Using Relativity to Decide Between Abraham and Minkowski Momenta
What goes on inside a dielectric of dielectric constant, e, and peremability, mu? In the vacuum, the Poynting vector, P=ExH, determines the energy flux density, S=c P, and the momentum density g=P/c. It follows that the energy-momentum tensor is symmetrical since T_(i4)=T_(4i)=P, the Poynting vector where i indicates the space components.
Minkowski, in contrast proposed that whereas the energy flux, S, should remain the same, the momentum density should change inside a dielectric to read g=(1/c)(DxB) where D=eE and B= mu H. Thus, the momentum density should be characteristic of the material through which like is propagating. Apart from the fact that T_(i4) is no longer symmetrical, the difference between the momentum densities is n^2, where n is the index of refraction.
The linear momentum of a photon in a dielectric medium of index, n, is p=mc/n, where m=hw/c^2 is the mass equivalent, h standing for Planck's constant, divided by 2 pi, and w is the angular speed. p, therefore appears as Abraham's choice. Yet, if we consider two momenta, one in the vacuum, p=mc, and the other in the dielectric medium, p'=mc/n. This is all well and good except for Snell's law which states that
n' sin theta' = n sin theta.
Since n=1 in the vaccuum, the conservation of transverse momentum,
p' sin theta' = p sin theta.
p'/p = n'.
Since p=mc=hw/c, in the vacuum, it would appear that p' increases with n (neglecting from now on the aposthrophe) in accordance with Minkowski's choice, and not Abraham's choice, p'=mc/n. To patch things up, Tangherlini ["On Snell's law and the gravitational deflection of light"] allowed the effective mass to increase as n^2 so as to make p' proportional to the index of refraction. Had the nineteenth century physicists realized such an enigma, it would have led to the dependency of the effective mass upon Coulomb's potential divided by c^2, long before the electron mass was found to increase with speed.
Using Hamilton's ray optics, the equation of motion with a Hamiltonian given by H=(c/n)p, where n=n(r) is
dp/dt=-dH/dr=c/n^2 dn/dr p = m GM d/dr (1/r).
If the effective mass m does increase with n, then p can be considered to increase with n. However, it is more natural to consider p=mc/n, and consider the effective mass constant. Either way we come out with the equation
dn/dr/n^3= -GM/r^2 c^2.
Upon integrating, and fixing the constant of integration such that as r-> infinity, n->1 (no gravitational field), we come out with the expression n^2=1/(1–2GM/rc^2). This tells us how the material, as seen through the index of refraction, depends on the gravitational field. As a bonus, all relativistic corrections can be derived by applying the same procedure to the angular momentum. If we choose Abraham’s prescription, r^2 w=h/n, where w is the angular speed and h is the angular momentum, the conservation of energy (dr/dt)^2 + (r w)^2 -2GM/r = 2E, where E is the total energy, leads directly to Einstein modification of the orbital equation that gave him the correct expression for the advance of the perihelion of Mercury. Introducing w=h/n r^2, into the the conservation of energy gives (1/2)v^2=(1/2)[(dr/dt)^2+h^2/r^2]=GM/r +h^2 GM/c^2 r^3, for E=0. Taking the derivative of the above equation gives the force per unit mass and introducing it into the equation of the orbit [Eq (2.45) in Seeing Gravity] u” + u =-f/(hu)^2=GM/h^2 +3GM/c^2 u^2, where u=1/r, and the primes stand for the differentiation with respect to the angle variable. It is important to bear in mind that the equation of the orbit is for a photon of effective mass, hw/c^2, and not some imaginary "test" particle which does not exist!
The orbital equation is precisely Einstein’s equation for the modified orbit. All the relativistic corrections that are obtained by general relativity can be obtained by switching between between the Abraham, h/n, and Minkowski’s definition, hn, of the angular momentum.
By taking into account the dielectric properties of the material, through the index of refraction, there is no need to specify a metric and solve for its unknown coefficients (gravitational potentials). Since there is an equivalence between this optical approach and general relativity, it clearly shows that the latter deals with the propagation of light in a static gravitation field which affects the dielectric properties of the medium.
Bergmann in his Introduction to the Theory of Relativity has compares the perihelion advance with Sommerfeld's classical analysis of the motion of a relativistic electron.
The relativistic conservation of energy according to him is
Rather, if we consider Born's Mechanics of the Atom, the same conservation law reads
It is clear from Born's equation that p~1/n and
However, the index of refraction does not tend to 1 as r-> infinity, but, rather, to a constant, 1+E/2mc^2.
The equation of the trajectory is
u"+ (1-e^2/mc^2h^2)u =(e/h)^2(1+E/mc^2).
Dividing through by (1-e^2/m c^2h^2) and using the fact that the second term is much smaller than 1, the angle theta is reduced by the amount
The advance of the perihelion is
2pi [(1+e^2/mc^2h^2)^(1/2) -1]= pi (e^2/mc^2h^2).
Making the substitution e^2->GMm, this is 1/6 th the value of what Gerber (1895) and Einstein (1916) found for the perihelion advance of Mercury after all the other effects of the planets were substracted from the total advance of the perihelion.
We have been using h as the angular momentum. However, if we identify h with Planck's constant, the ratio, (e/ch), becomes the fine structure constant. The gravitational analog is (GM/ch), where h is the angular momentum. All relativistic corrections should be a power of this gravitational fine structure constant.