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Semiclassical gravity in the far ﬁeld of stars and black holes

Jake Percival

December 18, 2018

D E Jake Percival φ2 far from stars and black holes December 18, 2018 1 / 22 Overview

1 Introduction: Semiclassical gravity

2 Expectation values of scalar ﬁelds

3 Examples of taking diﬀerences

4 Diﬀerences between spacetimes: Previous Work

5 Diﬀerences between spacetimes: New Work

D E Jake Percival φ2 far from stars and black holes December 18, 2018 2 / 22 This leads to the semiclassical Einstein Field Equations,

Gµν = 8π hTµνi

hTµνi is non-local. It requires information about global boundary conditions on the ﬁelds.

hTµνi is constructed out of products of matter ﬁeld operators. The expectation value of such quantities is, formally speaking, inﬁnite!

Introduction: Semiclassical gravity

Semiclassical gravity is a ﬁrst approximation to a theory of quantum gravity, where we quantize matter ﬁelds while leaving the background spacetime classical.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 3 / 22 hTµνi is non-local. It requires information about global boundary conditions on the ﬁelds.

hTµνi is constructed out of products of matter ﬁeld operators. The expectation value of such quantities is, formally speaking, inﬁnite!

Introduction: Semiclassical gravity

Semiclassical gravity is a ﬁrst approximation to a theory of quantum gravity, where we quantize matter ﬁelds while leaving the background spacetime classical. This leads to the semiclassical Einstein Field Equations,

Gµν = 8π hTµνi

D E Jake Percival φ2 far from stars and black holes December 18, 2018 3 / 22 hTµνi is constructed out of products of matter ﬁeld operators. The expectation value of such quantities is, formally speaking, inﬁnite!

Introduction: Semiclassical gravity

Gµν = 8π hTµνi

hTµνi is non-local. It requires information about global boundary conditions on the ﬁelds.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 3 / 22 Introduction: Semiclassical gravity

Gµν = 8π hTµνi

hTµνi is non-local. It requires information about global boundary conditions on the ﬁelds.

hTµνi is constructed out of products of matter ﬁeld operators. The expectation value of such quantities is, formally speaking, inﬁnite!

D E Jake Percival φ2 far from stars and black holes December 18, 2018 3 / 22 Rather than hTµνi we can consider the vacuum polarization of the scalar ﬁeld, φ2 = hφ(x)φ(x)i . This is calculated from the Green’s function of the ﬁeld,

2 0 0 φ = lim φ(x)φ(x ) = lim GE x, x x0→x x0→x

Scalar ﬁeld in a spherically symmetric spacetime

Toy model: Massless scalar ﬁeld φ, spherically symmetric, asymptotically ﬂat spacetime. After a Wick rotation τ = −it this has the Euclidean metric

ds2 = f (r) dτ 2 + h (r) dr 2 + r 2dΩ2.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 4 / 22 This is calculated from the Green’s function of the ﬁeld,

2 0 0 φ = lim φ(x)φ(x ) = lim GE x, x x0→x x0→x

Scalar ﬁeld in a spherically symmetric spacetime

Toy model: Massless scalar ﬁeld φ, spherically symmetric, asymptotically ﬂat spacetime. After a Wick rotation τ = −it this has the Euclidean metric

ds2 = f (r) dτ 2 + h (r) dr 2 + r 2dΩ2.

Rather than hTµνi we can consider the vacuum polarization of the scalar ﬁeld, φ2 = hφ(x)φ(x)i .

D E Jake Percival φ2 far from stars and black holes December 18, 2018 4 / 22 Scalar ﬁeld in a spherically symmetric spacetime

Toy model: Massless scalar ﬁeld φ, spherically symmetric, asymptotically ﬂat spacetime. After a Wick rotation τ = −it this has the Euclidean metric

ds2 = f (r) dτ 2 + h (r) dr 2 + r 2dΩ2.

