Black holes in massive gravity

Transcription

1 Black holes in massive gravity Eugeny Babichev LPT, Orsay IAP PARIS, MARCH REVIEW: in collaboration with: R. Brito, M. Crisostomi, C. Deffayet, A. Fabbri,P. Pani, ARXIV:

2 OUTLINE Introduction Classes of solutions Solutions with a source Black holes Perturbations and (in)stability of black holes Conclusions

3 INTRODUCTION

4 Why modify gravity? Why modify gravity? - cosmological constant problems, - non-renormalizability problem, - benchmarks for testing General Relativity - theoretical curiosity. Many ways to modify gravity: - f(r), scalar-tensor theories, - Galileons, Horndeski (and beyond) theory, KGB, Fab-four, - higher-dimensions, - DGP, - Horava, Khronometric - massive gravity - Naively, cancellation of the cosmological constant, because of the Yukawa decay; - Small cosmological constant due to small graviton mass

5 Old problems of massive gravity 1. Physical ghost [Boulware&Deser 72] 2. Extra propagating degrees of freedom. It is difficult to pass basic Solar system gravity tests. vdvz discontinuity [van Dam&Veltman 70, Zakharov 70]

6 Fierz-Pauli massive gravity Expand the Einstein-Hilbert action: g µ = µ + h µ S GR = M 2 P Z d 4 x p gr = Z d 4 x 1 2 hµ E µ h + O(h 3 ) E µ h = 1 µ@ h 1 2 h µ + µ h + h µ 1 2 µ h h) 2 propagating spin: 2 massless gravitons, spin-2 x! x +,h µ = µ; ;µ

7 Fierz-Pauli massive gravity Fierz-Pauli action (Fierz&Pauli 39): S PF = M 2 P Z d 4 x apple 1 2 hµ E µ h 1 4 m2 h µ h µ h 2 Linearized Einstein- Hilbert term mass term x! x +,h µ = µ; ;µ 5 healthy degrees of freedom (because of a particular choice of the potential, h=0) for a generic mass term 6 d.o.f., one is necessary Ostrogradski ghost Non-linear completion?

8 Non-linear massive gravity Introduce an extra metric to construct a mass term (in order to contract indices) Construct a potential, which is invariant under diffeomorphism (common for two metrics) + some technical conditions

9 Non-linear massive gravity potential for metric [Boulware & Deser 72 [Arkani-Hamed et al 03]

10 Vainshtein mechanism screens extra degree of freedom Non-linear effects restore General Relativity close to the source due to the non-linear effects [Vainshtein 72] [EB, Deffayet, Ziour 09 10] Problem 2 is solved

11 Non-linear massive gravity Boulware-Deser ghost Generically there are two propagating scalars: one is a ghost! [Boulware & Deser 72]

12 drgt massive gravity special case of non-linear massive gravity Massive gravity without Boulware-Deser ghost [de Rham, Gabadadze, Tolley 10 11, Hassan & Rosen 12] +many other works Y = p g 1 f S =M 2 P Z + applem 2 P g is physical metric; f is fixed (flat) or extra dynamical metric. d 4 x p Z g " d 4 x p k=4 R[g] m 2 X f R[f] k=0 ke k (Y ) # e 0 =1,e 1 =[X], e 2 = 1 2 [X]2 [X 2 ],e 3 = 1 6 [X]3 3[X][X 2 ] + 2[X 3 ], e 4 = 1 24 [X]4 6[X] 2 [X 2 ] + 3[X 2 ] 2 + 8[X][X 3 ] 6[X 4 ]

13 Equations of motion G µ = m 2 T µ + g µ p G µ = m 2 p g f T µ apple + f µ + T µ (matter) M 2 P Variation of mass term Non-derivative coupling of the two metrics

14 Spherical symmetry and beyond

15 Two types of (static) spherically symmetric solutions Bi-diagonal: When two metrics can be put in the diagonal form simultaneously. Non Bi-diagonal: When this is not the case A no-go theorem for bi-diagonal black holes [Deffayet, Jacobson 11]

