Sensitivity of Hawking radiation to superluminal dispersion relations luis j. garay 1 Universidad 2 Instituto
Complutense de Madrid
de Estructura de la Materia, CSIC
C.Barceló, ljg, G.Jannes,
Valencia, 3 February 2009
PRD 79 (2009)
intro standard-hr superluminal-hr results conclusions
Outline mod-disp superlum results
Outline
[Introduction]
Introduction Modified dispersion relations Superluminal dispersion relations Summary of results Standard Hawking radiation Geometry — collapse Wave equation — inner product Bogoliubov coefficient β Hawking radiation with superluminal dispersion Modified wave equation Scalar product Bogoliubov coefficient β Modified Hawking spectrum Results Dependence on ωc0 Frequency dependence of κω0 Dependence on time Conclusions Summary Remarks luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
2/26
intro standard-hr superluminal-hr results conclusions
Outline mod-disp superlum results
Modified dispersion relations
[Introduction]
Effects of (Lorentz breaking) superluminal dispersion relations on Hawking radiation produced by collapsing configurations. Hawking’s original derivation rested on the assumption that the low-energy laws of physics, and in particular Lorentz invariance, are preserved up to arbitrarily large scales. Robustness: Analyze effective field theories with high-energy modifications of the dispersion relations. X Subluminal modifications (under reasonable assumptions) dampen the influence of ultra-high frequencies; do not explore arbitrarily large frequencies. ? Superluminal modifications magnify the influence of ultra-high energies.
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
3/26
intro standard-hr superluminal-hr results conclusions
Outline mod-disp superlum results
Superluminal dispersion relations
[Introduction]
Superluminal is qualitatively different to subluminal: The horizon lies ever closer to the singularity for increasing frequencies. This causes the interior of the (zero-frequency) horizon to be exposed to the outside world. Boundary conditions at the horizon Ñ Ñ boundary conditions at the singularity! Moreover, if quantum effects remove the general relativistic singularity, a critical frequency might appear above which no horizon would be experienced at all.
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
4/26
intro standard-hr superluminal-hr results conclusions
Outline mod-disp superlum results
Superluminal dispersion relations
[Introduction]
Superluminal is qualitatively different to subluminal: The horizon lies ever closer to the singularity for increasing frequencies. This causes the interior of the (zero-frequency) horizon to be exposed to the outside world. Boundary conditions at the horizon Ñ Ñ boundary conditions at the singularity! Moreover, if quantum effects remove the general relativistic singularity, a critical frequency might appear above which no horizon would be experienced at all. Our approach: Hawking derivation through the relation between the asymptotic past and future in a collapsing configuration. No extra assumptions on the asymptotic regions (only standard ones: Minkowski geometry in the past and flatness at spatial infinity also in the future). luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
4/26
intro standard-hr superluminal-hr results conclusions
Outline mod-disp superlum results
Summary of results
[Introduction]
Differences in the late-time radiation (superluminal vs. relativistic) At any instant, above a critical frequency, there is no horizon. This induces a cutoff in the modes contributing to radiation. Intensity is lower even if the critical frequency is well above the Planck scale. Radiation will extinguish as time advances.