Rather than hTµνi we can consider the vacuum polarization of the scalar ﬁeld, φ2 = hφ(x)φ(x)i . This is calculated from the Green’s function of the ﬁeld,

2 0 0 φ = lim φ(x)φ(x ) = lim GE x, x x0→x x0→x

D E Jake Percival φ2 far from stars and black holes December 18, 2018 4 / 22 0 GE diverges as x → x. The divergence can be subtracted away using Hadamard counterterms. This is renormalisation. The counterterms do not depend on the global boundary conditions.

Scalar ﬁeld in a spherically symmetric spacetime

The Green’s function GE is the solution to the Klein-Gordon equation of motion for the ﬁeld, with appropriate global boundary conditions.

δ4 (x, x0) [g µν∇ ∇ − ξR (x)] G x, x0 = − √ µ ν E g

D E Jake Percival φ2 far from stars and black holes December 18, 2018 5 / 22 The counterterms do not depend on the global boundary conditions.

Scalar ﬁeld in a spherically symmetric spacetime

The Green’s function GE is the solution to the Klein-Gordon equation of motion for the ﬁeld, with appropriate global boundary conditions.

δ4 (x, x0) [g µν∇ ∇ − ξR (x)] G x, x0 = − √ µ ν E g

0 GE diverges as x → x. The divergence can be subtracted away using Hadamard counterterms. This is renormalisation.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 5 / 22 Scalar ﬁeld in a spherically symmetric spacetime

The Green’s function GE is the solution to the Klein-Gordon equation of motion for the ﬁeld, with appropriate global boundary conditions.

δ4 (x, x0) [g µν∇ ∇ − ξR (x)] G x, x0 = − √ µ ν E g

0 GE diverges as x → x. The divergence can be subtracted away using Hadamard counterterms. This is renormalisation. The counterterms do not depend on the global boundary conditions.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 5 / 22 Consider a quantum ﬁeld in Minkowski spacetime with and without two parallel mirrors. The diﬀerence in the energy density of the vacuum state can be calculated c ∆ T t = T t − T t ∼ ~ t t mirror t Mink d4 where d is the distance between the mirrors.

Example: The Casimir eﬀect

How important are these global boundary conditions?

D E Jake Percival φ2 far from stars and black holes December 18, 2018 6 / 22 The diﬀerence in the energy density of the vacuum state can be calculated c ∆ T t = T t − T t ∼ ~ t t mirror t Mink d4 where d is the distance between the mirrors.

Example: The Casimir eﬀect

How important are these global boundary conditions? Consider a quantum ﬁeld in Minkowski spacetime with and without two parallel mirrors.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 6 / 22 Example: The Casimir eﬀect

How important are these global boundary conditions? Consider a quantum ﬁeld in Minkowski spacetime with and without two parallel mirrors. The diﬀerence in the energy density of the vacuum state can be calculated c ∆ T t = T t − T t ∼ ~ t t mirror t Mink d4 where d is the distance between the mirrors.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 6 / 22 A diﬀerence in this between two scenarios can be written Z 2 ren 2 ren X ω` ω` φ 1 = φ 2 + dω F1 − F2 `

These two scenarios could be two diﬀerent quantum states, or the same quantum state on two diﬀerent spacetimes.

Taking diﬀerences

In general, the renormalised vacuum polarisation takes the form

ren Z X φ2 = dω F ω` − counterterms `

D E Jake Percival φ2 far from stars and black holes December 18, 2018 7 / 22 These two scenarios could be two diﬀerent quantum states, or the same quantum state on two diﬀerent spacetimes.

Taking diﬀerences

In general, the renormalised vacuum polarisation takes the form

ren Z X φ2 = dω F ω` − counterterms `

A diﬀerence in this between two scenarios can be written Z 2 ren 2 ren X ω` ω` φ 1 = φ 2 + dω F1 − F2 `

D E Jake Percival φ2 far from stars and black holes December 18, 2018 7 / 22 Taking diﬀerences

In general, the renormalised vacuum polarisation takes the form

ren Z X φ2 = dω F ω` − counterterms `

A diﬀerence in this between two scenarios can be written Z 2 ren 2 ren X ω` ω` φ 1 = φ 2 + dω F1 − F2 `

