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Gravity and Signature Change View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server gr-qc/9610063 30 September 1996 GRAVITY AND SIGNATURE CHANGE 1 Tevian Dray Dept. of Physics and Mathematical Physics, University of Adelaide, Adelaide, SA 5005, AUSTRALIA School of Physics and Chemistry, Lancaster University, Lancaster LA1 4YB, UK Department of Mathematics, Oregon State University, Corval lis, OR 97331, USA [email protected] George Ellis Department of Applied Mathematics, University of Cape Town, Rondebosch 7700, SOUTH AFRICA [email protected] 2 Charles Hellaby School of Physics and Chemistry, Lancaster University, Lancaster LA1 4YB, UK Department of Applied Mathematics, University of Cape Town, Rondebosch 7700, SOUTH AFRICA [email protected] 1 Corinne A. Manogue Dept. of Physics and Mathematical Physics, University of Adelaide, Adelaide, SA 5005, AUSTRALIA School of Physics and Chemistry, Lancaster University, Lancaster LA1 4YB, UK Department of Physics, Oregon State University, Corval lis, OR 97331, USA [email protected] du ABSTRACT The use of prop er \time" to describ e classical \spacetimes" which con- tain b oth Euclidean and Lorentzian regions p ermits the intro duction of smo oth (generalized) orthonormal frames. This remarkable fact p ermits one to describ e b oth a variational treatment of Einstein's equations and distribution theory using straightforward generalizations of the standard treatments for constant signature. 1 Permanent address is Oregon State University. 2 Permanent address is University of Cap e Town. - 2 - 1. INTRODUCTION 3 A signature-changing spacetime is a manifold which contains b oth Euclidean and Lorentzian regions. Signature-changing metrics must be either degenerate (vanishing de- terminant) or discontinuous, but Einstein's equations implicitly assume that the metric is 4 nondegenerate and at least continuous. Thus, in the presence of signature change, it is not obvious what \the" eld equations should be. For discontinuous signature-changing metrics, one can derive such equations from a suitable variational principle [2]. This turns out to follow from the existence in this case of a natural generalization of the notion of orthonormal frame. The standard theory of tensor distributions, as well as the usual variation of the Einstein-Hilb ert action, can b oth be expressed in terms of orthonormal frames, and thus generalize in a straightforward manner to these mo dels. No such derivation is known for continuous signature-changing metrics. Our key p oint is that although signature change requires the metric to exhibit some sort of degeneracy, there is in the discontinuous case a more fundamental eld, namely the (generalized) orthonormal frame, which remains smo oth. We intro duce here two simple examples in order to establish our terminology. A typical continuous signature-changing metric is 2 2 2 2 ds = tdt + a(t) dx (1) whereas a typical discontinuous signature-changing metric is 2 2 2 2 ds = sgn( ) d + a( ) dx (2) Away from the surface of signature change at = ft = 0g = f = 0g, these metrics are related by a smo oth co ordinate transformation, with denoting prop er \time" away p from . However, since d = jtj dt, the notions of smo oth tensors asso ciated with these co ordinates are di erent at , corresp onding to di erent di erentiable structures. We argue here in favor of the discontinuous metric approach, b oth physically and mathematically. Physically, b ecause of the fundamental role played by prop er time. Math- ematically, b ecause of the geometric invariance of the unit normal to the surface of signa- ture change. The resulting (generalized) orthonormal frames provide a clear path leading to a straightforward generalization of b oth Einstein's equations and the theory of tensor distributions. 2. PHYSICS A standard to ol in the description of physical pro cesses is the intro duction of an orthonormal frame. Physical quantities can b e expressed in terms of tensor comp onents in 3 Due to the frequent misuse of the word Riemannian to describ e manifolds with metrics of any signature, we instead use Euclidean to describ e manifolds with a p ositive-de nite metric and Lorentzian for the usual signature of relativity. This is not meant to imply atness in the former case, nor curvature in the latter. 