Stochastic Gravity in Conformally Flat Spacetimes Hing-Tong Cho Department of Physics, Tamkang University, Taiwan (Collaboration with Bei-Lok Hu, University of Maryland, USA) University of Witwatersrand - Feb 17, 2015 Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
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Stochastic Gravity in Conformally FlatSpacetimes
Hing-Tong Cho
Department of Physics, Tamkang University, Taiwan
(Collaboration with Bei-Lok Hu, University of Maryland, USA)
University of Witwatersrand - Feb 17, 2015
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Outline
I. Introduction
II. Brownian motion paradigm
III. Stochastic gravity
IV. Conformal transformation and influence functional
V. Conformally flat spacetimes
VI. Noise kernels of Robertson-Walker spacetimes
VII. Discussions
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
I. Introduction
Scattering problems:
⟨0, out|0, in⟩J = e iW [J]
=
∫Dϕ e iS[ϕ]+i
∫Jϕ
where W [J] is the generating function for n-point functions.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Time evolution problems or initial value problems in field theory:Schwinger’s in-in formalism
⟨0, in|ϕ(x1)ϕ(x2) . . . ϕ(xn)|0, in⟩
In-in generating functional
J−⟨0, in|0, in⟩J+ = e iW [J+,J−]
=∑α
J−⟨0, in|α, out⟩⟨α, out|0, in⟩J+
where |α, out⟩ is a complete set of out-states on some spacelikehypersurface Σ.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Using the path integrals
e iW [J+,J−]
=
∫dϕ′
∫Dϕ+Dϕ− e iS[ϕ+]+i
∫J+ϕ+−iS[ϕ−]−i
∫J−ϕ−
=
∫CTP
Dϕ+Dϕ− e iS[ϕ+]+i∫J+ϕ+−iS[ϕ−]−i
∫J−ϕ−
where ϕ+ = ϕ− = ϕ′ on Σ.
CTP means Closed-Time-Path.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
II. Brownian motion paradigm
A particle (system) coupled linearly to a set of harmonic oscillators(environment):
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Although both the closed and the open Einstein universes areconformally flat, their conformal vacua are not the same. Theconformal vacuum of the Einstein universe is the Minkowskivacuum, while that of the open Einstein universe is the Rindlervacuum.
Like the Minkowski and the Rindler vacua, the vacua of theEinstein and open Einstein universes are related by thermalization.
G+O (x , x ′)thermal =
∞∑n=−∞
∆s/a
4π2 sinh(∆s/a)[−(∆tO + inβ)2 +∆s2]
= G+E
∣∣a→ia
with β = 1/T = 1/2πa.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Conformally flat spacetimes with Minkowski conformal vacuum:
Spatially flat de Sitter, flat Robertson-Walker, Einstein universe,global de Sitter, closed Robertson-Walker
Conformally flat spacetimes with Rindler conformal vacuum:
Open Einstein universe, Milne universe, open Robertson-Walker,static de Sitter
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
VI. Noise kernels of Robertson-Walker spacetimes
(HTC and B.-L. Hu, CQG 32 (2015) 055006)
The new ingredient in stochastic gravity is the stochastic force ξµνinduced by the fluctuations of the quantum fields.
The correlation function of the stochastic force is the noise kernel,
Nµνα′β′(x , x ′) =⟨ξµν(x)ξα′β′(x ′)
⟩s
The noise kernel is also the correlation function of the stressenergy tensor Tµν of the quantum field
Nµνα′β′(x , x ′) =⟨tµν(x), tα′β′(x ′)
⟩q
where tµν = Tµν − ⟨Tµν⟩.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Hence, the noise kernel can be obtained from the second derivativeon the imaginary part of the influence action
Nµνα′β′(x , x ′) = g+µρ(x)g+νσ(x)g+α′ξ′(x′)g+β′ζ′(x
′)
4√g+(x)g+(x ′)
δ2ImSIF [g+, g−]
δg+ρσ(x)δg+ξ′ζ′(x ′)
∣∣∣∣g+=g−=g
Between two conformally related spacetimes gµν = Ω2gµν , onetherefore has
Nµνα′β′(x , x ′) = Ω−2(x)Nµνα′β′(x , x ′)Ω−2(x ′)
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
For a spatially homogeneous spacetime one can write
where si = ∇i (∆s) and sj ′ = ∇j ′(∆s) are the derivatives on thespatial geodesic distance ∆s between x and x ′. Also, gij ′ is theparallel transport bivector such that si = −gi
j ′sj ′ .