These two scenarios could be two diﬀerent quantum states, or the same quantum state on two diﬀerent spacetimes.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 7 / 22 Fabbri and Anderson (2007) applied this idea to the spacetimes of a star and a black hole. Outside a star and a Schwarzschild black hole, the spacetime is locally the same! Let’s deﬁne the diﬀerence,

2 2 ren 2 ren ∆ φ = φ star − φ Schw

Diﬀerences between spacetimes: Previous work

Birkhoﬀ’s Theorem (rough version) (Birkhoﬀ, 1923) Outside two spherically symmetric gravitational sources of the same mass M the spacetime metric is the same.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 8 / 22 Outside a star and a Schwarzschild black hole, the spacetime is locally the same! Let’s deﬁne the diﬀerence,

2 2 ren 2 ren ∆ φ = φ star − φ Schw

Diﬀerences between spacetimes: Previous work

Birkhoﬀ’s Theorem (rough version) (Birkhoﬀ, 1923) Outside two spherically symmetric gravitational sources of the same mass M the spacetime metric is the same.

Fabbri and Anderson (2007) applied this idea to the spacetimes of a star and a black hole.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 8 / 22 2 2 ren 2 ren ∆ φ = φ star − φ Schw

Diﬀerences between spacetimes: Previous work

Birkhoﬀ’s Theorem (rough version) (Birkhoﬀ, 1923) Outside two spherically symmetric gravitational sources of the same mass M the spacetime metric is the same.

Fabbri and Anderson (2007) applied this idea to the spacetimes of a star and a black hole. Outside a star and a Schwarzschild black hole, the spacetime is locally the same! Let’s deﬁne the diﬀerence,

D E Jake Percival φ2 far from stars and black holes December 18, 2018 8 / 22 Diﬀerences between spacetimes: Previous work

Birkhoﬀ’s Theorem (rough version) (Birkhoﬀ, 1923) Outside two spherically symmetric gravitational sources of the same mass M the spacetime metric is the same.

2 2 ren 2 ren ∆ φ = φ star − φ Schw

D E Jake Percival φ2 far from stars and black holes December 18, 2018 8 / 22 Z ∞ 0 1 0 GE x, x = 2 dω cos ω τ − τ 4π 0 ∞ X × (2` + 1) P` (cos γ) C p (r<) q (r>) `=0

The p(r) and q(r) are radial solutions with Dirichlet boundary conditions at the inner and outer boundaries respectively. C is a normalisation constant deﬁned using p and q.

Diﬀerences between spacetimes: Previous work

2 0 Recall that φ = GE (x, x )

D E Jake Percival φ2 far from stars and black holes December 18, 2018 9 / 22 The p(r) and q(r) are radial solutions with Dirichlet boundary conditions at the inner and outer boundaries respectively. C is a normalisation constant deﬁned using p and q.

Diﬀerences between spacetimes: Previous work

2 0 Recall that φ = GE (x, x )

Z ∞ 0 1 0 GE x, x = 2 dω cos ω τ − τ 4π 0 ∞ X × (2` + 1) P` (cos γ) C p (r<) q (r>) `=0

D E Jake Percival φ2 far from stars and black holes December 18, 2018 9 / 22 C is a normalisation constant deﬁned using p and q.

Diﬀerences between spacetimes: Previous work

2 0 Recall that φ = GE (x, x )

Z ∞ 0 1 0 GE x, x = 2 dω cos ω τ − τ 4π 0 ∞ X × (2` + 1) P` (cos γ) C p (r<) q (r>) `=0

The p(r) and q(r) are radial solutions with Dirichlet boundary conditions at the inner and outer boundaries respectively.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 9 / 22 Diﬀerences between spacetimes: Previous work

2 0 Recall that φ = GE (x, x )

Z ∞ 0 1 0 GE x, x = 2 dω cos ω τ − τ 4π 0 ∞ X × (2` + 1) P` (cos γ) C p (r<) q (r>) `=0

The p(r) and q(r) are radial solutions with Dirichlet boundary conditions at the inner and outer boundaries respectively. C is a normalisation constant deﬁned using p and q.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 9 / 22 pstar = αpSchw + βqSchw

This relation produces a simple expression for ∆ φ2

∞ 1 Z ∞ X β ∆ φ2 = dω (2` + 1)C q2 4π2 Schw α Schw 0 `=0

This sum and integral rapidly converge, so it is justiﬁed to look at the small ω behaviour of the ` = 0 mode.

