The usual statement of Snell's law, "in particle language", states that since there is no force acting in the transverse direction, there will be conservation of the transverse component. If A and A' are the angles with respect to the normal to the surface separating vacuum made by momenta p and p' respectively, then the conservation condition
p sin A = p' sin A'.
Comparing this with Snell's law
sin A = n' sin A',
where n' is the index of refraction in the medium, we get
p'/p = n'.
For a photon traveling in the medium, the momentum will be
p' = E/c n'. (1)
Squaring leads to
r^2 |p'|^2 = |rxp'|^2 + (r.p')^2,
so that
(E/c)^2 (rn')^2 = J'^2 + r^2 p'_r^2, (2)
where J is the angular momentum and p'_r is the radial momentum.
Snell's law rests on the fact that J=(E/c)h is a constant, where h is an arbitrary constant, interpreted as an impact parameter. But what if J undergoes a change when it enters the second medium? Analogous to (1), we would have
J' =E/c h n'_r, (3)
where n'_r is a "relativistic" index of refraction.
For what interests us, the index of refraction n' is given by the Eaton lens
n'^2= 2m/r + E, (4)
where m=GM, the gravitational parameter and E is the total energy. If we take for the the relativistic index of refraction
n'_r^2 = 1+2m/rc^2=1-2m/rc^2, (5)
then the orbital equation can be written as
(dr/d theta/r)^2= (1/h^2){(rn')^2 - h^2/n_r^2}. (6)
The effects we are dealing with are determined by the ratio
(h/r)^2 =(2m/r + E)/(1-2m/rc^2),
recalling that h, the angular momentum, differs from the impact parameter by a factor of c, i.e. h->hc.
Making the change of variable, r=1/u, and taking the derivative with respect to the angle theta, we get
u" +u = m/h^2 + 3m/c^2 u^2, (7)
where the primes stand for the derivatives with respect to the angle. Since the factor E/c cancels from the numerator and denominator, we can reinstate h as the constant angular momentum so that h^2/m is the semi-latus rectum of the ellipse. But because of the last term in (7) there will be a precession of the orbit.
The orbital equation (7) is precisely Einstein's modified equation that yields the missing value in the advance of the perihelion of Mercury. The transverse components of the momentum, J, and J' are no longer equal so that Snell's law is violated. Their difference is proportional to
(h/c)^2 2m/r,
which is a coupling between the angular momentum and the Newtonian gravitational potential. The symmetry between the indices of refraction (4) and (5) is surprising, and the orbital equation (6) is proportional to the difference of their squares.
What we have done is to consider a conformally equivalent metric to that of the space part of the Schwarzschild metric,
ds^2 = n'_r^(-2){dr^2 + (r d theta)^2 n'_r^2}. (8)
The conformal factor, 1/n'_r^2, does not change the fact that the optical instrument is perfect (i.e., every point in the plane has a perfect conjugate point in the plane), so that we can concentrate on the second factor in (8). Call
dG/dr=G'(r)=(1-2m/Gc^2)^(1/2).
The sectional curvatures will be given by the radial
G"/G = m/G^3,
and rotational
(1-G'^2)/G^2 = 2m/G^3
components. The condition for tidal equilibrium is
2G"/G- (1-G'^2)/G^2 = 0. (9)
The tidal deformations can be represented in terms of the semi-axies of a triaxial ellipsoid, a>=b=c, where a=(1-G'2)/G^2 and b=c=G"/G. This is a prolate ellipsoid.
Condition (9) was improperly referred to as the "condition of an empty universe" by Einstein. Rather, it is a compatibility condition that the vanishing of the eigenvalues of the Ricci tensor ensure the exactness conditions for the existence of a scalar potential are obeyed. The tidal forces do not deform the geometry of space; when they vanish the system returns to its original, undeformed state. Rather, if the deformations are plastic, then the geometry of space will be permanently deformed.
In the linearized form, the vanishing of the Einstein tensor is the compatibility condition that deformations are elastic, and, consequently, can be described by a symmetric stress tensor. Deformations which take the system from one state to another are thus path independent. Drawing the analogy with Maxell's electromagnetic field theory, elastic deformations would be caused by electrostatic forces. Plastic deformations require an electric field which is not the gradient of a scalar potential, i.e. the addition of the time-rate-of-change of the vector potential to the electric field definition. Then plastic deformations would be representative of the magnetic field, the curl of the vector potential. That the electric and magnetic fields are interdependent, being connected by derivatives of the vector potential, imply inductive phenomena. Nothing that observations of on gravitational phenomena would ever make us believe that there is a gravitational analog to a magnetic field, which general relativists refer to as a "gravitomagnetic" field. And without such a field there can be no transverse waves, i.e. gravitational waves don't exist.
We have been able to account for the advance of the perihelion without ever mentioning time dependent phenomena. This is substantiated by the fact that we can refer all sectional curvatures, or tidal force components, to a three-dimensional ellipsoid.
The generalization of the indefinite Minkowski metric to allow for the metric coefficients to become nonlinear did not mean that electromagnetic phenomena, described by light waves, has been transformed into gravitational waves. We may ask: At what point do light waves, even propagating through a medium with a different index of refraction than that of the vacuum, become gravitational waves?
