Tevian Dray 1. Robin W. Tucker. ABSTRACT

Transcription

1 25 January 1995 gr-qc/ BOUNDARY CONDITIONS FOR THE SCALAR FIELD IN THE PRESENCE OF SIGNATURE CHANGE PostScrit rocessed by the SLAC/DESY Libraries on 26 Jan GR-QC Tevian Dray 1 School of Physics & Chemistry, Lancaster University, Lancaster LA1 4YB, UK Deartment of Mathematics, Oregon State University, Corvallis, OR 97331, USA tevian@math.orst.edu Corinne A. Manogue 1 School of Physics & Chemistry, Lancaster University, Lancaster LA1 4YB, UK Deartment of Physics, Oregon State University, Corvallis, OR 97331, USA corinne@hysics.orst.edu Robin W. Tucker School of Physics & Chemistry, Lancaster University, Lancaster LA1 4YB, UK R.W.Tucker@lancaster.ac.uk ABSTRACT We show that, contrary to recent criticism, our revious work yields a reasonable class of solutions for the massless scalar eld in the resence of signature change. 1. INTRODUCTION In a recent letter [1], Hayward urorts to show that our earlier work on signature change, esecially [2] but also [3], is \mathematically inconsistent", \entails a nonuniqueness which destroys redictability", and does not \make sense of the relevant eld equations". He has made similar criticisms elsewhere [4]. Contrary to Hayward's claims, our aroach is comletely consistent and makes sense of our eld equation. In many situations in hysics one deals with equations that admit singular solutions. Sometimes one can simly excise those domains of the manifold where the singularity occurs and relace the eect of the singularity by a suitable set of arameters. But sometimes the comonents of tensor equations themselves are singular. In this case a strategy must be adoted to formulate the roblem before attemting a solution. This 1 Permanent address is Oregon State University. 1

2 is recisely the situation under discussion here, where a scalar eld equation is sought on a manifold with a degenerate metric which changes signature at a hyersurface. Our formulation of the roblem is dierent from Hayward's, leading to dierent equations with dierent solution saces. In the absence of exerimental guidance, both aroaches are viable. Hayward's aroach [1,5] is to give a global denition of the scalar eld equation on such a manifold and to demand that, since the eld equation is second order, its solutions be globally C 2. He concludes that the eld momentum must vanish at the surface of signature change. Our aroach [3] is to demand instead that the scalar eld equation be dened in a iecewise fashion on the manifold and to admit iecewise smooth solutions. We then choose conditions that match the iecewise solutions across the degeneracy hyersurface. Our conditions are simly that the eld and the unit normal derivative of the eld (or equivalently the canonical momentum) exist and be continuous. These conditions can be derived by romoting the iecewise formulation of the eld equation into a single global distributional equation [3]. Hayward [1] dismisses our aroach claiming that it leads to non-unique solutions and hence \destroys redictability". This claim is false. We show below how solutions may be comletely determined u to normalization. Our rocedure does however select a class of solutions that are not necessarily globally C 2. We note with interest that recent work [6,7] in the context of Kleinian signature change also argues in favor of a continuity condition on the unit normal derivative of the eld, rather than requiring that it vanish. We begin this rebuttal by giving the simle examle, rst introduced in [2], which Hayward uses in his attemt [1] to show that our solutions are non-unique. We demonstrate that our aroach does indeed determine the arbitrary constants. In Section 3, we examine several dierent generalizations of the standard action for the massless scalar eld to a signature changing background. For each such generalization, avariational rincile requires that the aroriate canonical momentum be continuous (and in one case, zero) at the boundary. In this way, we see how both our boundary conditions and Hayward's can be derived in a arallel manner, but from dierent actions. From this oint of view, it is not surrising that the theories obtained from these actions have dierent saces of classical solutions. We also discuss the imlications of each of the resulting theories. Contrary to Hayward's claims, our aroach can be derived from an exlicit eld equation using standard techniques. We discuss our eld equation in Section 4. The fact that our distributional eld equation is not the one that Hayward uses seems to have been lost in his claims that we do not make sense of \the" eld equations. To discuss a scalar eld equation on a manifold with a singular metric, one must rst formulate the roblem in an unambiguous manner. It is simly incorrect to refer to \the" scalar eld equation on such a manifold. In the Aendix, we further show that our aroach yields the standard junction conditions for discontinuous Maxwell elds. An imortant asect of quantum cosmology concerns semiclassical aroximations to ath integrals, which exloit articular classical solutions hence the desire of some to look at \real-tunneling" solutions. Our theory here is urely classical, and we wish to consider all classical solutions. We discuss these diering motivations in Section 5. 2

