Type N universal spacetime

Home Search Collections Journals About Contact us My IOPscience Type N universal spacetime This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Conf. Ser. 600 012065 (http://iopscience.iop.org/1742-6596/600/1/012065) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 152.94.27.12 This content was downloaded on 06/01/2016 at 12:53 Please note that terms and conditions apply. Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution IOP Publishing Journal of Physics: Conference Series 600 (2015) 012065 doi:10.1088/1742-6596/600/1/012065 Type N universal spacetimes S. Hervik1 , V. Pravda2 , A. Pravdová2 1 Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway; Institute of Mathematics, Academy of Sciences of the Czech Republic Žitná 25, 115 67 Prague 1, Czech Republic 2 E-mail: [email protected], [email protected], [email protected] Abstract. Universal spacetimes are vacuum solutions to all theories of gravity with the Lagrangian L = L(gab , Rabcd , ∇a1 Rbcde , . . . , ∇a1 ...ap Rbcde ). Well known examples of universal spacetimes are plane waves which are of the Weyl type N. Here, we discuss recent results on necessary and sufficient conditions for all Weyl type N spacetimes in arbitrary dimension and we conclude that a type N spacetime is universal if and only if it is an Einstein Kundt spacetime. We also summarize the main points of the proof of this result. 1. Introduction Due to the diffeomorphism invariance, correction terms in perturbative quantum gravity which are added to the Einstein action consist of curvature invariants constructed from the Riemann tensor and its covariant derivatives. The resulting modified field equations are in general very complicated. In the case where only correction terms quadratic in the Riemann tensor are added to the Einstein-Hilbert action - the so called quadratic gravity,   Z
1 n √ 2 2 2 2 2 , (1) S = d x −g (R − 2Λ0 ) + αR + βRab + γ Rabcd − 4Rab + R κ the field equations read [1]     1 1 1 Rab − Rgab + Λ0 gab + 2αR Rab − Rgab + (2α + β) (gab  − ∇a ∇b ) R κ 2 4  
1 2 2 +2γ RRab − 2Racbd Rcd + Racde Rbcde − 2Rac Rbc − gab Rcdef − 4Rcd + R2 4     1 1 +β Rab − Rgab + 2β Racbd − gab Rcd Rcd = 0. 2 4 (2) Interestingly, there exist vacuum solutions of Einstein’s gravity (with possibly non-zero cosmological constant) that are “immune” to these corrections, i.e they are vacuum solutions to both theories, the Einstein gravity and the quadratic gravity. For instance, it has been shown in [2] that for type N1 Einstein spacetimes, all terms in (2) are proportional to the metric and thus in arbitrary dimension all Weyl type N Einstein 1 Type N spacetimes in the algebraic classification of tensors introduced in [3], recently reviewed in [4] and briefly discussed in section 2. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution IOP Publishing Journal of Physics: Conference Series 600 (2015) 012065 doi:10.1088/1742-6596/600/1/012065 spacetimes with appropriately chosen effective cosmological constant Λ are exact solutions of quadratic gravity (2). It is a natural question to ask whether some of the type N Einstein spacetimes solve also more general theories than (2). Here, we will be interested in the so called universal spacetimes first introduced in [5]. Definition 1.1. A metric is called universal if all conserved symmetric rank-2 tensors constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order2 are multiples of the metric. Vacuum field equations of modified gravities obtained by varying a diffeomorphism invariant Lagrangian with respect to the metric are conserved, rank-2, and symmetric and