Abstract
The ghost-free bimetric theory describes interactions of gravity with another spin-2 field in terms of two Lorentzian metrics. However, if the two metrics do not admit compatible notions of space and time, the formulation of the initial value problem becomes problematic. Furthermore, the interaction potential is given in terms of the square root of a matrix which is in general nonunique and possibly nonreal. In this paper we show that both these issues are evaded by requiring reality and general covariance of the equations. First we prove that the reality of the square root matrix leads to a classification of the allowed metrics in terms of the intersections of their null cones. Then, the requirement of general covariance further restricts the allowed metrics to geometries that admit compatible notions of space and time. It also selects a unique definition of the square root matrix. The restrictions are compatible with the equations of motion. These results ensure that the ghost-free bimetric theory can be defined unambiguously and that the two metrics always admit compatible 3+1 decompositions, at least locally. In particular, these considerations rule out certain solutions of massive gravity with locally Closed Causal Curves, which have been used to argue that the theory is acausal.
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References
M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211.
D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].
C.J. Isham, A. Salam and J.A. Strathdee, F-dominance of gravity, Phys. Rev. D 3 (1971) 867 [INSPIRE].
B. Zumino, Effective Lagrangians and Broken Symmetries, in Lectures on Elementary Particles and Quantum Field Theory, volume 2, Brandeis University, Cambridge, MA, U.S.A., (1970), pp. 437-500.
N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B 597 (2001) 127 [hep-th/0007220] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
S.F. Hassan and R.A. Rosen, Resolving the Ghost Problem in non-Linear Massive Gravity, Phys. Rev. Lett. 108 (2012) 041101 [arXiv:1106.3344] [INSPIRE].
S.F. Hassan and R.A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].
S.F. Hassan and R.A. Rosen, Confirmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity, JHEP 04 (2012) 123 [arXiv:1111.2070] [INSPIRE].
N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].
P. Creminelli, A. Nicolis, M. Papucci and E. Trincherini, Ghosts in massive gravity, JHEP 09 (2005) 003 [hep-th/0505147] [INSPIRE].
M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres, (In French), Mem. Ac. St. Petersbourg 4 (1850) 385.
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
S.F. Hassan and R.A. Rosen, On Non-Linear Actions for Massive Gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [INSPIRE].
S.F. Hassan, R.A. Rosen and A. Schmidt-May, Ghost-free Massive Gravity with a General Reference Metric, JHEP 02 (2012) 026 [arXiv:1109.3230] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Proof of Consistency of Nonlinear Massive Gravity in the Stúckelberg Formulation, Phys. Lett. B 715 (2012) 335 [arXiv:1203.5283] [INSPIRE].
D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, Degrees of Freedom in Massive Gravity, Phys. Rev. D 86 (2012) 101502 [arXiv:1204.1027] [INSPIRE].
J. Kluson, Non-Linear Massive Gravity with Additional Primary Constraint and Absence of Ghosts, Phys. Rev. D 86 (2012) 044024 [arXiv:1204.2957] [INSPIRE].
C. Deffayet, J. Mourad and G. Zahariade, Covariant constraints in ghost free massive gravity, JCAP 01 (2013) 032 [arXiv:1207.6338] [INSPIRE].
T. Kugo and N. Ohta, Covariant Approach to the No-ghost Theorem in Massive Gravity, PTEP 2014 (2014) 043B04 [arXiv:1401.3873] [INSPIRE].
M. Kocic, Geometric mean of bimetric spacetimes, arXiv:1803.09752 [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, On Consistent Theories of Massive Spin-2 Fields Coupled to Gravity, JHEP 05 (2013) 086 [arXiv:1208.1515] [INSPIRE].
C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory, J. Phys. A 49 (2016) 183001 [arXiv:1512.00021] [INSPIRE].
P. Martín-Moruno, V. Baccetti and M. Visser, Massive gravity as a limit of bimetric gravity, in Proceedings, 13th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG13): Stockholm, Sweden, July 1-7, 2012, pp. 1270-1272, arXiv:1302.2687 [INSPIRE].