Diﬀerences between spacetimes: Previous work

C, p and q can be found in both the star and Schwarzschild spacetimes. Outside the gravitational source they are related by Birkhoﬀ’s theorem,

D E Jake Percival φ2 far from stars and black holes December 18, 2018 10 / 22 This relation produces a simple expression for ∆ φ2

∞ 1 Z ∞ X β ∆ φ2 = dω (2` + 1)C q2 4π2 Schw α Schw 0 `=0

This sum and integral rapidly converge, so it is justiﬁed to look at the small ω behaviour of the ` = 0 mode.

Diﬀerences between spacetimes: Previous work

C, p and q can be found in both the star and Schwarzschild spacetimes. Outside the gravitational source they are related by Birkhoﬀ’s theorem,

pstar = αpSchw + βqSchw

D E Jake Percival φ2 far from stars and black holes December 18, 2018 10 / 22 ∞ 1 Z ∞ X β ∆ φ2 = dω (2` + 1)C q2 4π2 Schw α Schw 0 `=0

This sum and integral rapidly converge, so it is justiﬁed to look at the small ω behaviour of the ` = 0 mode.

Diﬀerences between spacetimes: Previous work

C, p and q can be found in both the star and Schwarzschild spacetimes. Outside the gravitational source they are related by Birkhoﬀ’s theorem,

pstar = αpSchw + βqSchw

This relation produces a simple expression for ∆ φ2

D E Jake Percival φ2 far from stars and black holes December 18, 2018 10 / 22 This sum and integral rapidly converge, so it is justiﬁed to look at the small ω behaviour of the ` = 0 mode.

Diﬀerences between spacetimes: Previous work

C, p and q can be found in both the star and Schwarzschild spacetimes. Outside the gravitational source they are related by Birkhoﬀ’s theorem,

pstar = αpSchw + βqSchw

This relation produces a simple expression for ∆ φ2

∞ 1 Z ∞ X β ∆ φ2 = dω (2` + 1)C q2 4π2 Schw α Schw 0 `=0

D E Jake Percival φ2 far from stars and black holes December 18, 2018 10 / 22 Diﬀerences between spacetimes: Previous work

C, p and q can be found in both the star and Schwarzschild spacetimes. Outside the gravitational source they are related by Birkhoﬀ’s theorem,

pstar = αpSchw + βqSchw

This relation produces a simple expression for ∆ φ2

∞ 1 Z ∞ X β ∆ φ2 = dω (2` + 1)C q2 4π2 Schw α Schw 0 `=0

This sum and integral rapidly converge, so it is justiﬁed to look at the small ω behaviour of the ` = 0 mode.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 10 / 22 `+1 Fabbri and Anderson used in the far ﬁeld qSchw = ω k`(ωr) and −` pSchw = ω i`(ωr) where k` and i` are modiﬁed spherical Bessel functions. ∆ φ2 can then be integrated by parts to get a series expansion in r −1.

Diﬀerences between spacetimes: Previous work

The constants α and β can be determined if we know the radial modes, pstar ,pSchw and qSchw .

D E Jake Percival φ2 far from stars and black holes December 18, 2018 11 / 22 ∆ φ2 can then be integrated by parts to get a series expansion in r −1.

Diﬀerences between spacetimes: Previous work

The constants α and β can be determined if we know the radial modes, pstar ,pSchw and qSchw . `+1 Fabbri and Anderson used in the far ﬁeld qSchw = ω k`(ωr) and −` pSchw = ω i`(ωr) where k` and i` are modiﬁed spherical Bessel functions.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 11 / 22 Diﬀerences between spacetimes: Previous work

D E Jake Percival φ2 far from stars and black holes December 18, 2018 11 / 22 Mξ ∆ φ2 = − + O(r −4) 4π2r 3

3Mξ(ξ − 1 ) ∆ hT µi = − 6 diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5

M Question 1: Does this hold for a star model in which R is not small? Question 2: Does this result hold beyond the leading order in r −1 far from the gravitational source?

