16. Time Travel 1 1. Closed Timelike Curves and Time Travel

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1 16. Time Travel 1 1. Closed Timelike Curves and Time Travel Recall: The Einstein cylinder universe is a solution to the Einstein equations (with cosmological constant) that has closed spacelike curves. open timelike curve time closed spacelike curve space Are there solutions to the Einstein equations with closed timelike curves?

2 Recall: A timelike curve is a worldline of a physical object (an object traveling at a speed < c). Timelike curves "thread" lightcones. Def. A closed timelike curve (CTC) is a worldline of a physical object that intersects itself. A physical object following a CTC goes back in time! At some point in its history, it will reach a point on its worldline that it previously occupied.

3 Rolled-Up Minkowski Spacetime Simplest solution to the Einstein equations with CTCs. + time Minkowski Spacetime space Roll it up: Identify + and. CTC time Rolled-Up Minkowski Spacetime space

4 Time Travel Paradox in Rolled-Up Minkowski Spacetime time space Albert kills himself if and only if Albert does not kill himself. Possible Resolutions: (1) Universes with CTCs are physically impossible. (2) Universes with CTCs are physically possible, but they are constrained in ways that prevent violations of causality. If (2), are such constraints mysterious? Any more so than typical initial conditions for physical processes?

5 Constraints on Spacetimes with CTCs Simplest Case - Universe with only 1 particle Case 1: Spacetime with no CTCs Particle can have any initial velocity at t = t 0 (w.r.t. a given observer). t = t 0 t = t 0 t = t 0... All of these initial conditions lead to physically indistinguishable spacetimes! Velocities are relative in general relativistic (and Minkowski, and Galilean) spacetimes.

6 Case 2: Rolled-Up Minkowski Spacetime Particle is constrained to have only one initial velocity at t = t 0 (w.r.t. a given observer). t = t 0 leads to Any other initial velocity leads to contradiction! t = t 0 leads to But: This entails the initial setup is: Are there more general constraints that would result in more interesting spacetimes with CTCs?

7 2. Constraints on Time Machines Time travel is possible in GR: Spacetimes with CTCs are possible. Can we use them? Can we operate a time machine? Time Machine (TM) Characteristics (1) TM should be confined to finite region of space and operate for a finite time. (2) TM should cause CTCs to evolve: At time t, throw switch and time travel! Task: Translate these requirements into restrictions on the causal structure of candidate spacetimes.

8 Causal Structures in Spacetime Def 1. An initial data surface is a spacelike surface. Def. 2. The future domain of dependence D + () of consists of all points p such that every timelike curve through p with no past endpoint meets in one point. Def. 3. The past domain of dependence D () of consists of all points p such that every timelike curve through p with no future endpoint meets in one point. D + () p q q is not in D + () since there are timelike curves through q with no past endpoints that do not meet D ()

9 Def. 4. The domain of dependence D() of consists of the union D + () D (). D() = {all points that are uniquely determined by } D + () D ()

10 Def. 4. The domain of dependence D() of consists of the union D + () D (). D() = {all points that are uniquely determined by } Def. 5. The causal future J + () of consists of all points p such that there is at least one future-directed causal curve from to p. Def. 6. The causal past J () of consists of all points p such that there is at least one past-directed causal curve from to p. J + () p q J ()

11 D() is different from J + () J (). This means that the points that determines are in general different from those it can causally interact with! (Recall the discussion of the Horizon "Problem".). Example: p is a point that can causally effect but does not uniquely determine. J + () p D + () D () J ()

12 Let K = time machine region. Let V = chronology violating region = all points p such that there is a CTC through p. Let = time slice = time before TM is switched on. TM Characteristics: (1) TM should be confined to finite region of space and operate for a finite time. So: Require no CTCs before. Or: No future timelike curve intersects more than once. (Require that be a "partial Cauchy surface".) Def. A partial Cauchy surface is a spacelike surface such that no timelike curve intersects it more than once.

13 Let K = time machine region. Let V = chronology violating region = all points p such that there is a CTC through p. Let = time slice = time before TM is switched on. TM Characteristics: (2) TM should cause CTCs to evolve. Problem: Conditions on should specify how the TM operates. " But: Conditions on cannot always uniquely determine what goes on in V.