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
5/26
intro standard-hr superluminal-hr results conclusions
Outline mod-disp superlum results
Summary of results
[Introduction]
Differences in the late-time radiation (superluminal vs. relativistic) At any instant, above a critical frequency, there is no horizon. This induces a cutoff in the modes contributing to radiation. Intensity is lower even if the critical frequency is well above the Planck scale. Radiation will extinguish as time advances. Surface gravity is frequency-dependent and the radiation depends on the physics inside the black hole. The radiation spectrum undergoes a strong qualitative modification: High-frequency radiation is not negligible compared to the low-frequency thermal part, but can even become dominant. This effect becomes more important with increasing critical frequency. luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
5/26
intro standard-hr superluminal-hr results conclusions
geometry wave-eq bogoliubov
Standard Hawking radiation Geometry — collapse Painlevé-Gullstrand 1 + 1 spacetime ds2 = −[c2 − v 2 (t, x)]dt 2 − 2v(t, x)dtdx + dx 2 , regular at the horizon c = speed of light, v = velocity of free-fall acoustic models: c = speed of sound, v = flow velocity
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
6/26
intro standard-hr superluminal-hr results conclusions
geometry wave-eq bogoliubov
Standard Hawking radiation Geometry — collapse Painlevé-Gullstrand 1 + 1 spacetime ds2 = −[c2 − v 2 (t, x)]dt 2 − 2v(t, x)dtdx + dx 2 , regular at the horizon c = speed of light, v = velocity of free-fall acoustic models: c = speed of sound, v = flow velocity Schwarzschild-type velocity profile v¯ (x) (only qualitative features are relevant) s 2M/c2 v¯ (x) = −c x + 2M/c2 v¯ (ξ(t)), x ≤ ξ(t), v(t, x) = v¯ (x), x ≥ ξ(t). luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
6/26
intro standard-hr superluminal-hr results conclusions
geometry wave-eq bogoliubov
Wave equation — inner product
[Standard Hawking radiation]
Wave equation: (∂t + ∂x v)(∂t + v∂x )φ = c2 ∂2x φ Equivalent to 3 + 1 spherical symmetry if backscattering (grey-body factors) is ignored ↔ R Klein-Gordon product: (φ1 , φ2 ) ≡ −i Σt dx φ1 ∂t φ2∗
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
7/26
intro standard-hr superluminal-hr results conclusions
geometry wave-eq bogoliubov
Wave equation — inner product
[Standard Hawking radiation]
Wave equation: (∂t + ∂x v)(∂t + v∂x )φ = c2 ∂2x φ Equivalent to 3 + 1 spherical symmetry if backscattering (grey-body factors) is ignored ↔ R Klein-Gordon product: (φ1 , φ2 ) ≡ −i Σt dx φ1 ∂t φ2∗ Future null coordinates u(t, x) → t − x/c ,
w(t, x) → t + x/c,
when t, x → +∞
Independent of Σt . For t → ∞, Z +∞ ic (φ1 , φ2 ) = − du [φ1 ∂u φ2∗ − φ2∗ ∂u φ1 ]w=+∞ 2 −∞ Z +∞ ∗ ∗ + dw [φ1 ∂w φ2 − φ2 ∂w φ1 ]u=+∞ . −∞
Likewise for past null coordinates U, W and t → −∞. luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
7/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
geometry wave-eq bogoliubov
(i)
[Standard Hawking radiation]
Right-moving positive frequency past and future modes: ψω0 0 = √
1 2πc ω0
0
e−iω U ,
ψω = √
1 2πc ω
e−iωu .
Hawking radiation is encoded in βωω0 ≡ (ψω0 0 , ψω∗ ). Mode mixing happens in the right-moving sector. Therefore, we only need the first term of the previous KG expression: Z ic +∞ du ψω0 0 ∂u ψω − ψω ∂u ψω0 0 w=+∞ βωω0 = − 2 −∞ r Z 0 ω 1 = du e−iω U(u) e−iωu . 0 2π ω All the info is contained in U = U(u) ≡ U(u, w → +∞).
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
8/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
geometry wave-eq bogoliubov
(ii)
[Standard Hawking radiation]
At late times, U = UH − Ae−κu/c , where UH , A and the d¯ v are constants. surface gravity κ ≡ c dx xH We can define a threshold time uI at which an asymptotic observer will start to detect thermal radiation from the black hole. This retarded time corresponds to the moment at which the function U(u) enters the exponential regime. Rewrite as U = UH − A0 e−κ(u−uI )/c , valid for u > uI . Obtain βωω0 . [Dirac-delta normalization]
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
9/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
geometry wave-eq bogoliubov
(ii)
[Standard Hawking radiation]
At late times, U = UH − Ae−κu/c , where UH , A and the d¯ v are constants. surface gravity κ ≡ c dx xH We can define a threshold time uI at which an asymptotic observer will start to detect thermal radiation from the black hole. This retarded time corresponds to the moment at which the function U(u) enters the exponential regime. Rewrite as U = UH − A0 e−κ(u−uI )/c , valid for u > uI . Obtain βωω0 . [Dirac-delta normalization] Narrow wave packets: Pωj ,ul (ω) ≡
eiωul √ , ∆ω
0,
− 21 ∆ω < ω − ωj < 21 ∆ω, otherwise;
centered at ul ≡ u0 + 2πl/∆ω, with u0 an overall reference; central frequency: ωj ≡ j∆ω; width: ∆ω ωj . luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
9/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
geometry wave-eq bogoliubov
(iii)
[Standard Hawking radiation]
Define
Z βωj ,ul
;ω0
≡
dω βωω0 Pωj ,ul (ω),
z = (c∆ω/2κ) ln(ω0 A0 ) ,
zl = (∆ω/2)(ul − uI ).
