Strong Cosmic Censorship and Quantum Fields
Jochen Zahn

based on arXiv:1912.06047 [with S. Hollands & R.M. Wald]
1st virtual LQP workshop, June 2020

Determinism

Values beyond the Cauchy horizon CH(S) not
determined.

IThe strong cosmic censorship (sCC) conjecture
asserts that determinism generically holds inGR, given initial data which is, in a suitablesense, complete (e.g., asymptotically flat).

I

Cauchy horizons should be generically singular,
so that no observer may cross them.

(SI)

A field φ subject to a hyperbolic field equation,e.g. ( − µ2 )φ = 0, is determined by initial dataon S within the domain of dependence D + (S).

CHIxD + (S)SCH(S)
ΣEach past
inextendiblecausal curve
emanatingfrom x hits S.

Challenges to sCC: Reissner-Nordström spacetime
I

Reissner-Nordström spacetime (RN) has a regular
Cauchy horizon CHR , beyond which a field φ is
not determined by its data on a Cauchy surface Σ.

V+CHIVCHRLCHi+III+RHLH
Σ

H−I−I

Challenges to sCC: Reissner-Nordström spacetime
II

Reissner-Nordström spacetime (RN) has a regular
Cauchy horizon CHR , beyond which a field φ isnot determined by its data on a Cauchy surface Σ.

VWhat happens to the imprudent observer Bob who
falls into the black hole and reaches CHR in finite
proper time?

+CH
IV
CHRL

CH
II

i+

B

I+

RHL

H
I

H−−I

Challenges to sCC: Reissner-Nordström spacetimeIIII

Reissner-Nordström spacetime (RN) has a regular
Cauchy horizon CHR , beyond which a field φ is
not determined by its data on a Cauchy surface Σ.

VWhat happens to the imprudent observer Bob whofalls into the black hole and reaches CHR in finite
proper time?
Alice, who does not enter the black hole, sends
periodic signals to Bob. She needs ∞ proper time
to reach i + , so she may send ∞ many of those.
As Bob receives them in finite proper time, thefrequency diverges as he approaches CHR .
For generic perturbations of fields on RN one
expects a divergence of the stress tensor and thus
the curvature as CHR is approached [Penrose 1974].

+CHIVCHRLCH
IIi+BI+RHLHAIH−−IChallenges to sCC: Reissner-Nordström spacetimeIIIII

Reissner-Nordström spacetime (RN) has a regular
Cauchy horizon CHR , beyond which a field φ isnot determined by its data on a Cauchy surface Σ.

VWhat happens to the imprudent observer Bob who
falls into the black hole and reaches CHR in finite
proper time?
Alice, who does not enter the black hole, sends
periodic signals to Bob. She needs ∞ proper time
to reach i + , so she may send ∞ many of those.
As Bob receives them in finite proper time, thefrequency diverges as he approaches CHR .
For generic perturbations of fields on RN one
expects a divergence of the stress tensor and thus
the curvature as CHR is approached [Penrose 1974].
Christodoulou formulation: sCC holds if generically
1φ 6∈ Hloc
near CHR , i.e., divergence at least asTVV ∼ V −1 .

0IVCHRIIi+

BI+RHLHA
IH−−IChallenges to sCC: Reissner-Nordström-deSitter spacetimeI

With a positive cosmological constant Λ,
the blue-shift of the frequency is counteracted by the cosmological expansion, sothat [Hintz & Vasy 2017]
1 +β2φ ∈ Hloc ,TVV ∼ (−V )−2+2βwithαspectral gap of QNMs
β==κ−
surface gravity at CHCHLCH

Ri+II

HL

R

HLH

Σc

RHc

I

H−III−

Hci−

on Reissner-Nordström-deSitter (RNdS).
I

Near extremal RNdS, scalar fields: β > 12 [Cardoso et al 2017], [Dias et al 2018].

I

Near extremal RNdS, linearized Einstein-Maxwell: β > 2 [Dias at al 2018].

IsCC violated on RNdS (but not on Kerr-dS [Dias et al 2018]).

Challenges to sCC: Reissner-Nordström-deSitter spacetime
I

With a positive cosmological constant Λ,
the blue-shift of the frequency is counteracted by the cosmological expansion, sothat [Hintz & Vasy 2017]
1 +β2φ ∈ Hloc ,TVV ∼ (−V )−2+2βCHLCHRi+IIHLRHLHΣ

cα
spectral gap of QNMs
β==κ−
surface gravity at CHH−−Hci−on Reissner-Nordström-deSitter (RNdS).
I

Near extremal RNdS, scalar fields: β > 12 [Cardoso et al 2017], [Dias et al 2018].

I

Near extremal RNdS, linearized Einstein-Maxwell: β > 2 [Dias at al 2018].

