Relativistic raytracing

In a Lorentzian manifold like the spacetime of special relativity (which we'll focus on here) nonzero tangent vectors at spacetime point p (i.e., for any curve passing through p) can be in three separate categories: spacelike, null, and timelike. Details <https://en.wikipedia.org/wiki/Causal_structure>. Curves that physical objects travel are typically timelike at every point (for objects with inertial mass) or everywhere null (for objects with no inertial mass). When unaccelerated, i.e. in inertial motion, rest mass determines whether the object is on a timelike geodesic or a null geodesic.

Four-velocity is defined on a timelike geodesic - its four functions take proper time as a parameter.

Proper time is a parametrization of a timelike geodesic which has several useful properties: it is monotonic allowing for orientability, and gives a unique real value for every infinitesimal point on the geodesic. This in turn means the tangent vector is preserved when parallel-transported along the proper-time-labelled timelike geodesic, with the gross physical meaning that the direction and speed of travel (in local forward/left/up coordinates for example [1]) does not change as proper time advances.

But we can label any curve -- or segment of a curve -- any way we want as coordinates are not physical. It's just that proper time is exceptionally useful as a coordinate. In particular, it satisfies the geodesic equation, gives us a four-velocity, which multiplied by the object's Lorentz-invariant mass gives us a four-momentum.

Unfortunately proper time is undefined on non-timelike geodesics, so can't be used for light, which in vacuum is always on a null geodesic.

However, there are functions that have the same features as proper time, namely monotonicity, uniqueness, and the preservation of the parallel-transported tangent vector. These are affine parametrizations. Proper time is an affine parametrization for timelike geodesics, and is usually denoted τ; null affine parametrizations which satisfy the geodesic equation are usually denoted λ.

We just can't use τ on null geodesics, since it is zero everywhere along one (so non-unique, and non-advancing). Statements based on the use of τ like "light experiences no time" or "light's path is infinitely length-contracted" improperly imply the use of τ.

However, if we use affine time λ, we get uniqueness, monotonicity, and the preservation of the tangent vector under parallel transport. The first derivative with respect to affine time at a point on the null geodesic lets us define a 4-momentum so we can use the Planck-Einstein relation E = pc = hf = hν = ħω relating the frequencies and wavelengths of photons, and how they redshift and dilate from point to point. (This is especially handy in curved spacetimes, like the expanding one in cosmology or the collapsing one around black holes).

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[1] An image may help: drop two dimensions from 3+1 spacetime to give us 2-space we get a tangent plane instead of a tangent volume at each point along a geodesic on a 2-sphere (which meridians on Earth's surface approximate) <https://en.wikipedia.org/wiki/Local_tangent_plane_coordinate...> -- sliding the tangent plane south along the meridian in uniform motion is like advancing the proper time.

It does if you are ray tracing. They are trying to find the source of a theoretical photon that hits the screen at the present time, the source will therefore have been in the past so they need to run time backwards as the photon moves away from the camera.

From TFA:

> But wouldn't we want to see the black hole? One could send a ray through every pixel of a screen, and let them go around the black hole and hit its surroundings. And if the rays are sent back in time, it's as if one received the light emitted by the surrounding objects in the past!