Number of particles with frequency ωj detected at time ul by an asymptotic observer: Z Nωj ,ul =
+∞
dω0 |βωj ,ul ;ω0 |2 =
0
Z
+∞
dz
−∞
1 sin2 (z − zl ) 2 π(z − zl ) exp(2πcωj /κ) − 1
1 = . exp(2πcωj /κ) − 1
Hawking’s formula (in the absence of backscattering): Planckian spectrum with temperature TH = κ/(2πc).
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
10/26
intro standard-hr superluminal-hr results conclusions
mod-wave-eq scalar-prod bogoliubov spectrum
Hawking radiation with superluminal dispersion Modified wave equation
(i)
Quartic modification to wave equation: 1 4 2 2 (∂t + ∂x v)(∂t + v∂x )φ = c ∂x + 2 ∂x φ , kP Dispersion relation: (ω − vk)2 = c2 k2 1 + k2 /kP2 . kP : ‘Planck scale’ — non-relativistic deviations. In BEC, kP = 2/ξ (inverse of the healing length)
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
11/26
intro standard-hr superluminal-hr results conclusions
mod-wave-eq scalar-prod bogoliubov spectrum
Hawking radiation with superluminal dispersion Modified wave equation
(i)
Quartic modification to wave equation: 1 4 2 2 (∂t + ∂x v)(∂t + v∂x )φ = c ∂x + 2 ∂x φ , kP Dispersion relation: (ω − vk)2 = c2 k2 1 + k2 /kP2 . kP : ‘Planck scale’ — non-relativistic deviations. In BEC, kP = 2/ξ (inverse of the healing length) Modification in the phase and the group velocities, vk,ph ≡ ω/k = ck,ph + v ,
vk,g ≡ dω/dk = ck,g + v ,
due to k-dependent phase and group speeds of light/sound q 1 + 2k2 /kP2 ck,g = c q . ck,ph = c 1 + k2 /kP2 , 1 + k2 /kP2 luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
11/26
intro standard-hr superluminal-hr results conclusions
Modified wave equation
(ii)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
ck,g ck,ph c kP
k
Both speeds ck,g and ck,ph show the same qualitative behaviour. Our results are independent of the choice → ck . Frequency-dependent horizon when ck + v = 0.
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
12/26
intro standard-hr superluminal-hr results conclusions
Modified wave equation
(ii)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
ck,g ck,ph c kP
k
Both speeds ck,g and ck,ph show the same qualitative behaviour. Our results are independent of the choice → ck . Frequency-dependent horizon when ck + v = 0. Since ck becomes arbitrarily high for increasing wave number, there will be a critical ωc0 such that waves with an initial frequency ω0 > ωc0 do not experience a horizon at all. The only exception occurs when the velocity profile ends in a singularity v¯ → −∞, which implies ωc0 → ∞. luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
12/26
intro standard-hr superluminal-hr results conclusions
Scalar product
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
Same as before for t =constant. It is well-defined and conserved: Z ↔ ∂t (φ1 , φ2 ) = dx φ1 ∂4t φ2∗ = 0. There is a preferred time frame: the ‘laboratory’ time t. Perform the same change of coordinates (t, x → u, w) as before and evaluate at t → +∞. The relevant part (right-moving sector) of the inner product is Z ic +∞ du [φ1 ∂u φ2∗ − φ2∗ ∂u φ1 ]w=+∞ . − 2 −∞ Invariant under change of integration variable u → f(u).