Y. Akrami, S.F. Hassan, F. Könnig, A. Schmidt-May and A.R. Solomon, Bimetric gravity is cosmologically viable, Phys. Lett. B 748 (2015) 37 [arXiv:1503.07521] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Extended Weyl Invariance in a Bimetric Model and Partial Masslessness, Class. Quant. Grav. 33 (2016) 015011 [arXiv:1507.06540] [INSPIRE].
E. Gourgoulhon, 3+1 Formalism in General Relativity, Springer (2012), [https://doi.org/10.1007/978-3-642-24525-1].
Y. Choquet-Bruhat and R.P. Geroch, Global aspects of the Cauchy problem in general relativity, Commun. Math. Phys. 14 (1969) 329 [INSPIRE].
A.N. Bernal and M. Sanchez, On smooth Cauchy hypersurfaces and Geroch’s splitting theorem, Commun. Math. Phys. 243 (2003) 461 [gr-qc/0306108] [INSPIRE].
P.A.M. Dirac, The Theory of gravitation in Hamiltonian form, Proc. Roy. Soc. Lond. A 246 (1958) 333.
R.L. Arnowitt, S. Deser and C.W. Misner, Canonical variables for general relativity, Phys. Rev. 117 (1960) 1595 [INSPIRE].
N.J. Higham, Functions of Matrices: Theory and Computation, SIAM, (2008), [https://doi.org/10.1137/1.9780898717778].
N.J. Higham, Computing real square roots of a real matrix, Linear Algebra Appl. 88-89 (1987) 405.
R.A. Horn and C.R. Johnson. Topics in Matrix Analysis, Cambridge University Press, (1994), [https://doi.org/10.1017/CBO9780511840371].
F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices, Linear Algebra Appl. 7 (1973) 281.
F. Uhlig, A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil, Linear Algebra Appl. 14 (1976) 189.
F. Uhlig, A recurring theorem about pairs of quadratic forms and extensions: a survey, Linear Algebra Appl. 25 (1979) 219.
F.R. Gantmacher, The Theory of Matrices, volume 2, Chelsea (1959).
Y.P. Hong, R.A. Horn and C.R. Johnson, On the reduction of pairs of Hermitian or symmetric matrices to diagonal form by congruence, Linear Algebra Appl. 73 (1986) 213.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, (1990), [https://doi.org/10.1017/CBO9781139020411].
V. Baccetti, P. Martín-Moruno and M. Visser, Gordon and Kerr-Schild ansatze in massive and bimetric gravity, JHEP 08 (2012) 108 [arXiv:1206.4720] [INSPIRE].
S. Dey, A. Fring and T. Mathanaranjan, Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type, Int. J. Theor. Phys. 54 (2015) 4027 [INSPIRE].
R.A. d’Inverno and J. Smallwood, Covariant 2+2 formulation of the initial-value problem in general relativity, Phys. Rev. D 22 (1980) 1233 [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Bimetric theory and partial masslessness with Lanczos-Lovelock terms in arbitrary dimensions, Class. Quant. Grav. 30 (2013) 184010 [arXiv:1212.4525] [INSPIRE].
S.F. Hassan, M. Kocic and A. Schmidt-May, Absence of ghost in a new bimetric-matter coupling, arXiv:1409.1909 [INSPIRE].
L. Bernard, C. Deffayet and M. von Strauss, Massive graviton on arbitrary background: derivation, syzygies, applications, JCAP 06 (2015) 038 [arXiv:1504.04382] [INSPIRE].
K. Izumi and Y.C. Ong, An analysis of characteristics in nonlinear massive gravity, Class. Quant. Grav. 30 (2013) 184008 [arXiv:1304.0211] [INSPIRE].
S. Deser, K. Izumi, Y.C. Ong and A. Waldron, Massive Gravity Acausality Redux, Phys. Lett. B 726 (2013) 544 [arXiv:1306.5457] [INSPIRE].
S. Deser, K. Izumi, Y.C. Ong and A. Waldron, Superluminal Propagation and Acausality of Nonlinear Massive Gravity, in Proceedings, Conference in Honor of the 90th Birthday of Freeman Dyson: Singapore, Singapore, August 26-29, 2013, pp. 430-435, 2014, arXiv:1312.1115 [INSPIRE].