therefore universal spacetimes solve vacuum equations of all theories with the Lagrangian being a polynomial curvature invariant in the form L = L(gab , Rabcd , ∇a1 Rbcde , . . . , ∇a1 ...ap Rbcde ). (3) Obviously, all universal spacetimes are Einstein (i.e. Rab = (R/n)gab ), where n is the dimension of the spacetime. So far, necessary and sufficient geometrical conditions for universality are unknown. However, when considering only type N spacetimes, such conditions can be found3 . Theorem 1.2. A type N spacetime is universal if and only if it is an Einstein Kundt spacetime. Note that special examples of Ricci-flat type N Kundt spacetimes are plane waves that were identified as vacuum solutions of all theories given by (3) already in [8] and [9]. So far, we did not properly define type N spacetimes in arbitrary dimension and Kundt spacetimes. These terms together with other basic definitions will be briefly reviewed in the next section. In section 3, we outline the key points of the necessity and sufficiency parts of the proof of the theorem 1.2 and in section 4, we give explicit examples of type N universal metrics. 2. Preliminaries We employ the algebraic classification of the Weyl tensor [3] and higher dimensional generalizations of the Newman-Penrose [10, 11] and the Geroch-Held-Penrose formalisms [12]. We will follow the notation of [4, 12]. Here, let us give basic definitions just very briefly and refer to [4] for more information. We will work in a null frame with two null vectors ` and n and n − 2 spacelike vectors m(i) obeying `a `a = na na = 0, `a na = 1, m(i)a m(j) (4) a = δij , where coordinate indices a, b, . . . and frame indices i, j, . . . take values from 0 to n − 1 and 2 to n − 1, respectively. We say that a quantity q has a boost weight b if it transforms as q̂ = λb q (5) under boosts `ˆ = λ`, n̂ = λ−1 n, 2 m̂(i) = m(i) . (6) We consider only symmetric rank-2 tensors constructed as contractions of polynomials from the metric, the Riemann tensor and its covariant derivatives of arbitrary order, however, most of our results hold also for analytic functions of such invariants. 3 The necessary conditions for universality were found in [6], sufficient conditions were also first proven in [6], however, without a proof the result was stated already in [5] and brief general comments about some points of the proof were also included by one of us in v2 of [7]. 2 Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution IOP Publishing Journal of Physics: Conference Series 600 (2015) 012065 doi:10.1088/1742-6596/600/1/012065 By definition [3, 4], type N spacetimes are spacetimes for which the Weyl tensor (in an appropriately chosen frame (4)) admits only components of boost weight -2 and thus can be expressed as (i) (j) Cabcd = 4Ω0 ij `{a m b `c m d } , (7) where Ω0 ij is symmetric and traceless and for an arbitrary tensor Tabcd 1 T{abcd} ≡ (T[ab][cd] + T[cd][ab] ), 2 (8) so that Cabcd = C{abcd} . It can be shown [10] that for type N Einstein spacetimes, the multiple WAND4 ` is geodetic. If we choose an affine parameterization we can express the covariant derivative of ` as (i) (j) (i) `a;b = L11 `a `b + L1i `a m b + τi m(i) a `b + ρij m a m b . (9) Optical scalars of `, shear σ 2 , expansion θ and twist ω 2 can be expressed as σ 2 = `(a;b) `(a;b) − 1 n−2 `a;a