16 Spherically symmetric solutions Bi-diagonal Solutions with source. Vainshtein mechanism: GR is restored, tiny modification of GR Black holes - Bi-diagonal solutions: the two metrics are GR-like and equal or proportional (horizons coinside). - hairy black holes (numerics), non- GR Stability of black holes - Black holes are unstable (mild tachyon-like instability) Non Bi-diagonal Solutions with source. GR is perfectly restored Black holes - Non bi-diagonal solutions: the two metrics are GR-like and not proportional (horizons may not coinside). Stability of black holes - Black holes are stable Rotating solutions - Two GR-like equal metrics Rotating solutions - Two GR-like non-equal metrics

17 Solutions with a source

18 <latexit sha1_base64="b6ups/hw1cepejbl41rqnehna4i=">aaab7nicbvbnswmxej3ur1q/qh69bivgqe6kqmecf48vxftol5jns21okl2trfcw/gkvhls8+nu8+w9m2z1o64obx3szzmylusgn9bxvvfpzxvvfkg9wtrz3dveq+wcpjsk0zqfnrklbetfmcmucy61g7vq Vainshtein mechanism in bi-gravity Weak-field approximation [EB,Deffayet,Ziour 10] Vainshtein mechanism in bi-gravity [Volkov 12] [EB,Crisostomi 13] Linearized Einstein equations for metric g Linearized Einstein equations for metric f<latexit sha1_base64="nkrhn8k Inside the Compton length r 1/m µ =0 <latexit sha1_base64="eyw3c4sr4ufsuwvzfg1fhka160e=">aaab+hicbvbns8naej34wetx1koxxsiiqkhfrbeh4mvjbwmlbsyb7azdursju5tccf0nxjyoepwneppfug1z0nyha4/3zpizf6wcaep7387k6tr6xmzpq7y9s7u37x4cpuoku4qgjogjakvyu84kdqwznlzsrbgiog1gw9up3xxrpvkih8w4pahafclirrcxutd1oyj7qqfz5hkeuk Algebraic equation of 7th order on µ<latexit All other metric functions depend on µ<latexit <latexit sha1_base64="czlnz6ujv0 <latexit sha1_base64="czlnz6ujv0

19 Vainshtein mechanism in bi-gravity recovery of GR lêl GR n'ên' GR rêr v

20 Non-Vainshtein recovery of GR Z [EB unpublished] Physical metric in the EF form, ds 2 g = h(r)dv 2 +2k(r)dvdr + r 2 d 2, Ansatz for the second metric ds 2 f C 2 = dv2 +2Sdvdr+ For C such that (C 1) 2 2 (C 1) + 1 = 0. (rµ) 02 S 2 dr 2 +(rµ) 2 d 2, Recovery of GR up to a Lambda-term ~ m 2

21 BLACK HOLES

22 <latexit sha1_base64="n93ak0bldk5biiyjul7bawixb0o=">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</latexit> <latexit sha1_base64="gql0qwwel2qjtgi1a2cs9nkwwco=">aaacghicbvdlssnafj3uv42vqes3g0wpsetsrn0ihw5cvjc20mqwmu7aozmhmxohhp6gg3/fjqsvt935n07allt1wmdhnho5c4+fmcqkax5rpzxvtfwn8qa+tb2zu2fshzyioowy2dhmme/6sbbgi2jlkhnpjpyg0gek449aud95ilzqolqx44s4irpenkayssv5hsm9aty9gdwlhed3fcirrlzq1tljo9aadmljsbdortxnekbfrjszwgvifaqccrq9y+r0y5ygjjkyisf6lplin0ncuszirhdsqrker2haeopgkctczwaxtecjuvowill6kyqz9fdehkihxqgvkigsq7ho5ej/xi+vwbwb0shjjynwffgqmihjmnce+5qtlnlyeyq5vx+feig4wlkvqassrmwtl4ndqf/wzbulsrnztfegr+ayviefrkat3ii2saegz+avvimp7uv70z61r3m0pbuzh+aptokpklia6q==</latexit> Black holes Schwarzschild metric Non-bidiagonal BHs [Salam & Strathdee 77] [Isham & Storey 78] [Koyama, Niz, Tasinato 11]+many others Ansatz (bi-eddington-finkelstein form) [EB& Fabbri 13] ds 2 g = 1 r g ds 2 f = C 2 h 1 r dv 2 +2dvdr + r 2 d 2, r f dv 2 +2dvdr + r 2 d 2i r Two choices: r g = r f (C 1) 2 2 (C 1) + 1 = 0 bi-diagonal non-bidiagonal For these choices the extra mass energy-momentum tensor reduces to effective cosmological constant