3 2. EXAMPLE The dierences in the two aroaches can be best seen in terms of an examle, rst introduced in [2] and also discussed in [1]. Consider the singular dierential equation 2t = _ (1) This equation can be viewed as the massless scalar wave equation for the 1-dimensional signature-changing metric tdt 2. The general solution to (1) is = ( A( t) 3=2 + B (t <0) Ct 3=2 + D (t>0) (2) where A, B, C, D are constants. The requirement that be continuous at t = 0 xes D = B. Our aroach [3] is to demand in addition that the (unit) normal derivative of at t = 0, dened in terms of 1-sided limits, be continuous, namely _ lim t = lim t!0 + t!0 _ t (3) This xes C = A in (2), and the solution becomes 2 = ( A( t) 3=2 + B (t <0) At 3=2 + B (t >0) (4) The integration constant B can be determined, e.g. by the choice of (0), and A can be xed by aroriate normalization. If however, following Hayward [1], one instead demands that be globally C 2, then the only solution is = B. Each choice leads to its own articular class of solutions. Alternatively, in [1], Hayward examines the solutions (2) without imosing any further conditions such as (3). It is these solutions which Hayward claims exhibit non-uniqueness. We have never suggested using solutions of this form. The controversy, therefore, aarently comes down to whether or not it is reasonable to allow solutions to a second order dierential equation which are not globally C 2. This is commonly done in elementary hysics across regions which contain hysical discontinuities such as the electrostatic eld across a hollow charged conductor; see the Aendix. 2 As discussed in [3], an alternative boundary condition, corresonding to the freedom in choosing the relative orientation of the normal derivatives, is to insert a minus sign on one side of (3), which would lead to C = A in (2). Both of these (classes of) solutions were given in [2]. 3

4 3. VARIATIONAL APPROACH The standard action for a real massless scalar eld on an n-dimensional (non-degenerate) Lorentzian background is: S L = 1 2 g ; ; gd n 1 xdt (5) There are several ossible generalizations of this action which might be used to accommodate a background sacetime which changes signature from Lorentzian to Euclidean at a hyersurface = ft = constantg. It is illuminating to examine the consequences of varying these actions. One ossibility is to extend (5) to the Euclidean region by utting an absolute value in the square root. Such an action has the advantage that it naturally remains real even in the Euclidean region: S 1 = 1 2 g ; ; jgj d n 1 xdt g ; ; jgj d n 1 xdt (6) Lorentzian Euclidean We vary S 1 and demand that it be stationary for variations of that do not necessarily vanish on. The variation is S 1 = Lorentzian g ; jgj ; d n 1 xdt + g ; jgj ; d n 1 xdt g 0 ; jgj g 0 ; jgj d n d n 1 x 1 x (7) Euclidean and we immediately obtain the scalar eld equations in the Lorentzian and Euclidean regions searately from variations whose suort does not intersect. If we now seek solutions such that is continuous, it is natural to assume that is continuous at, and the remaining variations yield a boundary condition on the canonical momentum 3 1 := S 1 ; 0 = jgjg 0 ; (8) namely that 1 be continuous at. This is recisely the boundary condition whichwehave roosed [2,3]; it is equivalent to continuity of the normal derivative, as in (3). Note that this boundary condition is required by a variational rincile. Solutions of the equations of motion for which and 1 are continuous do not suer from any \non-uniqueness" which 3 For all of the actions in this section, we consider only elds for which the canonical momenta have bounded limits to ; this imlies that the integrands of the various actions are well-behaved near. The variation of each action then yields the usual scalar eld equations in the Lorentzian and Euclidean regions searately; it is the boundary conditions which dier in each case. 4