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
13/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
(i)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
With slowly varying profiles, the past and future right-moving positive-energy modes are (up to grey-body factors) ψω0 0 ≈ √
1 2πcω0
0
e−iω Uω0 (u,w) ,
ψω ≈ √
e−iωuω (u,w) ,
1 2πcω
where Uω0 (u, w) and uω (u, w) can be obtained by integration of the ray equation dx/dt = ck(ω0 ) (t, x) + v(t, x). Integration for an initial frequency ω0 starting from the past left infinity towards the right gives Uω0 (u, w) =constant. Starting from the future, we can define uω (u, w). Ditto for Wω0 and wω . Uω0 and uω are not null (geometric) coordinates, since they are frequency-dependent, but share many properties with them. Simple (piecewise) profile → explicit integration. luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
14/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
(ii)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
When calculating βω,ω0 , change integration variables u, w → uω (u, w), wω (u, w) w → ∞ implies wω → ∞,
uω → uω (u),
Uω0 → Uω0 (u).
Then, up to grey-body factors, Z ic +∞ duω [ψω0 0 ∂uω ψω − ψω ∂uω ψω0 0 ]wω =+∞ βωω0 = − 2 −∞ r Z 0 1 ω ≈ duω e−iω Uω0 (uω ) e−iωuω . 2π ω0
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
15/26
intro standard-hr superluminal-hr results conclusions
Bogoliubov coefficient β
(iii)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
Integration of the ray equation provides the relation between Uω0 and uω where ω0 is the initial frequency of a ray at the past left infinity and ω = ω(ω0 ) is its final frequency when reaching the future right infinity. Result: Uω0 = UH,ω0 − A0 e−κω0 (uω −u¯ I,ωc0 )/c ; valid for ω0 < ωc0 (for which a horizon is experienced; valid for times uω > uI,ωc0 , where uI,ωc0 is the largest threshold time (this induces a slight underestimation of the effect). The term carrying UH,ω0 is moduloed away, so the only relevant frequency-dependent factor that we are left with is the surface gravity κω0 .
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
16/26
intro standard-hr superluminal-hr results conclusions
Modified Hawking spectrum
(i)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
Smear with narrow packets. Change integration variable from ω0 to z: z=
c∆ω ln(ω0 A0 ), 2κ0
zl,ω0 =
[κ0 ≡ κω0 =0 ]
κω0 ∆ω (ul − u ¯ I,ωc0 ). κ0 2
Number of particles (with frequency ωj at time ul ): h i 2 κ0 κ0 sin κω0 (z − zl,ω0 ) 1 = dz . h i 2 κω 0 exp(2πcωj /κω0 ) − 1 −∞ π κκ00 (z − zl,ω0 ) Z
Nωj ,ul
zc
ω
Dependence on the critical frequency ωc0 (through zc ) Importance of the frequency-dependent κω0 . As ul increases, a smaller part of the central peak will be integrated over, so the radiation will die off. luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
17/26
intro standard-hr superluminal-hr results conclusions
Modified Hawking spectrum
mod-wave-eq scalar-prod bogoliubov spectrum
(ii)
[Hawking radiation with superluminal dispersion]
Given a concrete profile, we can explicitly deduce the relation between κω0 and ω0 as follows: The horizon for a particular initial frequency ω0 is formed when ck2 (xH,ω0 ) = v 2 (xH,ω0 ). This is an equation for xH,ω0 . Use this value in the expression for the surface gravity. For a Schwarzschild profile, κω0
d¯ v 1 ≡ c = κ0 √ dx xH,ω0 2 2
s 1+
ω02 1+4 2 2 c kP
!3/2 .
Everything is ready for evaluation.
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
18/26
intro standard-hr superluminal-hr results conclusions
Modified Hawking spectrum
(iii)
mod-wave-eq scalar-prod bogoliubov spectrum
[Hawking radiation with superluminal dispersion]
Frequencies ω0 > ωc0 do not contribute to the radiation at all, since they do not experience a horizon. This cut-off is not imposed ad hoc, but appears explicitly because of the superluminal character of the system at high frequencies. The critical frequency depends directly on the physics inside the horizon, i.e. on the velocity profile, and can be calculated from the dispersion relation.