I

sCC violated on RNdS (but not on Kerr-dS [Dias et al 2018]).

I

We find that on RNdS, in any state Ψ which is Hadamard around Σ,hTVV iΨ ∼ CV −2near CHR with C generically non-vanishing and state-independent.

I

sCC rescued by quantum effects.

RHcIwithIIIRNdS spacetime
If (r )IV IIIIIIMetric given byg = −f (r )dt 2 + f (r )−1 dr 2 + r 2 dΩ2 ,r−r+rc2f (r ) = 1 −Λ2M
Q+ 2 − r 2,rr
3I

Roots 0 < r− < r+ < rc of f are the Cauchy, event, cosmological horizon.

IThe corresponding surface gravities are κi = 12 |f 0 (ri )| for i ∈ {−, +, c}.

r

RNdS spacetime
If (r )IV IIIIII

Metric given byg = −f (r )dt 2 + f (r )−1 dr 2 + r 2 dΩ2 ,r−r+rcr

2f (r ) = 1 −Λ2MQ+ 2 − r 2,rr3I

Roots 0 < r− < r+ < rc of f are the Cauchy, event, cosmological horizon.

IThe corresponding surface gravities are κi = 12 |f 0 (ri )| for i ∈ {−, +, c}.

I

Introduce radial null coordinates u, v such thatVg = −f (r )dudv + r 2 dΩ2 .
IVCHv

RIIUH L III

RcHuIvVc

RNdS spacetime
If (r )IV IIIIII

Metric given byg = −f (r )dt 2 + f (r )−1 dr 2 + r 2 dΩ2 ,r−r+rcr2f (r ) = 1 −Λ
2MQ+ 2 − r 2,rr3I

Roots 0 < r− < r+ < rc of f are the Cauchy, event, cosmological horizon.

IThe corresponding surface gravities are κi = 12 |f 0 (ri )| for i ∈ {−, +, c}.

I

Introduce radial null coordinates u, v such thatVg = −f (r )dudv + r 2 dΩ2 .
I0In Kruskal coordinates U, V , Vc , we canextend the metric analytically over HR ,CHR , and HcL .

IVCHRIIUH L IIIRcHI00Vc

The 2d case
CHLCHI

2d toy model in which the angular directions
are suppressed: g = −f (r )dudv .

Tvv (U, v )IClassically, stress tensor conserved and traceless, so ∂u Tvv = 0, implying thatHLRHL

Rc

H

Tvv (U0 , v )κ2TVV (U, v ) = TVc Vc (U0 , v ) 2c (−V )−2+2κc /κ− .
κ−H−I

If field is regular at HcL , then generically singular at CHR .

IExponent dependent on spacetime parameters, coefficient state-dependent.

−HcThe 2d caseCHLCHI2d toy model in which the angular directions
are suppressed: g = −f (r )dudv .

Tvv (U, v )IClassically, stress tensor conserved and traceless, so ∂u Tvv = 0, implying thatHLRHLRcH

Tvv (U0 , v )κ2TVV (U, v ) = TVc Vc (U0 , v ) 2c (−V )−2+2κc /κ− .
κ−H−I

If field is regular at HcL , then generically singular at CHR .

IExponent dependent on spacetime parameters, coefficient state-dependent.

I

For a quantum field, trace anomaly: T = aR.

IIntegration of ∂u hTvv iΨ now yields, near CHR [Birrell & Davies 1978]hTVV iΨ = 2a (κ2c /κ2− − 1) V −2 + O((−V )−2+2κc /κ− )
|{z}
CRI

Power law singularity at CH , exponent universal, coefficient C dependent
on spacetime parameters and state-independent.

I

C 6= 0 up to special spacetime parameters and both signs possible.

−Hc

The 4d case+CHIVCHLCHII

In 4d, trace anomaly and conservation are
not sufficient to integrate the stress tensor: Unknown state-dependent tangentialpressures enter [Birrell & Davies 1978].

Ri+IIRHLH

We define a stationary Unruh state h·iU , which isHadamard in I ∪ II ∪ III, and a stationary comparison state h·iC , which is Hadamard in II ∪ IV.

H−ΣH L IIIcR

Hc

I−

Hci−
I To investigate the divergence of the stress tensor near CH , we compute
RhTVV iΨ = hTVV iΨ − hTVV iU + hTVV iU − hTVV iC + hTVV iC

Can be controlledusing results forthe classical case.
Yields ∼ (−V )−2+2β .

Can be evaluatednumerically.
Yields ∼ CV −2 .

Regularacross CHR

hTVV iΨ − hTVV iU
IIh·iU defined as the vacuum w.r.t. to the Unruh
modes [Hawking 1975, Unruh 1976]−ikVc
Ψink`m ∼ Y`m (θ, φ)eon Hc− ∩ HcR−ikU
Ψupk`m ∼ Y`m (θ, φ)eon H− ∩ HL

Well-definedness and Hadamard property frompropagation of singularities and decay
properties on RNdS [Hintz & Vasy 2017] similarly toSchwarzschild case [Dappiaggi, Moretti, Pinamonti 2011].