5 \destroys redictability". Quite the contrary, any real Euclidean solution is matched to recisely one Lorentzian solution by this rescrition. We showed in [3] that the Klein-Gordon roduct of such solutions is conserved even across the hyersurface of signature change. This is a necessary condition for a corresonding quantum theory to be unitary and is thus an attractive feature of this aroach. The action S 1 can be extended to the comlex scalar eld in the usual way, yielding the action S 2 = g ; ; jgj d n 1 xdt + g ; ; jgj d n 1 xdt (9) Lorentzian Euclidean Because S 2 is real, it is essentially two coies of the action for the real scalar eld, as usual. The boundary condition obtained from varying S 2 is just the continuity of the (now comlex) canonical momentum (8). Such an action is a natural candidate for extension, via minimal couling to an electromagnetic eld in a U(1) invariant way. A third ossibility involves extending (5) to the Euclidean region via analytic continuation of g in the variable t, yielding S 3 = g ; ; gd n 1 xdt + g ; ; gd n 1 xdt (10) Lorentzian Euclidean where we have imlicitly chosen a branch. We allow to take comlex values, even in the Lorentzian region, and discuss later the consequences of restricting to real. Varying the real art of S 3 yields the boundary condition that the aroriate canonical momentum, given by 3 := S 3 ; 0 = gg 0 ; (11) should be continuous at. Notice that this canonical momentum diers from the revious one (8) by a factor of i in the Euclidean region. The imaginary art of S 3 is redundant, yielding the same eld equations and the same boundary condition, (11), so that the real art of S 3 can be used alone as a real action for this theory. Unlike S 1 and S 2, the Klein-Gordon roducts of some solutions of the equations of motion matched by (11) are not conserved across the hyersurface of signature change, so that it would be dicult to build a unitary quantum theory from such an action. Also, it is not obvious how to coule the scalar eld in (10) to an electromagnetic eld in a U(1) invariant way,even in the Lorentzian region. If one makes the added restriction that solutions must be real in both the Lorentzian and Euclidean regions, it is imossible to satisfy continuity of 3 unless the canonical momentum is zero at the boundary. This is the junction condition which Hayward refers. Solutions of this tye do have conserved Klein-Gordon roduct, but the class of allowable solutions is restricted by this extra reality condition. We discuss in Section 5 why such articular solutions are of relevance in the context of a Euclidean aroach to the quantum theory. 5

6 A fourth ossibility involves starting with the conventional action for the comlex massless scalar eld in the Lorentzian region and using the analytic extension of g into the Euclidean region. Then the action becomes S 4 = g ; ; gd n 1 xdt + g ; ; gd n 1 xdt (12) Lorentzian Euclidean This results in a non-analytic integrand and a comlex action. By varying S 4 with resect to the real and imaginary arts of we obtain the equations of motion together with the following boundary conditions: lim t! lim t! gg 0 ; = lim t! + gg 0 ; = lim t! + gg 0 ; gg 0 ; (13) This action is comlex, so that the boundary condition on is not the conjugate of the condition on, yielding an additional constraint on the allowed solutions. We nd this roerty of comlex actions like S 4 rather disturbing, as it restricts the allowed solutions away from the boundary in a way which the other actions considered here do not. Solving both equations in (13) simultaneously requires both sides of each equation to be zero; this is again the junction condition which Hayward refers. However, as the resulting solution sace turns out to be a roer subset of that determined by S 1, it is clear that Klein- Gordon roducts are reserved. Furthermore, and desite the action being comlex, this theory is a candidate for extension, via minimal couling to an electromagnetic eld in a U(1) invariant way. Of the two real actions considered here, S 1 and the real art of S 3, only the boundary condition obtained from a variational rincile for S 1 matches real solutions to real solutions. Furthermore, only S 1 conserves Klein-Gordon roducts of solutions across the signature change unless the solution sace for S 3 is additionally restricted by a reality condition. Of the actions for the comlex eld considered here, both S 2 and S 4 conserve Klein-Gordon roducts of solutions and both aear to rovide candidates for minimal couling to an electromagnetic eld. However, in the case of S 4 our rocedure yields two boundary conditions, instead of the exected one, since the boundary condition on is not the comlex conjugate of the boundary condition on. The set of actions given here is by no means exhaustive. There are clearly many other ossibilities, such as other relative factors between the Lorentzian and Euclidean actions, or the addition of surface terms on, leading to discontinuous canonical momenta. Nevertheless, we feel that this language is a good one for illustrating how avariety of boundary conditions can be obtained. It also rovides a framework for discussing some of the considerations which might arise when one attemts to nd a hysical interretation of the extension of the theory of the massless scalar eld to a signature changing background. A recent aer by Embacher [8] considers similar generalizations of the Einstein-Hilbert action to accommodate signature change. 6

