Let γ: [s_{1}, s_{2}] → M be a smooth, future-directed timelike curve in M with tangent field ξ^{a}. We associate with it an elapsed proper time (relative to g_{ab}) given by

∥γ∥= ∫_{s1}^{s2} (g_{ab}ξ^{a}ξ^{b})^{1/2} ds

This elapsed proper time is invariant under reparametrization of γ and is just what we would otherwise describe as the length of (the image of) γ . The following is another basic principle of relativity theory:

**Clocks record the passage of elapsed proper time along their world-lines.**

Again, a number of qualifications and comments are called for. We have taken for granted that we know what “clocks” are. We have assumed that they have worldlines (rather than worldtubes). And we have overlooked the fact that ordinary clocks (e.g., the alarm clock on the nightstand) do not do well at all when subjected to extreme acceleration, tidal forces, and so forth. (Try smashing the alarm clock against the wall.) Again, these concerns are important and raise interesting questions about the role of idealization in the formulation of physical theory. (One might construe an “ideal clock” as a point-size test object that perfectly records the passage of proper time along its worldline, and then take the above principle to assert that real clocks are, under appropriate conditions and to varying degrees of accuracy, approximately ideal.) But they do not have much to do with relativity theory as such. Similar concerns arise when one attempts to formulate corresponding principles about clock behavior within the framework of Newtonian theory.

Now suppose that one has determined the conformal structure of spacetime, say, by using light rays. Then one can use clocks, rather than free particles, to determine the conformal factor.

Let g′_{ab} be a second smooth metric on M, with g′_{ab} = Ω^{2}g_{ab}. Further suppose that the two metrics assign the same lengths to timelike curves – i.e., ∥γ∥g′_{ab} = ∥γ∥g_{ab} ∀ smooth, timelike curves γ: I → M. Then Ω = 1 everywhere. (Here ∥γ∥g_{ab} is the length of γ relative to g_{ab}.)

Let ξ^{oa} be an arbitrary timelike vector at an arbitrary point p in M. We can certainly find a smooth, timelike curve γ: [s_{1}, s_{2}] → M through p whose tangent at p is ξ^{oa}. By our hypothesis, ∥γ∥g′_{ab} = ∥γ∥g_{ab}. So, if ξ^{a} is the tangent field to γ,

∫_{s1}^{s2} (g’_{ab} ξ^{a}ξ^{b})^{1/2} ds = ∫_{s1}^{s2} (g_{ab}ξ^{a}ξ^{b})^{1/2} ds

∀ s in [s_{1}, s_{2}]. It follows that g′_{ab}ξ^{a}ξ^{b} = g_{ab}ξ^{a}ξ^{b} at every point on the image of γ. In particular, it follows that (g′_{ab} − g_{ab}) ξ^{oa} ξ^{ob} = 0 at p. But ξ^{oa} was an arbitrary timelike vector at p. So, g′_{ab} = g_{ab} at our arbitrary point p. The principle gives the whole story of relativistic clock behavior. In particular, it implies the path dependence of clock readings. If two clocks start at an event p and travel along different trajectories to an event q, then, in general, they will record different elapsed times for the trip. This is true no matter how similar the clocks are. (We may stipulate that they came off the same assembly line.) This is the case because, as the principle asserts, the elapsed time recorded by each of the clocks is just the length of the timelike curve it traverses from p to q and, in general, those lengths will be different.

Suppose we consider all future-directed timelike curves from p to q. It is natural to ask if there are any that minimize or maximize the recorded elapsed time between the events. The answer to the first question is “no.” Indeed, one then has the following proposition:

Let p and q be events in M such that p ≪ q. Then, for all ε > 0, there exists a smooth, future directed timelike curve γ from p to q with ∥γ ∥ < ε. (But there is no such curve with length 0, since all timelike curves have non-zero length.)

If there is a smooth, timelike curve connecting p and q, there is also a jointed, zig-zag null curve connecting them. It has length 0. But we can approximate the jointed null curve arbitrarily closely with smooth timelike curves that swing back and forth. So (by the continuity of the length function), we should expect that, for all ε > 0, there is an approximating timelike curve that has length less than ε.

