×

Predictability of subluminal and superluminal wave equations. (English) Zbl 1416.83016

Summary: It is sometimes claimed that Lorentz invariant wave equations which allow superluminal propagation exhibit worse predictability than subluminal equations. To investigate this, we study the Born-Infeld scalar in two spacetime dimensions. This equation can be formulated in either a subluminal or a superluminal form. Surprisingly, we find that the subluminal theory is less predictive than the superluminal theory in the following sense. For the subluminal theory, there can exist multiple maximal globally hyperbolic developments arising from the same initial data. This problem does not arise in the superluminal theory, for which there is a unique maximal globally hyperbolic development. For a general quasilinear wave equation, we prove theorems establishing why this lack of uniqueness occurs, and identify conditions on the equation that ensure uniqueness. In particular, we prove that superluminal equations always admit a unique maximal globally hyperbolic development. In this sense, superluminal equations exhibit better predictability than generic subluminal equations.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35L05 Wave equation
53Z05 Applications of differential geometry to physics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams, A., Arkani-Hamed, N., Dubovsky, S., Nicolis, A., Rattazzi, R.: Causality, analyticity and an IR obstruction to UV completion. JHEP 0610, 014 (2006). https://doi.org/10.1088/1126-6708/2006/10/014, arXiv:hep-th/0602178
[2] Babichev, E., Mukhanov, V., Vikman, A.: k-Essence, superluminal propagation, causality and emergent geometry. JHEP 0802, 101 (2008). https://doi.org/10.1088/1126-6708/2008/02/101, arXiv:0708.0561 [hep-th]
[3] Geroch, R.: Faster than light. AMS/IP Stud. Adv. Math. 49, 59 (2011). arXiv:1005.1614 [gr-qc]
[4] Papallo, G., Reall, H.S.: Graviton time delay and a speed limit for small black holes in Einstein-Gauss-Bonnet theory. JHEP 1511, 109 (2015). https://doi.org/10.1007/JHEP11(2015)109, arXiv:1508.05303 [gr-qc] · Zbl 1388.83494
[5] Barbashov, B., Chernikov, N.: Solution and quantization of a nonlinear two-dimensional model for a Born-Infeld type field. Sov. Phys. JETP 23(5), 861 (1966)
[6] Barbashov, B., Chernikov, N.: Solution of the two plane wave scattering problem in a nonlinear scalar field theory of the Born-Infeld type. Sov. Phys. JETP 24(2), 437 (1967)
[7] Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2007) · Zbl 1138.35060 · doi:10.4171/031
[8] Hawking, S.W.: Chronology protection conjecture. Phys. Rev. D 46, 603 (1992). https://doi.org/10.1103/PhysRevD.46.603 · doi:10.1103/PhysRevD.46.603
[9] Choquet-Bruhat, Y., Geroch, R.P.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329 (1969). https://doi.org/10.1007/BF01645389 · Zbl 0182.59901 · doi:10.1007/BF01645389
[10] Sbierski, J.: On the existence of a maximal Cauchy development for the Einstein equations: a dezornification. Ann. Henri Poincare 17(2), 301 (2016). https://doi.org/10.1007/s00023-015-0401-5, arXiv:1309.7591 [gr-qc] · Zbl 1335.83009
[11] Taniuti, T.: On wave propagation in non-linear fields. Prog. Theor. Phys. 9, 69 (1959) · Zbl 0086.21801 · doi:10.1143/PTPS.9.69
[12] Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974) · Zbl 0373.76001
[13] Sogge, C.: Lectures on Non-linear Wave Equations, 2nd edn. Int. Press of Boston Inc, Boston (2008) · Zbl 1165.35001
[14] Ringström, H.: On the Topology and Future Stability of the Universe. Oxford University Press, Oxford (2013) · Zbl 1270.83005 · doi:10.1093/acprof:oso/9780199680290.001.0001
[15] O’Neill, B.: Semi-Riemannian Geometry. Academic Press Inc, London (1983) · Zbl 0531.53051
[16] Beem, J.K., Ehrlich, P., Easley, K.: Global Lorentzian Geometry. CRC Press, London (1996) · Zbl 0846.53001
[17] Wald, R.M.: General Relativity. Chicago University Press, Chicago (1984) · Zbl 0549.53001 · doi:10.7208/chicago/9780226870373.001.0001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.