7 4. DISTRIBUTIONAL WAVE EQUATION An alternate aroach is to use the language of tensor distributions. We summarize here our aroach to the massless scalar eld in this language, as resented in detail in [3], and give a 1-dimensional examle. Suose one is given two regions U of a manifold M sharing a common boundary given by f =0g. Suose further that the eld equations df = 0 hold searately on the 2 regions, where F are dierential forms on U.Introduce the distributional eld F = + F + + F (14) in terms of the Heaviside distributions with suort in U and such that d = (15) where = () d in terms of the Dirac delta \function" (). We shall call a dierential form F on U + [ U regularly discontinuous if the restrictions F = F j U are smooth and the (1-sided) limits F j = lim t! F exist. It follows that df = + df + + df + ^ [F ] (16) where [F ]:=F + j F j is the discontinuityinf.ifwenowostulate the distributional eld equation df =0 (17) then we obtain both the original eld equations df =0 (18) and the boundary condition ^ [F ]=0 (19) This formalism is valid irresective of whether the metric signature changes at. An examle with constant signature is given in the Aendix. Consider now the manifold M = R with signature-changing metric ds 2 = tdt 2 (20) Away from ft =0g, the Hodge dual oerator associated with this metric is given by 1 = jtjdt jtj dt = sgn(t) (21) The massless wave equation for a 0-form on a region of M where the metric is nondegenerate may be written df =0 (22) 7

8 where F = d in terms of the Hodge ma dened by the metric. Setting F = d on U := ft >0g, where = j U,wethus require that (22) be satised on U, i.e. that df =0. Wenow seek distributional solutions to (17) where F is dened as above. In order for this to make sense, we admit only solutions such that the tensor F, dened for t 6= 0byFj U =F =d, is regularly discontinuous at ft =0gso that [F ]=[d] is well-dened. 4 Using (21) we see that d = t (23) jtj t = jtj is bounded as t!0 then d is regularly discontinuous at ft t [d] = lim + + lim t!0 + jtj t (24) jtj Then (19) imlies dt ^ [d] =0 (25) which for this simle examle is equivalent to [d] =0 (26) since M is 1-dimensional. In summary, the distributional eld equation (17) requires satisfaction not only of the wave equations on each side, namely ddj U =0 (27) 4 In [4], Hayward argues that, in the resence of signature change, must be interreted as a distribution. He further argues that d must be C 1 so that can act on it. There is no need for such a requirement; it is only necessary that [d] be welldened. Furthermore, if T are tensors on U such that the tensor W, dened by W j U = T, is regularly discontinuous at, then a natural Hodge dual oerator ^ can be dened via ^ + T + + T = + T + + T where refers to the Hodge dual oerators on U as aroriate. If we now imose the natural condition that be continuous, then F = + d + + d ^d where = equation., so that in this case (17) takes the usual form of the wave 8

9 but also of the boundary conditions (26). Introducing new \time" arameters for t < 0 and for t>0by t = tdt 0 t (28) = tdt allows us to rewrite (26) as 0 ; j = ; j (29) (Changing the relative orientations of U amounts to inserting a factor of sgn(t) into both of equations (21), resulting in (29) without the minus sign; this gives (3) for the 1-dimensional examle discussed reviously.) 5. GENESIS OF EUCLIDEAN METHODS IN QUANTUM THEORY It is of interest to recall the genesis of the use of Euclidean methods in quantum theory. A convenient way to determine the sectrum fe n g of the quantum Hamiltonian H and its associated energy eigenfunctions jni in the osition reresentation jxi is to use the generating function X hx f je HT jx i i = e EnT hx f jnihnj x i i (30) n where T is a real arameter. The leading term in this exression for large T gives the energy and lowest energy wavefunction. The left hand side of this equation may be reresented as a ath integral over all trajectories x() satisfying x(0) = x i and x(t )=x f hx f je HT jx i i = A 0 e S T [dx] (31) where in terms of the classical otential V S T = T 0 (dx=d) 2 =2+V(x()) d (32) This reresentation has the advantage that it can be aroximated in the semi-classical limit where the integral is dominated by the stationary aths x of S T. For otentials with a double-well shae the stationary aths x of S T are the instanton solutions of the classical Euclidean equations of motion. By considering the uctuations about the stationary oints one can derive the standard transmission amlitude for a article to tunnel through a otential barrier. In this case the stationary ath x is known as the \bounce". For such a solution the modulus of the transmission amlitude through a otential barrier of width x 2 x 1 is given aroximately by ex S 1 (x) ex x2 2V (x) dx in full accord with the WKB aroximation. It should be noted that the bounce is the zero \energy" Euclidean solution corresonding to vanishing \kinetic energy" at T = 1 9 x 1 (33)

