Traversable Acausal Retrograde Domains In

Ben Tippett, UBCO CCGRRA 2016 “” Closed Timelike Curves

Timelike curves which are closed

t

“The Time Machine” HG Wells 1895 Popular Geometries with Closed Timelike Curves

• Gödel 1949 • angular momentum cosmology

• Tipler 1974 • infinite rotating cylinder

• Tomimatsu Sato 1973 • weird rotating kerr thing

• Friedman, Morris et al. 1990 • + twin paradox Piecemeal Refutation of Closed Timelike Curves

• Gödel 1949 • angular momentum cosmology Asymptotically • Tipler 1974 weird • infinite rotating cylinder

• Tomimatsu Sato 1973 • weird rotating kerr thing

• Friedman, Morris et al. 1990 • wormholes + twin paradox Piecemeal Refutation of Closed Timelike Curves

• Gödel 1949 • angular momentum cosmology Asymptotically • Tipler 1974 weird • infinite rotating cylinder

• Tomimatsu Sato 1973 How do we even • weird rotating kerr thing make this thing?

• Friedman, Morris et al. 1990 • wormholes + twin paradox Piecemeal Refutation of Closed Timelike Curves

• Gödel 1949 • angular momentum cosmology Asymptotically • Tipler 1974 weird • infinite rotating cylinder

• Tomimatsu Sato 1973 How do we even • weird rotating kerr thing make this thing?

• Friedman, Morris et al. 1990 • wormholes + twin paradox Unphysicality of Morris-Thorne

• Violates Topological Censorship • Friedman, Schleich, Witt 1993 • Required

• Compactly Generated Cauchy Horizon

• Unstable to perturbations

Morris, Thorne, Yutsever 1988 General Refutations of CTC

• Manifold is not time Orientable • Convenient assumption • ad hoc

• Chronology Protection Conjecture • Hawking 1991 • Compactly Generated Cauchy Horizons are unstable • Some CTC are CGCH free

• with CTC and no CGCH violate energy conditions or are geodesically incomplete • Maeda, Ishibashi, Narita 1998 Alcubierre t=3 t=2 v>1 t=0

• Vacuum inside and outside • “Bubble” geometry • observer inside, comoving with walls has same time coord as an external observer • Violates Classical Energy conditions 2 Warp bubbles=CTC

t=2 T1 V2 V1 T2

• time coordinate can’t be “retrograde” Causal structure of a “Time Machine” bubble 10

t t

x x

x

t

x t

t

x t B

A x

t

x

Figure 9: This illustrates how lightcones will be oriented around the TARDIS geometry. Outside of the TARDIS, time is aligned in the vertical direction. Inside of the bubble, lightcones tip along a circular loop, and massive objects can travel backwards in time. Part 2: ExplainingToroidal the TARDIS Traversable Acausal Tuesday, October 22, 2013 RetrogradeTHE TARDIS GEOMETRY Domain In Spacetime Our TARDIS geometry can be described as a bubble of spacetime geometry which carries its contents backwards and forwards through space and time as it endlessly tours a large circular path in spacetime9. The inside and outside of the bubble are a flat vacuum, and the two regions(T.A.R.D.I.S) are separated with a boundary of spacetime curvature. In this case TARDIS stands for Traversable Achronal Retrograde Domain In Spacetime. The name refers to a bubble (a Domain) which moves through the spacetime at speeds greater than the speed of light (it is Achronal); it moves backwards in time (Retrograde to the arrow of time outside the bubble); and finally, it can transport massive objects (it is Traversable)10. Fig. 9 is a spacetime diagram which shows the orientation of lightcones inside and outside of the TARDIS bubble. Outside of the bubble, the lightcones are all oriented upwards, pushing everyone steadily towards the future. Inside the bubble, lightcones tip and turn, allowing massive objects to move along a