2 , θ= 1 a n−2 ` ;a , ω 2 = `[a;b] `a;b . (10) Now, we are ready to define Kundt spacetimes. Definition 2.1. Kundt spacetimes are spacetimes admitting a null geodetic congruence ` with vanishing shear, expansion and twist. Kundt metrics in higher dimensions were introduced in [13, 14]. 3. Main points of the proof of theorem 1.2 Let us briefly mention the main points of the proof [6] of the theorem 1.2. 3.1. Sufficiency First, let us discuss the proof of the sufficiency part of the theorem 1.2 , i.e. the proof of the statement that all Einstein type N Kundt spacetimes are universal. Thus, we want to show that in this case, all rank-2 tensors constructed from the Weyl5 tensor and its covariant derivatives are proportional to the metric (in fact, they vanish). For rank-2 tensors constructed from the Weyl tensor only (without covariant derivatives), the proof is very simple. Any rank-2 tensor has only terms of boost weight ≥ -2 and the type N Weyl tensor admits only boost weight -2 terms. Therefore, all rank-2 tensors constructed from the type N Weyl tensor which are quadratic or of a higher order in the Weyl tensor vanish and, due to the tracelessness of the Weyl tensor, rank-2 tensors linear in the Weyl tensor vanish as well. Thus, it is not possible to construct a non-vanishing rank-2 tensor from the type N Weyl tensor. For covariant derivatives of the Weyl tensor, the proof is more involved. The key point is Proposition 3.1. For type N Einstein Kundt spacetimes, the boost order of ∇(k) C (a covariant derivative of an arbitrary order of the Weyl tensor) with respect to the multiple WAND is at most −2. 4 Weyl aligned null direction [3]. Obviously, for Einstein spacetimes, the Ricci tensor is proportional to the metric and its covariant derivatives vanish. 5 3 Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution IOP Publishing Journal of Physics: Conference Series 600 (2015) 012065 doi:10.1088/1742-6596/600/1/012065 The proof of the above proposition [6] using balanced scalar approach introduced in [15] is rather technical, and it relies on the special form of the Bianchi and Ricci identities for this class of spacetimes. A direct consequence is Lemma 3.2. For type N Einstein Kundt spacetimes, rank-2 tensors constructed from ∇(k) C, which are quadratic or of higher order in ∇(k) C, vanish. Using the expression for the commutator of covariant derivatives, the above results and the Bianchi identities, one can generalize the above lemma also to the case of rank-2 tensors constructed from ∇(k) C, which are linear in ∇(k) C (see [6]). This completes the sufficient part of the proof of the theorem 1.2. 3.2. Necessity The proof of the necessity part of the theorem 1.2, i.e. the statement that all type N universal spacetimes are Einstein and Kundt is based on another result of [6] that will be discussed in more detail elsewhere in this volume Theorem 3.3. A universal spacetime is necessarily a CSI spacetime. CSI (constant curvature invariant) spacetimes are spacetimes for which all curvature invariants constructed from the metric, the Riemann tensor and its derivatives of arbitrary order are constant, see e.g. [14]. Let us study the simplest non-trivial curvature invariant for type N spacetimes [16] IN ≡ C a1 b1 a2 b2 ;c1 c2 Ca1 d1 a2 d2 ;c1 c2 C e1 d1 e2 d2 ;f1 f2 Ce1 b1 e2 b2 ;f1 f2 . (11) In terms of higher dimensional GHP quantities, it can be shown [15] that IN is proportional (via a numerical constant) to  2 IN ∝ (Ω0 22 )2 + (Ω0 23 )2 (S 2 + A2 )4 , (12) where S and A are closely related to the optical scalars (see [6]). The invariant above is nonconstant unless the type N Einstein spacetime is Kundt [15] and thus, in this class of spacetimes, only Kundt spacetimes are CSI. From theorem 3.3, it follows that type N universal spacetimes are Kundt. 4. Explicit examples of universal type N Kundt metrics By theorem 1.2, all type N Einstein Kundt metrics are universal. In four dimensions, all type N Einstein Kundt metrics can be expressed as [17]
