Bi-metric pseudo-Finslerian spacetimes
aa r X i v : . [ g r- q c ] A ug Bi-metric pseudo–Finslerian spacetimes
Jozef Skakala and Matt Visser
School of Mathematics, Statistics, and Operations Research,Victoria University of Wellington, PO Box 600, Wellington, New ZealandE-mail: [email protected], [email protected]
Abstract.
Finsler spacetimes have become increasingly popular within the theoretical physicscommunity over the last two decades. Because physicists need to use pseudo –Finslerstructures to describe propagation of signals, there will be nonzero null vectors in boththe tangent and cotangent spaces — this causes significant problems in that many ofthe mathematical results normally obtained for “usual” (Euclidean signature) Finslerstructures either do not apply, or require significant modifications to their formulationand/or proof. We shall first provide a few basic definitions, explicitly demonstratingthe interpretation of bi-metric theories in terms of pseudo–Finsler norms . We shall thendiscuss the tricky issues that arise when trying to construct an appropriate pseudo-Finsler metric appropriate to bi-metric spacetimes. Whereas in Euclidian signaturethe construction of the Finsler metric typically fails at the zero vector, in Lorentziansignature the Finsler metric is typically ill-defined on the entire null cone.Keywords: Finsler norm, Finsler metric, bimetric theories.4 August 2010; L A TEX-ed 1 November 2018
Contents i-metric pseudo–Finslerian spacetimes
1. Introduction
Over the last two decades, Finsler norms and Finsler metrics have become increasinglyutilized in various extensions of general relativity, and sometimes in reinterpretations ofmore standard situations. However the fact that physicists need to work in Lorentziansignature ( − + ++) instead of the Euclidean signature (+ + ++) more typically used bythe mathematicians leads to many technical subtleties (and can sometimes completelyinvalidate naive conclusions). After very briefly presenting the basic definitions, weshall interpret bi-metric theories in a Finslerian manner, this being one of the simplestnontrivial Finsler structures one could consider. While there is a very natural way ofmerging the two signal cones into a “combined” pseudo-Finsler norm , we shall see thatthe situation with regard to Finsler metrics is considerably more complicated. To setthe stage, we point out that in Bernhard Riemann’s 1854 inaugural lecture [1], he madesome brief speculations about possible extensions of what is now known as Riemanniangeometry:The next case in simplicity includes those manifolds in which the line-elementmay be expressed as the fourth root of a quartic differential expression.The investigation of this more general kind would require no really differentprinciples, but would take considerable time and throw little new light on thetheory of space, especially as the results cannot be geometrically expressed. . .. . . A method entirely similar may for this purpose be applied also to themanifolds in which the line-element has a less simple expression, e.g. , thefourth root of a quartic differential. In this case the line-element, generallyspeaking, is no longer reducible to the form of the square root of a sum ofsquares, and therefore the deviation from flatness in the squared line-elementis an infinitesimal of the second order, while in those manifolds it was of thefourth order.In more modern language, Riemann was speculating about distances being defined byexpressions of the formd s = g abcd d x a d x b d x c d x d . (1)That is d s = p g abcd d x a d x b d x c d x d . (2)Such manifolds, and their generalizations, have now come to be called Finslergeometries [2]. (More specifically, this particular case corresponds to a so-called 4th-root Finsler geometry.) Finsler geometries are extremely well-known in the mathematicscommunity, with key references being [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], but areconsiderably less common within the physics community [14, 15, 16, 17, 18, 19, 20, 21,22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Perhaps the most extensive use of pseudo-Finslergeometries has been within the “analogue spacetime” community [32] where Finsler-like structures have arisen in the context of normal mode analyses [33, 34, 35], and inmulti-component BEC acoustics [36, 37, 38, 39]. i-metric pseudo–Finslerian spacetimes
2. Finsler basics
