Time and World-Lines

Let γ: [s1, s2] → M be a smooth, future-directed timelike curve in M with tangent field ξa. We associate with it an elapsed proper time (relative to gab) given by

∥γ∥= ∫s1s2 (gabξaξb)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 = Ω2gab. Further suppose that the two metrics assign the same lengths to timelike curves – i.e., ∥γ∥g′ab = ∥γ∥gab ∀ smooth, timelike curves γ: I → M. Then Ω = 1 everywhere. (Here ∥γ∥gab is the length of γ relative to gab.)

Let ξoa be an arbitrary timelike vector at an arbitrary point p in M. We can certainly find a smooth, timelike curve γ: [s1, s2] → M through p whose tangent at p is ξoa. By our hypothesis, ∥γ∥g′ab = ∥γ∥gab. So, if ξa is the tangent field to γ,

s1s2 (g’ab ξaξb)1/2 ds = ∫s1s2 (gabξaξb)1/2 ds

∀ s in [s1, s2]. It follows that g′abξaξb = gabξaξb at every point on the image of γ. In particular, it follows that (g′ab − gab) ξoa ξob = 0 at p. But ξoa was an arbitrary timelike vector at p. So, g′ab = gab 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.)

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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 , ∀ s1, s2 ∈ I with s1 ≤ s ≤ s2, if the image of γ′ = γ|[s1, s2] is contained in O, then γ′ (and its reparametrizations) are longer than all other timelike curves in O from γ(s1) to γ(s2). (Here γ|[s1, s2] is the restriction of γ to the interval [s1, s2].)

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)

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Two-dimensional Minkowski spacetime rolledup into a cylindrical spacetime. Three timelike curves are displayed: γ1 and γ3 are geodesics; γ2 is not; γ1 is longer than γ2; and γ2 is longer than γ3.

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 ξa of the curve has unit length (ξaξa = 1). And if a particle happens to have the image of the curve as its worldline, then, at any point, ξa is called the particle’s four-velocity there.

Causal Isomorphism as a Diffeomorphism. Some further Rumination on Philosophy of Science. Thought of the Day 82.0

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes that are both future- and past-distinguishing, and let φ : M → M′ be a ≪-causal isomorphism. Then φ is a diffeomorphism and preserves gab up to a conformal factor; i.e. φ⋆(g′ab) is conformally equivalent to gab.

Under the stated assumptions, φ must be a homeomorphism. If a spacetime (M, gab) is not just past and future distinguishing, but strongly causal, then one can explicitly characterize its (manifold) topology in terms of the relation ≪. In this case, a subset O ⊆ M is open iff, ∀ points p in O, ∃ points q and r in O such that q ≪ p ≪ r and I+(q) ∩ I(r) ⊆ O (Hawking and Ellis). So a ≪-causal isomorphism between two strongly causal spacetimes must certainly be a homeomorphism. Then one invokes a result of Hawking, King, and McCarthy that asserts, in effect, that any continuous ≪-causal isomorphism must be smooth and must preserve the metric up to a conformal factor.

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The following example shows that the proposition fails if the initial restriction on causal structure is weakened to past distinguishability or to future distinguishability alone. We give the example in a two-dimensional version to simplify matters. Start with the manifold R2 together with the Lorentzian metric

gab = (d(at)(db)x) − (sinh2t)(dax)(dbx)

where t, x are global projection coordinates on R2. Next, form a vertical cylinder by identifying the point with coordinates (t, x) with the one having coordinates (t, x + 2). Finally, excise two closed half lines – the sets with respective coordinates {(t, x): x = 0 and t ≥ 0} and {(t, x): x = 1 and t ≥ 0}. Figure shows, roughly, what the null cones look like at every point. (The future direction at each point is taken to be the “upward one.”) The exact form of the metric is not important here. All that is important is the indicated qualitative behavior of the null cones. Along the (punctured) circle C where t = 0, the vector fields (∂/∂t)a and (∂/∂x)a both qualify as null. But as one moves upward or downward from there, the cones close. There are no closed timelike (or null) curves in this spacetime. Indeed, it is future distinguishing because of the excisions. But it fails to be past distinguishing because I(p) = I(q) for all points p and q on C. For all points p there, I(p) is the entire region below C. Now let φ be the bijection of the spacetime onto itself that leaves the “lower open half” fixed but reverses the position of the two upper slabs. Though φ is discontinuous along C, it is a ≪-causal isomorphism. This is the case because every point below C has all points in both upper slabs in its ≪-future.

Causal Isomorphism as Homeomorphism, or Diffeomorphism or a Conformal Isometry? Drunken Risibility.

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes.

We say that a bijection φ : M → M′ between their underlying point sets is a ≪-causal isomorphism if, ∀ p and q in M,

p ≪ q ⇐⇒ φ(p) ≪ φ(q).

