Field Theory As Free Fall. Blegdamsvej 17. Denmark. Abstract. It is shown that the classical eld equations pertaining to gravity coupled to

Transcription

1 NBI-HE-95-4 gr-qc/ August 995 Fiel Theory As Free Fall J. Greensite The Niels Bohr Institute Blegamsvej 7 DK-00 Copenhagen Denmark Abstract It is shown that the classical el equations pertaining to gravity couple to other bosonic els are equivalent to a single geoesic equation, escribing the free fall of a point particle in superspace. Some implications for quantum gravity are iscusse. Permanent aress: Physics an Astronomy Dept., San Francisco State University, San Francisco CA 943 USA greensit@stars.sfsu.eu

2 Introuction In the canonical formulation of general relativity, it is customary to view the timeevolution of the three metric g ij (x), as well as the evolution of any other nongravitational els A (x), as tracing out a trajectory in the "space of all els" known as superspace. In this article I will show that the classical equations of motion an constraints, which govern the ynamics of g ij an A, are equivalent to a geoesic equation in superspace, with a certain supermetric that will be specie. Fiel theory in general relativity may therefore be regare as escribing the free fall of a point particle in a (super) gravitational el. Connections to Jacobi's principle in mechanics, implications for quantization (in particular, the problem of time in quantum gravity), an possible generalizations, will also be iscusse below. Geoesics in Superspace Let fq a (x);p a (x);a=;; :::; n f g enote the canonical variables of a set of integerspin els incluing gravity, i.e. fq a (x)g = fg ij (x); A (x)g, with the non-gravitational els scale by an appropriate power of Newton's constant so as to be imensionless. 3 The rst-orer ADM action has the form S = 4 x [p t q a NH x N i H i x] H x = G ab p a p b + p gu(q) H i x = O ia x ]p a () leaing to the ynamical equations an constraints " t q a (x) = 3 x 0 N p a (x) H + N i p a (x) Hi " t p a (x) = 3 x 0 N q a (x) H + N i q a (x) Hi H = 0 H i = 0 () As pointe out by Moncrief an Teitelboim [], the supermomentum constraints H i = 0 nee not be impose inepenently. These constraints are implie by the requirement that the Hamiltonian constraints H = 0 are preserve by the time evolution, which emans the vanishing of the Poisson brackets fh(x); H(y)g. Since these Poisson brackets turn out to be linear in H i, the momentum constraint follows. 3 I will work here entirely in the metric (g ) formalism; spinor els will not be iscusse.

3 To help x the notation, we note that for pure gravity, where the conjugate variables are the three metric g ij an corresponing momenta p ij, the various expressions in the ADM action are as follows: fa = 6g $ f(i; j); ijg q a (x) $ g ij (x) ( p ij (x) (i = j) p a (x) $ p ij (x) (i <j) where G ijkl is the DeWitt superspace metric G ab (x) $ G ijnm (x) p gu = p g 3 R H i = p ik ;k (3) G ijkl = p g (g ikg jl + g il g jk g ij g kl ) (4) In this case, the Hamilton equations plus constraints () are equivalent to the vacuum Einstein equations. The lapse an shift functions N an N i foliate spacetime into space + time, an set the coorinates on each constant-time hypersurface. If one sets N i =0,it is still possible to choose arbitrary foliations using the lapse function, although the coorinates on each time-slice are then xe. It is also possible, without aecting the freeom to choose arbitrary foliations, to limit the lapse functions N to a subset N = ~ N satisfying 3 x ~ N p gu(q) = (5) where is an arbitrary parameter with imensions of mass. Equivalently, ~N = N R 3 xn p gu(q) where N is unconstraine. The global constraint (5), like the choice N i =0,oes not limit the choice of constant-time hypersurfaces; it only aects the value of the label t assigne to each hypersurface. Making the gauge choices (6) N i =0 an N = ~ N (7) the st-orer equations of motion an constraints t q a (x) = N ~ G ab p b t p a (x) = 3 x 0 N ~ q a (x) H H = G ab p a p b + p gu =0 (8) 3

