\prime}\dot{r}^2-r\dot{\varphi}^2-\frac{A^{\prime}}{2A^2}\dot{t}^2=0,$$

for the radial coordinate in the plane $\vartheta=\pi/2$, and the first integrals, $r^2\dot{\varphi}=L$ and $\dot{t}/A=\kappa$. 

Note that in the definition of the Legendre transform we have used the definition of $\dot{t}=A\dot{r}$, whereas the variational equations identified $\dot{t}/A=\mbox{const.}$ so it is natural to set that constant equal to the limiting speed $\kappa$. Introducing  $c$ into the radial equation of motion results in

$$\ddot{r}-r\dot{\varphi}^2=-\frac{A^{\prime}}{2}\left(\dot{r}^2-c^2\right).$$

Since the radial acceleration must be negative, $A^{\prime}<0$. For the Weber force, $A=1/r$ and the vanishing of the force field is

$$\ddot{r}-\frac{1}{2}\frac{\dot{r}^2}{r^2}+\frac{c^2}{r}=0,$$

where $\kappa=\surd 2 c$, is Weber's constant. The index of refraction is $n=c\surd(2/r)$, as those for light orbits.

The vanishing of the Weber force is compatible with the geodesic equations resulting from the Euler-Lagrange equation

$$\frac{d}{d\tau}\frac{\partial\mathcal{L}}{\partial\dot{r}}-\frac{\partial\mathcal{L}}{\partial r}=0.$$

It is to be emphasized that $\tau$ is not proper time, but just some affine parameter that measures the advance along the trajectory $r$. Similarities can often times be very deceiving!

Now, the index of refraction has its foot in two worlds: the static world of light propagation in a homogeneous, or inhomogeneous, medium, described by the variation of one affine parameter with respect to another, and the kinetic world because of its definition of speed in that medium, 

$$n=c/v,$$

where $v$ is the velocity of light in a medium with an index of refraction $n$. The angular momentum has been defined with respect to the affine parameter, $\tau$. Let us use $\lambda$ and $\lambda^{\prime}$ for the two affine parameters, and $t$ for time--this time. We can write for any affine parameter $\lambda$ as

$$\frac{c}{n^2}=\frac{d\lambda}{dt}=\frac{d\lambda}{d\lambda^{\prime}}\frac{d\lambda^{\prime}}{dt}.$$

One definition of the  index of refraction is $n=d\lambda^{\prime}/d\lambda$, while the other is $n=c/v=c/d\lambda^{\prime}/dt$. If we consider the angular momentum

$$n^2r^2\frac{d\varphi}{dt}=L,$$

we find for the Weber field that the angular velocity $rd\varphi/dt$ is a constant of the motion. This agrees with the interpretation of Evans and Rosenquist.

Regarding the first definition of the index of refraction as the variation of one affine parameter with respect to the other, we have

$$d\lambda^{\prime}=n d\lambda,$$

indicating that a shift in the affine parameter occurs for $n\neq 1$. 

For the Schwarzschild metric, the coefficient $A=(1-\mathcal{R}/r)^{-1}$ so that the index of refraction varies as $n=1/(1-\mathcal{R}/r)^{1/2}$. This will cause a shift in the affine parameters by an amount

$$d\lambda^{\prime}=\left(1-\mathcal{R}/r\right)^{-1/2}d\lambda.$$

In a medium of  varying index of refraction, the affine parameter, which can be measured in a number of wavelengths of light that is used, will vary. However, nothing can be said about the frequency since in a medium of varying index of refraction the wavelength varies but the frequency does not. If the gravitational shift of spectral lines is determined in terms of wavelength, everything is fine. But, no reddening can occur because the frequency of light is invariant. 

In other words, there is no mechanism that will allow the frequency to vary, as in the case of a moving mobile, while light propagates through a completely static field whose index of refraction changes according to the strength of the gravitational potential.