Why a timelike geodesic maximizes path length?

Question

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is this statement easy to prove? I mean, does it follow straightly from the locally minimizing property of geodesics in Riemannian manifolds? If yes, please explain it to me, otherwise suggest me some reference where I can found the proof.

It seems that those minimizing/maximizing properties depend upon the geodesics' causal structure (timelike, spacelike or null-like) and, in general, nothing can be said about those geodesic's properties without knowing its causal structure. Is this correct?

Answer 1

First we sketch a proof that a timelike geodesic is a maximum of proper time. (We exclude saddle points for now.) Let $\gamma$ be a curve satisfying the geodesic equation, i.e. it is an extremum of proper time defined by $\tau[\gamma]:=\int\sqrt{-\langle\dot\gamma,\dot\gamma\rangle}\,\mathrm{d}t$. It is fairly simple to show that there always exists a curve $\mu$ for which $\tau[\mu]<\tau[\gamma]$, implying $\gamma$ is not a minimum. Construct along $\gamma$ a "tube" which is arbitrarily wide. Let $\mu$ be a curve which has the same start and end points as $\gamma$. Let $\mu$ be confined to the tube along $\gamma$. Now wind $\mu$ along the tube so that it is almost null, i.e. the curve's tangent approaches the null cone at every point on the tube. Thus we have constructed a curve with $\tau[\mu]$ arbitrarily close to zero, which can be made less than $\tau[\gamma]$.

This implies that a geodesic is not a minimum, but cannot determine that a timelike geodesic is not a saddle. However, this is not entirely true either. Here we quote Theorem 9.9.3 in [1]$^1$.

Let $\gamma$ be a smooth timelike curve connecting two points $p,q$. Then the necessary and sufficient condition that $\gamma$ locally maximize the proper time between $p$ and $q$ over smooth one parameter variations is that $\gamma$ be a geodesic with no point conjugate to $p$ between $p$ and $q$.

So a timelike geodesic is not necessarily a maximum of proper time. The study of geodesics does tie in to causal structure, Refs. [1] and [2] are highly recommended for this purpose.

Two standard references on causal structure are:

[1] R.M. Wald, General Relativity (1984).

[2] S.W. Hawking & G.F.R. Ellis, The large scale structure of space-time (1973).

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$^1$This is in turn quoted from Proposition 4.5.8 in [2], but I prefer [1]'s wording. Note that the full proof is found in [2].

Answer 2

The proof is elementary in Minkowski spacetime. It extends to general Lorentzian manifolds relying on the so-called Gauss' lemma and it is by no means obvious.

You can find the proof in several books of Lorentzian geometry or (mathematucally minded) GR. Also in my lecture notes https://moretti.maths.unitn.it/manifolds.pdf end of section 7.2, Proposition 7.29.

Answer 3

Think of the twin paradox, in which one twin travels out into space in a spaceship at high speed and the returns, while the other twin remains stationary on Earth. In the end, the twin traveling in a spaceship will have aged less because of time dilation. Arguably, the twin who remains on Earth travels along a geodesic, and he will have aged more than his twin (who deviated from the geodesic), i.e., his timelike path will have a greater proper time (as the “length” of a timelike path is its proper time). This is exactly the path maximization principle in general relativity.

Answer 4

In my opinion the thought demonstration of a (very large) rotating space-station, with many occupation levels, is well suited for putting into context that a geodesic is a path that maximizes proper time.

Let there be a space-station, rotating. Let there be multiple levels of occupation. At every level there is a G-load, as a consequence of the centripetal acceleration. As we know: the closer to the axis of rotation, the smaller the G-load

Let there be clocks at each level. The observation will be that as described by special relativity: for clocks further away from the axis of rotation a smaller amount of proper time elapses.

Now imagine that you throw an object vertically upwards, having it fall back into your hands. That is, you and the object part ways, and rejoin later on. When that object lands back in your hands: it has traveled a shorter spatial distance through Minkowski spacetime than you have. Once the object is released it is in inertial motion. On the other hand: your motion (through Minkowski spacetime) is curvilinear since you are co-moving with the space-station.

At this point we bring in the Principle of Equivalence. Let it be granted that the nature of spacetime and gravitational interaction is such that a local experiment cannot distinguish between velocity time dilation and gravitational time dilation.

If the principle of equivalence holds good then we will observe that for clocks that are higher up a gravitational potential a larger amount of proper time elapses than for clocks lower down the gravitational potential.

Reminder: On a rotating space-station going down the gravitational potential means that you move away from the axis of rotation.

From here we interpret the observation in terms of gravitational time dilation.

Imagine that you throw an object vertically upwards. Once the object is released it is in inertial motion. During its flight the object is traversing regions of space that are higher up the gravitational potential. As a consequence, when that object lands back in your hands: for that object a larger amount of proper time has elapsed than for you.

So: if it is granted a local setup cannot tell the difference between velocity time dilation and gravitational time dilation then the following must hold good:
When two clocks part ways, rejoining later on, and one moves along a trajectory of inertial motion, and the other clock moves along a trajectory that involves pulling G's, then for the clock moving in inertial motion the amount of proper time that has elapsed will be larger.

That is:
While it is the case that you can get to some destination faster than inertial motion: the acceleration (relative to inertial motion) means that along the way you are pulling G's, and accordingly for you a smaller amount of proper time will have elapsed.

Inertial motion is unbeatable; when two clocks part ways and arrive at a destination along different paths, then for the clock that has moved in inertial motion the largest amount of proper time has elapsed.

Further reading:

There is an article by Andrew J. S. Hamilton and Jason P. Lisle, The river model of black holes (2006) in which an interesting heuristic is described. The heuristic is referred to as 'River model of spacetime' It's not a new theory, or even a new interpretation. The heuristic offers a way of thinking of velocity time dilation and gravitational time dilation in relation to each other.

In terms of the river model the visualization is that spacetime is flowing down gravitational potential. The flow accelerates down a gravitational potential. Objects in spacetime do not respond to the velocity of the spacetime flow, but they do respond to the acceleration of the spacetime flow; they co-accelerate.

In terms of the river model: for a clock deeper down a gravitational potential spacetime is rushing by faster than for a clock higher up the gravitational potential, since the flow of spacetime accelerates. That way the river model supplies a way of thinking of gravitational time dilation as a form of velocity time dilation.

The river model does not require some thought demonstration being local; if the river model is set up for, say, the Earth-Moon sytem then the model covers the entire system in a single comprehensive representation.

The fact that the river model supplies a way of thinking of gravitational time dilation as a form of velocity time dilation expresses a core property of GR. Among the implications of the Principle of Equivalence is assertion that there is only a single form of time dilation. Velocity time dilation and gravitational time dilation should not be thought of as distinct phenomena. In terms of GR: only a single form of time dilation exists.