4 The supermomentum constraints have been roppe, since they are implie by the other equations. We go to the n-orer form by solving the rst of these equations for p a p a = G N ~ t q b (9) an inserting into the secon two equations of (8), N ~ G t q b 4 ~ tq t q + 3 x 0 ~ N q a (x) (p gu) =0 (0) an G 4 N ~ t q t q b + p gu =0 () as the classical el equations with lapse/shift conitions (7). Let us now introuce some notation. Dene a mixe iscrete/continuous inex (; x) as a "coorinate inex" in superspace ( N(x) =0 q (x) q (x) = q a (x) =a6=0 () Apart from notation, we are enlarging the enition of superspace to inclue the el N (x), which appears in eq. (6). The summation convention for tensor inices is then X V ::(x)::w ::(x):: 3 xv ::(x)::w ::(x):: (3) We are now reay to state the main results: I. The equation of motion (0) an Hamiltonian constraint () are the equations of a geoesic in superspace, with the time label t an ane parameter in superspace, proportional to the proper-time along the geoesic; II. Both the equation of motion (0) an the constraint () are obtaine by extremizing the proper-time of the path in superspace, such that the ane parameter t, given by t = s G (x)(y) q (x) q (y) (4) is stationary with respect to variations of q (x) (), where the (egenerate) metric of superspace is proportional to G (x)(y) = ( hr 3 x 0 N p gui G 4N (x) ab (x) 3 (x y) = a; = b 0 =0 an/or =0 (5) 4

5 The equation of motion (0) is obtaine from the stationarity conition t q (ax) () =0 (6) with iscrete inex a 6= 0. Taking the inicate functional erivative of (4), we n 8 0 = >< hr 3x 00 N p = 9 gui q >= b G ab N + >: h R 3 x0 4N G q q b i = hr 3x 00 N p gui = h R 3 x0 4N G q q b i = h R 3x 0 4N G q q bi = hr 3 x00 N p gui c q c q a >; x 000 N q a(p gu) (7) Dening ~ N accoring to eq. (6), this becomes 0 = + 8 >< >: an then using we obtain hr 3 x00 N p i =h R gu 3 x0 4N G q q b i = hr 3 x00 N p i =h R i gu 3 x0 G 4N q q b = 3 x 00 N p = gu t = t = s G (x)(y) ( ) q b N ~ G ab t 4 ~ N 3 x 0 4N G q q b = q (x) q (y) 3 x 00 N p gu c q c q 4 N a t t + q b N ~ G a x 000 q c q 9 >= >; ~ N q a(p gu) (8) 3 x 0 4N G q q b = (9) x 0 ~ N q a(p gu) =0 (0) which is ientical to the classical equation of motion (0). The Hamiltonian constraint is obtaine from the remaining stationarity conition t q (0x) () = t =0 () N(x; ) 5

6 Again taking the inicate functional erivative gives hr 3x 00 N p = gui h R i 3 x0 G 4N q q b = h R 3x 0 + 4N G q q bi = hr 3 x00 N p gui = 4N G q q b p gu =0 () an using (6) an (9) we n q a q b G 4 N ~ ab t t + p gu =0 (3) which is simply the Hamiltonian constraint (). Consistency of the Hamiltonian constraint with the equations of motion (0) then implies the supermomentum constraints. In this way, the full set of classical el equations an constraints are obtaine from the requirement t q (x) () =0 (4) verifying the results (I) an (II) state above. Note that equations (0) an () are covariant with respect to a change in the mass parameter ; the increment of proper time on the 4-manifol, namely ~ Nt,is-inepenent. The constant is completely arbitrary, an has been introuce only to give the evolution parameter t the imensions of time. From the point of view of the ADM equations of motion, achange of is just a relabeling of the time variable t; from the point of view of the geoesic conition, it is simply a rescaling of the ane parameter. If, instea of using the explicit form (5) of the supermetric G (x)(y), the supermetric is left arbitrary, then variation of t by q (x) leas, by stanar manipulations, to the equation G (x)(y) q (y) t + G (x)(y) q (z) + G (x)(z) q (y) G (y)(z) q (x)! q (y) t q (z) t =0 (5) Inserting the supermetric (5), it is not icult to verify explicitly that the =0 component of this equation is the Hamiltonian constraint (), while the = a 6= 0 components are just the el equations (0). If the metric were invertible, one coul multiply this expression by the reciprocal metric, an obtain the usual form of the geoesic equation q (x) q (y) q (z) + (x) =0 (6) t (y)(z) t t 6

