# Variational principle and the medium equation for photon paths in general relativity

I am reading a 2015 paper by Rogers on Frequency-dependent effects of gravitational lensing within plasma. He gives the relation between the components of the photon four-momentum and the refractive index of the medium, $$n^2=1+\frac{p_i p^i}{\left(p_j V^j\right)^2}\tag{3}$$ where $$p^i$$ is the linear momentum of the photon, $$V^j$$ is the velocity of the plasma medium, and $$n$$ is the refractive index of the medium.

He then goes on to write

We describe photon paths through the plasma in the Schwarzschild space–time by the Hamiltonian in the geometric optics limit. Using the variational principle and the medium equation we have $$H\left(x^i, p_i\right)=\frac{1}{2}\left[g^{i j} p_i p_j+\left(n^2-1\right)\left(p_j V^j\right)^2\right]=0\,.\tag{4}$$

Please, can you explain how the above equation is derived?

• I cannot say where the 1/2 enters, but inside the []'s is just the first expression you showed. The metric tensor $g^{ij}$ is merely used to contract the dyad $p_{i} p_{j}$, I think. In your first expression, it's just the inner product which gives a scalar. A GR expert may know more but it looks like there's nothing sneaky going on... Mar 13, 2023 at 13:02

1. We can wlog consider a reference frame where the plasma medium is static, i.e. $$V^{\mu}$$ has only a temporal component: $$-1 = ||V||^2~=~g_{00} V^0V^0 \qquad \Rightarrow \qquad V^0 ~=~ (-g_{00})^{-1/2}. \tag{A}$$

2. We assume that the metric $$g_{\mu\nu}$$ does not have any mixed temporal-spatial components.

3. Eq. (3) in Ref. 1 goes to the center of the Abraham-Minkowski controversy$$^1$$. Eq. (3) is equivalent to Minkowski's proposal that the photon 3-wavevector in a refractive medium is $$|{\bf k}|~=~n\frac{\omega}{c},\tag{1.10}$$ cf. eq. (1.10) in Ref. 2. Equivalently$$^2$$ $$n^2~\stackrel{(1.10)}{=}~\frac{g_{ab}k^ak^b}{-g_{00}(k^0)^2} ~\stackrel{p=\hbar k}{=}~1+\frac{p^2}{(p^0\sqrt{-g_{00}})^2} ,\qquad p^2~:=~ g_{\mu\nu}p^{\mu}p^{\nu},\tag{B}$$ which is eq. (3).

4. The refractive index $$n$$ is by definition the reciprocal phase velocity. Assuming no dispersion, we can identify it with the reciprocal group velocity $$|{\bf v}|~=~\frac{c}{n}.\tag{C}$$ Equivalently $$\frac{1}{n^2}~\stackrel{(C)}{=}~\frac{g_{ab}\dot{x}^a\dot{x}^b}{-g_{00}(\dot{x}^0)^2},\qquad \dot{x}^{\mu} ~=~\frac{dx^{\mu}}{d\lambda}.\tag{D}$$

5. Eq. (1.10) and the speed of light condition (C) suggests that in adapted coordinates $$\dot{x}^a~=~ep^a, \qquad \dot{x}^0~=~ en^2p^0, \tag{E}$$ where $$e$$ is an einbein/Lagrange multiplier. In covariant form eq. (E) reads $$\frac{\dot{x}^{\mu}}{e}~=~p^{\mu} -(n^2-1)V^{\mu} (p\cdot V)~=~G^{\mu\nu} p_{\nu}, \tag{F}$$ where we have introduced an effective metric tensor $$G_{\mu\nu}~=~g_{\mu\nu} + (1-n^{-2}) V_{\mu}V_{\nu}\qquad\Leftrightarrow\qquad G^{\mu\nu}~=~g^{\mu\nu} - (n^2-1) V^{\mu}V^{\nu}. \tag{G}$$

6. We next consider the Hamiltonian Lagrangian $$L_H~=~p\cdot \dot{x}- H,\tag{H}$$ where the Hamiltonian is of the form Lagrange multiplier times constraint $$H~:=~\frac{e}{2} p_{\mu}G^{\mu\nu}p_{\nu} ~=~\frac{e}{2}\left( p^2 - (n^2-1) (p\cdot V)^2 \right).\tag{I}$$ In the gauge $$e=1$$ the Hamiltonian $$H$$ becomes eq. (17) in Ref. 3 and eq. (4) in Ref. 1. Note that eq. (4) contains a sign mistake in the second term.

7. The EL equations of $$L_H$$ wrt. $$p_{\mu}$$ and $$e$$ reproduce eq. (F) and eq. (1.10), respectively, as they should.

8. The Hamiltonian action principle $$S_H ~=~\int d\lambda~L_H \tag{J}$$ agrees with the variational principle (16) in Ref. 3.

9. It is straightforward to integrate out $$p_{\mu}$$ and/or $$e$$ to arrive at corresponding Lagrangian formulations, cf. e.g. this Phys.SE post. The light trajectories are null-like geodesics wrt. the $$G_{\mu\nu}$$ metric.

References:

1. A. Rogers, Frequency-dependent effects of gravitational lensing within plasma, arXiv:1505.06790.

2. S.M. Barnett & R. Loudon, The enigma of optical momentum in a medium, Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences 368 (2010).

3. O.Yu. Tsupko & G.S. Bisnovatyi-Kogan, Gravitational lensing in plasma: Relativistic images at homogeneous plasma, arXiv:1305.7032.

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$$^1$$ Ref. 2 claims in section 5 that the resolution of the controversy is that the Abraham 3-momentum is the kinetic momentum of the light in the medium, while the Minkowski 3-momentum is the canonical momentum.

$$^2$$ Let $$\mu,\nu\in\{0,1,2,3\}$$ be spacetime indices while $$a,b\in\{1,2,3\}$$ are spatial indices.

• Thanks, 1. please could you let me know where I can read more about the speed of light condition. 2. Also, is $n^{−2}$ in (G) meant to be $n^2$? 3. Additionally, is $k^\mu=p^\mu$? Mar 15, 2023 at 0:41
• 1+3. I updated the answer. 2. No. Mar 15, 2023 at 7:54