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The BTZ black hole violates strong cosmic censorship

Dias, Oscar J.C., Reall, Harvey S., Santos, Jorge E.·2019·Journal of High Energy Physics
DOI 10.1007/jhep12(2019)097
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Abstract

We investigate the stability of the inner horizon of a rotating BTZ black hole. We show that linear perturbations arising from smooth initial data are arbitrarily differentiable at the inner horizon if the black hole is sufficiently close to extremality. This is demonstrated for scalar fields, for massive Chern-Simons fields, for Proca fields, and for massive spin-2 fields. Thus the strong cosmic censorship conjecture is violated by a near-extremal BTZ black hole in a large class of theories. However, we show that a weaker \rough" version of the conjecture is respected. We calculate the renormalized energymomentum tensor of a scalar field in the Hartle-Hawking state in the BTZ geometry. We show that the result is finite at the inner horizon of a near-extremal black hole. Hence the backreaction of vacuum polarization does not enforce strong cosmic censorship.

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Figures

Figure 1 Penrose diagram for the non-extremal BTZ black hole. Region I is the black hole exterior and region II the interior. The asymptotic timelike boundary is I R , the future event horizon is H + R and the past event horizon is at H -R . Region III is the white hole region. Region IV is another asymptotically AdS region with timelike boundary I L and future event horizon H + L . CH + L,R are the 'left' and 'right' future Cauchy horizons, respectively.
Figure 1 Penrose diagram for the non-extremal BTZ black hole. Region I is the black hole exterior and region II the interior. The asymptotic timelike boundary is I R , the future event horizon is H + R and the past event horizon is at H -R . Region III is the white hole region. Region IV is another asymptotically AdS region with timelike boundary I L and future event horizon H + L . CH + L,R are the 'left' and 'right' future Cauchy horizons, respectively.CC-BY · From the PDF
Figure 2 The conformal dimension ∆ as a function of the scalar field mass µ 2 L 2 . The upper branch (blue) corresponds to ∆ + , the lower branch (orange) to ∆ -. Note ∆ ± ( µ BF ) = 1 and ∆ -(0) = 0.
Figure 2 The conformal dimension ∆ as a function of the scalar field mass µ 2 L 2 . The upper branch (blue) corresponds to ∆ + , the lower branch (orange) to ∆ -. Note ∆ ± ( µ BF ) = 1 and ∆ -(0) = 0.CC-BY · From the PDF
Figure 3 Our (characteristic) initial-boundary data for solving the Klein-Gordon equation consists of smooth (outgoing) wavepackets on H + L and H -R and a smooth wavepacket source at I R . We will determine the solution in regions I and II and investigate its smoothness at CH + R .
Figure 3 Our (characteristic) initial-boundary data for solving the Klein-Gordon equation consists of smooth (outgoing) wavepackets on H + L and H -R and a smooth wavepacket source at I R . We will determine the solution in regions I and II and investigate its smoothness at CH + R .CC-BY · From the PDF
Figure 4 Contours of integration C and C m in the complex ω plane and the relevant poles: poles marked in red contribute to the integral and poles marked in black don't. The pole with smallest | Im( ω ) | on the left is the fundamental retrograde quasinormal mode ϖ m and on the right we show the poles with ω -m Ω -= -inκ -which only give a smooth contribution at CH + R .
Figure 4 Contours of integration C and C m in the complex ω plane and the relevant poles: poles marked in red contribute to the integral and poles marked in black don't. The pole with smallest | Im( ω ) | on the left is the fundamental retrograde quasinormal mode ϖ m and on the right we show the poles with ω -m Ω -= -inκ -which only give a smooth contribution at CH + R .CC-BY · From the PDF