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
19/26
intro standard-hr superluminal-hr results conclusions
dep-ωc0 freq-dep-κω0
dep-time
Results Dependence on ωc0 Profile with κω0 = κ0 constant 10
Important decrease even when ωc0 ≥ kP
ωc0 /kP : ∞ 1061 1 10−39 10−78 10−139 10−339
ω3 N
8
6
E
4
Contributions from extremely wide range of frequencies
2
0 0
5
10
luis j. garay (UCM)
ω
15
ω
20
25
30
Superluminal sensitivity of Hawking radiation
Valencia, 2009
20/26
dep-ωc0 freq-dep-κω0
intro standard-hr superluminal-hr results conclusions
dep-time
Frequency dependence of κω0
[Results]
Schwarzschild profile (ωc0 ∼ kP ) 10
50 600
Standard HR
8
6
E
500
κω0 /κ0
ω3 N
ωc0 /kP : 13 10 7 4 0.1
κ
400
300
4
200 2
100 00 0
0 0
20
40
ω
60
ω
80
100
120
10
20
30
0 /k ) log(ωlogω P
401
Important ultraviolet contribution ‘Interior’ of black hole is explored luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
21/26
dep-ωc0 freq-dep-κω0
intro standard-hr superluminal-hr results conclusions
dep-time
Dependence on time
[Results]
For a solar-mass black hole: 10
Standard HR 8
ω3 N
time ul (ms): 0 3.5 5.6 8.0 80
6
E
4
2
0 0
20
40
ω
60
ω
80
100
120
As time increases, radiation dies off Decay rate ∼ 0.3 ms−1 luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
22/26
intro standard-hr superluminal-hr results conclusions
summary remarks
Conclusions Summary
(i)
Collapsing configuration with superluminal dispersion: horizon, surface gravity. . . become frequency-dependent interior of the (zero-frequency) horizon is probed at every moment of collapse: critical ωc0 above which there is no horizon (unless we allow for an untamed singularity)
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
23/26
intro standard-hr superluminal-hr results conclusions
summary remarks
Conclusions Summary
(i)
Collapsing configuration with superluminal dispersion: horizon, surface gravity. . . become frequency-dependent interior of the (zero-frequency) horizon is probed at every moment of collapse: critical ωc0 above which there is no horizon (unless we allow for an untamed singularity) Hawking radiation: ωc0 -dependent: radiation fainter than standard HR radiation dies off κω0 -dependent: high-frequency contribution
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
23/26
intro standard-hr superluminal-hr results conclusions
summary remarks
Conclusions Summary
(i)
Collapsing configuration with superluminal dispersion: horizon, surface gravity. . . become frequency-dependent interior of the (zero-frequency) horizon is probed at every moment of collapse: critical ωc0 above which there is no horizon (unless we allow for an untamed singularity) Hawking radiation: ωc0 -dependent: radiation fainter than standard HR radiation dies off κω0 -dependent: high-frequency contribution Superluminal dispersion leads to strong modification of standard Hawking spectrum, even if ωc0 kP Schwarzschild profile does not reproduce standard spectrum luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
23/26
intro standard-hr superluminal-hr results conclusions
Summary
summary remarks
(ii)
[Conclusions]
Recovering standard Hawking radiation If the velocity profile is such that the surface gravity is frequency independent, then the thermal form is preserved. If we do not regularize the singularity, there is no critical frequency and the Hawking spectrum is unchanged: full intensity stationarity
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
24/26
intro standard-hr superluminal-hr results conclusions
summary remarks
Remarks
[Conclusions]
Conditions for robustness of Hawking radiation 1. freely falling frame is preferred 2. high-frequency excitations start off in ground state at the horizon (w.r.t. freely falling frame) 3. adiabatic evolution
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
25/26
intro standard-hr superluminal-hr results conclusions
summary remarks
Remarks
[Conclusions]
Conditions for robustness of Hawking radiation 1. freely falling frame is preferred 2. high-frequency excitations start off in ground state at the horizon (w.r.t. freely falling frame) 3. adiabatic evolution Superluminal dispersion: Lorentz breaking Ï preferred frame: the lab frame Assumption 1 is not satisfied Horizon approaches singularity as ω0 increases Conditions at horizon conditions at singularity! Low-frequency modes couple to the collapsing geometry Ultrahigh-frequency modes couple to the lab frame Assumption 2 is not satisfied
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
25/26
intro standard-hr superluminal-hr results conclusions
summary remarks
theend
luis j. garay (UCM)
Superluminal sensitivity of Hawking radiation
Valencia, 2009
26/26