23 Charged Black holes Electromagnetic field coupled to g [EB& Fabbri 13] ds 2 g = ds 2 f = C 2 " 1 r g r + r2 Q r 2 r 2 1 r f r r 2 l 2 f l 2 g dv 2 +2dvdr + r 2 d 2,! dv 2 +2dvdr + r 2 d 2 #. A µ = Q r, 0, 0, 0

24 Rotating Black holes Original Kerr metric ds 2 g = 1 r g r 2 dv + a sin 2 d +2 dv + a sin 2 d dr + a sin 2 d + 2 d 2 +sin 2 d 2 2 = r 2 + a 2 cos 2 f is flat, but unusual form ds 2 f =C 2 dv 2 +2dvdr +2a sin 2 drd + 2 d 2 + r 2 + a 2 sin 2 d 2 2 Obtained from ds 2 M = dt 2 + dx 2 + dy 2 + dz 2 by: t = v r, x + iy =(r ia)e i sin, z = r cos r! Cr, v! Cv, a! Ca

25 Hairy bi-diagonal black holes Asymptotically AdS hairy solutions exist [Volkov 12] g µ dx µ dx = Q 2 dt 2 + N 2 dr 2 + R 2 d 2, f µ dx µ dx = a 2 dt 2 + b 2 dr 2 + U 2 d 2, Numerical integration of a system of coupled ODEs 8 >< >: N 0 = F 1 (r, N, Y, U, µ, apple, 3, 4 ) Y 0 = F 2 (r, N, Y, U, µ, apple, 3, 4 ) U 0 = F 3 (r, N, Y, U, µ, apple, 3, 4 ) a/a 0 N/N Y/Y 0 Q/Q ln(r/r h )

26 Hairy bi-diagonal black holes Asymptotically flat hairy solutions [Brito,Cardoso,Pani 13]!1 C 1 N =1 2r + C 2(1 + rµ) e rµ, 2r C 1 C 2 (1 + rµ) Y =1 e rµ, 2r 2r U = r + C 2(1 + rµ + r 2 µ 2 ) e rµ, µ 2 r 2 Yukawa decay α singular at small µ No solutions for µ<µ c Regular for any µ No solutions for µ<µ c α 3 =-α 4 For generic potential only for large BH mass. r s 1/H -2-3 No solutions for µ<µ c α 3

27 PERTURBATIONS of BLACK HOLES

28 <latexit sha1_base64="eo5t7dpd/bxac/enou+7qx4obls=">aaacsxicfvblswmxgmy2pmp9rxr0eixci1j2rvqqoedfywvrc91asmm2dc1mlzyesuzv8+ljmz/ciwcvt6btgrwka4hjzdd8yfgxo1i5zrovyy8sli0xvoqra+sbm/bw9q2mtmckgsmwizapjgguk4aiipfwlagkfuaa/vby7dfviza04jdqfjnoipqcbhqjzasujfrdxau1x3ukl+d35s4po5x0ydar9cvpoxcoi8fsjpg3eltsrl1yqs4e8ddxm1icgepd+8nrrvihhcvmkjrt14lvj0fcucxiwvs0jdhcq9qnbum5consjjmqurhvlb4mimeov3ciziysfeo5cn0zgsi1kppewpzla2svnhusymotcmftryfmuevw3cvsuugwyindebbuvbxiariik9n+0ztgzn/5n2kcvu+qzvvxqvbl2iiaxbahysafp6agrkadnaagd+afvif369f6tt6sz+lozsoyo+ahcvkvszoyra==</latexit> <latexit sha1_base64="njvslz6/ca+rottpadq+uourpz0=">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</latexit> <latexit sha1_base64="q7h5kzlsky6jodneeo9qwincs+g=">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</latexit> Perturbations spherically symmetric ansatz for perturbations [EB & Fabbri 14] Perturbations of both metrics g µ = g µ (0) + h (g) µ,f µ = f µ (0) + h (f) µ G µ = m 2 T µ, G µ = m2 apple h µ (g) = e v 0 h vv (g) (r) h vr (g) (r) hvr p p g T µ f (g) (r) 0 0 (g) (r) 0 0 hrr 0 0 h (g) (r) r 2 0 h µ ( ) hµ (g) C 2 h µ (f) i.e. h vv ( )(r) =h vv (g)(r) h (g) (r) r 2 sin 2. 1 C A h vv (f)(r) ( > 0)

