Topology And Time Reversal

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Topology And Time Reversal

Transcript Of Topology And Time Reversal

arXiv:gr-qc/9510006v1 4 Oct 1995

University of Cambridge Silver Street
Cambridge CB3 9EW U.K.
In this lecture we address some topological questions connected with the existence on a general spacetime manifold of diffeomorphisms connected to the identity which reverse the time-orientation.
1 Introduction
If one regards Quantum Gravity as an attempt to unify two distinct but equally fundamental physical theories; quantum mechanics on the one hand and general relativity on the other, one can ask what elements of either theory is it most likely that one will have to sacrifice in the eventual unification. Perhaps the most fundamental innovations of general relativity relate to its treatment of the notion of time. One of most striking features of quantum mechanics is its use of complex amplitudes. One may argue that the introduction of complex numbers into the basic structure of quantum mechanics is closely connected to the treatment in that theory of the notion of change and of time evolution. It therefore seems reasonable to regard the use of complex numbers in conventional quantum mechanics as a potential casuality. More precisely, one may argue that if, as is commonly supposed in quantum cosmology, the classical idea of time is an emergent concept, valid only at late times, low energies and large distances, then so too is our usual idea of a quantum mechanical Hilbert space with its attendant complex structure. In other words, the complex numbers in quantum mechanics should be thought of as having an essentially historical origin. Some ideas along these lines

were discussed within the context of the semi-classical approach to quantum cosmology in [16–17].
A related question is to ask: how in a theory in which one assumes that spacetime has an everywhere well-defined Lorentzian metric are the properties of quantum fields in those spacetimes affected by such global properties of the spacetime as the existence of closed timelike curves (‘CTCs ’), a lack of time-orientability or some other pathology which would normally be excluded in a globally hyperbolic spacetime? Are there restrictions on the possible spacetimes for example? One possible restriction comes about by demanding that spacetime admit a spin or pin structure [6]. Another possible restriction arises by demanding that the spacetime has a time-orientation. If it does not, one may argue that one may not be able to construct a quantum mechanical Hilbert space endowed with a complex structure. This suggestion was made some time ago [25] and it has received further support from the work of Bernard Kay [7].
One motivation for asking this question is to try to extend the range of applicability of quantum field theory in a fixed background. Another motivation might be to answer questions about what possibilities the laws of physics in principle allow. This has provided much of the impetus behind recent work on CTCs . Another, and possibly more cogent, reason for considering non-globally-hyperbolic spacetimes is that in the path integral approach to quantum gravity in which one sums over all possible Lorentzian metrics there is a priori no good reason for excluding them. One might attempt to perform the functional integral by first freezing the metric and integrating over all matter fields on that spacetime, and then summing over all spacetimes. The first part of the integral is then tantamount to quantizing matter fields on a fixed background. It is customary in the Euclidean formulation to replace the sum over Lorentzian metrics by a sum over Riemannian metrics but one may ask what happens if one tries to avoid this step.
In the Euclidean version one is often concerned with anomalies that may arise when functional determinants fail to be well-defined, for example they may not be invariant under spacetime diffeomorphisms. The diffeomorphisms in question may either be continuously connected to the identity or not. The latter type of global anomalies are closely related to discrete symmetries, or lack of them, such as parity or orientation. They may also be investigated from an Hamiltonian point of view . However, this does not address the possibility of anomalies of a purely Lorentzian kind which manifest them-

