Acausal - SFU.ca - Simon Fraser University · 2016. 7. 6. · Ben Tippett, UBCO CCGRRA 2016 In Spacetime “Time Travel” Closed Timelike Curves Timelike curves which are closed

Traversable Acausal Retrograde Domains

Ben Tippett, UBCO CCGRRA 2016

In Spacetime

“Time Travel” 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 • wormholes + twin paradox

Piecemeal Refutation of Closed Timelike Curves





Asymptotically  weird

Piecemeal Refutation of Closed Timelike Curves





Asymptotically  weird

How do we evenmake this thing?

Unphysicality of Morris-Thorne Wormhole

Morris, Thorne, Yutsever 1988

• Violates Topological Censorship

• Friedman, Schleich, Witt 1993 • Exotic Matter Required

• Compactly Generated Cauchy Horizon

• Unstable to perturbations

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

• Spacetimes with CTC and no CGCH violate energy conditions or are geodesically incomplete

• Maeda, Ishibashi, Narita 1998

Alcubierre Warp drive

t=0

t=2

t=3

v>1

• 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

V1V2

T1

T2

t=2

• time coordinate can’t be “retrograde”

Causal structure of a “Time Machine” bubble

10

x

t

x

t

x

t

xt

x

t

x

t

Tuesday, October 22, 2013

Figure 9: This spacetime diagram 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: Explaining the TARDISTHE TARDIS GEOMETRY

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 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

9 The geometry has a spacetime line-element: ds2 =h1 � h(x, y, z, t)

⇣2t2

x2+t2

⌘i(�dt2 + dx2)+ h(x, y, z, t)

⇣4xt

x2+t2

⌘dxdt + dy2 + dz2. For

more specific information, read the original paper.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.

t

x

AB

Toroidal Traversable Acausal Retrograde Domain In Spacetime

(T.A.R.D.I.S)

Bubble Geometryds2 =

�1� h(xa)

�2t2

x2 + t2

��(�dt2 + dx2) + h(xa)

�4xt

x2 + t2

�dxdt + dy2 + dz2

ds2 =�

x2 � t2

x2 + t2

�(�dt2 + dx2) +

�4xt

x2 + t2

�dxdt + dy2 + dz2

Inside Bubble: Rindler Vacuum h(x)=1

ds2 = �dt2 + dx2 + dy2 + dz2

Outside Bubble: Minkowski Vacuum h(x)=0

t = R sin(�) , x = R cos(�) ds2 = �R2d�2 + dR2 + dy2 + dz2

t

�

From The Outside:ds2 =

�1� h(xa)

�2t2

x2 + t2

��(�dt2 + dx2) + h(xa)

�4xt

x2 + t2

�dxdt + dy2 + dz2

2

t

x

AB

Figure 1: A spacetime schematic of the trajectory of theTARDIS bubble. Arrows denote the local arrow of time.Observers Amy (A) and Barbara (B) experience the same

events very di↵erently.

(a) TARDIS boundaries atT=0

(b) TARDIS boundaries atT=±50

(c) TARDIS boundaries atT=±75

(d) TARDIS boundaries atT=±100

Figure 2: Evolution of the boundaries of the bubble, asdefined in section II B, as seen by an external observer. At

T = �100 the bubble will suddenly appear, and split in to twopieces which will move away from one another

(T = �75, T = �50). At T = 0, the two bubbles will come torest, and then begin accelerating towards one-another

(T = +75, T = +50). Whereupon, at T = +100 the twobubbles will merge and disappear.

within the bubble.An observers travelling within the bubble will tell a dramat-

ically di↵erent story from observers outside of it. To illustratethe general properties of this spacetime, let us imagine a pairof observers: one named Amy (A) who travels along a closedtimelike curve 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 mustfeel a consistent acceleration in (what she perceives as) onedirection. Amy will see the hands of her own clock moveclockwise the entire time. When she looks out at Barbara’sclock, she will see it alternate between moving clockwise andmoving 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.The two bubbles will slow down, stop (at T = 0), and thenbegin moving back towards each other, until they merge (atT = +100) . Amy will appear in both of the bubbles: thehands of the clock will turn clockwise in one bubble, andcounter-clockwise in the other.

B. Metric

The TARDIS geometry has the following metric:

ds2 =⇤1 � h(x, y, z, t)

�2t2

x2 + t2

⇥⌅(�dt2 + dx2)

+ h(x, y, z, t)�

4xtx2 + t2

⇥dxdt + dy2 + dz2

(1)

Similar to the Alcubierre bubble, this metric relies on a top-hat function h(x, y, z, t) to demarcate the interior of the bubbleh(x, y, z, t) = 1 from the exterior spacetime h(x, y, z, t) = 0.