Figure 5 Plot of L 3 〈 0 | T rr | 0 〉 as a function of log 10 ( ζ -χ ) computed for ∆ = 1, α = 2 π and β = 3 / 2. The black dot denotes L 3 t rr .
Figure 5 Plot of L 3 〈 0 | T rr | 0 〉 as a function of log 10 ( ζ -χ ) computed for ∆ = 1, α = 2 π and β = 3 / 2. The black dot denotes L 3 t rr .CC-BY · From the PDF
Figure 6 Left panel: Logarithmic plot of L 3 t rr as a function of β . Right panel: L 3 ( β -1) t rr as a function of β . Both plots are for ∆ = 2 and α = 2 π . The vertical red dashed line marks β = 1. The plots show that lim r → r -〈 0 | T rr | 0 〉 diverges as 1 / ( β -1) as β → 1 + .
Figure 6 Left panel: Logarithmic plot of L 3 t rr as a function of β . Right panel: L 3 ( β -1) t rr as a function of β . Both plots are for ∆ = 2 and α = 2 π . The vertical red dashed line marks β = 1. The plots show that lim r → r -〈 0 | T rr | 0 〉 diverges as 1 / ( β -1) as β → 1 + .CC-BY · From the PDF
Figure 7 Left panel: Plot of ( ζ -χ ) 1 -β L 3 〈 0 | T rr | 0 〉 as a function of log 10 ( ζ -χ ) showing the complete range of ζ that we probed. Right panel: Plot of ( ζ -χ ) 1 -β L 3 〈 0 | T rr | 0 〉 in the range -200 < log 10 ( ζ -χ ) < -180 corresponding to the orange region on the left panel. Both panels are for ∆ = 1, β = 1 / 2 and α = 2 π .
Figure 7 Left panel: Plot of ( ζ -χ ) 1 -β L 3 〈 0 | T rr | 0 〉 as a function of log 10 ( ζ -χ ) showing the complete range of ζ that we probed. Right panel: Plot of ( ζ -χ ) 1 -β L 3 〈 0 | T rr | 0 〉 in the range -200 < log 10 ( ζ -χ ) < -180 corresponding to the orange region on the left panel. Both panels are for ∆ = 1, β = 1 / 2 and α = 2 π .CC-BY · From the PDF
Figure 8 Left panel: | σ N | computed for several values of N . Right panel: | ( β -N ) σ N | as a function of β computed for several values of N . In both panels, the solid blue curve corresponds to N = 1, the orange dashed line to N = 2 and the dotted green line to N = 3. Finally, the vertical lines represent β = 1 , 2 , 3, from left to right. In all cases we fixed α = 2 π .
Figure 8 Left panel: | σ N | computed for several values of N . Right panel: | ( β -N ) σ N | as a function of β computed for several values of N . In both panels, the solid blue curve corresponds to N = 1, the orange dashed line to N = 2 and the dotted green line to N = 3. Finally, the vertical lines represent β = 1 , 2 , 3, from left to right. In all cases we fixed α = 2 π .CC-BY · From the PDF
Figure 9 Left panel: | Ξ N ( r ) | + N -1 , computed for several values of N , as a function of log 10 ( ζ -χ ). The N -1 offset has been added to prevent the curves lying on top of each other. Right panel: ∣ ∣ ∣ Ξ N ( r ) -Ξ N ( r ) ∣ ∣ ∣ in the region -190 < log 10 ( ζ -χ ) < -185, corresponding to the red band on the left panel. In both panels, the blue disks corresponds to N = 1, the orange squares to N = 2 and the green diamonds to N = 3. In all cases we we have α = 2 π and β = 1 / 2.
Figure 9 Left panel: | Ξ N ( r ) | + N -1 , computed for several values of N , as a function of log 10 ( ζ -χ ). The N -1 offset has been added to prevent the curves lying on top of each other. Right panel: ∣ ∣ ∣ Ξ N ( r ) -Ξ N ( r ) ∣ ∣ ∣ in the region -190 < log 10 ( ζ -χ ) < -185, corresponding to the red band on the left panel. In both panels, the blue disks corresponds to N = 1, the orange squares to N = 2 and the green diamonds to N = 3. In all cases we we have α = 2 π and β = 1 / 2.CC-BY · From the PDF

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Metadata

Authors
Dias, O. J. C., Reall, H. S., & Santos, J. E.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2019-12
Volume
2019
Issue
12
ISSN
1029-8479
DOI
10.1007/jhep12(2019)097
Cited by
60
References
1
Language
en
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