selves only in non-globally-hyperbolic spacetimes. An example is a breakdown of spin structure. If one assumes that spacetime is both time and space-orientable, this can only occur in a spacetime which is not globally hyperbolic . If one drops the requirement of space-orientability, however, there may exist pin structure even though the spacetime is globally hyperbolic. An example is provided by RP2 × R2, endowed with the product metric formed from the standard ‘round’ metric on RP2 and the Minkowski metric on R2 (with either signature) [6].
One possible viewpoint on the difficulties experienced with non-timeorientable spacetimes is precisely that there is some sort of anomaly. Roughly speaking, for each complex amplitude in the functional sum one must, if there is no global time-orientation, include its complex conjugate which is associated to the time-reversed amplitude. The result must then necessarily be real and so no true quantum interference is possible. It is interesting to note that this sort of problem would also arise in some attempts to generalize the usual quantum formalism being made by Gell-Mann and Hartle [22] since they also make use of complex amplitudes and they incorporate a rule relating complex conjugation to time reversal of a sequence of observables.
The purpose of this lecture is to explore some of these issues in more depth. In particular, we will discuss the relation between the topology of a time-orientable spacetime {M, g} and the existence and properties of various kinds of time-reversing diffeomorphisms. We shall, for the sake of mathematical precision, mainly concentrate on spacetime manifolds M which are compact and without boundary, but we will comment on the case of non-compact spacetimes and spacetimes with boundaries.
As well as the motivations given above, our results are also relevant to suggestions like that of Sakharov [18] that the early universe may simply be a time reflection of the late universe. Such a viewpoint is essentially a Lorentzian version of the (historically later) no-boundary proposal or the idea of a universe born from nothing [23].
2 Compact Spacetimes
An assumption of compactness in spatial directions is quite natural when discussing topological questions because one has in mind a situation where

the non-trivial topology can be localized, at least to the extent that it is not

allowed to escape from the spacetime altogether. Compactness in the time

direction is less easy to justify (unless there are spacelike boundaries) because

it necessarily implies the existence of closed timelike curves . Formerly this

was thought to rule out consideration of such spacetimes but more recently,

with the advent of studies of the properties of time machines, this view has

been abandoned and so we shall not be put off by this feature.

In fact the Euler number χ(M) of a compact spacetime of arbitrary

dimension must vanish:

χ(M) = 0,


and in four dimensions:

χ = 2 − 2b1 + b2,


where bi are the Betti numbers. Thus a compact spacetime must have an

even second Betti number and infinite fundamental group, and so its universal

covering space is non-compact. In this sense it may be thought of as a non-

compact spacetime which has been periodically identified and this is indeed

typically how examples of time machines are constructed in the literature.

However, the reader is cautioned that there is, as we shall see later, no logical

connection between whether or not a curve is closed and timelike and whether

or not it is homotopically trivial. In general, one expects the fundamantal
group π1(M) to be non-Abelian. This is what one expects in the case of two or more time machines, for example if the spacetime has in a connected sum decomposition two summands of the form S1 × S3 with time running around the S1 factors.

In the exceptional case that the fundamental group is Abelian, it may be shown [8–11] that the possible Betti numbers (b1, b2) must belong to the set: {(1,0), (2,2), (3,4), (4,6)} . This is because for any closed orientable

manifold of any dimension which has an Abelian fundamental group one has

the inequality:

1 b1(b1 − 1) ≤ b2



The result follows from (1) which holds for any spacetime dimension and

(2) which holds in four dimensions.

The significance of this non-Abelian-ness in the case that homotopically

non-trivial time machines are present is presumably that some physical effects

may depend upon the order in which one enters the time machines. It would


be interesting to explore this point further. In that connection, it is perhaps

worth recalling why it is that non-simply-connected four-manifolds are not

classifiable [26]. The point is that by taking the connected sum #kS1×S3 of k

copies of S1 ×S3 one obtains a four-manifold whose fundamental group is the

free group on k generators (which of course is maximally non-Abelian). One

may now perform surgery on this manifold to obtain a new manifold whose

fundamemtal group has k generators and r arbitrarily chosen relations. Since

there is no algorithm for deciding whether two different presentations give

an isomorphic group there can be no algorithm for deciding whether two

four-manifolds are homeomorphic .

The process of surgery can be described as follows. Given an element g ∈

π1(M′) of a four-manifold M′ one can represent it by a closed curve γ ∈ M′.