29 <latexit sha1_base64="eo5t7dpd/bxac/enou+7qx4obls=">aaacsxicfvblswmxgmy2pmp9rxr0eixci1j2rvqqoedfywvrc91asmm2dc1mlzyesuzv8+ljmz/ciwcvt6btgrwka4hjzdd8yfgxo1i5zrovyy8sli0xvoqra+sbm/bw9q2mtmckgsmwizapjgguk4aiipfwlagkfuaa/vby7dfviza04jdqfjnoipqcbhqjzasujfrdxau1x3ukl+d35s4po5x0ydar9cvpoxcoi8fsjpg3eltsrl1yqs4e8ddxm1icgepd+8nrrvihhcvmkjrt14lvj0fcucxiwvs0jdhcq9qnbum5consjjmqurhvlb4mimeov3ciziysfeo5cn0zgsi1kppewpzla2svnhusymotcmftryfmuevw3cvsuugwyindebbuvbxiariik9n+0ztgzn/5n2kcvu+qzvvxqvbl2iiaxbahysafp6agrkadnaagd+afvif369f6tt6sz+lozsoyo+ahcvkvszoyra==</latexit> <latexit sha1_base64="njvslz6/ca+rottpadq+uourpz0=">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</latexit> <latexit sha1_base64="g+s85vmxqj32p0l++b+x665ezes=">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</latexit> Perturbations spherically symmetric ansatz for perturbations [EB & Fabbri 14] Perturbations of both metrics g µ = g µ (0) + h (g) µ,f µ = f µ (0) + h (f) µ G µ = m 2 T µ, G µ = m2 apple h µ (f) = e v C 2 0 h vv (f) (r) h vr (f) (r) hvr p p g T µ f (f) (r) 0 0 (f) (r) 0 0 hrr 0 0 h (f) (r) r 2 0 h µ ( ) hµ (g) C 2 h µ (f) i.e. h vv ( )(r) =h vv (g)(r) h. (f) (r) r 2 sin 2 h vv (f)(r) 1 C A ( > 0)

30 Spherical Perturbations Regularity at horizons and infinity!

31 Perturbations for bidiagonal case E µ h (+) =0 is massive is massless

32 Bi-diagonal case GL instability A system of equations of second order plus 2 constraints on Playing with equations we can obtain a single equation on ' 0 (a combination of H tt, H rr and H tr ) d 2 dr 2 ' 0 +! 2 V (r) ' 0 = V 0 = 1 r g r apple 2M r 3 + m M(M r)m02 +6r 3 (r 4M)m 04 (2M + r 3 m 02 ) 2

33 Bi-diagonal case: Instability Instability Confirmed independently by [Brito, Cardoso, Pani 13]

34 Instability of black holes rate of instability Rate of instability Very slow instability! Approximately linear dependance

35 <latexit sha1_base64="kr8cupwieblo+mqm48hrlwhbdgm=">aaaclxicbzdlsgmxfiyz9dbw26hln8eivjqyi6juhikilqs4ttdwkkkzbwisgzkmuiz5ije+ii4el7j1nuwvily9kpdz/eeqnn+pgfxacd6sznz8wujsnpdfxlldw7c3nu9ugetmpbyyunz8paijgniaakzqkssi+4xu/d75wk8+eklokg51pyjnjjqcbhqjbvdlvsh175mgjxsitltjsxmq7kxwdmjfckeslvzephb/bi9yc9sfp+qmc04ldywkyfyvlv3saic45krozjbsddejddnbulpmsjpvxipecpdqh9snfigt1uyg66zw15a2dejpjtbwsp9ojigr1ee+6erid9wkn4czvhqsg9nmqkuuaylw6kegzlchcjadbfnjsgz9ixcw1pwv4i6scgutcn6e4e6upc28w9jxybk+kptl4zsyybvsgcjwwqkogytqar7a4be8g3fwyt1zr9an9tvqzvjjms3wr6zvhzmepmc=</latexit> Non-bidiagonal case General solution for perturbations [EB & Fabbri 14] h µ (g,f) = hµ (g,f) GR + h µ (g,f) (m) h µ GR = rµ r µ <latexit sha1_base64="stnblatoennduztg4+8sbgw9ddg=">aaacixicbzdlsgmxfiyzxmu9jbp0eyycg8tuxmtckljqzrvrc512yksznjtjdlmizeizupfv3lhq6u58gdnpqw09epj5/nm4ox+ymkq05306c/mli0vluzx86tr6xqa7tx2vyimxqekyxbieikuyfasqqwaknkicemhilexdjvzaa5gkxujo9xps5kgjaeqx0hyf7nm3lfrc+mimgvtqdgavidz0bqozalko/ufasuypeyzd3aruwst6wcfzuzqiaphujxchfjvghhohmunknupeopspkppirgz53yisinxdhdkwuibovdpnthzafuvamiqlfuldjp6esbfxqs9d28mr7qppbwt/8xpgr2fnlireaclweffkgnqxhouf21qsrfnfcoqltx+fuiskwtqmmrchlkzpnhxvo+jj0bs5lptlkzryybfsgqnqaqegdk5bbvqbbk/gbbybd+fzexu+