Diﬀerences between spacetimes: Previous work

M For R 1 at leading order, where R is the radius of the star, Fabbri 2 and Anderson showed universality: ∆ φ = 0 and ∆ hTµνi = 0 1 when ξ = 0 (minimal) and ∆ hTµνi = 0 when ξ = 6 (conformal).

D E Jake Percival φ2 far from stars and black holes December 18, 2018 12 / 22 3Mξ(ξ − 1 ) ∆ hT µi = − 6 diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5

M Question 1: Does this hold for a star model in which R is not small? Question 2: Does this result hold beyond the leading order in r −1 far from the gravitational source?

Diﬀerences between spacetimes: Previous work

M For R 1 at leading order, where R is the radius of the star, Fabbri 2 and Anderson showed universality: ∆ φ = 0 and ∆ hTµνi = 0 1 when ξ = 0 (minimal) and ∆ hTµνi = 0 when ξ = 6 (conformal). Mξ ∆ φ2 = − + O(r −4) 4π2r 3

D E Jake Percival φ2 far from stars and black holes December 18, 2018 12 / 22 M Question 1: Does this hold for a star model in which R is not small? Question 2: Does this result hold beyond the leading order in r −1 far from the gravitational source?

Diﬀerences between spacetimes: Previous work

3Mξ(ξ − 1 ) ∆ hT µi = − 6 diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5

D E Jake Percival φ2 far from stars and black holes December 18, 2018 12 / 22 Question 2: Does this result hold beyond the leading order in r −1 far from the gravitational source?

Diﬀerences between spacetimes: Previous work

3Mξ(ξ − 1 ) ∆ hT µi = − 6 diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5

M Question 1: Does this hold for a star model in which R is not small?

D E Jake Percival φ2 far from stars and black holes December 18, 2018 12 / 22 Diﬀerences between spacetimes: Previous work

3Mξ(ξ − 1 ) ∆ hT µi = − 6 diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5

M Question 1: Does this hold for a star model in which R is not small? Question 2: Does this result hold beyond the leading order in r −1 far from the gravitational source?

D E Jake Percival φ2 far from stars and black holes December 18, 2018 12 / 22 The radial equation can be solved on the surface of the star by a power series, to ﬁnd pstar .

Finding qSchw and pSchw on the surface of the star requires the method of matched asymptotic expansions (for an analytical approximation), or numerical methods.

Uniform density star

Suppose our star spacetime is that of a uniform density star of radius R and mass M.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 13 / 22 Finding qSchw and pSchw on the surface of the star requires the method of matched asymptotic expansions (for an analytical approximation), or numerical methods.

Uniform density star

Suppose our star spacetime is that of a uniform density star of radius R and mass M. The radial equation can be solved on the surface of the star by a power series, to ﬁnd pstar .

D E Jake Percival φ2 far from stars and black holes December 18, 2018 13 / 22 Uniform density star

Suppose our star spacetime is that of a uniform density star of radius R and mass M. The radial equation can be solved on the surface of the star by a power series, to ﬁnd pstar .

Finding qSchw and pSchw on the surface of the star requires the method of matched asymptotic expansions (for an analytical approximation), or numerical methods.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 13 / 22 Mξ ∆ φ2 = − S ( M ) + O(r −4) 4π2r 3 0 R

6 12 S (x) = 1 − (2ξ + 1) x + 17ξ2 + 10ξ − 4 x2 + O x3 . 0 5 35

Some additional work allows this technique to be generalized to the stress-energy tensor,

3Mξ(ξ − 1 ) ∆ hT µi = − 6 S ( M )diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5 0 R

Uniform density star

I ﬁnd that the leading order term of ∆ φ2 between a uniform density star of radius R and a black hole is,