Mathematically, a Finsler function (Finsler norm, Finsler distance function) is definedas a function F ( x, v ) on the tangent bundle to a manifold such that F ( x, κ v ) = κ F ( x, v ) . (3)This then allows one to define a notion of distance on the manifold, in the sense that S ( x ( t i ) , x ( t f )) = Z t f t i F (cid:18) x ( t ) , d x ( t )d t (cid:19) d t (4)is now guaranteed to be independent of the specific parameterization t . In the particularcase of a (pseudo–)Riemannian manifold with metric g ab ( x ) one would take F ( x, v ) = p g ab ( x ) v a v b , (5)but a general (pseudo–)Finslerian manifold the function F ( x, v ) is completely arbitraryexcept for the linearity constraint in v . In Euclidean signature, the function F ( x, v ) istaken to be smooth except at v = 0. (This is most typically phrased in terms of thefunction F ( x, v ) being smooth on the “slit tangent bundle”; the tangent bundle with thezero vector deleted.) In Lorentzian signature however, we shall soon see that F ( x, v ) istypically non-smooth for all null vectors — so that non-smoothness issues have typicallygrown to affect the entire null cone. Sometimes a suitable power , F n , of the Finsler normis smooth. It is standard to define the (pseudo–)Finsler metric as g ab ( x, v ) = 12 ∂ [ F ( x, v )] ∂v a ∂v b (6)which then satisfies the constraint g ab ( x, κ v ) = g ab ( x, v ) . (7)This can be viewed as a “direction dependent metric”, and is clearly a generalization ofthe usual (pseudo–)Riemannian case. Almost all of the relevant mathematical literaturehas been developed for the Euclidean signature case (where g ab ( x, v ) is taken to be apositive definite matrix). Herein we wish to raise some cautionary flags with regard tothe Lorentzian signature pseudo-Finsler case.
3. Bimetric theories
Bi-metric theories contain two distinct metrics g ± ab , so we can define two distinct“elementary” Finsler norms F ± ( x, v ) = p g ± ab v a v b . Suppose one now wants a combinedFinsler norm that simultaneously encodes both signal cones — then the natural thingto do is to implement Bernhard Riemann’s original suggestion and take F ( x, v ) = p F + ( v ) F − ( v ); g abcd = g +( ab g − cd ) . (8)This construction for F ( x, v ) is automatically linear in v , and the vanishing of F ( x, v )correctly encodes the two signal cones. So this definition of F ( x, v ) provides a perfectlygood Finsler norm . The Finsler metric is however quite ill behaved, and the technical i-metric pseudo–Finslerian spacetimes F ( x, v ) has the interesting “feature” that it picks up non-trivial complex phases: Since F ± ( x, v ) is always real, (positive inside the propagationcone, negative outside), F ± ( x, v ) is either pure real or pure imaginary. But then, thanksto the additional square root in defining F ( x, v ), one has: • F ( x, v ) is pure real inside both propagation cones. • F ( x, v ) is proportional to √ i = i √ between the two propagation cones. • F ( x, v ) is pure imaginary outside both propagation cones.Thus in a bi-metric Lorentzian signature situation the particular “natural” pseudo-Finsler norm considered above cannot be smooth as one crosses the propagation cones— what was in Euclidean signature a complicating technical feature that only arose atthe zero vector of each tangent space has in Lorentzian signature grown to affect all nullvectors. This pseudo-Finsler norm is at best “smooth on the tangent bundle excludingthe null cones”. (Note that individually the F ± ( x, v ) are smooth across the propagationcones, but in the bi-metric case one has to go to F ( x, v ) to get a smooth function.)By extension, it is clear that similar phenomena will occur whenever one encountersmulti-sheeted signal cones (bi-refringence, multi-metric spacetimes, multi-refringence).Second, when attempting to bootstrap this Finsler norm to a pseudo-Finsler metric one encounters additional and more significant complications. The Finsler metric willhave (at least some) infinite components — g ab ( x, v ) has infinities on both signal cones.To see this consider g ab ( x, v ) = 12 ∂ a ∂ b q F F − (9)= 14 ∂ a (cid:20) ∂ b [ F ] F − F + + ∂ b [ F − ] F + F − (cid:21) (10)= 14 (cid:20) ∂ a ∂ b [ F ] F − F + + ∂ b ∂ b [ F − ] F + F − (cid:21) + 12 (cid:20) ∂ a F + ∂ b F − + ∂ a F − ∂ b F + − ∂ a F + ∂ b F + F − F + − ∂ a F − ∂ b F − F + F − (cid:21) . (11)That is, tidying up: g ab ( x, v ) = 12 (cid:20) g + ab F F + g − ab F F (cid:21) + 12 (cid:20) ∂ a F + ∂ b F − + ∂ a F − ∂ b F + − ∂ a F + ∂ b F + F − F + − ∂ a F − ∂ b F − F + F − (cid:21) . (12)The problem is that this “unified” and “natural” Finsler metric g ab ( x, v ) has singularitieson both of the signal cones.The good news is that the quantity g ab ( x, v ) v a v b = F ( x, v ), and so since on eitherpropagation cone F ( x, v ) →
