It seems that many users find the homework tag "demeaning", e.g. here. There have been many other instances of this, which has also lead to various posts being locked. The users refute that "This is not homework!!!", often, too. I suggest that we rename it (for example, as proble...

The proper time of an observer is the time shown on that observer's clock. This is because the observer's frame looks locally like flat space so: $$ d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2 $$ and (obviously) in the observer's frame the observer is at rest, so $dx$, $dy$ and $dz$ are all zero and th...

I think Jerry's point is that if you have a metric that looks like: $$ ds^2 = a(dx^0)^2 + f(dx^1, dx^2, dx^3) $$ where $a$ is a constant and $f$ is any function, then an observer moving in the $x^0$ direction follows a geodesic of the type you describe. An obvious example is the FLRW metric whe...

How about homework-and-exercises? It is self-explanatory, and that way we still call a spade a spade while simultaneously leave open the possibility that it might not be actual homework.