Now surround this closed curve γ by a tube or collared neighbourhood N of

the form N = D3 ×γ ≡ D3 ×S1 where D3 is a closed 3-dimensional disc. The

boundary ∂N of this tube has topology ∂N ≡ S1 ×S2. One now removes the

tube N from M′ and replaces it with the simply connected manifold D2 × S2 which has the same boundary. The result is a new manifold M′′ whose

fundamental group differs from that of M′ only by the imposition of the

relation g = 0. This process is called ‘killing an element of the fundamental

group’. It may be shown that by a succesion of such killings one may obtain

from #kS1 × S3 a manifold with any desired finitely generated fundamental


From a physical point of view it is interesting to note two things. Firstly

that the undecidability problem reviewed above may give rise to limitations

on what is ‘in principle’ allowed by the laws of physics when it comes to the

sort of wormhole and time machine engineering envisaged by Thorne and

others. The possibility arises of having two sets of instructions for building

a multiple time machine but having no algorithm for deciding whether the

two spacetimes have the same topology. Whether or not this is true is not

obvious from the general result quoted above because a compact spacetime

must have vanishing Euler number. We do not know whether such manifolds

are classifiable or not.

The second point is that the process of surgery gives rise to a manifold

which physically looks rather like one containing the creation and annihi-

lation of an extra Einstein-Rosen throat. If the 2-disc D2 has coordinates

X + iT = r exp


iπ 2

where the cyclic ‘time’ coordinate t which param-

eterizes the original curve γ runs between 0 and 2π then ‘half-way round’,


i.e. on the real axis T = 0, the interior of the tube N has been replaced by a manifold which has the same topology as the Kruskal manifold of a black hole and therefore it has embedded in it a three-manifold which has the topology of a bridge, i.e. of R × S2. If these sorts of manifolds do arise in a Lorentzian form of quantum gravity it seems reasonable to think of them as containing ‘virtual black holes’.
This interpretation receives some support from the observation that the Riemannian manifolds used as instantons or real tunnelling geometries in the Euclidean approach to vacuum instability and black hole pair creation may be obtained by surgery on a circle , which we would like to associate with the world line of a virtual black hole, from the corresponding false vacuum spacetime. Thus the Euclidean Schwarzschild manifold (R2 × S2) may be obtained from the hot flat space manifold S1 ×R3, the Ernst instanton manifold (S2 × S2 − {pt}) for the creation of pairs of oppositely charged non-extreme black holes from a constant electromagnetic field (topology R4), and the Nariai and Mellor-Moss Instantons (both with topology S2 × S2) are obtained from the De Sitter manifold (S4). In Kaluza-Klein theory, Witten [27] has argued that the five-dimensional Schwarzschild solution (topology R2 × S3) is the bounce solution which mediates the decay of the Kaluza-Klein vacuum (topology S1 × R4 ). The five-dimensional manifold corresponding to a magnetic field also has topology S1 × R4. This may decay via Witten’s instability but it may also decay into a monopole-anti-monopole pair. The instanton for this process has topology S5 − S1 ≡ R2 × S3 and so may also be obtained by surgery on a circle from the false vacuum spacetime manifold.
3 Time Reversal in a General Spacetime
Let {M, gL} be a time-orientable spacetime. Thus the bundle of timeoriented frames SO↑(n − 1, 1)(M, gL) falls into two connected components. One typically thinks of time reversal Θ as a diffeomorphism:
which reverses time-orientation, whose lift to SO↑(n−1, 1)(M, gL) exchanges the two connected components and is an involution of order two:
Θ2 = id.