0, we see that F ( x, v ) itself has a well defined limit. Butnow let v a be the vector the Finsler metric depends on, and let w a be some otheri-metric pseudo–Finslerian spacetimes g ab ( x, v ) v a w b = 12 v a w b ∂ a ∂ b [ F ] = 12 w b ∂ b [ F ] = 12 w b ∂ b p F + F − = 14 w b (cid:20) ∂ b [ F ] F − F + + ∂ b [ F − ] F + F − (cid:21) = 12 (cid:26) ( g + ab v a w b ) F − F + + ( g − ab v a w b ) F + F − (cid:27) . (13)The problem now is this: g + ab and g − ab are both by hypothesis individually well definedand finite. But now as we go to propagation cone “+” we have g ab ( x, v ) v a w b →
12 ( g + ab v a w b ) F − ∞ , (14)and as we go to the other propagation cone “ − ” we have g ab ( x, v ) v a w b →
12 ( g − ab v a w b ) F + ∞ . (15)So at least some components of this “unified” Finsler metric g ab ( x, v ) are unavoidablysingular on the propagation cones. Related singular phenomena have been encounteredin multi-component BECs, where multiple phonon modes can interact to produceFinslerian propagation cones [36]. Things are just as bad if we pick u and w to be two vectors distinct from v . Then g ab ( x, v ) u a w b = 12 (cid:20) g + ( u, w ) F − F + + g − ( u, w ) F + F − + g + ( u, v ) g − ( w, v ) + g + ( w, v ) g − ( u, v ) F + F − − g + ( u, v ) g + ( w, v ) F − F − g − ( u, v ) g − ( w, v ) F + F − (cid:21) . (16)Again, despite the fact that g + and g − are by hypothesis regular on the signal cones. the“unified” Finsler metric g ab ( x, v ) is unavoidably singular there — unless, that is, you only choose to look in the vv direction. By extension, it is clear that similar phenomena willoccur whenever one encounters multi-sheeted signal cones (bi-refringence, multi-metricspacetimes, multi-refringence).
4. Discussion and Conclusions
On the one hand we have seen how bi-metric theories are good exemplars for providinga clean physical implementation of the mathematical notion of a pseudo-Finsler norm,and how they naturally lead to a prescription for building a pseudo-Finsler metric.On the other hand we have also seen how this rather straightforward physical modelnevertheless leads to significant technical mathematical difficulties.It is when one tries to “unify” the two metrics into a single structure that the mostsignificant problems arise — the pseudo-Finsler norm is certainly well defined (and isextremely close to Riemann’s original conception of what a 4th-order geometry should i-metric pseudo–Finslerian spacetimes metric is singular on the entire signal cone. Thisproblematic feature is intimately related to the fact that we are dealing with Lorentziansignature pseudo -Finsler geometries — it is a “divide by zero” problem, associatedwith non-zero null vectors on the signal cone, that leads to singular values for metriccomponents. This appears to us to be an intrinsic and unavoidable feature of bi-metric pseudo -Finsler spacetimes.In earlier work [40, 41] we had investigated this point within the technically morecomplicated context of biaxial birefringent crystal optics. The many extra technicaldetails required to deal with biaxial crystals somewhat obscured the generality of thepoint we wish to make. The considerably simpler framework of bi-metric theories issufficient to make our point with clarity: pseudo-Finsler metrics are typically not smoothover the entire null cone.In closing we wish to emphasize that this does not mean that all attempts atconstructing pseudo-Finsler metrics are intrinsically ill conceived. The situation is moresubtle. While the situation with multiple signal cones is clearly diseased, at least ifone wishes to encode all signal cones in one Finsler structure in any straightforwardmanner, and while one cannot blindly carry Euclidean signature Finsler results overto Lorentzian signature, the case of a single (geometrically distorted) signal cone maystill be of interest — one will just have to check explicitly that all Euclidean signatureconstructions can be generalized to Lorentzian signature. While this step is relativelystraightforward for the Riemannian → pseudo-Riemannian transition, it is much moresubtle for the Finsler → pseudo-Finsler transition. Acknowledgments
This research was supported by the Marsden Fund administered by the Royal Society ofNew Zealand. JS was also supported by a Victoria University of Wellington postgraduatescholarship.