7 where (x) (y)(z) is the connection corresponing to the supermetric G (ax)(by). This supermetric is not invertible, however, an there is no term involving a secon erivative N =t. As a result, there is not one, but rather a continuously innite set of trajectories fq a (x; t); N (x; t)g which extremize the proper time in superspace, proportional to the ane parameter t,between given initial q (x) in an nal q (x) f con- gurations. This is as it shoul be. Each geoesic (a solution of the el equations), satisfying given initial an nal bounary conitions, represents a ierent foliation, corresponing to a particular choice of lapse, of a certain 4-manifol. The set of all such geoesics, corresponing to all possible foliations of the same 4-manifol, forms an equivalence class. 4 Thus the non-invertibility of the supermetric, which leas to an innite egeneracy in solutions for a geoesic, is just a consequence of the (orinary) time ieomorphism invariance in four-imensional space, which allows for an innite number of possible foliations. If a set of stationary paths between two points in superspace are just ierent representations of the same 4-manifol (+ non-gravitational els), then one woul expect that the "proper time" interval in superspace along each path woul be the same. In fact, the proper time in superspace, along a geoesic joining two congurations fqin a (x)g an fqa f (x)g is proportional to the ieomorphism invariant action p S = 4 x gr + Lnon grav (g ; A ) + bounary terms (7) evaluate on a solution of the equations of motion boune by the given initial an nal congurations. To erive the proportionality of tan S, begin with the enition of the ane parameter t in superspace = G (ax)(by) t = 3 x 4 N ~ q (ax) q (by) t q a G ab t Solving the Hamiltonian constraint () for ~ N gives ~N = vu u t 4 p gu G q a q b ab t t an substituting this expression into (8) we have = 3 x s q b t p q a q b gugab 4 t t (8) (9) (30) 4 There may be more than one equivalence class, since there many be more than one 4-manifol solving the equations of motion, boune by q in an q f. 7

8 or Then t = t = 3 x 4 x s s 4 p gugab q a q b (3) p q a q b gugab (3) 4 x 0 x 0 The integral in eq. (3) is the Baierlein-Sharp-Wheeler form of the action (7), in "shift gauge" N i = 0; it is obtaine from the ADM action by solving for the momenta an the lapse function in terms of the velocities []. The value of the BSW action, evaluate along a stationary path, is equal to the ieomorphisminvariant action (7) evaluate along the same path. Because of ieomorphism invariance, any geoesic in an equivalence class, subject to given initial an nal bounary conitions, will have the same action S. Therefore, since t = S (33) evaluate along the geoesic, all geoesics between given en-points in superspace, which ier (in 4-space) only by a foliation, have the same interval of proper time in superspace. 3 Jacobi's Principle The stationarity oft in (4) is closely relate to Jacobi's principle in classical non-relativistic mechanics. Consier a particle of energy E, whose motion in a D-imensional space is governe by the Hamiltonian H = m K ab (x)p a p b + V (x) (34) Accoring to Jacobi's principle, the path trace out in x-space by the particle trajectory x() extremizes the quantity s = s G ab x a x b (35) where G ab =(E V)mK ab (36) an where K ab is inverse to K ab. The similarity of these expressions to (4) an (5) is obvious. Despite these similarities, one woul not say that a non-relativistic particle moving in an arbitrary potential is in free fall. For one thing, the Euler- Lagrange equations for a non-relativistic particle involve the mass parameter m, 8