2 9 The geometry has a spacetime line-element: ds2 = 1 h(x, y, z, t) 2t ( dt2 + dx2) + h(x, y, z, t) 4xt dxdt+ dy2 + dz2. For x2+t2 x2+t2 more specific information, read the original paper. h ⇣ ⌘i ⇣ ⌘ 10 You might be asking “Why aren’t you concentrating on making it bigger on the inside?” The short answer is that it is very easy to make a spacetime which is curved so that a very large volume sits inside a very small box. You can even use curved spacetime geometries to make very large objects appear very very small [11]! Long story short: Bigger on the inside is too easy to bother. Bubble Geometry

2t2 4xt ds2 = 1 h(xa) ( dt2 + dx2)+h(xa) dxdt + dy2 + dz2 x2 + t2 x2 + t2

Outside Bubble: Minkowski Vacuum h(x)=0 ds2 = dt2 + dx2 + dy2 + dz2 t Inside Bubble: Rindler Vacuum h(x)=1

x2 t2 4xt ds2 = ( dt2 + dx2)+ dxdt + dy2 + dz2 x2 + t2 x2 + t2 t = R sin() ,x= R cos() ds2 = R2d2 + dR2 + dy2 + dz2 Bubble Geometry

2t2 4xt ds2 = 1 h(xa) ( dt2 + dx2)+h(xa) dxdt + dy2 + dz2 x2 + t2 x2 + t2

Outside Bubble: Minkowski Vacuum h(x)=0 ds2 = dt2 + dx2 + dy2 + dz2 t Inside Bubble: Rindler Vacuum h(x)=1

x2 t2 4xt ds2 = ( dt2 + dx2)+ dxdt + dy2 + dz2 x2 + t2 x2 + t2 t = R sin() ,x= R cos() ds2 = R2d2 + dR2 + dy2 + dz2 2

t within the bubble. An observers travelling within the bubble will tell a dramat- x ically di↵erent story from observers outside of it. To illustrate the general properties of this spacetime, let us imagine a pair of observers: one named Amy (A) who travels along a within the bubble; and the other named Barbara (B), who views the bubble from the outside (see Fig. 1). Amy’s worldline evolves in a counter-clockwise in space- time. To maintain her place within the bubble, Amy must feel a consistent acceleration in (what she perceives as) one direction. Amy will see the hands of her own clock move clockwise the entire time. When she looks out at Barbara’s B clock, she will see it alternate between moving clockwise and A From The Outside: moving counter-clockwise. Barbara, looking upon Amy, will see something rather pe- culiar. At some initial time (T = 100 in Fig. 2), two bub- bles will suddenly appear, moving away from one-another. 2 The two bubbles will slow down, stop (at T = 0), and then Figure 1: A spacetime schematic of the trajectory of the 2 a 2t 2 2 a 4xt 2 2 begin moving back towards each other, until they merge (at TARDIS bubble. Arrows denote the local arrow of time. ds = 1 h(x ) ( dt + dx )+h(x ) dxdt + dy + dz T =+100) . Amy will appear in both of the bubbles: the 2 2 Observers Amy2 (A) and2 Barbara (B) experience the same x + t x + t hands of the clock will turn clockwise in one bubble, and 11 events very di↵erently. counter-clockwise in the other.