The spacetime outside of the bubble (where h(x, y, z, t) = 0)is a Minkowski vacuum. The interior of the bubble (whereh(x, y, z, t) = 1) is a Rindler vacuum. Note that within thisregion the metric can be re-written:

ds2 =�

x2 � t2x2 + t2

⇥(�dt2 +dx2)+

�4xt

x2 + t2

⇥dxdt+dy2 +dz2 (2)

which, under the coordinate transformation:

t = ⇥ sin(�) , x = ⇥ cos(�)

becomes Rindler spacetime.

ds2 = �⇥2d�2 + d⇥2 + dy2 + dz2 (3)

The Rindler geometry within this context has a modifiedtopology, amounting to identifying the surfaces � = 0 with� = 2⇤. Thus, trajectories described with constant ⇥ =Â, y = Ŷ , z = Ẑ coordinates will be closed timelike curves.Note that while such curves are not geodesic, the acceler-ation required to stay on this trajectory can be modest. If

11

Backwards In Time

Forwards In Time

1. Bubble Appears

2. Bubble splits, and moves apart

3. Bubbles slow and stop

4. Bubbles acceleratetogether

5. Bubbles Merge and Dissapear

Thursday, October 24, 2013

Figure 10: An external observer will see two bubbles suddenly emerge from one another, one whose contents appear

to move backwards in time.

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!

A person travelling within the TARDIS would describe it as a room which is constantly accelerating

forwards. Furthermore, any events which occur inside of the TARDIS bubble must satisfy Novikov’s self

consistency condition11.

A person outside the bubble would describe an entierly di�erent scene (see Fig. 10). Due to its closed

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 Tensor6

(a) Gxz along the slice t = 0, x = 100

(b) Gxy along the slice t = 0, x = 100

Figure 9: Nonzero terms of the stress energy tensor along theslice x = 100, t = 0.

sophically problematic and unphysical. Most theorems con-cerning CTC do not even consider manifolds which are nottime 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.

Also note that, in light of CPT symmetry, errant time trav-ellers who leaves the machine midway through the journeymy find themselves composed of antimatter.

The singularities are, perhaps, the most glaring issue withour geometry. Although it is not uncommon for geome-tries with CTC and no compactly generated Cauchy horizonsto have singularities [22][17]: the CTC spacetime generatedthrough a sequence of accelerating warp bubbles do not pos-sess such singularities [6]. It seems from the causal structureof our spacetime that the singularities are necessitated by thetopology of the system. Is the CTC spacetime generated bywarp bubbles geodesically complete? Could a geodesicallycomplete CTC geometry be derived where the acausal seg-ments of our geometry have been excised and replaced withAlcubierre warp bubbles?

[1] M. Alcubierre. The warp drive: Hyper-fast travel within generalrelativity. Class. Quantum. Grav., 11(L73-L77), 1994.

[2] A. Chamblin, G. W. Gibbons, and A. R. Steif. Phys. Rev. D,50(R2353), 1994.

[3] S. Deserr, R. Jackiw, and ’t Hooft, G. Physical cosmic stringsdo not generate closed timelike curves. Phys. Rev. Lett.,68(3):267–269, 1992.

[4] D. Deutsch. Phys. Rev. D, 44(3197), 1991.[5] F. Echeverria, G. Klinkhammer, and K. S. Thorne. Billiard balls

in wormhole spacetimes with closed timelike curves: Classicaltheory. Phys. Rev. D, 44(4):1077–1099, 1991.

[6] A. E. Everett. Warp drive and causality. Phys. Rev.,53(12):7365–7368, 1996.

[7] A. E. Everett and T. A. Roman. Superluminal subway: The

5

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

(b) Gyy along the slice y = 0, z = 0

Figure 5: Nonzero terms of the stress energy tensor along theslice y = 0, z = 0.

The violation of the classical energy conditions is unsur-prising, since the derivation of this metric mirrored that of theAlcubierre warp drive, and the spacetime bubble is meant totravel superluminaly.

In spite of the particle physics community’s exuberance atgenerously interpreting CPT symmetry in terms of electronsmoving backwards in time, the relativity community under-stands that geometries which are not time orientable are philo-

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

(b) Gyy along the slice y = 0, t = 0

(c) Gxz along the slice y = 0, t = 0

Figure 7: Nonzero terms of the stress energy tensor along theslice y = 0, t = 0

7

(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

(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

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.[8] J. Friedman, M. S. Morris, I. D. Novikov, F. Echeverria,

G. Klinkhammer, K. S. Thorne, and U. Yurtsever. Cauchy prob-lem in spacetimes with closed timelike curves. Phys. Rev. D,42(6):1915–1930, 1990.

[9] J. Friedman, K. Schleich, and D. Witt. Topological censorship.Phys. Rev. Lett., 71(1486), 1993.

[10] G. W. Gibbons and S. W. Hawking. Selection rules for topologychange. Commun. Math. Phys., 148:345–352, 1992.

[11] K. Godel. An example of a new type of cosmological solutionof einstein’s field equations of gravitation. Rev. Mod. Phys.,21(3):447–450, 1949.