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
From the traceless condition of the noise kernel is given by
C11 − C31 − 3C32 = 0
C21 + C51 + 3C52 − 2C53 = 0
C31 − C61 − 3C62 + 4C63 = 0
C32 − C62 − 2C64 − 3C65 = 0.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
The conservation condition of the noise kernel is given by
∂C11
∂∆η+
∂C21
∂∆s+ 2AC21
∂C21
∂∆η+
∂C31
∂∆s+
∂C32
∂∆s+ 2AC31 = 0
∂C21
∂∆η− ∂C41
∂∆s+
C42
∂∆s− 2AC41 + 2(A+ C )C42 = 0
∂C31
∂∆η− ∂C51
∂∆s+ 2
C53
∂∆s− 2AC51 + 2(2A+ 3C )C53 = 0
∂C32
∂∆η− C52
∂∆s− 2AC52 − 2CC53 = 0
∂C41
∂∆η+
∂C51
∂∆s+
∂C52
∂∆s− ∂C53
∂∆s+ 2AC51 + CC52 − (A+ 2C )C53 = 0
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
∂C42
∂∆η+
∂C53
∂∆s+ CC52 + 3AC53 = 0
∂C51
∂∆η− ∂C61
∂∆s− ∂C62
∂∆s+ 2
∂C63
∂∆s
−2AC61 − 2CC62 + 2(A+ 3C )C63 = 0
∂C52
∂∆η− ∂C62
∂∆s− ∂C65
∂∆s− 2AC62 − 2CC63 + 2(A+ C )C64 = 0
∂C53
∂∆η− ∂C63
∂∆s+
∂C64
∂∆s− CC62 − 3AC63 + 3(A+ C )C64 = 0
where A = 1/∆s and C = −1/∆s for R3, A = cot(∆s) andC = − csc(∆s) for S3, and A = coth(∆s) and C = −csch(∆s) forH3
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
From the Wightman function one can calculate the noise kernel(Phillips and Hu 2001).
For the Minkowski spacetime we have
(C11)M =3∆s4 + 10∆s2∆η2 + 3∆η4
12π4(−∆η2 +∆s2)6; (C21)M =
2∆s∆η(∆s2 +∆η2)
3π4(−∆η2 +∆s2)6
(C31)M =4∆η2∆s2
3π4(−∆η2 +∆s2)6; (C32)M =
1
12π4(−∆η2 +∆s2)4
(C41)M = −∆s2(3∆η2 +∆s2)
3π4(−∆η2 +∆s2)6; (C42)M = −
∆η2 +∆s2
6π4(−∆η2 +∆s2)5
(C51)M = −4∆η∆s3
3π4(−∆η2 +∆s2)6; (C52)M = 0 ; (C53)M = −
∆η∆s
3π4(−∆η2 +∆s2)5
(C61)M =4∆s4
3π4(−∆η2 +∆s2)6; (C62)M = 0 ; (C63)M =
∆s2
3π4(−∆η2 +∆s2)5
(C64)M =1
6π4(−∆η2 +∆s2)4; (C65)M = −
1
12π4(−∆η2 +∆s2)4
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
For the flat Robertson-Walker spacetime,
ds2 = a2(η)(−dη2 + dr2 + r2dθ2 + r2 sin2 θdϕ2
)the corresponding coefficients for the noise kernel are given by
(Cij)fFRW = a−2(η) (Cij)M a−2(η′)
For a(η) = −1/Hη one has the de Sitter in spatially flatcoordinates.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
One can obtain the noise kernel for the static de Sitter case if firsta conformal transformation with
Ω(χ) =1
α coshχ
and a further coordinate transformation with
tanhχ = αr
are made on the open Einstein noise kernel.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
The coefficient functions (Cij)sdS of the noise kernel in static deSitter spacetime are given by
(Cij)sdS = (1− r2)−1(Cij)O(1− r ′2)−1,
where the geodesic distance
∆s = cosh−1(1− r2)−1/2(1− r ′2)−1/2[1− rr ′(cos θ cos θ′ + sin θ sin θ′ cos(ϕ− ϕ′))
] .
