How to define the proper time of a photon?

Question

I'm writing a paper about the motion of photons near a Schwarzschild black hole. At some point there's a derivative of the Hamiltonian of the system with respect to time $\tau$. I need to explain what the proper time is $\tau$, but it's quite odd because photons don't have any proper time.

The Hamiltonian that I have is

$$H = - \left( 1-\frac{2M}{r} \right)^{-1} \frac{p_{t}^2}{2}+\left( 1-\frac{2M}{r} \right) \frac{p_{r}^2}{2}+\left( \frac{p_{ \theta}^2}{2r^2}+\frac{p_{\phi}^2}{2r^2sin^2\theta} \right).$$

• So what would be the definition in this case?
• "the proper time is the time for the photon although he doesn't have one?"

Does anyone know?

Answer 1

I actually asked a question about the four velocity of the photon.

Four-velocity vector of light

where Chiral Anomaly says:

Short answer: - If the term "four-velocity" is used in the strict sense of $d x^\mu/d\tau$ where $\tau$ is the object's proper time, then four-velocity is undefined for light because the elapsed proper time is always zero ($d\tau=0$) along a lightlike worldline. - If the term "four-velocity" is used in the generalized sense of $dx^\mu/d\lambda$ where $\lambda$ is an affine parameter that increases monotonically along the lightlike worldline, then the four-velocity is perfectly well-defined for light.

So photons do not have a proper time, but you can use a $\lambda$ affine parameter that increases monotonically along the lightlike worldline.