CHLCHRi+IIHLRHLH

Σc−Hci−RHcIH−IIIhTVV iΨ − hTVV iU
IIh·iU defined as the vacuum w.r.t. to the Unruh
modes [Hawking 1975, Unruh 1976]−ikVc
Ψink`m ∼ Y`m (θ, φ)eon Hc− ∩ HcR−ikU
Ψupk`m ∼ Y`m (θ, φ)eon H− ∩ HLLCHCH0Rx xi+

HL

Well-definedness and Hadamard property frompropagation of singularities and decay
properties on RNdS [Hintz & Vasy 2017] similarly toSchwarzschild case [Dappiaggi, Moretti, Pinamonti 2011].

ΣRHc

OH−−Hci−000W (x, x ) = hφ(x)φ(x )iΨ − hφ(x)φ(x )iUsmooth in I ∪ II ∪ III.
Decay estimates [Hintz & Vasy 2017]
and Sobolev embedding
P thms:ktµν (−, y )kLp (R− ) . j kbj k2C m
for 1/p > 2 − 2β with β > 21 .

[Verch 1994]

PW (x, x 0 ) = j ±ψ̄j (x)ψj (x 0 )2∞(P − µ 2)ψj = bj ∈ C0 (O)j kbj kC m < ∞ ∀mhTVV iΨ − hTVV iU . (−V )−2+2β .

hTVV iU − hTVV iC+CHIh·iC as the vacuum state w.r.t. Unruh modes on CHL ∪ CH+ .

I

Is Hadamard in II ∪ IV.

IVL

CHCHRIIRHLH

hTVV iU − hTVV iC+CHI

h·iC as the vacuum state w.r.t. Unruh modes on CHL ∪ CH+ .

I

Is Hadamard in II ∪ IV.

I

Compute C̃ = hTvv iU − hTvv iC on CHL :Z ∞Xdω ωn` (ω).
C̃ ∼(2` + 1)
`II

IVL

CH

R

By stationarity, we have the same value on CH , so−2hTVV iU − hTVV iC ∼ C̃ κ−2.
− VRIIR

HHL0The “density of states” n` (ω) expressable in termsof transmission and reflection coefficients Tω` , Rω`for Boulware modes e −iωu . Must be computed
numerically.

CHCHLCHRIHLIRIITHLc

RI
H−I

T−Hc

hTVV iU − hTVV iC+CH
I

h·iC as the vacuum state w.r.t. Unruh modes on CHL ∪ CH+ .

IIs Hadamard in II ∪ IV.

I

Compute C̃ = hTvv iU − hTvv iC on CHL :Z ∞Xdω ωn` (ω).
C̃ ∼(2` + 1)
`IIR

By stationarity, we have the same value on CH , soRCHLCHRIICompatible with results on RN [Zilberman et al 2019].

RIIITHLcHLRI˜
CIT

H−

8.× 10-8

Generically C̃ 6= 0, both signs possible.

RH

HL

0

The “density of states” n` (ω) expressable in termsof transmission and reflection coefficients Tω` , Rω`for Boulware modes e −iωu . Must be computed
numerically.

CH
II−2hTVV iU − hTVV iC ∼ C̃ κ−2.
− VIIVLCH−Hc

6.× 10-8
4.× 10-8
2.× 10-8

0.96

0.97

0.98

-2.× 10-8-4.× 10-8

[Hollands, Klein, Z. 2020]

0.99

1.00

Q/M

Conclusion
I

Strong quantum effects near the Cauchy horizon inside BHs!

I

Global effect!

Conclusion
I

Strong quantum effects near the Cauchy horizon inside BHs!

I

Global effect!

I

RVV ∼ CV −2 corresponds to strong curvature singularity [Tipler 1977].

I

Area spanned by spacelike Jacobi vector fields Z along light-like geodesic
γ approaching CH vanishes (C > 0) or diverges (C < 0) on CH.

I

Infinite crushing (C > 0) or stretching (C < 0) of observer.

C >0C <0CHZγCHZγConclusion
I

Strong quantum effects near the Cauchy horizon inside BHs!

IGlobal effect!

I

RVV ∼ CV −2 corresponds to strong curvature singularity [Tipler 1977].

I

Area spanned by spacelike Jacobi vector fields Z along light-like geodesic
γ approaching CH vanishes (C > 0) or diverges (C < 0) on CH.

I

Infinite crushing (C > 0) or stretching (C < 0) of observer.

C >0

C <0CHZγCHZγTHANK YOU FOR YOUR ATTENTION!