9 while the geoesic equation erive from Jacobi's principle oes not. Jacobi's principle only concerns the parametrize orbit x a (), while the Euler-Lagrange equations eal with the trajectory x a (T ) in terms of the Newtonian time T.To obtain the Euler-Lagrange equations, it is necessary to introuce an aitional conition mk x a x b ab T T + V = E (37) which enes the Newtonian time T. The mass parameter enters the equations of motion at this point. Since the geoesic equation by itself is not equivalent to the Euler-Lagrange equations, the motion of a non-relativistic particle in an arbitrary potential is not equivalent to free fall. On the other han, consier the relativistic action S = m s g (x) x In this case, the parametrize trajectory x () contains all there is to know about the particle's motion; the Euler-Lagrange equations an the geoesic equation are equivalent, an o not involve the particle mass m. The action (38) therefore escribes a particle in free fall. Free fall, for a relativistic particle, can be reformulate as the motion of a particle in a certain kin of potential. To see this, consier an arbitrary factoring of g into two parts Then the rst-orer form of the action (38) is given by S = x (38) g (x) =(x)g (x) (39) x NH] H = m G p p + m(x) (40) This is verie by rst solving for the momenta in terms of the velocities x = N@H=@p, then solving for the lapse function N from the Hamiltonian constraint H = 0, an nally substituting the results into the rst-orer action in (40). The square-root action (38) follows. Note, however, that the Hamiltonian constraint can be written as G p p + m (x) =0 (4) This equation can be interprete as referring to a particle moving q in a spacetime of backgroun metric G, with a "position-epenent mass" m (x). Therefore, the motion of a particle in a manifol of metric G, with a position-epenent 9

10 mass, is simply a ierent way of formulating the free fall of particle in a manifol of metric g, with a constant mass. Note again the similarity to the Jacobi form (35) an (36). In this case, G ab ;K ab ;mv(x) in (36) correspon to g ; G ;(x), respectively, with E = 0. Now in gravitation theory, the metric of superspace is usually taken to be the ultralocal DeWitt metric G (ax)(by) = G ab [g ij (x)] 3 (x y) (4) an the term p gu in the Hamiltonian is viewe as the potential. In close analogy to the position-epenent mass term of a relativistic particle, the potential term in the ADM Hamiltonian can be absorbe into a reenition of the metric (eq. (5)), an we arrive at the action principle (4). Taking G (x)(y), rather than G (ax)(by), as the supermetric, the ynamical el equations ecribe the free fall of a point particle through superspace. As in the case of the relativistic particle, the geoesic equation is equivalent to the classical equations of motion, an a given geoesic q (x; t) through superspace provies a complete escription of the ynamics. No aitional information, analogous to the enition of Newtonian time in (37), is require, an no constants (analogous to mass) not appearing in the geoesic equation are neee. It shoul be note that there has been some previous work on the topic of Jacobi's principle in general relativity, by Brown an York [3]. In their approach the cosmological constant is interprete as being analogous to an energy parameter. This entails a moication of classical relativity (the unimoular theory [4]) in which the cosmological constant is taken to be a ynamical egree of freeom, much like the energy E of a non-relativistic theory, which is conjugate to another variable interprete as a time-evolution parameter. The motivation for introucing such a parameter was to aress the problem of time in quantum gravity. As in the nonrelativistic case, the equations of motion in this particular time parameter are not geoesic equations. The reaer is referre to ref. [3] for further etails concerning this approach. 4 Quantization We next consier the rst-orer formulation of free fall in superspace; the object is to n the quantity whose Poisson brackets evolve the system in the ane parameter t. 0

11 In orinary 4-space, the geoesic equation is obtaine by variation of s = with respect to x, an then using q x (43) s = q x (44) to replace the arbitrary parameter with the proper-time s in the resulting equations of motion. As is well known [5], the geoesic equation is also obtaine by varying the action S = m 0 t g x x (45) t t with respect to x (t). This leas irectly to a geoesic equation x + x x =0 (46) t t t where the ane parameter t is proportional to the interval in proper time s. Going to a rst-orer formulation of the action (45) we see that the quantity S = t t x m 0 g p p (47) E p m 0 x x = g t t is a constant of motion, an therefore, using (44), (48) s = p Et (49) This is the relationship between the evolution parameter t, an the proper time s. Note that the action (45) has no constraint on the mass m = p = Em 0 of the particle, which is treate simply as a constant of motion. This mass, of course, cannot be etermine from the particle trajectory (a geoesic) in conguration space. In the same way, the equations of motion (0) an constraint () in superspace are obtaine by variation of the action S = q (x) q (y) t G (x)(y) 0 t t = q (ax) q (by) t G (ax)(by) 0 t t (50)