36 <latexit sha1_base64="jd4aqrmcsbnzr9xtrh2tvbqb9fy=">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</latexit> <latexit sha1_base64="1ila0xcj1lfnsgcxlqdr+pikku0=">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</latexit> <latexit sha1_base64="9fmjcmsg5z9vsvwgjeb6kiieiv4=">aaab/nicbvbns8naen34wetxvpdizbeiglbsefuiflx4rgbtoqlls9m0szebudsphnidf8wlbxwv/g5v/hu3bq7a+mdg8d4mm/p8rhanjvntlswula+sltbk6xubw9v2zu69jlnfwzpgilztn2gmugrn4cbyo1gmrl5glx9wpfzbq6y0j+udzanzitktposugjg69r5yicyulyfk7knkajy8ghpvtsto1zkaz5naqsqoqknrf7lbtnoisaccan2poql4ovhaqwcjsptqlha6id3wmvssigkvn9w/wkdgcxayk1ms8et9pzgtsoss8k1nrkcvz72x+j/xssg89hiukxsypnnfysqwexkcbg64yhrezgihip Non-Bi-diagonal case explicit solution for perturbations h µ (f) GR =0 <latexit sha1_base64="c7alixiafdx+ti6kuup+n7idlp4=">aaab/nicbvbns8naen34wetxvpdizbei9vjsefuifdzosyqxhsagzxbtlt3dhn2nugio/huvhls8+ju8+w/ctjlo64obx3szzmwle0avdpxva25+yxfpubrsxl1b39i0t7bvvjxktfwcs1i2q6qio4k4mmpg2okkiiemtmlbxchvprcpacxu9ta h µ (g) GR = e v 0 0 c r c 1 c 0 g 2r c 0 r c 0 csc 2 ( )r 3 1 C A h rr(g) (m) = A(r g r f )e v 4 h rr(f) (m) = apple 1 h rr(g) (m). m 2 h ( ), Since at r!1 v = t + r the perturbations are not regular at infinity. NO unstable modes Non-bidiagonal solution is stable against radial perturbations

37 Non-Bi-diagonal case Non-radial perturbations: QNM [EB,Brito,Pani 15] Decomposition of perturbations in axial and polar modes The quasinormal modes are the same as those of a Schwarzschild BH in GR! - QNM are vibration of a relativistic self-gravitating object. - The boundary conditions are important. For a mode to be QN, the perturbation must behave as ingoing wave near the horizon, perturbations be e i(ωt+k r ) i(k r ωt) and outgoing at the infinity, zon, e e i(k +r ωt).

38 Non-Bi-diagonal case Non-radial perturbations: general perturbations In general we do not have to assume outgoing behavior at infinity. Then the modes are not quasinormal. GR bi-diagonal non bi-diagonal monopole pure gauge dynamical (2nd order PDE) free propagating mode Polar dipole l=1 pure gauge dynamical free propagating mode Polar and axial l>=2 dynamical dynamical (double GR) dynamical (double GR + source term)

39 CONCLUSIONS It is possible to construct non-bidiagonal solutions in massive gravity, which are analogues of corresponding GR solutions (Schwarzschild, charged, rotating). There are hairy massive gravity black holes The non-bidiagonal black holes in massive gravity are stable The bi-diagonal spherically symmetric BHs are unstable due to the helicity-0 mode instability. The rate of instability is extremely small. Superradiant instability for rotating BHs in massive gravity. The fate of unstable BHs? The endpoint of gravitational collapse? Rotating hairy BHs? ds hairy black holes? Do perturbations around black holes contain ghosts?