I'm reading Caroll's Lectures on GR 2 on pages 71-72, he states:
Let’s now explain the earlier remark that timelike geodesics are maxima of the proper
time. The reason we know this is true is that, given any timelike curve (geodesic or not), we
can approximate it to arbitrary accuracy by a null curve. To do this all we have to do is to
consider “jagged” null curves which follow the timelike one:
As we increase the number of sharp corners, the null curve comes closer and closer to the
timelike curve while still having zero path length. Timelike geodesics cannot therefore be
curves of minimum proper time, since they are always infinitesimally close to curves of zero
proper time; in fact they maximize the proper time.
My question is, if the geodesic is infinitesimally close to a null curve, shouldn't it also have zero path length? Why does this imply maximizing proper time?
Best Answer
The fact that the curve doesn't have zero path length is identical to the following 'proof' that $\pi=4$.
A detailed explanation can be found in this link, but the main idea is that the black line doesn't become a tangent line in the limit. This means the perimeter of the circle and jagged line aren't equal in the limit.
To why this implies it is maximal: a regular function has the property that in its maximum $f(x+\delta x)\leq f(x)$. Here $x$ maximizes $f$ and $\delta x$ is a small (or infinitessimal) quantity. For the proper time this argument is less obvious because it depends on the entire path. You can define it as a functional: an object which takes a function as input and outputs a scalar. $$\Delta \tau[x^\mu]=\int d\lambda\sqrt{-\eta_{\mu\nu}\frac {d x^\mu}{d\lambda}(\lambda)\frac {d x^\nu}{d\lambda}(\lambda)}$$
Our argument can then be extended to
$$\cases{\Delta \tau[x^\mu+\delta x^\mu]<\Delta \tau[x^\mu] & $x^\mu$ is a maximum\\ \Delta \tau[x^\mu+\delta x^\mu]>\Delta \tau[x^\mu] & $x^\mu$ is a minimum}$$
Now $\delta x^\mu(\lambda)$ it not a constant anymore but a function. In our case it is the offset between our geodesic and the jagged approximation of the geodesic. Since the proper time is positive and $\Delta \tau[x^\mu+\delta x^\mu]=0$ we have that $x^\mu$ must be a maximum.
Note: forgive me if I made mistakes, has been a while since I did any GR.