4 This can be weakened [1] to allow lo cally integrable metrics admitting a square- integrable weak derivative. Discontinuous metrics do not satisfy this condition. - 3 - an orthonormal frame, corresp onding to measurements using prop er distance and prop er time. For example, when studying a scalar eld on signature-changing backgrounds such as (1) or (2), it is imp ortant to know the value of the canonical momentum at the b oundary, which is essentially the derivative of the eld with resp ect to prop er time. Furthermore, the well-p osedness of the initial-value problem in the Lorentzian region tells us that the canonical momentum will b e well-b ehaved at if it is well-b ehaved at early times. Continuous signature-changing metrics necessarily havevanishing determinant at the surface of signature change, which prevents one from de ning an orthonormal frame there. The situation is di erent for signature-changing metrics such that prop er \time" is an ad- missible co ordinate. Although the metric is necessarily discontinuous, 1-sided orthonormal frames can be smo othly joined at . Remarkably, the resulting generalized orthonormal frame is smo oth, and is as orthonormal as p ossible. In fact, requiring not only that the 1-sided induced metrics on , but also the 1-sided orthonormal frames, should agree at implies that either the full metric is continuous (and nondegenerate) or that the signature changes. Such frames can b e used to derive Einstein's eld equations from the Einstein-Hilb ert action, obtained for constant signature by integrating the Lagrangian density a b c ^ e ^ e (3) L = g R b ac where c a a a (4) ^ ! + ! = d! R b c b b are the curvature 2-forms and denotes the Ho dge dual. Varying this action with resp ect to the metric-compatible connection ! leads to the further condition that ! b e torsion-free, while varying with resp ect to the (arbitrary) frame e leads to Einstein's equations. In the 5 presence of a b oundary, one obtains (in vacuum) the Darmois junction condition [3], namely that the extrinsic curvature of the b oundary must be the same as seen from each side. (In general, one obtains the usual Lanczos equation [4] relating the stress tensor of the b oundary to the discontinuity in the extrinsic curvature.) The ab ove derivation of Einstein's equations requires that the connection 1-forms admit (1-sided) limits to the b oundary. For continuous signature-changing metrics, the connection 1-forms typically blow up at the b oundary, but for discontinuous signature- 6 changing metrics in a (1-sided) orthonormal frame they don't. It thus seems reasonable to prop ose that \Einstein's equations" for signature-changing manifolds should b e obtained 7 with resp ect to the (generalized) byvarying the (piecewise extension of ) the ab ove action 5 A surface term (the trace of the extrinsic curvature) must be added to the Einstein- Hilb ert action in the presence of b oundaries; this has nothing to do with signature change. 6 This will be the case if each 1-sided manifold-with-b oundary has a well-de ned con- nection, as for instance when glueing manifolds together or, on the Lorentzian side, when starting from well-p osed initial data. 7 There are a numb er of relative sign ambiguities b etween regions of di erent signature, so that the relative sign in the action | and hence in the b oundary conditions | can be chosen arbitrarily. - 4 - orthonormal frame. As exp ected, one obtains Einstein's equations separately in the two 8 regions together with the Darmois junction conditions at the b oundary [2]. 3. MATHEMATICS Theories involving internal b oundaries are typically formulated using distribution the- ory. The standard theory of hyp ersurface distributions is based on a nondegenerate volume element, which is usually taken to be the metric volume element if available. It is a re- markable prop erty of signature-changing spacetimes for which is an admissible co ordinate that, even though the metric is discontinuous, the (continuous extension of ) the metric volume element is smo oth. This of course follows immediately from the smo othness of the generalized orthonormal frame, from which the volume element can b e constructed. Thus, standard distribution theory can be used with no further ado [6]. Smo oth signature-changing metrics, on the other hand, have metric volume elements which vanish at . In fact, the combined requirements that the metric volume element be used where p ossible and that smo oth tensors be distributions result in this case in a theory [6] in which the Dirac delta distribution is identically zero! To illustrate these results, consider the following informal example. Consider rst the discontinuous signature-changing metric (2) with metric volume element ! = d ^ dx (5) de ned initially away from = 0, then continuously extended. Let V = V @ b e a smo oth vector eld, and let = d = ( ) d (6) be the standard hyp ersurface distribution asso ciated with = 0, namely the derivative of the Heaviside distribution . Then Z Z h ;V i = V ()! = V dx (7) M t=0 Now rep eat the ab ove construction for the smo oth signature-changing metric (1) with metric volume element p !^ = jtj dt ^ dx (8) again de ned initially away from t = 0, then continuously extended.
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