B. Metric 5. Bubbles Merge and Dissapear The TARDIS geometry has the following metric:

4. Bubbles accelerate 2t2 together ds2 = 1 h(x, y, z, t) ( dt2 + dx2) x2 + t2 ⇤ ⇥⌅ (1) 4xt + h(x, y, z, t) dxdt + dy2 + dz2 x2 + t2 3. Bubbles slow ⇥ and stop (a) TARDIS boundaries at (b) TARDIS boundaries at Similar to the Alcubierre bubble, this metric relies on a top- T=0 T= 50 hat function h(x, y, z, t) to demarcate the interior of the bubble ± h(x, y, z, t) = 1 from the exterior spacetime h(x, y, z, t) = 0. 2. Bubble splits, and The spacetime outside of the bubble (where h(x, y, z, t) = 0) moves apart is a Minkowski vacuum. The interior of the bubble (where h(x, y, z, t) = 1) is a Rindler vacuum. Note that within this Backwards In Forwards In region the metric can be re-written: 1. Bubble Appears Time Time x2 t2 4xt ds2 = ( dt2 + dx2) + dxdt+ dy2 + dz2 (2) x2 + t2 x2 + t2 ⇥ ⇥ Figure 10: An external observer will see two bubbles suddenly emerge from one another, one whose contents appear which, under the coordinate transformation: to move backwards in time. Thursday, October 24, 2013 t = ⇥ sin() , x = ⇥ cos() (c) TARDIS boundaries at (d) TARDIS boundaries at T= 75 T= 100 becomes Rindler spacetime. closed circle in spacetime. If a person were to be transported within the bubble, she would be moving ± ± alternatively forwards, sideways and even backwards in time! Figure 2: Evolution of the boundaries of the bubble, as defined in section II B, as seen by an external observer. At ds2 = ⇥2d2 + d⇥2 + dy2 + dz2 (3) A person travelling within the TARDIS would describe it as a room which is constantly acceleratingT = 100 the bubble will suddenly appear, and split in to two pieces which will move away from one another The Rindler geometry within this context has a modified forwards. Furthermore, any events which occur inside of the TARDIS bubble must satisfy Novikov’s(T = self75, T = 50). At T = 0, the two bubbles will come to topology, amounting to identifying the surfaces = 0 with rest, and then begin accelerating towards one-another = 2⇤. Thus, trajectories described with constant ⇥ = 11 ˆ ˆ ˆ consistency condition . (T =+75, T =+50). Whereupon, at T =+100 the two A, y = Y, z = Z coordinates will be closed timelike curves. bubbles will merge and disappear. Note that while such curves are not geodesic, the acceler- A person outside the bubble would describe an entierly dierent scene (see Fig. 10). Due to its closed ation required to stay on this trajectory can be modest. If trajectory, there will be a time before the bubble (and its contents) exists. Abruptly, we would see a bubble appear and split into two, the two boxes moving away from one another. Initially, they will be moving at superluminal speeds, but they will decelerate to a stand-still. The pair of boxes will then begin accelerating towards one another until they exceed the speed of light, merge and disappear. Mysteriously, the contents of one bubble will be appear to evolve forwards in time, while the other will evolve backwards. To illustrate how time is experienced inside and outside of the bubble, imagine that there are two people in our spacetime: Amy, who is travelling inside the bubble; and Barbara, who has been left behind. Suppose that the two women are holding large clocks, and that the walls of the bubble are transparent, so that the two women can see one another (See Fig. 11). Amy will only ever see the hands of her own clock move in a clockwise direction. When she looks out at Barbara, she will see the hands of Barbara’s clock moving clockwise at some times and counterclockwise

11 the Doctor and Romana were stuck in a loop of repeating events in the “Megalos.” The episode also involved a cactus as the villain, so... Stress Energy Tensor 5 7

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sophically problematic and unphysical. Most theorems con- cerning CTC do not even consider manifolds which are not time orientable. This is a bubble spacetime, e↵ectively gen- erated by stitching together two geometries across a boundary [28]. In our case, the choices we made for an internal geome- try has rendered the overall manifold non time orientable.

(a) Gzz along the slice y = 0, t = 0 (a) Gtt along the slice z = 30, y = 0 (b) Gty along the slice z = 30, y = 0 (c) Gxx along the slice z = 30, y = 0

(a) Gzz along the slice y = 0, z = 0

Also note that, in light of CPT symmetry, errant time trav- ellers who leaves the machine midway through the journey (a) Gxz along the slice t = 0, x = 100 my find themselves composed of antimatter.