12 Going over to the rst-orer formulation in the usual way, we n S = t p (ax) q (ax) t! (5) where an G (ax)(by) = = 4 0G (ax)(by) p (ax)p (by) (5) R 3 x0 N p gu 4N G ab 3 (x y) (53) The quantity is a constant of motion, enote = ynamical equations E 0, of the corresponing q (ax) t = p (ax) ; p (ax) t = ; 0= q (ax) N(x) (54) It is straightforwar to verify that eq. (54) reprouces the equations of motion (0) an constraint (), upon solving for the momenta in terms of velocities, an ientifying = p E 0 in ~ N. In contrast to the usual Hamiltonian of general relativity, the quantity[q; p;n] is not constraine to be zero, although the constraint equation () is obtaine from variation with respect to N. The value = 0 E is a constant of motion, which, however, cannot be etermine from the trajectory followe in superspace. For = 0 E, the constraint =N = 0 is equivalent to H E x= p G ab p a p b + p E p gu =0 (55) E This looks like the usual Hamiltonian constraint, apart from the presence of a free parameter E. In fact, the parameter E is irrelevant to the conguration-space equations of motion. This is seen by starting from the Hamiltonian equations of motion base on H E x, q (ax) = 3 x 0 N(x 0 ) HE x 0 p (ax) p (ax) = 3 x 0 N(x 0 ) HE x 0 q (ax) H E x = 0 (56) an solving for the momenta in terms of velocities. One then ns that the parameter E rops out of the resulting secon-orer equations of motion an constraint.

13 This means that the constant E, like the mass of a particle in free fall, or the tension of the Nambu string, or the value of Newton's constant invacuum gravity, 5 oesn't appear in the Euler-Lagrange equations of the theory, an hence cannot be etermine from the classical trajectories in conguration space. In this sense, E is an unetermine constant. The existence an implications of such constants in Hamiltonian constraints has been iscusse in ref. [8]. The ynamical equations (54) can be rewritten in Poisson bracket form Q[q; p] =fq; g ; t =0 (57) N (x) Since is a constant of motion whose value at the classical level is unconstraine, an since the equations (54) are equivalent to the el equations of general relativity, 6 it seems reasonable to base the canonical quantization of gravity on the Schroinger equation corresponing to the Poisson bracket of eq. (57), i.e. ih t = (any N ) (58) thereby avoiing the notorious "problem of time" in quantum gravity [6]. In fact, this proposal has been mae an evelope by Carlini an the author in a series of papers [7]-[9], following a ierent line of reasoning. One of the avantages of this proposal, iscusse in more etail in the cite references, is that it allows for the existence of physical states which are eigenstates of non-stationary observables. Because a physical state can only epen on the conguration-space variables fq a (x)g, an not on N (x), such states must satisfy (58) for any choice of N. Expaning a time-epenent solution in terms of stationary states X [q; t]= c E; E; [q]e i 0Et=h E; (59) an inserting into (58), the conition of N -inepenence implies " # h E Gab q a q + p gu b E; [q] =0 (60) which is a Wheeler-DeWitt equation (with the usual orering ambiguity) parametrize by E. This is the operator version of the constraint (55) corresponing to =N = 0; the label istinguishes among a linearly inepenent set of solutions of this equation. The physical Hilbert space is thereby spanne by the solutions 5 In vacuum gravity, where p gu = p gr=, the unetermine constant E can be absorbe, by a scaling new = = p E,into an (unetermine) Newton's constant. 6 The el equations are in N i = 0 gauge, but with no restriction on foliation. 3