It need not necessarily be an isometry (in general the spacetime will not
admit any isometries). One could imagine considering a more general finite
group action but presumably one could always find a Z2 subgroup and we shall assume that this can be done.
In a general non-globally-hyperbolic spacetime it is not obvious whether Θ should reverse space-orientation, or total orientation (assuming {M, gL}
to be space or time-orientable respectively) , whether it should act freely on M or fix a three-surface for example, or whether it should belong to the identity component Diff0(M). The existence and uniqueness and other properties of Θ depends both on the topology of the manifold M and on the Lorentz metric gL.
To illustrate these subtleties, consider even the simplest globally hyperbolic spacetime M ≡ R × Σ with coordinates t, x, t being timelike and Σ being an orientable (n − 1)-manifold. Naively we might take

ΘT : (t, x) → (−t, x)

but nothing prevents us from considering

ΘJ : (t, x) → (−t, x∗)


J : x → x∗

is an involution on the (n − 1)-manifold Σ. Clearly ΘT fixes the threemanifold Σ and reverses total orientation. It therefore lies outside the identity component Diff0(M). On the other hand, we might arrange for J to act freely on Σ, possibly reversing or not reversing space-orientation.
These seemingly rather artificial examples actually arise in some applications. In quantum field theory in De Sitter spacetime, dSn, Σ is the (n − 1)-sphere and J its antipodal map. This preserves space-orientation if the spacetime dimension n is even. The map ΘJ is an isometry and is the centre of the isometry group O(n, 1). One may identify points under the action of ΘJ to obtain the ‘elliptic interpretation’. This then provides a possible non-singular realisation of Sakharov’s ideas of a Lorentzian model of a universe born from nothing. The idea immediately generalizes to a Friedman model whose scale factor is an even function of time. Sakharov’s idea was in fact to impose some sort of time-reflection symmetry about a singular big


bang at which the scale factor vanishes. He did not use the involution J. In spatially closed models the scale factor often starts from a zero value at the big bang, t = 0, rises to a maximum at t = tmax say, and then symmetrically decreases to a vanishing value at the big crunch at t = 2tmax. This has led Gold [2] to conjecture that the ‘arrow of time reverses’ in the contracting phase. In effect he proposed that the entire quantum state is invariant under a time-reversing involution whose action on spacetime is given by:
ΘG : t → tmax − t.
By contrast Davies [1] (see also Albrow [4]) prefers to continue through the Big Bang and Big Crunch to get a model in which the arrow of time reverses in sucessive cycles. In other words, one imposes invariance under the action of semi-direct product Z ⊙ Z2 given by
t → t + 2tmax
and t → −t.
It is clear that similar options are available for non-singular periodic models in which there is neither a Big Bang nor a Big Crunch. Thus for example, in the case of Anti-De Sitter spacetime AdeSn, the scale factor is a sinusoidal function of cosmic time but the vanishing of the scale factor is an artefact of a poor choice of coordinates. In fact M ≡ S1 × Hn−1 where Hn−1 ≡ Rn−1 is hyperbolic space and time t runs around the circle, 0 ≤ t < 2π. The center of the isometry group O(n − 1, 2) does not reverse time (it sends (t, x) to (t + π, −x)). Intuitively, it seems clear that time reversal must have fixed points since we must reverse t and compose with an involution J which may be thought to act on Euclidean space.
In the examples so far (at least if we wish to maintain the boundary conditions) there was no natural choice of Θ in the identity component Diff0(M). However in more exotic situations, as we shall see in detail shortly, this seemingly paradoxical situation can occur. Now if no possible Θ lies in the identity component Diff0(M) it is reasonable to say that the spacetime {M, gL} has an intrinsic sense of the passage of time (even though time itself may not be defined!). If however there exists a Θ which does lie in the identity component this is not reasonable. The general situation with respect to Diff(M)