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
For the Rindler space
ds2 = −ξ2dτ2 + dξ2 + dy2 + dz2
one can obtain the noise kernel after the series of conformal andcoordinate transformations.
It is however easier to work directly with the Rindler Wightmanfunction
G+R =
1
4π2
(α
ξξ′ sinhα
)(1
−(τ − τ ′)2 + α2
),
where
coshα =ξ2 + ξ′2 + (y − y ′)2 + (z − z ′)2
2ξξ′.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
For example,
Nτττ′τ′ =G2
9α2ξ3ξ′3
ξ3ξ′3 + (1 − ξ
2ξ′2)(ξ2 + ξ
′2) cothαcschα
+ξξ′[(1 + 5α2) + (2 − 3α2)ξ2ξ′2 − 2α(4 − ξ
2ξ′2) cothα
]csch
2α
+α(1 − ξ2ξ′2)(ξ2 + ξ
′2)(1 − 2α cothα)csch3α
+α2ξξ
′(7 − 4ξ2ξ′2)csch4
−8π2G3 sinhα
9α3ξ2ξ′2
ξ3ξ′3
[(2 + 5α2) − 6α cothα
]−α(ξ2 + ξ
′2)(3 − ξ2ξ′2 − 2α cothα)cschα
+2α2ξξ
′[4 + ξ
2ξ′2 − α(4 − ξ
2ξ′2) cothα
]csch
2α
+α3(1 − ξ
2ξ′2)(ξ2 + ξ
′2)csch3α
+64π4G4 sinh2 α
9α2ξξ′
3ξξ′
[3 + (3 + α
2)ξ2ξ′2 − 2αξ2ξ′2 cothα
]−α(ξ2 + ξ
′2)(5 − α cothα)cschα + 5α2ξξ
′csch
2
−1024π6G5 sinh3 α
9α
6ξξ′(1 + 2ξ2ξ′2) − α(ξ2 + ξ
′2)cschα
+16384π8G6 sinh4 α
9
ξ2ξ′2(1 + 3ξ2ξ′2)
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
VII. Discussions
1. The noise kernel is the two point correlation function of thestress-energy tensor. It represents the backreaction of thequantum fluctuations of the matter field onto the backgroundspacetime. Therefore, it is interesting to investigate thebehaviors of the noise kernel near horizons as well as initialsingularities of various FRW spacetimes.
2. Other than the noise kernel we need to consider the conformaltransformation of the Einstein-Langevin equation in detail. Inparticular, we should also investigate the transformation of theterms related to the dissipation kernel.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
3. Next we shall try to solve the Einstein-Langevin equation
Gµν [g + h] = κ (⟨Tµν [g + h]⟩+ ξµν [g ])
Here gµν is the background Robertson-Walker spacetime withthe scaling factor a(η) being a solution to the semiclassicalEinstein equation. To avoid solutions that are not physical,one might resort to consistent procedures like the orderreduction method of Parker and Simon.
4. Subsequently, one could solve for hµν using standardperturbation methods around the Robertson-Walkerbackgrounds. Here ξµν acts like an external force. Thecorrelator ⟨hµν(x)hα′β′(x ′)⟩s can therefore be evaluated withthe appropriate noise kernels.
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes