The elliptic interpretation of black holes and quantum mechanics

Nuclear PhysicsB271 (1986) 497-508 North-Holland, Amsterdam

T H E ELLIPTIC I N T E R P I ~ T A ~ O N OF BLACK H O L E S AND QUANTUM MECHANICS G.W. GIBBONS DAMTP, Universityof Cambridge, Silver Street, CambridgeCB3 9EW, UK

Received30 December1985 The old "ellipticinterpretation"of the Kruskalmanifoldis reviewedin the light of quantum mechanics. It is arguedthat its adoptionwouldrequire givingup the complexstructure of quantum mechanicsand workingwith real Hilbert spaces. This may also be neededin cosraolo~cal theories in which the universe is CPT invariam. The cases of de Sitter and anti-de Sitter space are also discussed.

I. Introduction

Recently there has been renewed interest in the quantum properties of black holes [3-6]. The original derivation by Hawking [2, 7] of the fact that black holes formed by gravitational collapse should radiate thermally with a temperature T = 1/8~rGM where M is the mass of the black hole was obtained by considering a background spacetime which contains the collapsing body producing the black hole. The spacetime is neither everywhere time independent nor is it symmetrical with respect to time reversal. There is a natural definition for the incoming no-particle state 10) which is determined in terms of an appropriate and natural notion of positive frequency at past infinity. Similarly there is a natural definition of outgoing particles at future infinity. There is no natural definition of particles falling through the future horizon but Hawking showed that this did not affect the predicted outward flux of particles at infinity. Given these assumptions (and ignoring the back reaction of the geometry) Hawking's derivation follows unambiguously at least as far as non-interacting fields are concerned. Inclusion of the back reaction must of course modify the result substantially at later times but is unlikely to make a great deal of difference at times much larger than the collapse time but short compared with the evaporation time. They would of course introduce non-thermal correlations to the outgoing flux would no longer be exactly thermal but the qualitative picture should remain unchanged. Some authors [4,5, 8] have however preferred to work not with the dynamic spacetime representing the formation of a black hole in gravitational collapse and 0550-3213/86/$03.50©Elsevier SciencePublishers BN. (North-Holland PhysicsPublishingDivision)

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G.W. Gibbons / Black holes

the subsequent occurrence of a spacetime singularity inside a future event horizon but rather with the exact maximally extended vacuum solution. The geometry of this spacetime, -'¢'/K, is very different from that of the dynamic collapse. For example it is symmetric under time reversal (1.1)

r:

in standard Kruskal null coordinates. This means that ~ / ~ contains both future and past horizons, and singularities both in the future and in the past. The extended manifold is also invariant under the spatial reflection (1.2)

n:

This means that ~¢K contains not one but two asymptotically flat regions. Furthermore the behaviour of fields on -//gK is not determined by initial data posed at either or both past infinities but requires also that data be specified on both past horizons. The spacetime J / K is sometimes referred to as that of an "eternal black hole" - however this description is not really accurate. It actually represents a black hole and a "white hole" in the same spacetime. [A "white hole" is usually defined as the time reversal of a black hole.] It is clear that the behaviour of quantum or classical fields on J / K is much less obvious than on the dynamical spacetime of gravitational collapse. Depending upon ones boundary conditions one could obtain virtually any result. Furthermore we have no good reason to believe that objects described by this spacetime actually will occur in nature. Purely as a model however it may provide interesting insights into how general relativity and quantum mechanics may be combined. I want to point out some consequences of a model in which the two asymptotic regions and the two singularities are regarded as identical. This model has something in common with the ideas of 't Hooft [4], discussions with him indicate that it is not the same as his. For the sake of a name I shall refer to it as the elliptic interpretation.

2. The identified spacetime Attempts have been made in the past to identify the two asymptotic regions of J [ K by identifying point x and Jx where J is an isometry of the Schwarzschild metric whose square is the identity (i.e. an involution). In order to obtain a non-singular quotient manifold ~ ' K =J/#K/J it is essential that J has no fixed points. This requirement is not insisted upon in all the references in [9] and is not used explicitly in [4]. If we demand that J commute with the continuous symmetries of ~ ' K , which will thus descend to ~ ' K , one finds that J must be given by

J:

(v, v,

o,

--, ( - v , - v ,

- o, ¢ +

(2.1)

G.W. Gibbons / Black holes

499

[The identification (U, V, 0, ~) ~ ( - U, - V, 0, q~) used in [9] would give a singularity at the "Boyer axis" U = V= 0.] One may check that J is space orientation preserving but time orientation reversing. Thus ~¢K is space orientable but not time orientable- there is no continuous assignment of future directed and past directed timelike vectors on ~ KThis ambiguity does not arise if one remains outside the horizon. Even if one penetrates the horizon one can still not find closed smooth non-spacelike closed curves in ~ K , for suppose one could: such a curve "/would correspond in JC'K to a non-closed curved joining two distinct points x and .Ix. If one traverses ~, twice in ~¢K one gets a smooth closed curve F in ~//K going from x to "Ix to x. Since 2I is everywhere non-spacelike F will be everywhere non-spacelike but "gK, unlike ~'K, certainly contains on everywhere non-spacelike closed curves so ~, can't exist. This simple argument makes no essential use of the detailed geometry of the Schwarzschild metric. A simple calculation, which will not be repeated here, shows that ~ K does have closed timelike and null curves starting from a point x and going into the local future and returning to x from the local future. They are of course not smooth at x. Because the fundamental group, ~rl(J~K) equals Z2, tensor fields on ~gtK need not be single valued, they may be sections of a twisted bundle over ~ K " Since we have a lorentzian metric the appropriate structural group is the full Lorentz group. This has 4 connected components related by P, T, and PT. Depending upon whether one represents these elements by ___1 we get ordinary tensors, time pseudotensors, space pseudo-tensors and spacetime pseudo-tensors. These correspond to the 4 possible representations of the group Z 2 ® Z 2 generated by P and T. Since ~ ' K is neither time orientable nor spacetime orientable tensor fields of the second or fourth kind will be double valued, changing sign as one traverses a non-trivial loop in ~rl(~K). On the covering space .A¢K they must be antisymmetric with respect to the involution J while ordinary tensors or space-pseudo tensors must be symmetric with respect to J. In flat spacetime one usually regards the electromagnetic field strength F~ as a time pseudo-tensor. This is because under time reversal the 4-current j r = (#, j ) (p, - j ) and F~, is taken to have the same properties as j~ under time reversal. If we maintain this convention in curved space we see that Maxwell fields on ~¢[K must be double valued or alternatively must be antisymmetric under J on the covering space ~¢/K- This has an important consequence for electromagnetic black holes in the elliptic interpretation. Consider the Reissner-Nordstrom black holes. If one does not identify under the obvious generalization from Schwarzschild of the inversion J the electromagnetic field can have any "duality complexion", i.e. it may be pure electric, pure magnetic or any duality rotation of these. If on identifies however only the pure magnetic case is allowed since one may easily check that only in this case will F,~ be odd with respect to J. Thus a black hole can't have an electric charge in the elliptic

500

(7. W. Gibbons / Black holes

interpretation. It should be noted that in the case of extreme Reissner-Nordstrom black holes (which correspond in certain theories to solitons) the analogue of the inversion map J does not exist. Since Weyl neutrinos transform in a well-defined way under time reversal there is no problem in introducing them on ~'K- This would not have been true if J had reversed parity. A reversal of parity is required in the anti-de Sitter case and does indeed give rise to difficulties as will be discussed later. Scalar propagators on ~¢K may be obtained by taking any propagator on the covering space ..l/K, G(x, y) satisfying

(-V~ + m2)G(x, y ) = 8(x, y)

(2.2)

and averaging with respect to both arguments using J. Thus G(x, y) will be an appropriate propagator on ~¢K if

O(x,Y)=½{G(x,y)+G(Jx, y)+_ G(x, Jy)+G(Jx, Jy)}, where the minus signs allow for twisted fields. If sense that

(2.3)

G(x, y) is invariant under J in the

G(x, y) = G(Jx, Jy).

(2.4)

G(x, y) is given by the simple formula G(x, y) =

G(x, y) ++_G(x, Jy),

(2.5)

so that we merely add an antipodal source in addition to the original one. If G(x, y) is the usual retarded propagator on the Kruskal manifold which satisfies:

aret(jx, Jy) = aadv(x, y ) ,

(2.6)

we find that the resulting propagator on the identified space is

~1 , ' G a d v Z~x , y)\

__ GaaV(x, Jy) + Gret(x, Jy)). ~ , y) +

+ Gret[ x

(2.7)

If x and y are both outside the horizon the 3rd and 4th terms will not influence them and so the net result of the elliptic interpretation in the exterior region is to force the use of half-advanced plus half-retarded Green functions. A different approach to field theory on ~¢K is to pose Cauchy data on a t = constant hypersurface, e.g. U + V = 0. Call this hypersurface 2~. The involution J takes points on ~ to point on 2. Call its restriction to 2? I. The metric restricted to is ds 2 -- 1 + - ~ p

dx 2 + dy2 + dz2),

(2.8)

G. IV. Gibbons / Black holes

where 0 2= x 2 + y 2 + z 2. The horizon is at O > 0, I is given by I:

50t

p = ½GM, and 2 is all points such that

xi~ -

Oz .

(2.9)

If the data on N are such that the fields on 2; are invariant under I and their time derivatives are odd under I they will evolve on the Kruskal manifold ..//K to be invariant under J. This means that they are well defined on the identified space ~ K . For twisted fields one merely interchanges even and odd in the above statements. It is clear from this that one may impose whatever Cauchy data one pleases outside the horizon and still be consistent with the elliptic interpretation. As far as the outside of the horizon is concerned it makes no difference. Using an analogous construction for the initial data for Einstein's equations one can easily convince oneself that there is an entire class of time dependent spacetimes possessing an invohitive isometry J (but no continuous isometrics) in which the elliptic interpretation is possible. They represent as it were a different "sector" of the theory from what is usually considered in gravitational collapse. Although perhaps of limited physical or astrophysical interest it is certainly a mathematically consistent sector.

3. Quantum mechanics on the identified space

The question we wish to answer in this section is how to do quantum field theory on the identified space ~¢K- One approach might be to define Feynman propagator using the averaging technique described in the last section. Thus one could take the Hattie-Hawking propagator [10] on ~¢K, G~Qx, y) and use on ~¢K.

GnU(x, y)= ½(Gn2(x, y) + GH~(x, Jy)).

(3.1)

This would come from analytically continuing from the euclidean Schwarzschild metric

2GM~ ds 2= 1

r

)dr

2

ds2 + 1-2GM/r

+ r2(d0 2 + sin20 d,2).

(3.2)

The euclidean Schwarzschild solution is periodic in the imaginary time ~- with a period 8¢rGM [10,11]. To ensure invariance under J one would further identify under the euclidean version of J, are" Je:

(r,r,O,,)~(~'+4~rGM, r,~r-O,,+~r).

(3.3)

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G.IV. Gibbons/ Blackholes

On the face of it this suggests that one gets finite temperature Green functions with twice the usual Hawking temperature since the period is halved. However this is not correct. The Green function GH2(x, y) does not arise as the expectation value of field operators in a Fock space. To see this we follow the analysis given by Allen [11] in the closely related de Sitter case. He wasn't considering the elliptic interpretation directly but his arguments are relevant here. Consider the symmetric scalar propagator G(:)(x, y) corresponding to (3.1). If pi(x), n,(x) are the positive and negative frequency functions defining the putative Fock vacuum we have G(1)(x, y) = Z p i ( x ) p i ( y ) + n i ( x ) n i ( y ) .

(3.4)

fz{ G(:'*(x, y)3~p~(x) ) dY~~,

(3.5)

By forming

where Z is the t = constant surface used in sect. 2 and using the assumed completeness and orthogonality of p:(x) we obtain:

p~(x) = +_p,(Jx).

(3.6)

But this means that (no sum on i)

fp*( x) .pi( x ) d ~ = 0 .

(3.7)

Thus our putative positive function have zero norm. Our Green function fails to satisfy the correct positivity properties. Usually one has an ambiguity in building up a Fock basis in curved space because one could choose any set of "positive frequency" functions pi(x) which can be normalised to unity. Any such choice determines a Fock basis. In the present case however insisting on invariance under J as in (3.6) implies that the norm of p~(x) vanishes. From a mathematical point of view the real vector space of initial data for wave equation is endowed with a natural simplectic structure [13] given by:

(3.8) Picking a positive frequency function gives the space of Cauchy data a complex structure [13] and enables us to construct a complex Hilbert space. Such a construction breaks down if we use a function satisfying (3.6) since the norm vanishes. One may express this in more physical terms. J reverses the sense of time. Therefore in quantum mechanics we anticipate that J corresponds to an antiunitary operator J.

G. ~ Gibbons / Black holes

503

3 may be thought of as complex conjugation followed by the action of some unitary operator U say. Since J is involuntary we have that on any state [~b) J(3lq~)) = c l ~ ) ,

(3.9)

uu*l,) = I¢),

(3.10)

whence

where * denotes complex conjugation. It follows that either c = + 1 or c = - 1 . [In flat space, where J is replaced by CPT, c = ( - 1 ) e where F is the fermion number. This can be used in flat space to give the fermion superselection rate since if one superposes two states they must have the same value of c if CPT is to make sense.] Working on the identified space ~¢~: with Green functions invariant under J means effectively that all physical measurements are invariant under J and so in effect one is imposing on ones states the condition

Jl ) = < ¢ ) -

(3.11)

1 12= _+1,

(3.12)

Acting with J again shows that

so our states must have c = + 1 [presumably this means the ferrnion number must be zero mod2]. By multiplying Iq~) by a -1/2 one can set a = 1 with no loss of generality. The set of states ([~p)} satisfying (3.11) does not form a complex Hilbert space because J is antilinear and so does not commute with multiplication by a complex number. It does however constitute a real Hilbert space. Thus the elliptic interpretation seems to force us to use a real Hilbert space. Since in any complex Hilbert space there is natural antilinear isomorphism between kets and bras we see that the correspondence (3.1) using a~ provides a linear map between bras and kets which are in effect identified in a real Hilbert space. Thus the elliptic interpretation defined by (3.11) makes contact with the ideas of 't Hooft [4]. The general principles of quantum mechanics are compatible with the restriction to real numbers though it makes it difficult to talk about Heisenberg's uncertainty principle and Heisenberg equations of motion. In the present case this is perhaps not surprising. Real quantum mechanics has some unusual features. For example continuous symmetries no longer give rise to observables. This is because continuous symmetries are effected by orthgonal operators. These are obtained by exponentiating skew-symmetric operators which can't of course be diagonalized over the reals. Thus if we have global U(1) symmetry like the baryon number in p r e - G U T times, the elliptic boundary conditions tell us that the total charge Q corresponding to a conserved current must vanish Q = f j ~ d Z~' = 0.

(3.13)

504

G. IV. Gibbons / Black holes

We saw earlier that the total electric charge of a black hole must also vanish in the elliptic interpretation which is consistent.

4. More than one black hole

A natural question to ask is what happens if one has more than one black hole. Are there many asymptotic regions of just one? We don't have exact solutions describing many black holes however some years ago Misner [14] described how one could construct initial data for many black holes, though in those days they weren't called black holes. In the simplest case one considers time symmetric data of the form ds 2 = q~4dx_2,

(4.1)

V'2q~= 0,

(4.2)

where

where V "2 is the usual laplacian in flat euclidean 3-space. Setting GM

q~= 1 + - 21xl

(4.3)

gives the Schwarzschild data (c.f. eq. (2.8)). Let g be a conformal transformation of the flat metric dx 2 such that under g: x_--*_g(_x), d x 2 ~ I24(x)dx 2. Since (4.2) is conformally invariant, if q~(_x) solves (4.2) so will

U(g),.

(4.4)

1, defined by (2.9), is such a conformal transformation and one has GM

U(I)I = -21xl

(4.5)

Consider now N black holes at positions x, and mass M,, n = 1, 2 .... N. There are N inversion maps 11... I N. Consider the discrete Schottky-like group G generated by 11... I s subject to the obvious relations I~ = 1.

(4.6)

Let g be a typical element. The conformal factor (assuming convergence) ~b(x) = • g~G

U(g)l

(4.7)

G. W. Gibbons / Black holes

505

is invariant under the discrete group G. A fundamental domain for G will be the exterior of the balls in R 3 or the points X, with radius ½GM,,. We assume these balls don't overlap. This domain D will be repeated infinitely often in the interior of the balls, i.e. the sets { gD}, g ~ G will lie inside these balls. The time symmetric data using (4.7) in (4.1) now have infinitely many isometric regions related by elements of the group G. These may however all be identified if one pleases, since G acts without fixed points. The data constructed in this way are similar to but different from the data constructed by Misner [14] (see also [15]) in which there are 2 isometric asymptotic regions even though there are N black holes. The isometry joining these 2 regions has fixed points on the horizon and so would be unsuitable for the identification proposed here. In fact Misner's construction is to use the maps

[GM]2x' x' ~ + [-~---] ~5"

(4.8)

This has fixed points on the sphere of radius ½GM. It follows that the boundary of D, i.e. the surfaces of the N spheres is a totally goedesic submanifold. Thus Misner was able to take another copy of D, call it D 1 and join it to D across the spheres forming the "double". Misner constructed a 3-manifold for N black holes with 2 asymptotically flat regions. One cannot identify these regions however without getting singularities. The manifold with just two asymptotic regions is used by Professor 't Hooft in his work [4]. Finally I should mention that if one simply takes as data n=N

q~=l+

~

n~l

GM,, 21x_x, I ,

(4.9)

one has just N asymptotic regions which cannot be identified. Further possibilities (" wormholes") arise if two of the spheres have equal radius. From the point of view of the elliptic interpretation I believe that the model with just one asymptotic region is the most natural. On the covering space one would presumably identify points to future in those regions obtained by an even number of inversions with points in the past in those regions obtained by an odd number of inversions. Thus we see that the elliptic interpretation can deal satisfactorily with the case of more than one horizon.

5. de Sitter and anti-de Sitter spacetime

Historically the elliptic interpretation was first discussed in cosmology. Eddington pointed out the possibility of identifying points which were spatial antipodes of one another on the Einstein-static universe. The spatial cross section would thus be R P 3

506

G. IV. Gibbons / Black holes

or SO(3) rather than S3 on SU(2). In fact other identifications are possible but these would not commute with the isometry group. If the cosmological model has a moment of time symmetry, e.g. a k = 1 model, on may compose this with time reversal to get a fixed point free involutive isometry. This shares with our previous examples the property of reversing time orientation but not space orientation. A special case would be de Sitter space. The Robertson-Walker form of the line element is ds 2 = dt 2 - cosh2~ f ~ t ( d x 2 + sin2x (dO 2 + sin20 d,2))

(5.2)

and we identify under J:

( t , x , 6, q J ) - - * ( - t , l r - x , rr-O,q~+~r).

