Hunter Monroe- Topology and Closed Timelike Curves I: Dynamics

8/3/2019 Hunter Monroe- Topology and Closed Timelike Curves I: Dynamics

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arXiv:gr

-qc/0609054v1

15Sep2006

Topology and Closed Timelike Curves I: Dynamics

Hunter Monroe

International Monetary Fund, Washington, DC 20431

(Dated: September 18, 2006)

Abstract

No closed timelike curve (CTC) on a Lorentzian manifold can be continuously deformed as a CTC to

a point. Otherwise, any of the points neighborhoods would contain a CTC, a contradiction as Lorentzian

manifolds are locally causally well-behaved. Every CTC must pass through some topological feature, to

be called a timelike wormhole, which prevents the it from being deformed to a point. Tipler showed that a

closed timelike geodesic exists in a compact Lorentzian manifold if the covering space contains a compact

Cauchy surface. A test particle free falling along this geodesic transits the timelike wormhole; in the

test particles frame of reference, the wormhole propagates toward the test particle, exhibiting photon-like

behavior. In addition, a space-time that violates chronology at every point can experience topology change

consistent with theorems of Geroch and Tipler.

PACS numbers: 04.20.-q,02.40.-k, 04.20.Gz

In honor of the retirement from Davidson College of Dr. L. Richardson King, an extraordinary teacher and

mathematician. Email: relpap at huntermonroe.com. The views expressed in this paper are those of the author

and should not be attributed to the International Monetary Fund, its Executive Board, or its management.

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I. INTRODUCTION

The issue of when and whether a manifolds topology can change over time has arisen in many

contexts (see Borde [1]). Geroch [2] shows that topology change implies the existence of CTCs,

which are viewed by most as pathological, and Tipler [3] shows that the emergence of CTCs leads

to a singularity. Globally hyperbolic space-times have the topology R S that does not changeover time, and causal curves cannot probe the topology of such a space-time due to topological

censorship (Friedman et al [4]). No way has been found to relax assumptions to maintain causality

and to permit topology change, but not arbitrary topology change (see Borde [1]).

The approach is to show CTCs, which arise in physically realistic situations (Bonnor [5]), face

certain constraints. As Hawking [6] notes, strong causality assumptions risk ruling out something

that gravity is trying to tell us. It is conceivable that CTCs have a dissipative effect[ 7] such thatspace-time appears nearly causal.

The next three sections show no CTC can be deformed to a point; analyze the the dynamic

properties of the topological feature that prevents this; and observes that Tiplers theorem does not

apply if there are CTCs through every point.

II. CTCS AND TOPOLOGY

Following the literature (for instance Avez [8] and Galloway [9]), say that two CTCs 1 and 2

are timelike homotopic if there is a homotopy (without base-point) which deforms 1 into 2 via

CTCs. The second curve may be trivial (a point) rather than a CTC. Say that a Lorentzian manifold

which contains CTCs not timelike homotopic to point is timelike multiply connected, and likewise

for closed spacelike curves (CSCs).

Example 1. A two-dimensional Minkowski space-time with periodic x coordinate is spacelike

multiply connected. A two-dimensional Minkowski space-time with periodic t coordinate is

timelike multiply connected. If both coordinates are periodic, the space-time is timelike and

spacelike multiply connected.

It is straightforward to show that any Lorentzian manifold is timelike multiply connected.

Theorem 1. In any Lorentzian manifoldM , no CTC is timelike homotopic to a point. IfM

contains a CTC, M is timelike multiply connected. IfM is compact, it is timelike multiply

connected.

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Proof. Suppose there is a CTC which is timelike homotopic to a point p. Every neighborhood

U ofp contains a CTC F(t, 1 ) for > 0 sufficiently small. A basic property of a Lorentzianmanifold is that it is locally causally well behaved (see for instance Garcia-Parrado and Senovilla

Proposition 2.1 [10]).

The property timelike multiply connected is easier to satisfy than multiply connected (by

any type of curve), because the former allows homotopies only over those curves which are

timelike. In fact, it follows immediately from Theorem 1 that:

Corollary 1. A simply connected (by any type of curve) Lorentzian manifold containing a CTC is

nevertheless timelike multiply connected. All simply connected compact Lorentzian manifolds are

timelike multiply connected.

Proof. By Theorem 1, a Lorentzian manifold containing a CTC is timelike multiply connected.

Any compact Lorentzian manifold contains a CTC.

A contrived example of this behavior due to Geroch can be found in Flaherty [11]; the corollary

shows this can occur broadly, as seen in the following more natural example.

Example 2. The 3-sphere S3 is simply connected (by any type of curve) but timelike multiply

connected. EmbedS3 in Euclidean space E4 as x20

+ x21

+ x22

+ x23

= 1. The continuous non-

vanishing vector fieldV = (x1,x0, x3,x2) defines Lorentzian metric such thatV is everywheretimelike. Contracting the CTC (r cos , r sin , 0,

1 r2) by lowering r below 1 , the curve

becomes everywhere null at r =

2/2. Observe thatS3 has a Heegaard split into two solid

tori.

A. Can timelike curves transit vacuum wormholes?

The finding in Theorem 1 that no CTC is timelike homotopic to a point indicates, in the

absence of topological defects creating by excising points to create rips or holes, that some smooth

topological feature, to be called a timelike wormhole, prevents the CTC from being deformed to a

point and is transited by timelike curves.