S. Deser, M. Sandora, A. Waldron and G. Zahariade, Covariant constraints for generic massive gravity and analysis of its characteristics, Phys. Rev. D 90 (2014) 104043 [arXiv:1408.0561] [INSPIRE].
S. Deser, K. Izumi, Y.C. Ong and A. Waldron, Problems of massive gravities, Mod. Phys. Lett. A 30 (2015) 1540006 [arXiv:1410.2289] [INSPIRE].
R. Geroch, Faster Than Light?, AMS/IP Stud. Adv. Math. 49 (2011) 59 [arXiv:1005.1614] [INSPIRE].
E. Babichev, V. Mukhanov and A. Vikman, k-Essence, superluminal propagation, causality and emergent geometry, JHEP 02 (2008) 101 [arXiv:0708.0561] [INSPIRE].
M. Düll, F.P. Schuller, N. Stritzelberger and F. Wolz, Gravitational closure of matter field equations, Phys. Rev. D 97 (2018) 084036 [arXiv:1611.08878] [INSPIRE].
I.T. Drummond, Quantum field theory in a multimetric background, Phys. Rev. D 88 (2013) 025009 [arXiv:1303.3126] [INSPIRE].
M.S. Volkov, Stability of Minkowski space in ghost-free massive gravity theory, Phys. Rev. D 90 (2014) 024028 [arXiv:1402.2953] [INSPIRE].
M.S. Volkov, Energy in ghost-free massive gravity theory, Phys. Rev. D 90 (2014) 124090 [arXiv:1404.2291] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Particular Solutions in Bimetric Theory and Their Implications, Int. J. Mod. Phys. D 23 (2014) 1443002 [arXiv:1407.2772] [INSPIRE].
K. Hinterbichler and R.A. Rosen, Interacting Spin-2 Fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].
M. Bojowald, Canonical Gravity and Applications, Cambridge University Press, (2010), [https://doi.org/10.1017/CBO9780511921759].
X.O. Camanho, G. Lucena Gómez and R. Rahman, Causality Constraints on Massive Gravity, Phys. Rev. D 96 (2017) 084007 [arXiv:1610.02033] [INSPIRE].
S. Alexandrov, Canonical structure of Tetrad Bimetric Gravity, Gen. Rel. Grav. 46 (2014) 1639 [arXiv:1308.6586] [INSPIRE].
S.F. Hassan and A. Lundkvist, Analysis of constraints and their algebra in bimetric theory, arXiv:1802.07267 [INSPIRE].
L. Bernard, C. Deffayet, A. Schmidt-May and M. von Strauss, Linear spin-2 fields in most general backgrounds, Phys. Rev. D 93 (2016) 084020 [arXiv:1512.03620] [INSPIRE].
C. Deffayet, J. Mourad and G. Zahariade, A note on ‘symmetric’ vielbeins in bimetric, massive, perturbative and non perturbative gravities, JHEP 03 (2013) 086 [arXiv:1208.4493] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Metric Formulation of Ghost-Free Multivielbein Theory, arXiv:1204.5202 [INSPIRE].
L. Apolo and S.F. Hassan, Non-linear partially massless symmetry in an SO(1,5) continuation of conformal gravity, Class. Quant. Grav. 34 (2017) 105005 [arXiv:1609.09514] [INSPIRE].
C. Cheung and G.N. Remmen, Positive Signs in Massive Gravity, JHEP 04 (2016) 002 [arXiv:1601.04068] [INSPIRE].
D. Comelli, M. Crisostomi, K. Koyama, L. Pilo and G. Tasinato, New Branches of Massive Gravity, Phys. Rev. D 91 (2015) 121502 [arXiv:1505.00632] [INSPIRE].
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Hassan, S.F., Kocic, M. On the local structure of spacetime in ghost-free bimetric theory and massive gravity. J. High Energ. Phys. 2018, 99 (2018). https://doi.org/10.1007/JHEP05(2018)099
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DOI: https://doi.org/10.1007/JHEP05(2018)099