Found Phys (2008) 38: 1065–1069DOI 10.1007/s10701-008-9254-9

Are Causality Violations Undesirable?

Hunter Monroe

Received: 11 January 2008 / Accepted: 21 October 2008 / Published online: 29 October 2008© Springer Science+Business Media, LLC 2008

Abstract Causality violations are typically seen as unrealistic and undesirable fea-tures of a physical model. The following points out three reasons why causality viola-tions, which Bonnor and Steadman identified even in solutions to the Einstein equa-tion referring to ordinary laboratory situations, are not necessarily undesirable. First,a space-time in which every causal curve can be extended into a closed causal curveis singularity free—a necessary property of a globally applicable physical theory.Second, a causality-violating space-time exhibits a nontrivial topology—no closedtimelike curve (CTC) can be homotopic among CTCs to a point, or that point wouldnot be causally well behaved—and nontrivial topology has been explored as a modelof particles. Finally, if every causal curve in a given space-time passes through anevent horizon, a property which can be called “causal censorship”, then that space-time with event horizons excised would still be causally well behaved.

Keywords General relativity · Differential geometry · Spacetime topology

1 Introduction

Causality violations are typically seen as unrealistic and undesirable features of aphysical model. The occurrence of causality violations in Bonnor-Steadman solutions

In honor of the retirement from Davidson College of Dr. L. Richardson King, an extraordinaryteacher and mathematician. An earlier version (gr-qc/0609054v2) was selected as co-winner of theCTC Essay Prize set by Queen Mary College, University of London. The views expressed in thispaper are those of the author and should not be attributed to the International Monetary Fund, itsExecutive Board, or its management. This paper was not prepared using official resources.Comments are appreciated from anonymous referees and from participants in seminars at theUniversidad Nacional Autónoma de México and Davidson College.

H. Monroe (�)International Monetary Fund, Washington, DC 20431, USAe-mail: editorialmanager.com@huntermonroe.com

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to the Einstein equations referring to laboratory situations, for instance, two spinningballs, suggests that the breakdown of relativity’s explanatory power is not limited toblack hole and big bang singularities.1 This essay points out several respects in whichcausality violations are not necessarily undesirable features of a physical theory. First,certain conditions entailing causality violations rule out singularities (Sect. 2). Sec-ond, causality-violating space-times may exhibit rich topological behavior (Sect. 3).Third, causality-violating space-times can nevertheless behave in a “nearly causal”manner (Sect. 4).

2 Ruling Out Singularities

Three typical assumptions of singularity theorems are the absence of causal viola-tions, a nonzero energy density, and the occurrence of a closed trapped surface ortrapped set. A natural question is what violations of these assumptions suffice to ruleout singularities, with the focus below on violations of causality assumptions. Forinstance, any compact space-time contains a closed timelike curve (CTC) (Hawkingand Ellis [4] Proposition 6.4.2) and is singularity-free (Senovilla [5] Proposition 3.2).

Another more intuitive condition is that every timelike curve be extendible to forma CTC. This condition obviously rules out inextendible timelike curves as well asthe existence of trapped regions such as black holes which can be entered but notexited by timelike curves. Thus, the condition simultaneously violates two of typicalassumptions mentioned above on causality and trapped surfaces. If this conditionholds, gravitational collapse must fail to create a trapped surface separating space-time into exterior and interior regions, just as the Jordan curve theorem fails on atorus. If particles have a topological nature as discussed in the next section, it maybe impossible to circumscribe any massive particle by a space-like surface separatingspace-time into two disconnected components.

An attraction of such conditions for ruling out singularities is that they arepurely topological. Furthermore, showing that a causality-violating space-time issingularity-free also shows that the universal cover, which is causally well behaved,is singularity free (Hawking and Ellis [4]).

Several results in the literature provide a rationale for disallowing CTCs in phys-ical theories. Tipler [6] showed that the emergence of CTCs from regular initial dataleads to a singularity. Krasnikov [7] argues that human beings cannot create a timemachine, which he defines as a causal loop l lying in the future of a region U suchthat a causal loop in the future of U must exist in any maximal extension of U .However, neither of these results applies in a totally vicious space-time with a CTCthrough each point, as Tipler states explicitly. Another objection to CTCs is that mov-ing a massive particle around a CTC to meet its younger self violates conservationof mass-energy; this can be overcome by focusing on vacuum space-times in whichparticles have a purely topological nature. A vacuum space-time in which all timelikecurves are extendible into CTCs violates simultaneously all three typical assumptionsof singularity theorems.

1See Bonnor [1] and Bonnor and Steadman [2, 3].

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3 Allowing Rich Topological Behavior

CTCs are required for there to be interesting topological behavior, by well-knowntheorems. Globally hyperbolicity of a space-time implies that it has the uninterest-ing topology R × S (Geroch [8])2 and that causal curves cannot probe the topologyof S by the topological censorship theorem (Friedman et al. [10]). Topology changeis potentially desirable, for instance, to model the creation and destruction of parti-cles geometrically in a physical theory based upon empty curved space (Misner andWheeler [11]). A useful theory must rule out some phenomena while permitting oth-ers, but in fact any two compact 3-manifolds are Lorentz cobordant regardless oftopology (Lickorish [12]). A “topology selection rule” is needed, such as the result ofGibbons and Hawking [13, 14] that the existence of a spinor structure implies worm-holes can be created and destroyed only in multiples of 2. However, no convincingway has been found to relax assumptions to maintain causality and to permit topol-ogy change, but not arbitrary topology change (Borde [15]). This provides a rationalefor relaxing causality assumptions.