10 and the width of the Euclidean domain is determined by the otential. One might roceed in the semi-classical mode of thought and retend that the classical article materialises after barrier enetration from a Euclidean domain and evolves according to Newton's laws of motion starting with zero momentum. However no-one would demand that quantum mechanics requires that all classical solutions have such a restrictive initial condition. The above methodology exloits the analyticity of the standard Lagrangian for Newtonian mechanics in the evolution arameter to Wick rotate to imaginary time and dominate the continued action with articular real solutions to the continued equations of motion. With obvious caveats this hilosohy has been generalised to eld systems that ossess the required analyticity. When one deals with curved sace-time metrics the above Wick rotation is not available and the lack of analyticity of the Riemannian volume element requires somewhat ad hoc continuation methods. For roblems in quantum cosmology where the trajectories are 3-geometries, the insistence on real tunneling solutions across degenerate geometries that dominate the action gives rise to a behaviour analogous to that of the zero energy bounce solutions in article mechanics. The imortant oint to stress is that in all these cases it is quantum amlitudes that are being dened by nite action contributions to the ath integral. The roerties of articular Euclidean congurations that enter into such a descrition are not shared by the general classical congurations that can be atched across degeneracy hyersurfaces to Lorentzian signature domains. It is our urose to consider the so called theory of \classical signature change", i.e. to treat eld theories on a manifold with a degenerate metric tensor eld in a coherent manner without recourse to any ath integral technology. The quantisation of such congurations is a searate issue and deserves indeendent attention. 6. CONCLUSION We have shown that Hayward's criticisms of our work are untenable. Our eld equations make sense and our solutions do not suer from a lack of \redictability". The question of what conditions are the most aroriate to describe the hysics arising from the roagation of classical and quantum elds in the resence of a background which exhibits signature change can ultimately only be resolved by observation. Until then, the imlications of all reasonable, internally consistent aroaches should be investigated. APPENDIX: Discontinuities in electromagnetic elds The rst two of Maxwell's equations for a stationary electromagnetic eld are de =0=dB where is now the Hodge dual oerator on Euclidean R 3.At a boundary f =0gin R 3 we write each of the elds F = B; E as in (14). Postulating the distributional versions (17) of Maxwell's equations leads to Maxwell's equations on each side of the boundary together with the boundary conditions d ^ [F ]=0 10

11 This gives the standard boundary conditions: d ^ [E] =0 d ^ [B] =0 The remaining two equations are dd = dh = J where is the electric charge density and J the electric current. A similar argument including sources shows that if J and contain distributional comonents J and, resectively, then they must be related to the discontinuities of H and D by the standard boundary conditions: d ^ [H] =J ^d d ^ [D] = ^d in terms of surface distributions of charge and current. ACKNOWLEDGMENTS We gratefully acknowledge useful discussions with David Hartley and Phili Tuckey. TD and CAM would like to thank the School of Physics & Chemistry at Lancaster University for kind hositality during their sabbatical visit. This work was artially suorted by NSF Grant PHY (CAM & TD), a Fulbright Grant (CAM), and the Human Caital and Mobility Programme of the Euroean Union (RWT). REFERENCES 1. Sean A. Hayward, Weak Solutions Across a Change of Signature, Class. Quantum Grav. 11, L87 (1994). 2. Tevian Dray, Corinne A. Manogue, and Robin W. Tucker, Particle Production from Signature Change, Gen. Rel. Grav. 23, 967 (1991). 3. Tevian Dray, Corinne A. Manogue, and Robin W. Tucker, The Scalar Field Equation in the Presence of Signature Change, Phys. Rev. D48, 2587 (1993). 4. Sean A. Hayward, Junction conditions for signature change, gr-qc/ Sean A. Hayward, Class. Quantum Grav. 9, 1851 (1992); erratum: Class. Quantum Grav. 9, 2543 (1992). 6. L. J. Alty, Kleinian Signature Change, Class. Quantum Grav. 11, 2523 (1994). 7. L. J. Alty and C. J. Fewster, Initial Value Problems and Signature Change, DAMTP rerint R95/1, gr-qc/ (1995). 8. Franz Embacher, Actions for signature change, University of Vienna rerint UWThPh , gr-qc/ (1995). 11