[12] P. F. Gonzalez-Diaz. Warp drive space-time. Phys. Rev. D,62(044005), 2000.

[13] J. R. Gott. Closed timelike curves produced by pairs of movingcosmic strings: Exact solutions. Phys. Rev. Lett., 66(9):1126–1129, 1991.

[14] S. W. Hawking. Chronology protection conjecture. Phys. Rev.D, 46(2):603–610, 1991.

[15] SW. Kim and K. S. Thorne. Do vacuum fluctuations prevent thecreation of closed timelike curves. Phys. Rev. D, 43(12):3929–3947, 1991.

[16] S. Krasnikov. No time machines in classical general relativity.Class. Quantum. Grav., 19:4109–4129, 2002.

[17] S. V. Krasnikov. Time machines with non-compactly generatedcauchy horizons and “handy singularities”.

[18] S. V. Krasnikov. On the classical stability of a time machine.Class. Quantum. Grav., 11:2755–2759, 1994.

[19] S. V. Krasnikov. Hyperfast travel in general relativity. Phys.Rev. D, 57(8):4760–4766, 1998.

[20] F. S. N. Lobo. Closed timelike curves and causality violation.[21] R. J. Low. Time machines without closed causal geodesics.

Class. Quantum. Grav., 12:L37–L38, 1995.[22] K. Maeda, A. Ishibashi, and M. Narita. Chronology protection

and non-naked singularity.[23] K. Maeda, A. Ishibashi, and M. Narita. Chronology protection

and non-naked singularity. Class. Quantum. Grav., 15:1637–1651, 1998.

[24] M. S. Morris, K. S. Thorne, and U. Yurtsever. Wormholes,time machines, and the weak energy condition. Phys. Rev. Lett.,61(13):1446–1449, 1988.

[25] K. D. Olum. Superluminal travel requires negative energies.Phys. Rev. Lett., 81(17):3567, 1998.

[26] A. Ori. Must time-machine construction violate the weak en-

• Violates CEC

• Curvature Singularities

Causal Structure: Null Geodesics

• Light cones tilt

• Cauchy Horizons

• Manifold Non-Orientable

• CH not compactly generated

• (CPC does not apply)

4

Figure 4: Null geodesics travelling through a cross section y = 0, z = 0. The intersection of null curves can indicate theorientation of the light cone. The insertion of such a bubble geometry into an external spacetime renders it nonorientable. Notethat 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

To give a sense of the matter required to transition betweenthe interior and the exterior vacuums of the bubble, we haveplotted the various non-zero components of the stress energytensor taken along cross sections of spacetime geometry. Aswe would expect, the classical energy conditions are violated.

If we set y = 0, z = 0 we see a cross-section of the bubbleas it travels along its circular course through spacetime. Theonly non-zero elements of the stress energy tensor along thisslice are the non-zero pressures in the y and z directions, asseen in Fig. 6a and 6b respectively.

If we set y = 0, t = 0 we see a cross-section of the bubble(s)at rest with respect to the exterior coordinate system. Thereare non-zero pressure and also shear terms, as seen in Fig. 8a,8b, and 8c.

If we set t = 0, x = 100 we see a cross-section of thebubble over the y-z plane when the bubble is at rest relative tothe exterior coordinate system. In this case, the only non-zero

elements of the stress energy tensor along this slice are shearterms, as seen in Fig. 10a and 10b.

The energy density is not zero everywhere. If we look atcross-sections which do not lay on the x � t axis (across theslice z = 30, y = 0 in Fig. 11), we see a more complicatedstress-energy tensor.

Additionally, there are naked curvature singularities in theKretschmann scalar and in the first principal invariant of theyWeyl tensor at points: x = 0 where the top hat function hasvalues h(x, y, z, t) = 12 .

III. DISCUSSION

The purpose of building a spacetime geometry which anylayperson would describe as a “time machine” from popularfiction is a pedagogical exercise, rather than a physical one.It is not surprising that it is unphysical. The exercise is indetermining the specific ways in which it is unphysical.

Causal Structure: Geodesically Incomplete

CauchyHorizon

Ends at

Ends at

Ends atsingularity

I +I ±

Sunday, October 6, 2013

• Incomplete null geodesics terminate at curvature singularities

• Some of these incomplete curves generate Cauchy Horizon

Conclusion: Fun!

CauchyHorizon

Ends at

Ends at

Ends atsingularity

I +I ±

Sunday, October 6, 2013

• Bubble geometry has all desired properties

• Non Orientable manifold

• Physically Problematic: • Violates Classical EC • Curvature Singularities • Geo. Incomplete

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

Documents

Acausal - SFU.ca - Simon Fraser University · 2016. 7. 6. · Ben Tippett, UBCO CCGRRA 2016 In Spacetime “Time Travel” Closed Timelike Curves Timelike curves which are closed