14 of a one-parameter (E) family of Wheeler-DeWitt equations. A solution of any given Wheeler-DeWitt equation, with xe E, is a stationary state; it cannot be an eigenstate of non-stationary observables, such as 3-geometry or extrinsic curvature. However, a superposition of states E;, with ierent E, is a non-stationary state. Such states can inee be eigenstates of non-stationary observables, an therefore have the possibility of, e.g., escribing the outcome of a measurement process. All of this has been iscusse in some etail in ref. [7]-[9]. But what we now see, from the work of the previous section, is that the time parameter for quantum gravity in eq. (58) is, at the classical level, proportional to the proper-time of the trajectory of the Universe in superspace. 5 Beyon Free Fall The geoesic equation of motion in general relativity is the statement that the nongravitational force on a particle is zero. The equivalent statement, in superspace, is that all ynamics is free fall; there no other forces in superspace that act on the Universe, viewe as a point particle. We have no motivation, from phenomenology, to go beyon this statement. Still, it is intriguing to consier what might be the form of the equations of motion if there woul be some non-(super)gravitational forces in superspace, presumably so weak as to have gone unetecte. In other wors, what is the analog of F = ma in superspace, an are there any consistency conitions that must be satise by such "superforces"? The motion of a point particle in orinary spacetime is governe by the equation g x = m F (6) The irect generalization to superspace is an equation of the form q (y) G (x)(y) + G (x)(y) + G! (x)(z) G (y)(z) q (y) t q (z) q (y) q (x) t q (z) = F(x) (6) whereis a constant, an F (x) is the "superforce." As before, the = 0 components of this equation are the equations of constraint, while the 6= 0 components are the equations of motion. The requirement that the equations of constraint are preserve by the motion leas to certain conitions on the form of the superforce. In orinary spacetime, the electromagnetic force acting on a charge particle preserves the mass-shell conition; i.e. the Hamiltonian constraint H = + m =0 (63) m p 4 t

15 is unchange. Let us assume that this is also true in superspace, which woul insure that the number of inepenent ynamical egrees of freeom is unaecte by the force term. This requires F (0x) = 0. Then eq. (6) for = a 6= 0 becomes " # q bc q b q c G t N ~ ab t 4 N a t t + 3 x 0 N ~ q a (x) (p gu)= F(ax)(64) while the constraint, obtaine from (6) with =0, is unchange. Again ene 4 ~ N G t q t q b + p gu =0 (65) H x = G ab p a p b + p gu (66) Then the equation of motions (64) an constraint (65) are equivalent to q (ax) t p (ax) = 3 x 0 ~ N(x0 ) H x 0 p ax = 3 x 0 N(x0 ~ ) H x 0 t q + ax F(ax) H x = 0 (67) Consistency of the Hamiltonian constraint H x motion in (67) then requires = 0 with the other equations of 0 = H x t = H x = q (ax0 ) 3 x 0 q (ax0 ) t + H x p (ax 0 ) p (ax 0 ) t ~ N(x0 )fh x ; H x 0g + 4 Gab p a (x)f b (x) (68) Since the Poisson Bracket fh x ; H x 0g is linear in the supermomentum ensity H i x, consistency is obtaine by imposing the usual supermomentum constraint H i x =0 (69) on the canonical momenta, as well as the \orthogonality conition" on the force p a (x)f a (x) G ab p a (x)f b (x) =0 (70) at each point x. But then, we also nee to ensure that the supermomentum constraint is maintaine by the equation of motion. This requires 0 = Hi x t = 3 x 0 N(x0 ~ )fh i x ; H x 0g + H i x F (ax p 0 ) (7) (ax 0 ) 5