Consider several objections to the finding that there exist timelike wormholes transited by

timelike curves. First, a key finding of Kruskal-Szekeres coordinates is that only spacelike curves

may cross the wormhole between the two asymptotically flat regions; these coordinates do not

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seem appropriate analyzing a space-time with CTCs. Second, although traversable wormholes

seem to require the existence of exotic matter violating energy conditions, traversable wormholes

are defined not to have an event horizon so as to allow two-way travel. No such assumption is made

regarding timelike wormholes, which in a vacuum space time would typically have a horizon; if

so there is chronology protection against those CTCs that are not censored by passing through

a horizon, and a Cauchy-like surface may nevertheless exist through which all timelike curves

pass once and only once before crossing a horizon (see Monroe [12]). Finally, the topological

censorship theorem of Friedman et al [4] states that every causal curve in a globally hyperbolic

space-time from past null to past future infinity can be deformed into the trivial causal curve, but

a space-time containing a CTC is not globally hyperbolic.

III. DYNAMIC BEHAVIOR OF TIMELIKE WORMHOLES

Suppose some portion of a CTC transits a relatively flat region. Relative to this region, the

CTC defines an orientation to the timelike wormhole it also transits, such that one mouth can be

distinguished as the inward mouth, and the other as an outward mouth.

Tipler [13] shows that in a compact Lorentzian manifold such that the covering space has

a compact Cauchy surface, a closed timelike geodesic exists. A hypothetical test particle free

falling along this geodesic will fall toward, through, and out the timelike wormhole transited by

the geodesic. In the frame of reference of the test particle, the timelike wormhole will propagate

toward the test particle.

Example 3. Take a 2-sphere, and attach a handle between two nearby locations on the equator.

This manifold with zero Euler characteristic has a Lorentzian metric; choose a vector field pointing

through the handle to produce a Lorentzian metric. A hypothetical free-falling test particle will

fall along the equator through handle. In the test particles reference frame, the handle propagates

around the equator toward the test particle (somewhat like a jet engine).

IV. REVISITING TOPOLOGY CHANGE

Tiplers [3] theorem that the emergence of CTCs (which Geroch showed must exist when there

is topology change on a Lorentzian manifold) implies the existence of a singularity. However,

Tipler himself points out that his theorem does not apply to a manifold with CTCs through

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every point; consider such manifolds. Another problem arises: any two compact 3-manifolds

manifolds are Lorentz cobordant. A theory that permits arbitrary topology change cannot provide

a physically realistic model that rules out some phenomena and permits others.

However, it would appear that a propagating timelike wormhole has nonzero momentum. If so,

then timelike wormholes could only be created or destroyed in multiples of 2, to conserve energy-

momentum. Gibbons and Hawking [14][15] find that the existence of a spinor structure allows

wormholes to be created only in multiples of 2.

It would also appear that a pair of timelike wormholes has nonzero energy, and that this energy

would need to be radiated away in some form if this pair is destroyed. Under the Einstein vacuum

equation, the only possible way to release that energy is gravitational waves. The time-reverse

of this pairs destruction would be a gravitational wave collision that creates a pair of timelike

wormholes.[17]

V. CONCLUSION

The occurrence of CTCs in solutions of Einsteins equations referring to laboratory situations

appear to be a black mark against general relativity. However, the analysis above provides limited,

initial evidence to support the opposite conclusion: Einstein space-times which incorporate CTCs

have greater explanatory power than those which do not. An arbitrary assumption that CTCs do

not occur prevents interesting lines of research from being pursued, in the same way that parallel

postulate prevented the development of non-Euclidean geometry.

[1] A. Borde (1994), gr-qc/9406053.

[2] R. Geroch, J. Math. Phys. 8, 782 (1967).

[3] F. J. Tipler, Ann. Phys. 108, 1 (1977).

[4] J. L. Friedman, K. Schleich, and D. M. Witt, Phys. Rev. Lett. 71, 1486 (1993), gr-qc/9305017.

[5] W. B. Bonnor, Int. J. Mod. Phys. D12, 1705 (2003), gr-qc/0211051.

[6] S. Hawking and R. Penrose, The nature of space and time (Princeton University Press, Princeton,

1996), hep-th/9409195.

[7] G. t Hooft, Class. Quant. Grav. 16, 3263 (1999), gr-qc/9903084.

[8] A. Avez, Annales de linstitut Fourier 13, 105 (1963).

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[9] G. J. Galloway, Transactions of the American Mathematical Society 285, 379 (1984).

[10] A. Garcia-Parrado and J. M. M. Senovilla, Class. Quant. Grav. 22, R1 (2005), gr-qc/0501069.

[11] F. J. Flaherty, Proceedings of the American Mathematical Society 48, 199 (1975).

[12] H. Monroe (2006), gr-qc/0607134.

[13] F. J. Tipler, Proceedings of the American Mathematical Society 76, 145 (1979).

[14] G. W. Gibbons and S. W. Hawking, Commun. Math. Phys. 148, 345 (1992).

[15] G. W. Gibbons and S. W. Hawking, Phys. Rev. Lett. 69, 1719 (1992).

[16] J. B. Griffiths, Colliding plane waves in general relativity, Oxford Mathematical Monographs

(Clarendon Press, Oxford, 1991).

[17] The colliding plane wave literature [16] predicts that a singularity results from any collision.

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Hunter Monroe- Topology and Closed Timelike Curves I: Dynamics

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