! 2 2 Q2 ,u
Q Q 1 Q v − H du2 + 2 dx2 + dy 2 , (13) ds2 = 2 2 dudv + 2k 2 v 2 + 2 P P P P P where
Λ 1 Λ β(u)2 + γ(u)2 , P = 1 + (x2 + y 2 ), k = α(u)2 + 12 6 2   Λ 2 Λ Q = 1 − (x + y 2 ) α(u) + β(u)x + γ(u)y, H = 2f1,x − (xf1 + yf2 ), 12 3P where α(u), β(u), γ(u) are free functions (see [18] for the canonical forms) and f1 = f1 (u, x, y) and f2 = f2 (u, x, y) obey f1,x = f2,y , f1,y = −f2,x . 4 Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution IOP Publishing Journal of Physics: Conference Series 600 (2015) 012065 doi:10.1088/1742-6596/600/1/012065 Higher dimensional examples of type N universal metrics can be obtained by warping (13). Another higher-dimensional example is (A)dS-wave [6]
ds2 = e−pw 2dudv + H(u, w, xM )du2 + δM N dxM dxN + dw2 , (14)
−pw = 0. Further with p being a constant and H obeying H,KL δ KL + H,ww − n−1 2 pH,w e explicit examples can be found in [6]. Acknowledgments V.P. and A.P. acknowledge support from research plan RVO: 67985840 and research grant GAČR 13-10042S. References [1] I. Gullu and B. Tekin. Massive higher derivative gravity in D-dimensional anti-de Sitter spacetimes. Phys. Rev., D80:064033, 2009. [2] T. Málek and V. Pravda. Type III and N solutions to quadratic gravity. Phys. Rev. D, 84:024047, 2011. [3] A. Coley, R. Milson, V. Pravda, and A. Pravdová. Classification of the Weyl tensor in higher dimensions. Class. Quantum Grav., 21:L35–L41, 2004. [4] M. Ortaggio, V. Pravda, and A. Pravdová. Algebraic classification of higher dimensional spacetimes based on null alignment. Class. Quantum Grav., 30:013001, 2013. [5] A.A. Coley, G.W. Gibbons, S. Hervik, and C.N. Pope. Metrics with vanishing quantum corrections. Class. Quantum Grav., 25:145017, 2008. [6] S. Hervik, V. Pravda and A. Pravdová, Type III and N universal spacetimes. Class. Quantum Grav., 31:215005, 2014. [7] M. Gurses, T. C. Sisman, B. Tekin and S. Hervik, AdS-Wave Solutions of f(Riemann) Theories. Phys. Rev. Lett. 111:101101, 2013 [arXiv:1305.1565v2 ]. [8] D. Amati and C. Klimčik. Nonperturbative computation of the Weyl anomaly for a class of nontrivial backgrounds. Phys. Lett. B, 219:443–447, 1989. [9] G. T. Horowitz and A. R. Steif. Spacetime singularities in string theory. Phys. Rev. Lett., 64:260–263, 1990. [10] V. Pravda, A. Pravdová, A. Coley, and R. Milson. Bianchi identities in higher dimensions. Class. Quantum Grav., 21:2873–2897, 2004. [11] M. Ortaggio, V. Pravda, and A. Pravdová. Ricci identities in higher dimensions. Class. Quantum Grav., 24:1657–1664, 2007. [12] M. Durkee, V. Pravda, A. Pravdová, and H. S. Reall. Generalization of the Geroch-Held-Penrose formalism to higher dimensions. Class. Quantum Grav., 27:215010, 2010. [13] A. Coley, R. Milson, N. Pelavas, V. Pravda, A. Pravdová, and R. Zalaletdinov. Generalizations of pp –wave spacetimes in higher dimensions. Phys. Rev. D, 67:104020, 2003. [14] A. Coley, S. Hervik, and N. Pelavas. On spacetimes with constant scalar invariants. Class. Quantum Grav., 23:3053–3074, 2006. [15] A. Coley, R. Milson, V. Pravda, and A. Pravdová. Vanishing scalar invariant spacetimes in higher dimensions. Class. Quantum Grav., 21:5519–5542, 2004. [16] J. Bičák and V. Pravda. Curvature invariants in type-N spacetimes. Class. Quantum Grav., 15:1539–1555, 1998. [17] I. Ozsváth, I. Robinson and K. Rózga. Plane-fronted gravitational and electromagnetic waves in spaces with cosmological constant. J. Math. Phys., 26:1755-1761, 1985. [18] J. B. Griffiths and J. Podolský. Exact Space-Times in Einstein’s General Relativity. Cambridge University Press, Cambridge, 2009. 5