The answer to the second question (“Can one maximize recorded elapsed time between p and q?”) is “yes” if one restricts attention to local regions of spacetime. In the case of positive definite metrics, i.e., ones with signature of form (n, 0) – we know geodesics are locally shortest curves. The corresponding result for Lorentzian metrics is that timelike geodesics are locally longest curves.

Let γ: I → M be a smooth, future-directed, timelike curve. Then γ can be reparametrized so as to be a geodesic iff ∀ s ∈ I there exists an open set O containing γ(s) such that , ∀ s_{1}, s_{2} ∈ I with s_{1} ≤ s ≤ s_{2}, if the image of γ′ = γ|[s_{1}, s_{2}] is contained in O, then γ′ (and its reparametrizations) are longer than all other timelike curves in O from γ(s_{1}) to γ(s_{2}). (Here γ|[s_{1}, s_{2}] is the restriction of γ to the interval [s_{1}, s_{2}].)

Of all clocks passing locally from p to q, the one that will record the greatest elapsed time is the one that “falls freely” from p to q. To get a clock to read a smaller elapsed time than the maximal value, one will have to accelerate the clock. Now, acceleration requires fuel, and fuel is not free. So the above proposition has the consequence that (locally) “saving time costs money.” And proposition before that may be taken to imply that “with enough money one can save as much time as one wants.” The restriction here to local regions of spacetime is essential. The connection described between clock behavior and acceleration does not, in general, hold on a global scale. In some relativistic spacetimes, one can find future-directed timelike geodesics connecting two events that have different lengths, and so clocks following the curves will record different elapsed times between the events even though both are in a state of free fall. Furthermore – this follows from the preceding claim by continuity considerations alone – it can be the case that of two clocks passing between the events, the one that undergoes acceleration during the trip records a greater elapsed time than the one that remains in a state of free fall. (A rolled-up version of two-dimensional Minkowski spacetime provides a simple example)

*Two-dimensional Minkowski spacetime rolledup into a cylindrical spacetime. Three timelike curves are displayed: γ _{1} and γ_{3} are geodesics; γ_{2} is not; γ_{1} is longer than γ_{2}; and γ_{2} is longer than γ_{3}.*

The connection we have been considering between clock behavior and acceleration was once thought to be paradoxical. Recall the so-called “clock paradox.” Suppose two clocks, A and B, pass from one event to another in a suitably small region of spacetime. Further suppose A does so in a state of free fall but B undergoes acceleration at some point along the way. Then, we know, A will record a greater elapsed time for the trip than B. This was thought paradoxical because it was believed that relativity theory denies the possibility of distinguishing “absolutely” between free-fall motion and accelerated motion. (If we are equally well entitled to think that it is clock B that is in a state of free fall and A that undergoes acceleration, then, by parity of reasoning, it should be B that records the greater elapsed time.) The resolution of the paradox, if one can call it that, is that relativity theory makes no such denial. The situations of A and B here are not symmetric. The distinction between accelerated motion and free fall makes every bit as much sense in relativity theory as it does in Newtonian physics.

A “timelike curve” should be understood to be a smooth, future-directed, timelike curve parametrized by elapsed proper time – i.e., by arc length. In that case, the tangent field ξ^{a} of the curve has unit length (ξ^{a}ξ_{a} = 1). And if a particle happens to have the image of the curve as its worldline, then, at any point, ξ^{a} is called the particle’s four-velocity there.

Really great post on mathematical physics of relativity! It strikes me that the gap between wall-clock-measurable velocity and relativistic rapidity is particularly curious in the case of the photon, i.e., for lightlike curves (worldlines traversable at finite speed (c) but which are relativistically traversed with an infinite rapidity; so that travel, from a photon’s perspective, takes “zero time”…)

I am planning a writeup on energy-momentum tensor, which would yield a better perspective in the case. Another mathematical treatment in the pipe 🙂