References [1] Bernhard Riemann, “On the hypotheses which underlie the foundations of geometry”, (“ ¨Uber dieHypothesen, welche der Geometrie zu Grunde liegen”), Nature, Vol. VIII. Nos. 183, 184, pp.14–17, 36, 37. (Translation by William Kingdon Clifford, transcription by D. R. Wilkins.)[2] P. Finsler, “ ¨Uber Kurven und Flachen in allgemeinen Raumen”, Dissertation, Gottingen, 1918,published by Verlag Birkhauser Basel, 1951.[3] H. Buseman, “The geometry of Finsler space”, Bull. Amer. Math. Soc. , Number 1, Part 1(1950), 5-16.[4] H. Rund, The differential geometry of Finsler spaces , (Springer, Berlin, 1959).[5] D. Bao, S.S. Chern, and Z. Shen,
An introduction to Riemann–Finsler geometry , (Springer, NewYork, 2000).[6] D. Bao, S.S. Chern, and Z. Shen (editors),
Finsler Geometry , Proceedings of the JointSummer Research Conference on Finsler Geometry, July 1995, Seattle, Washington, (AmericanMathematical Society, Providence, Rhode Island, 1996). i-metric pseudo–Finslerian spacetimes [7] S.S. Chern, “Finsler geometry is just Riemannian geometry without the quadratic restriction”,Not. Amer. Math. Soc. (1996) 959–963.[8] S.S. Chern, and Z. Shen, Riemann–Finsler Geometry , (World Scientific, Singapore, 2005).[9] Z. Shen,
Differential Geometry of Spray and Finsler Spaces , (Kluwer, Dordrecht, 2001).[10] Z. Shen,
Lectures on Finsler Geometry , (World Scientific, Singapore, 2001).[11] P.L. Antonelli, R.S. Ingarden, M. Matsumoto,
The theory of sprays and Finsler spaces withapplications in physics and biology , (Springer [Kluwer], Berlin, 1993).[12] P.L. Antonelli and B.C. Lackey (editors),
The theory of Finslerian Laplacians and applications ,(Kluwer, Dordrecht, 1998).[13] A. Bejancu,
Finsler geometry and applications , (Ellis Horwood, Chichester, England, 1990).[14] G. W. Gibbons, J. Gomis and C. N. Pope, “General Very Special Relativity is Finsler geometry”,Phys. Rev. D (2007) 081701 [arXiv:0707.2174 [hep-th]].[15] A. P. Kouretsis, M. Stathakopoulos and P. C. Stavrinos, “The General Very Special Relativity inFinsler cosmology”, Phys. Rev. D (2009) 104011 [arXiv:0810.3267 [gr-qc]].[16] S. Siparov, “Introduction to the problem of anisotropy in geometrodynamics”, arXiv 0809.1817[gr-qc].N. Brinzei and S. V. Siparov, “Equations of electromagnetism in some special anisotropicspaces”, arXiv: 0812.1513 [gr-qc].S. Siparov and N. Brinzei, “Space-time anisotropy: theoretical issues and the possibility of anobservational test”, arXiv: 0806.3066 [gr-qc].[17] Z. Chang and X. Li, “Modified Newton’s gravity in Finsler space as a possible alternative to darkmatter hypothesis”, Phys. Lett. B (2008) 453 [arXiv: 0806.2184 [gr-qc]].Z. Chang and X. Li, “Modified Friedmann model in Randers–Finsler space of approximateBerwald type as a possible alternative to dark energy hypothesis”, Phys. Lett. B (2009) 173[arXiv: 0901.1023 [gr-qc]].[18] G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick and M. C. Werner, “Stationary metrics andoptical Zermelo–Randers–Finsler geometry”, Phys. Rev. D (2009) 044022 [arXiv:0811.2877[gr-qc]].