16 The vanishing of the Poisson bracket fh i x ; H x0g is guarantee by the usual Dirac algebra, an H x = 0, so the above conition reuces to a supermomentum constraint on the force term H i x [p a (x)! F a (x)] = 0 (7) where the notation means that the momenta in the supermomentum constraint are replace by the corresponing components of the superforce. In short, a consistent force term in the superspace equations of motion which preserves the form of the constraints must: () be velocity-epenent, in the sense that the force is orthogonal to the canonical momenta at every point in 3-space; an () obey a supermomentum constraint. Altogether: p a (x)f a (x) = 0 H i x [F a (x)] = 0 (73) The rst of these conitions is reminiscent of the Lorentz force of electromagnetism, F = eg (@ A ) x s (74) which is, in fact, orthogonal to the particle 4-momentum, i.e. p F = 0, in orinary spacetime. This orthogonality is ue to the antisymmetry of the electromagnetic el tensor, an it guarantees that the electromagnetic force leaves the 4-momentum ofacharge particle on its mass shell: H =0 ) p +m =0. The secon conition, a super-momentum constraint on the force, has no obvious analog in particle ynamics. 6 Conclusions The fact that the Einstein (+ other integer-spin) el equations can be reformulate as a single geoesic equation may be of interest aesthetically. More importantly, the formalism also suggests a natural evolution operator (an evolution parameter) for the corresponing quantum theory, which has obvious application to the problem of time. It woul be interesting to see if this geoesic reformulation can also be extene to inclue spinor els. Beyon this, it is tempting to speculate that superspace shoul be regare, rather literally, as a true arena of ynamics. The Universe is a point particle in this space, an it moves, uner the inuence of a (super)gravitational el, along a geoesic. If one particle can fall freely in superspace, why not others? Interactions among such particles woul result in eviations from geoesic motion, as iscusse 6

17 in the last section. This suggests that the geoesic equation in superspace, an its possible extensions, might be a natural starting point for constructing classical theories of multi-universe ynamics. The many-universe concept is not new in quantum gravity; in particular, it has been argue that wormhole processes are best escribe in the framework of thir quantization [0]. A classical theory of multi-universe ynamics, if it coul be constructe consistently, might well be the "particle limit" (in the sense of ref. []) of a corresponing thir-quantize el theory, associate with topology-changing processes. Acknowlegements I am grateful for the hospitality of the theory group at the Lawrence Berkeley Laboratory, where much of this work was carrie out; I woul also like to thank Alberto Carlini for helpful iscussions. This work is supporte in part by the U.S. Department of Energy, uner Grant No. DE-FG03-9ER407; support has also been provie by the Danish Natural Science Research Council. 7

18 References [] V. Moncrief an C. Teitelboim, Phys. Rev. D6 (97) 966. [] R.F. Baierlein, D.H. Sharp, an J.A. Wheeler, Phys. Rev. 6 (96) 864; J. A. Wheeler, in Relativity, Groups, an Topology, e. C. DeWitt an B. DeWitt (Goron an Breach, New York, 964). [3] J. D. Brown an J. W. York, Phys. Rev. D40 (989) 33. [4] W. G. Unruh, Phys. Rev. D40 (989) 048; M. Henneaux an C. Teitelboim, Phys. Lett. B (989) 95. [5] C. Misner, K. Thorne an J. Wheeler, Gravitation, (Freeman, San Francisco, 973). [6] K. Kuchar, Time an interpretations of quantum gravity, Proceeings of the 4th Canaian Conference on General Relativity an Astrophysics, e. G. Kunstatter et. al. (Worl Scientic, Singapore, 99); C. J. Isham, Imperial College preprint IMPERIAL-TP-9-9-5, archive: grqc/900. [7] A. Carlini an J. Greensite, Square Root Actions, Metric Signature, an the Path-Integral of Quantum Gravity, Norita preprint NORDITA-94/7 P, archive: gr-qc/ [8] A. Carlini an J. Greensite, Phys. Rev. D5 (995) 936. [9] J. Greensite, Phys. Rev. D49 (994) 930. [0] S. Giings an A. Strominger, Nucl. Phys. B3 (989) 48; V. Rubakov, Phys. Lett. B4 (988) 503; T. Banks, Nucl. Phys. B3 (989) 48. [] M. B. Halpern an W. Siegel, Phys. Rev. D6 (977)