(d) Gxz along the slice z = 30, y = 0 (e) Gyy along the slice z = 30, y = 0 (f) Gzz along the slice z = 30, y = 0

(b) Gyy along the slice y = 0, t = 0 Figure 11: Nonzero elements of the stress energy tensor along the slice y = 0, z = 30.

krasnikov tube. Phys. Rev. D, 56(4):2100–2108, 1997. [16] S. Krasnikov. No time machines in classical . [8] J. Friedman,• M. S. Morris, I. D. Novikov, F. Echeverria, Class. Quantum. Grav., 19:4109–4129, 2002. The singularities are, perhaps, the most glaring issue with G. Klinkhammer, K. S.Violates Thorne, and U. Yurtsever. Cauchy prob- CEC[17] S. V. Krasnikov. Time machines with non-compactly generated our geometry. Although it is not uncommon for geome- lem in spacetimes with closed timelike curves. Phys. Rev. D, cauchy horizons and “handy singularities”. 42(6):1915–1930, 1990. [18] S. V. Krasnikov. On the classical stability of a time machine. tries with CTC and no compactly generated Cauchy horizons (b) Gyy along the slice y = 0, z = 0 [9] J. Friedman, K. Schleich, and D. Witt. Topological censorship. Class. Quantum. Grav., 11:2755–2759, 1994. to have singularities [22][17]: the CTC spacetime generated Phys. Rev. Lett., 71(1486), 1993. [19] S. V. Krasnikov. Hyperfast travel in general relativity. Phys. Figure 5: Nonzero terms of the stress energy tensor along the through a sequence of accelerating warp bubbles do not pos-[10] G. W. Gibbons and S. W. Hawking. Selection rules for topology Rev. D, 57(8):4760–4766, 1998. slice y = 0, z = 0. sess such singularities [6]. It seems from the causal structure change. Commun. Math. Phys., 148:345–352, 1992. [20] F. S. N. Lobo. Closed timelike curves and violation. of our spacetime that the singularities are necessitated by the[11] K. Godel. An example of a new type of cosmological solution [21] R. J. Low. Time machines without closed causal geodesics. topology of the system. Is the CTC spacetime generated by of einstein’s• field equationsCurvature of gravitation. Rev. Mod. Phys., Class. Quantum. Grav., 12:L37–L38, 1995. (b) Gxy along the slice t = 0, x = 100 warp bubbles geodesically complete? Could a geodesically 21(3):447–450, 1949. [22] K. Maeda, A. Ishibashi, and M. Narita. Chronology protection The violation of the classical energy conditions is unsur- complete CTC geometry be derived where the acausal seg-[12] P. F. Gonzalez-Diaz. Warp drive space-time. Phys. Rev. D, and non-naked singularity. prising,Figure since 9: Nonzero the derivation terms of of the this stress metric energy mirrored tensor that along of the the ments of our geometry have been excised and replaced with 62(044005), 2000. [23] K. Maeda, A. Ishibashi, and M. Narita. Chronology protection Alcubierre warp drive,slice andx the= 100 spacetime, t = 0. bubble is meant to Alcubierre warp bubbles? [13] J. R. Gott. Closed timelike curves produced by pairs of moving and non-naked singularity. Class. Quantum. Grav., 15:1637– travel superluminaly. cosmic strings: ExactSingularities solutions. Phys. Rev. Lett., 66(9):1126– 1651, 1998. In spite of the particle physics community’s exuberance at 1129, 1991. [24] M. S. Morris, K. S. Thorne, and U. Yurtsever. Wormholes, [14] S. W. Hawking. Chronology protection conjecture. Phys. Rev. time machines, and the weak energy condition. Phys. Rev. Lett., generously interpreting CPT symmetry in terms of electrons (c) Gxz along the slice y = 0, t = 0 D, 46(2):603–610, 1991. 61(13):1446–1449, 1988. moving backwards in time, the relativity community under- Figure 7: Nonzero terms of the stress energy tensor along the[15] SW. Kim and K. S. Thorne. Do vacuum fluctuations prevent the [25] K. D. Olum. Superluminal travel requires negative energies. [1] M. Alcubierre. The warp drive: Hyper-fast travel within general [4] D. Deutsch. Phys. Rev. D, 44(3197), 1991. stands that geometries which are not time orientable are philo- slice y = 0, t = 0 creation of closed timelike curves. Phys. Rev. D, 43(12):3929– Phys. Rev. Lett., 81(17):3567, 1998. relativity. Class. Quantum. Grav., 11(L73-L77), 1994. [5] F. Echeverria, G. Klinkhammer, and K. S. Thorne. Billiard balls 3947, 1991. [26] A. Ori. Must time-machine construction violate the weak en- [2] A. Chamblin, G. W. Gibbons, and A. R. Steif. Phys. Rev. D, in wormhole spacetimes with closed timelike curves: Classical 50(R2353), 1994. theory. Phys. Rev. D, 44(4):1077–1099, 1991. [3] S. Deserr, R. Jackiw, and ’t Hooft, G. Physical cosmic strings [6] A. E. Everett. Warp drive and causality. Phys. Rev., do not generate closed timelike curves. Phys. Rev. Lett., 53(12):7365–7368, 1996. 68(3):267–269, 1992. [7] A. E. Everett and T. A. Roman. Superluminal subway: The Causal Structure: Null Geodesics