In addition, causality violations imply a nontrivial topology in that no CTC canbe shrunk to a point, as formalized below. Following the literature (for instance Avez[16] and Galloway [17]), say that two CTCs γ1 and γ2 are (freely) timelike homotopicif there is a homotopy without base-point which continuously deforms γ1 into γ2 viaCTCs; γ2 may also be a point rather than a CTC. Say that a Lorentzian manifoldwhich contains a CTC not timelike homotopic to any point is timelike multiply con-nected. It is known among some practitioners as a folk theorem that any Lorentzianmanifold containing a CTC is timelike multiply connected:

Theorem 1 No CTC on a Lorentzian manifold M is timelike homotopic to a point;any M containing a CTC is timelike multiply connected.

Proof Suppose there is a CTC γ which is timelike homotopic to a point p via thetimelike homotopy F : I × I → M, where the curve F(t, s) for fixed s is timelike,F(0, s) = F(1, s) (the timelike curve is closed), and s parameterizes the homotopyfrom F(t,0) = γ (t) to F(t,1) = p. Every neighborhood U of p contains the CTCF(t,1 − ε) for sufficiently small ε > 0. However, Lorentzian manifolds are locallycausally well behaved at any point (Garcia-Parrado and Senovilla Proposition 2.1[18]). �

Consider two examples of the above theorem.

Example 1 Construct a torus by taking the unit square in a 2-D Minkowski space,and identify t = 0 with t = 1 and x = 0 with x = 1.

In this case, the conclusion of the Theorem 1 is obvious; the torus is multiply con-nected, so it is also timelike multiply connected with respect to the strict subset of

2Bernal and Sanchez [9] have recently shown that the traditional definition of globally hyperbolic can besimplified.

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all homotopies which are timelike. However, it also follows immediately from Theo-rem 1 that a compact simply connected Lorentzian manifold is nevertheless, becauseit contains a CTC, timelike multiply connected. A contrived example of a manifoldwhich is simply connected (by any type of curve) but timelike multiply connectedcan be found in Flaherty [19]. However, by Theorem 1, the Gödel space-time is alsoan example, as it contains CTCs and is based upon the simply connected manifoldR

4. The following example suggests what prevents a CTC from being contracted toa point on a simply-connected manifold:

Example 2 The 3-sphere S3, which is compact, is simply connected but by The-orem 1 is timelike multiply connected. Embed S3 in Euclidean space R

4 as x20 +

x21 + x2

2 + x23 = 1. The continuous non-vanishing vector field V = (x1,−x0, x3,−x2)

defines a Lorentzian metric such that V is everywhere timelike. Consider the CTC(r cos θ, r sin θ,0,

√1 − r2) with parameter θ for fixed r . Contracting the curve by

lowering r below 1, it becomes a null curve at r = √2/2 and then becomes a space-

like curve as r is contracted further until it reaches the point (0,0,0,1) when r = 0.

This example follows from the Hopf fibration which has fibre S1; Theorem 1, whichsays that no CTC is null timelike homotopic, parallels the result that the Hopf mapη : S3 → S2 is not null homotopic.

Example 2 suggests that a CTC cannot be timelike homotopic to a point for oneof two reasons: (1) the CTC is not homotopic to a point even allowing homotopieswhich are not timelike; (2) the CTC is homotopic to a point, but under every suchhomotopy which is initially timelike, the curve as it is deformed becomes null atsome point and the homotopy is no longer a timelike homotopy.

A taxonomy of CTCs could distinguish those which are homotopic to a point andthose which are not. Within each category, the set of CTCs can be further dividedinto equivalence classes under timelike homotopy. These equivalence classes are theelements of the fundamental group under timelike homotopy described by Smith [20].

4 Behaving in a Nearly Causal Manner

This section considers one respect in which a causality-violating space-time can beseen as nearly causal. Consider which subsets S of M have the property that M\Scontains no closed causal curves, where the set S is well behaved in some sense, forinstance, it is compact, locally spacelike, and edge free. This is the case for the curvex = 1/2 in Example 1, which is a genus-reducing cut that slices open the handle.One may also ask whether there exists a physically significant S, for instance, the setof points at which neighboring geodesics orthogonal to S are neither converging nordiverging (a condition which holds at r = 2M in Schwarzschild coordinates). Sucha surface would provide “causal censorship” by placing causality violations behindhorizons where they cannot be seen from a region of space-time that appears causalto an observer. Thus, excluding causality violations from physical theories simplybecause we do not see them in practice is risky; a more sophisticated analysis isneeded.

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In any case, the standard tools of causal analysis can be applied to Mc = M\S. Inparticular, the definitions and results of Hawking and Ellis [4] Chap. 6 can be appliedto Mc, and Mc may be causally well-behaved. For instance, Mc may be globallyhyperbolic, with a global Cauchy surface through which all timelike curves in Mc

pass one and only once. Unfortunately, sufficient conditions for causal censorship tohold are not known.

5 Conclusion

Hawking [21] notes that strong causality assumptions risk “ruling out something thatgravity is trying to tell us” and that it would be preferable to deduce that causalityconditions hold in some region of a given space-time, for instance, Mc. Thus, causal-ity assumptions, like Euclid’s parallel postulate, risk closing off interesting lines ofinvestigation.

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