14 Deutsch-Politzer Spacetime chronologyviolating region V L + time identify space L Consider a particle traveling to the right at constant speed that registers on the surface and enters the chronology violating region V. " Its point of intersection with records its position and its velocity. " This initial data can then be used to predict the path of the particle to the future of. Does the initial data on uniquely determine the path of the particle to the future of?

15 Deutsch-Politzer Spacetime time chronologyviolating region V L + space L Prediction I: No collisions in V. (1) Particle travels to right with constant speed and enters region V. (2) Hits L + and goes back in time to L. (3) Continues to right at constant speed. (4) Steps (2) and (3) repeat three more times. (5) Particle leaves region V.

16 Deutsch-Politzer Spacetime time chronologyviolating region V p L + q space L Prediction II (same initial conditions): Two collisions in V. (1) Particle travels to right with constant speed and enters region V. (2) At p, hits an older version of self and comes to rest. (3) Travels back in time and is hit by a younger version of self (again at p). (4) Scatters to right at contant speed, travels back in time three more times. (5) At q, hits older version of self and comes to rest. (6) Travels back in time and is hit by younger version of self (again at q). (7) Scatters to right, travels back in time, and then finally leaves region V.

17 So: Initial conditions on do not uniquely determine what goes on in V. Now: "K causes V" means V J + (K) (V is in the causal future of K). And: " determines operation of K" means K D + () (K is inside the domain of dependence of ). And: " does not determine V" means V lies outside D + () (V is outside the future domain of dependence of ). CTC V J + (K) K D + ()

18 Problems (A) No time travel is possible to the past of the initial time. (B) In what sense does K cause the CTCs in V? V inside J + (K) and outside D + () allows for other influences on CTCs besides K. Example: CTC V J + (K) remove K D + ()

19 Problems (A) No time travel is possible to the past of the initial time. (B) In what sense does K cause the CTCs in V? V inside J + (K) and outside D + () allows for other influences on CTCs besides K. Example: CTC V J + (K) causal influence emerging from singularity that doesn't register on remove H + () = future Cauchy horizon of. K D + ()

20 Hawking's (1992) Condition for TMs In order for a TM to exist, the future Cauchy Horizon H + () of must be compactly generated (its generators cannot emerge from singularities or infinity). Claim: If TMs are possible, then H + () must be compactly generated. Or: If H + () is not compactly generated, then TMs are not possible. CTC V J + (K) causal influence emerging from singularity that doesn't register on remove K D + () H + () = future Cauchy horizon of. generator of H + ()

21 No-Go Theorems for TMs in GR Argue for a particular TM condition and then show that it does not hold for physically relevant general relativistic spacetimes. Hawking's (1992) Chronology Protection Conjecture If the following hold for a partial Cauchy surface, (a) is spatially open ("non-compact"), (b) the Einstein equations hold, (c) the "weak energy condition" holds (no negative energy densities), then H + () is not compactly generated, and thus (according to Hawking's Condition), a TM cannot exist. Problems: (1) Is Hawking's Condition necessary for the existence of TMs? (Just because there may be influences on V that don't register on, doesn't mean there actual are such influences.) (2) Is Hawking's Condition sufficient for the existence of TMs? (If no influences on V get swallowed up by holes, does this guarantee that completely determines what goes on in V?)

22 Def. An extension of D + () is an embedding of D + () as a proper subset of another "larger" spacetime. Earman, Smeenk & Wüthrich's (2008) Potency Condition for TMs In order for a TM to exist, every smooth, maximal, "hole-free" extension of D + () that satisfies the Einstein equations and energy conditions must contain CTCs. There are many ways of mathematically extending D + () into the future. " Some may contain holes. If so, we can't say conditions on are uniquely responsible for CTCs that may exist. " Some may not be "maximal" (i.e., they themselves may be further extended). (Technically, a non-maximal extension may contain hole-like structures. If so, we can't say conditions on are uniquely responsible for CTCs that may exist.) But: If all maximal and hole-free extensions contain CTCs, then we can reasonably say that conditions on must be responsible for these CTCs!

23 D + () K M!

24 V D + () K M 1!