appears to be quite difficult to analyse and so we shall restrict attention here to a simpler question. Is there a homotopy rather than a diffeomorphism carrying the metric gL with one time-orientation to the same metric with the opposite time-orientation? If there does exist a suitable Θ in the identity component Diff0(M) then a homotopy will certainly exist (simply pull back gL by a curve fs, 0 ≤ s ≤ 1 of diffeomorphisms joining f1 = Θ to the identity f0 = id). However the converse is not necessarily true. Given a homotopy gtL of Lorentz metrics there may exist no diffeomorphism producing it. Now from the point of view of homotopy theory, a closed time-oriented Lorentzian spacetime {M, gL} contains no more information than a Riemannian manfold M equipped with a unit vector field V. The spacetime with the opposite time-orientation corresponds homotopically to the same manifold equipped with the negative unit vector field −V.
4 Mathematical Interlude
This following mathematical interlude follows some conversations with Graeme Segal.
4.1 Linear and General Homotopies
We suppose that M is a closed, n-dimensional time-orientable Lorentzian manifold. We may, in the standard way, endow M with a Riemannian metric and hence deduce that M admits a global section V of the bundle S(M) of unit vectors over M. At each point x in M the fibre Sx of S(M) is an n − 1 sphere.
Pulling the Lorenzian metric back under the action of diffeomorphisms induces an action on V and we would like to know whether there exists a diffeomorphism f : M → M which takes V to its negative, i.e. which reverses the direction of time. In particular we would like to know whether there exits such a diffeomorphism f contained in the identity component Diff0(M) of the diffeomorphism group Diff(M). An easier question to ask is whether there exists a homotopy taking V to −V since if there exists a diffomorphism in the identity component a homotopy is given by a curve ft in Diff0(M) joining

f to the identity. The converse is however not sufficient because, as we shall see, if one considers M = S1 × S2n−1 with the vector field running around the S1 factor one finds that this cannot be reversed by a diffeomorphism but
it may be reversed by a homotopy A homotopy Vt between V and −V thus gives at each point x in M
a continuous path γx(t) from the north pole to the south pole of Sn−1. In other words a general homotopy Vt provides a global section sZ of a bundle Z(M) whose fibres Zx are the space of paths from the north to the south pole of Sn−1. Since any path from the north pole to the south pole of Sn−1 is homotopic to a closed path on Sn−1 one sees that from the point of view of homotopy the fibre Zx is equivalent to the loop space Ω(Sn−1) of based loops on Sn−1 .
Consider now a special or linear homotopy from V to −V. By definition this is one for which, at each point x in M, V(x)t lies in a an oriented two plane π(x) spanned say by the vectors V0 and Vt1 where 0 < t1 < 1. A linear homotopy gives a particular kind of path γx(t) from the north to the south pole of Sn−1, one which is along a great circle in the 2-plane defined
by by the vectors V0 and Vt1. The set of such great circles is parameterized by where the great circle intersects the equatorial Sn−1.
The existence of a linear homotopy is thus equivalent to the existence of a global section sY of the Sn−2 bundle Y (M) of unit vectors orthogonal to the vector field V (x). One may think of this Sn−2 fibre Yx as the equatorial Sn−2 in the Sn−1 fibre Sx of the bundle S(M). It follows that the bundle Y (M) is a sub-bundle of the bundle Z(M). The question of whether every homotopy
can be deformed into a linear homotopy then reduces to the question whether every section sZ may be deformed to a section sY .
It should also be clear that the existence of the vector field and a linear homotopy is equivalent to a non-vanishing section of the bundle Vn,2(M) of dyads, i.e of ordered pairs of linearly independent vectors e1 and e2 say. The fibre of the dyad bundle Vn.2(M ) is the Stiefel manifold Vn,2 of dyads. In addition a linear homotopy provides a global section sG of the bundle Gn,2(M ) of oriented 2-planes whose fibre is the Grassman manifold Gn,2. The existence of a section sG is, in fact, the necessary and suffient condition that a manifold admit a metric of signature (n − 2, 2).
We note en passant the following Lemma If M is even dimensional a sufficient condition for M to admit a linear homotopy is that it admit an almost complex structure J. In four