For null trajectories (such as those of photons) in general relativity, you cannot use proper time (\(\tau\)) as a parameter along the geodesic. This is because, for light-like (null) paths, the spacetime interval is exactly zero (\(ds^2 = 0\)), meaning proper time does not advance for the photon.
Instead, you use an affine parameter (commonly denoted \(\lambda\)) to parametrize the path of a photon along its geodesic. The geodesic equation for a trajectory \(x^\mu(\lambda)\) is then written as:
\[ \frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu \rho} \frac{dx^\nu}{d\lambda} \frac{dx^\rho}{d\lambda} = 0 \]
where \(\Gamma^\mu_{\nu \rho}\) are the Christoffel symbols.
An affine parameter has the property that it relates linearly to parallel transport and momentum along the geodesic. It has no intrinsic physical meaning like proper time but is mathematically required for the geodesic equations to work for massless particles.
For photons (null geodesics), use an affine parameter \(\lambda\) to trace their paths through spacetime because proper time is not defined for light.
In the context of decoherence theory, understanding null geodesics becomes important when considering how information propagates through spacetime. The affine parameter formalism may be relevant for understanding how decoherence effects propagate at the speed of light, particularly in gravitational contexts where spacetime curvature affects the causal structure.
Note: This appears to be foundational material for understanding how information and causal influences propagate in curved spacetime, which may be relevant to gravitational decoherence mechanisms.