Then we can ask the following: Does a ≪-causal isomorphism have to be a homeomorphism? A diffeomorphism? A conformal isometry? (We know in advance that a causal isomorphism need not be a (full) isometry because conformally equivalent metrics gab and Ω2gab on a manifold M determine the same relation ≪. The best one can ask for is that it be a conformal isometry – i.e. that it be a diffeomorphism that preserves the metric up to a conformal factor.) Without further restrictions on (M, gab) and (M′, g′ab), the answer is certainly “no” to all three questions. Unless the “causal structure” of a spacetime (i.e., the structure determined by ≪) is reasonably well behaved, it provides no useful information at all. For example, let us say that a spacetime is causally degenerate if p ≪ q for all points p and q. Any bijection between two causally degenerate spacetimes qualifies, trivially, as a ≪-causal isomorphism. But we can certainly find causally degenerate spacetimes whose underlying manifolds have different topologies. But a suitably “rolled-up” version of Minkowski spacetime is also causally degenerate, and the latter has the manifold structure S1 × R3.

There is a hierarchy of “causality conditions” that is relevant here. Hawking and Ellis impose, with varying degrees of stringency, the requirement that there exist no closed, or “almost closed,” timelike curves. Here are three.

Chronology: There do not exist smooth closed timelike curves. (Equivalently, for all p, it is not the case that p ≪ p.)

Future (respectively, past) distinguishablity: ∀ points p, and all sufficiently small open sets O containing p, no smooth future-directed (respectively, past-directed) timelike curve that starts at p, and leaves O, ever returns to O.

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Strong causality: For all points p, and all sufficiently small open sets O containing p, no smooth future-directed timelike curve that starts in O, and leaves O, ever returns to O.

It is clear that strong causality implies both future distinguishability and past distinguishability, and that each of the distinguishability conditions (alone) implies chronology.

The names “future distinguishability” and “past distinguishability” are easily explained. For any p, let I+(p) be the set {q: p ≪ q} and let I(p) be the set {q : q ≪ p}. It turns out that future distinguishability is equivalent to the requirement that, ∀ p and q,

I+(p) = I+(q) =⇒ p = q.

And the counterpart requirement with I+ replaced by I is equivalent to past distinguishability.

Conformal Factor. Metric Part 3.

Part 1 and Part 2.

Suppose gab is a metric on a manifold M, ∇ is the derivative operator on M compatible with gab, and Rabcd is associated with ∇. Then Rabcd (= gam Rmbcd) satisfies the following conditions.

(1) Rab(cd) = 0.

(2) Ra[bcd] = 0.

(3) R(ab)cd = 0.

(4) Rabcd = Rcdab.

Conditions (1) and (2) follow directly from clauses (2) and (3) of proposition, which goes like

Suppose ∇ is a derivative operator on the manifold M. Then the curvature tensor field Rabcd associated with ∇ satisfies the following conditions:

(1) For all smooth tensor fields αa1…arb1 …bs on M,

2∇[cd] αa1…arb1 …bs = αa1…arnb2…bs Rnb1cd +…+ αa1…arb1…bs-1n Rnbscd – αna2…arb1…bs Ra1ncd -…- αa1…ar-1nb1…bs Rarncd.

(2) Rab(cd) = 0.

(3) Ra[bcd] = 0.

(4) ∇[mRa|b|cd (Bianchi’s identity).

And by clause (1) of that proposition, we have, since ∇agbc = 0,

0 = 2∇[cd]gab = gnbRnacd + ganRnbcd = Rbacd + Rabcd.

That gives us (3). So it will suffice for us to show that clauses (1) – (3) jointly imply (4). Note first that

0 = Rabcd + Radbc + Racdb

= Rabcd − Rdabc − Racbd.

(The first equality follows from (2), and the second from (1) and (3).) So anti-symmetrization over (a, b, c) yields

0 = R[abc]d −Rd[abc] −R[acb]d.

The second term is 0 by clause (2) again, and R[abc]d = −R[acb]d. So we have an intermediate result:

R[abc]d = 0 —– (1)

Now consider the octahedron in the figure below.

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Using conditions (1) – (3) and equation (1), one can see that the sum of the terms corresponding to each triangular face vanishes. For example, the shaded face determines the sum

Rabcd + Rbdca + Radbc = −Rabdc − Rbdac − Rdabc = −3R[abd]c = 0

So if we add the sums corresponding to the four upper faces, and subtract the sums corresponding to the four lower faces, we get (since “equatorial” terms cancel),

4Rabcd −4Rcdab = 0

This gives us (4).

We say that two metrics gab and g′ab on a manifold M are projectively equivalent if their respective associated derivative operators are projectively equivalent – i.e., if their associated derivative operators admit the same geodesics up to reparametrization. We say that they are conformally equivalent if there is a map : M → R such that

g′ab = Ω2gab

is called a conformal factor. (If such a map exists, it must be smooth and non-vanishing since both gab and g′ab are.) Notice that if gab and g′ab are conformally equivalent, then, given any point p and any vectors ξa and ηa at p, they agree on the ratio of their assignments to the two; i.e.,

(g′ab ξa ξa)/(gab ηaηb) =  (gab ξa ξb)/(g′ab ηaηb)

(if the denominators are non-zero).

If two metrics are conformally equivalent with conformal factor, then the connecting tensor field Cabc that links their associated derivative operators can be expressed as a function of Ω.