[19] S. Liberati, “Quantum gravity phenomenology via Lorentz violations”, PoS P2GC (2007) 018[arXiv:0706.0142 [gr-qc]].[20] F. Girelli, S. Liberati and L. Sindoni, “Phenomenology of quantum gravity and Finsler geometry”,Phys. Rev. D (2007) 064015 [arXiv:gr-qc/0611024].[21] L. Sindoni, “The Higgs mechanism in Finsler spacetimes”, Phys. Rev. D , 124009 (2008)[arXiv:0712.3518 [gr-qc]].[22] V. Perlick, “Fermat Principle in Finsler Spacetimes”, Gen. Rel. Grav. (2006) 365[arXiv:gr-qc/0508029].[23] Y. Takano, “Theory of fields in Finsler spaces. 1”, Prog. Theor. Phys. (1968) 1159.[24] G. S. Asanov, Finsler geometry, relativity and gauge theories , (Kluwer, Amsterdam, 1985).[25] H. E. Brandt, “Finsler space-time tangent bundle”, Found. Phys. Lett. (1992) 221.[26] R. G. Beil, “Finsler gauge transformations and general relativity”, Int. J. Theor. Phys. (1992)1025.[27] G. Y. Bogoslovsky, “Finsler model of space-time”, Phys. Part. Nucl. (1993) 354 [Fiz. Elem.Chast. Atom. Yadra (1993) 813].[28] H. F. Gonner and G. Y. Bogoslovsky, “A class of anisotropic (Finsler-)space-time geometries”,Gen. Rel. Grav. (1999) 1383 [arXiv: gr-qc/9701067].[29] J. G. Vargas and D. G. Torr, “Marriage of Clifford algebra and Finsler geometry: A lineage forunification?”, Int. J. Theor. Phys. (2001) 275.[30] R. G. Beil, “Finsler geometry and relativistic field theory”, Found. Phys. (2003) 1107.[31] S. Mignemi, “Doubly special relativity and Finsler geometry”, Phys. Rev. D (2007) 047702[arXiv: 0704.1728 [gr-qc]].[32] C. Barcel´o, S. Liberati and M. Visser, “Analogue gravity”, Living Rev. Rel. (2005) 12 i-metric pseudo–Finslerian spacetimes [arXiv:gr-qc/0505065].[33] C. Barcel´o, S. Liberati and M. Visser, “Analog gravity from field theory normal modes?”, Class.Quant. Grav. (2001) 3595 [arXiv:gr-qc/0104001].[34] C. Barcel´o, S. Liberati and M. Visser, “Refringence, field theory, and normal modes”, Class. Quant.Grav. (2002) 2961 [arXiv:gr-qc/0111059].[35] M. Visser, C. Barcel´o and S. Liberati, “Bi-refringence versus bi-metricity”, Essays to celebrateProfessor Mario Novello jubilee (Frontier Group, 2003) ISBN: 2914601085 pp 397–429.[arXiv:gr-qc/0204017].[36] S. Weinfurtner, S. Liberati and M. Visser, “Analogue spacetime based on 2-component Bose-Einstein condensates”, Lect. Notes Phys. (2007) 115 [arXiv:gr-qc/0605121].[37] M. Visser and S. Weinfurtner, “Analogue spacetimes: Toy models for quantum gravity”, FromQuantum to Emergent Gravity: Theory and Phenomenology, PoS (QG-Ph) 042 [arXiv:0712.0427[gr-qc]].[38] S. Weinfurtner, “Emergent spacetimes”, PhD Thesis, arXiv:0711.4416 [gr-qc].[39] M. Visser, “Emergent rainbow spacetimes: Two pedagogical examples”, Proceedings of the secondconference on Time and Matter. Edited by M. O’Loughlin, S. Stanic, and D. Veberic Universityof Nova Gorica Press, 2008, pp 191–205. [arXiv:0712.0810 [gr-qc]].[40] J. Skakala and M. Visser, “Birefringence in pseudo-Finsler spacetimes”, NEB XIII conference, J.Phys. Conf. Ser. (2009) 012037 [arXiv:0810.4376 [gr-qc]].[41] J. Skakala and M. Visser, “Pseudo-Finslerian spacetimes and multi-refringence”, InternationalJournal of Modern Physics D19