Since de Sitter can be isometrically embedded as a hyperboloid in 5-dimensional Minkowski space jg5

(x°) 2 - (xl) 2 - (x2) 2 - (x3) 2 -

= -3/a.

(5.3)

J has the simple expression as inversion in vg 5: J:

X A ~ - X A.

(5.4)

The identification under J was discussed by SchriSdinger [16]. Since x and J x are always spacelike-separated in de Sitter space he felt the identification to be rather natural but recognised the difficulties with time orientability. Our previous discussion would apply. Classically, we use half-advanced plus half-retarded propagators (the last 2 terms in (2.7) vanish). Quantum mechanically we use Feynman propagators satisfying C(x, Jy) = _+C(x, y ) .

(5.5)

As Allen [12] has pointed out there is now no Fock vacuum and our previous remarks about real quantum mechanics follow. In de Sitter space there is no unique de Sitter invariant Fock vacuum for scalar fields. This is essentially because we can always add antipodal source. There is a unique Green function Ge(x, y ) which is regular on the euclidean section, S4. Then there is a one-parameter family of vacua of the form G°(x, y) --- cosh0 Ge(x, y) + sinh0 Ge(x, J y ) ,

(5.6)

the symmetric or antisymmetric Green functions are not members of this family. Only the euclidean propagator corresponds to a finite temperature propagator and is the one usually used in inflationary studies. In principle predictions using the elliptic

G.W. Gibbons / Black holes

507

interpretation might differ from the standard results. This has not, to my knowledge, been investigated. One might suspect that these effects should be small since the antipodal region would now be according to standard views almost infinitely far away from us. A related case is anti-de Sitter space which arises in gauged supergravity. Quantum field theory in anti-de Sitter space has received a lot of attention recently [17,18]. Anti-de Sitter space is again a hyperboloid but now in a flat space with 2 timelike directions. (X°) 2 + (X4) 2 - (X1) 2 - (X2) 2 - (X3) 2 =

-3/A.

(5.7)

The inversion J (still defined by (5.4)) now preserves time orientation but reverses spatial parity. Since time is not reversed there is nothing funny about quantum mechanics in anti-de Sitter space. However there is something funny about parity. Avis, Isham and Storey [17] and Brietenlohner and Freedman [18] found that certain boundary conditions needed to be imposed on the fields in anti-de Sitter space to give a well-defined quantum theory. One may express these boundary conditions in terms of the inversion map. They amount to demanding that the fields be even or odd under the action of J. Thus in a sense the elliptic interpretation is forced on one in anti-de Sitter spacetime. Since J reverses parity this has the effect of ruling out chiral fermions in anti-de Sitter space (c.f. [19, 20]). The differences between the two cases may be understood in a more group theoretic way. In both cases J lies in the centre of the isometry group which is 0(4,1) or O(3,2), respectively. Both groups have 4 connected components. In the case of 0(3, 2) J lies in the same component as time reversal. In the case of 0(4,1) it lies in the same component as parity.

6. Conclusion I hope I have convinced the reader that the elliptic interpretation is mathematically consistent. As far as black holes formed in gravitational collapse is concerned it seems to me to be irrelevant. It is conceivable that primordial black holes should be described in this way and it may be important at a deeper level. The general idea that the demanding symmetry under time reversal may require rethinking our ideas about quantum mechanics seems to be of some interest. This may be relevant to various speculations about the origin of time asymmetry in the universe [21,22]. Possibly the most interesting aspect of work is the possible relation between the global topology of spacetime and the structure of quantum mechanics. Topological effects concerned with spatial topology will never, it seems to me, lead to a substantial alteration of our ideas about quantum mechanics. If general relativity is to cause any changes in the overall structure of quantum theory it seems most likely

508

G. IV. Gibbons / Black holes

that they will come about because of the modifications of our understanding of the nature of time that Einstein's theory brings about. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [171 [18] [19] [20] [21] [22]

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