# Why these null coordinates are sometimes called “time”?

In Minkowski spacetime, if we introduce spherical coordinates the metric becomes

$$\eta = dt^2-dr^2-r^2d\Omega^2$$

with $d\Omega^2$ the $S^2$ round metric. It is then common to introduce

$$u = t-r,\quad v=t+r$$

which sometimes are called retarded and advanced time, specially to study the infinity of spacetime. I've seem one analogous thing being done in Schwarzschild spacetime.

What we know about those is:

1. They are null. Hence their coordinate lines are paths that could be taken by photons.

2. If we pick the surfaces $u = u_0$ and $v = v_0$, these surfaces have $\partial_u$ and $\partial_v$ respectively as normals. The normals are lightlike, hence the surface is lightlike. Such surface doesn't qualify (as far as I know of course) as a "space at fixed time" locus differently from a surface $t = t_0$ for some timelike coordinate $t$.

The point is that in all these cases we have two null coordinates $u,v$, in other words, coordinates with null worldlines, which end up being called retarded and advanced time.

I can't see how these coordinates can get the name "time". They clearly doesn't seem to be "the time measured by some observer". So why are these coordinates referred as time coordinates, and why retarded/advanced anyway?

## 1 Answer

First note, that both $u$ and $v$ are monotonically increasing functions along any timelike geodesic. Consequently, they can serve as "time" parameter along those geodesics.

Their usefulness comes (in part) from the fact that "retarded time" is the "time" coordinate for an observer at future null infinity, e.g. an outgoing wave at future null infinity will oscillate with $\exp(i\omega u)$. Similarly, "advanced time" is a time coordinate on past null infinity.

These properties make $u$ and $v$ extremely useful when describing how a system "looks" a distant observer.

The "retarded" and "advanced" comes from the fact that the retarded time is the time minus the signal time from the origin. Consequently, at any event in Minkowski space, it is the local time at the origin at which a signal emitted from the origin will reach the event.

Similarly, the "advanced time" of an event is it the local time at the origin at which a signal emitted from the event reaches the origin.