Not unsurprisingly, the latter travels at the same speed as that of light! The effects of gravity on the propagation of light can fully be described as offering a medium with a different index of refraction for the light waves to propagate through and observing the effects that it produces (e.g., the bending of light, precessional phenomena that light undergoes, etc).
In fact, the bending of light has the index of refraction in (4) unity! This means that the Newtonian attraction has vanished and we are left with a small perturbation coming from (5).
Alternatively, if (5) were unity, we would get the equation for a regular Keplerian ellipse
r theta' =h/[(rn')^2-h^2]^(1/2), (10)
where the prime stands for differentiation with respect to r. Equation (10) easily follows from Fermat's principle,
int n'[1+r^2 theta'2] dr = extremum.
In contrast, the index of refraction (5) is not derivable from Fermat's principle; the equation of the orbit is
(h du/d theta)^2 =1- (h.n'.u)^2,
or upon differentiating with respect to theta,
u" +u = 3m/c^2 u^2.
An interesting derivation of the Schwarzschild metric was given by Lenz in 1944. He considered a Lorentz contraction on going from Cartesian to spherical coordinates. It is only necessary to consider the spatial metric with
dx= dr/[1-v^2/c^2]^(1/2) and dz= r d phi,
since we are consider motion in the plane, theta= pi/2. The metric is
ds^2=dr^2/(1-v^2/c^2) + r^2d phi^2.
He then introduces the "transform"
(1/2)v^2= m/r,
which amounts to Kepler's III. Now, metrics which differ by a conformal factor show that skew rays can be treated as primary rays by considering
ds^2 = n_r^2[dr^2 + r^2d phi^2 n_r^(-2)]
where
n_r^2 = 1/[1-2m/r c^2]=1+2m/rc^2 (11)
to first order.
In terms of the trajectory, the orbital equation,
d phi/dr= h/r^2(n^2-h^2/r^2)^(1/2), (12)
becomes
d phi/dr= h/r^2[1-(h^2/r^2)(1-2m/r^2)]^(1/2). (13)
The first orbital equation integrates to
r(phi) = hc^2/m/[1-e. sin(phi)],
with an eccentricity
e =[1-(h c^2)^2/m^2]^(1/2),
with h having the dimension of an impact parameter.
Thus, by factoring out the conformal factor, (11), the equation of an ellipse (12) is "straightened" out and becoming a "mere" deflection about a massive body whose gravitational parameter is m.
Thanks for the place where the index of refraction appears in the conservation of momentum, we can determine the action of the tidal forces on a planet, which we take as a slightly deformed so that it has the appearance of a rod. This will allow us to distinguish be radial and tangential directions. If the effect of the gravitational field is to create another medium of index of refraction (11), then Fermat's principle
int n_r [1+r^2 phi'^2]^(1/2) dr = extremum,
where the prime denotes differentiation with respect to r, then
p_r^2 +h^2/r^2 = |p|^2
where
|p| =n_r E/c.
In such a medium the speed of light is c/n_r, less than that in the vacuum. The effect is to give a deflection of 2m/h c^2, where h is the impact parameter. It differs from the constant angular momentum by a factor of c.
Alternatively, if the tidal forces are operating, the rod will be elongated in the radial direction. This will have the effect of increasing the radial momentum, p_r->n_r p_r, so that Fermat's principle now reads
int [n_r^2 +r^2 phi'^2]^(1/2) dr = extremum.
The extremum condition is
n_r^2 p_r^2 + h^2/r^2 = |p|^2 = (E/c)^2.
This leads to a deflection half that that the contraction in the horizontal direction would give, i.e., 4 m/hc^2. This only confirms what we know from equation (9) that twice the radial sectional curvature is equal to the tangential curvature.
The two effects are equivalent of the single effect where the rod is contracted in the tangential direction, thus violating Snell's law. The deflection problem has a a single index of refraction given by (11). Rather, the advance of the perihelion makes use of two indices of refraction, (4) and (5). The relative magnitude of the effects is determined by the ratio, h/r.
As a final thought, we can determine the tidal forces for the classical index of refraction (4).
Set
G' = [2m/G+E]^(1/2).
The radial sectional curvature is
G"/G= -m/G^3, (14)
while the tangential sectional curvature is
(1-G'^2)/G^2 = (1-E -2m/G)/G^2. (15)
The condition of tidal equilibrium is now
-2G"/G +(1-G'^2)/G^2 = -(E-1)/G^2. (16)
For E-1>0, the difference is the constant sectional curvature of hyperbolic space. In contrast for E-1<0, the radius of curvature is a sphere, while if E=1, it vanishes, and the space is Euclidean. However, taken in the present context, the vanishing of (16) would be a condition for tidal equilibrium, and there would be no sectional curvature of a conic section.
Both equations, following Eq. (8) and the one before (14), highlight the intimate relationship between sectional curvatures, indices of refraction, and tidal forces. It are these indices of refraction that cause the plastic deformations--which are completely static---that give rise to tidal forces.