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• Light cones tilt

• Cauchy Horizons

• Manifold Non- Orientable

• CH not compactly generated

• (CPC does not apply)

Figure 4: Null geodesics travelling through a cross section y = 0, z = 0. The intersection of null curves can indicate the orientation of the . The insertion of such a bubble geometry into an external spacetime renders it nonorientable. Note that some geodesics (green) have both endpoints in what an external observer might call the “future”, and other geodesics (red) have both endpoints in the “past”.

C. Stress Energy Tensor and Curvature Invariants elements of the stress energy tensor along this slice are shear terms, as seen in Fig. 10a and 10b. To give a sense of the matter required to transition between The energy density is not zero everywhere. If we look at cross-sections which do not lay on the x t axis (across the the interior and the exterior vacuums of the bubble, we have plotted the various non-zero components of the stress energy slice z = 30, y = 0 in Fig. 11), we see a more complicated tensor taken along cross sections of spacetime geometry. As stress-energy tensor. we would expect, the classical energy conditions are violated. Additionally, there are naked curvature singularities in the If we set y = 0, z = 0 we see a cross-section of the bubble Kretschmann scalar and in the first principal invariant of they = as it travels along its circular course through spacetime. The Weyl tensor at points: x 0 where the top hat function has values h(x, y, z, t) = 1 . only non-zero elements of the stress energy tensor along this 2 slice are the non-zero pressures in the y and z directions, as seen in Fig. 6a and 6b respectively. If we set y = 0, t = 0 we see a cross-section of the bubble(s) III. DISCUSSION at rest with respect to the exterior coordinate system. There are non-zero pressure and also shear terms, as seen in Fig. 8a, The purpose of building a spacetime geometry which any 8b, and 8c. layperson would describe as a “time machine” from popular If we set t = 0, x = 100 we see a cross-section of the fiction is a pedagogical exercise, rather than a physical one. bubble over the y-z plane when the bubble is at rest relative to It is not surprising that it is unphysical. The exercise is in the exterior coordinate system. In this case, the only non-zero determining the specific ways in which it is unphysical. Causal Structure: Geodesically Incomplete

Cauchy Horizon Ends at I ± • Incomplete null geodesics Ends at I + terminate at curvature Ends at singularity singularities

• Some of these incomplete curves generate Cauchy Horizon

Sunday, October 6, 2013 Conclusion: Fun!

• Bubble geometry has all Cauchy Horizon desired properties Ends at I ± • Non Orientable manifold Ends at I + Ends at singularity • Physically Problematic: • Violates Classical EC • Curvature Singularities • Geo. Incomplete

• Why aren’t these issues present in the Warp Drive CTC geometry?

Sunday, October 6, 2013