D E Jake Percival φ2 far from stars and black holes December 18, 2018 14 / 22 6 12 S (x) = 1 − (2ξ + 1) x + 17ξ2 + 10ξ − 4 x2 + O x3 . 0 5 35

Some additional work allows this technique to be generalized to the stress-energy tensor,

3Mξ(ξ − 1 ) ∆ hT µi = − 6 S ( M )diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5 0 R

Uniform density star

I ﬁnd that the leading order term of ∆ φ2 between a uniform density star of radius R and a black hole is, Mξ ∆ φ2 = − S ( M ) + O(r −4) 4π2r 3 0 R

D E Jake Percival φ2 far from stars and black holes December 18, 2018 14 / 22 Some additional work allows this technique to be generalized to the stress-energy tensor,

3Mξ(ξ − 1 ) ∆ hT µi = − 6 S ( M )diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5 0 R

Uniform density star

I ﬁnd that the leading order term of ∆ φ2 between a uniform density star of radius R and a black hole is, Mξ ∆ φ2 = − S ( M ) + O(r −4) 4π2r 3 0 R

6 12 S (x) = 1 − (2ξ + 1) x + 17ξ2 + 10ξ − 4 x2 + O x3 . 0 5 35

D E Jake Percival φ2 far from stars and black holes December 18, 2018 14 / 22 3Mξ(ξ − 1 ) ∆ hT µi = − 6 S ( M )diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5 0 R

Uniform density star

I ﬁnd that the leading order term of ∆ φ2 between a uniform density star of radius R and a black hole is, Mξ ∆ φ2 = − S ( M ) + O(r −4) 4π2r 3 0 R

6 12 S (x) = 1 − (2ξ + 1) x + 17ξ2 + 10ξ − 4 x2 + O x3 . 0 5 35

Some additional work allows this technique to be generalized to the stress-energy tensor,

D E Jake Percival φ2 far from stars and black holes December 18, 2018 14 / 22 Uniform density star

I ﬁnd that the leading order term of ∆ φ2 between a uniform density star of radius R and a black hole is, Mξ ∆ φ2 = − S ( M ) + O(r −4) 4π2r 3 0 R

6 12 S (x) = 1 − (2ξ + 1) x + 17ξ2 + 10ξ − 4 x2 + O x3 . 0 5 35

Some additional work allows this technique to be generalized to the stress-energy tensor,

3Mξ(ξ − 1 ) ∆ hT µi = − 6 S ( M )diag [2, −2, 3, 3] + O(r −6) ν 4π2r 5 0 R

D E Jake Percival φ2 far from stars and black holes December 18, 2018 14 / 22 2 1 ∆ φ , ξ = 6

D E Jake Percival φ2 far from stars and black holes December 18, 2018 15 / 22 t 1 ∆ hTt i, ξ = 12

D E Jake Percival φ2 far from stars and black holes December 18, 2018 16 / 22 Search for a solution to the radial equation of the form

∞ 2Mω 2Mω X n qSchw = ω (r − 2M) (2Mω) Yn (ωr) n=0

This is the method of Poisson and Sasaki (1994).

Substituting this in gives a set of 2nd order ODE’s, one for each Yn.

The equations up to Y1 can be solved for ` = 0, giving us pSchw and qSchw to sub-leading order in ω.

Beyond the leading order

2 µ Now focus on the case ξ = 0, when both ∆ φ and ∆ hTν i vanish.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 17 / 22 This is the method of Poisson and Sasaki (1994).

Substituting this in gives a set of 2nd order ODE’s, one for each Yn.

The equations up to Y1 can be solved for ` = 0, giving us pSchw and qSchw to sub-leading order in ω.

Beyond the leading order

2 µ Now focus on the case ξ = 0, when both ∆ φ and ∆ hTν i vanish. Search for a solution to the radial equation of the form

∞ 2Mω 2Mω X n qSchw = ω (r − 2M) (2Mω) Yn (ωr) n=0

D E Jake Percival φ2 far from stars and black holes December 18, 2018 17 / 22 Substituting this in gives a set of 2nd order ODE’s, one for each Yn.

The equations up to Y1 can be solved for ` = 0, giving us pSchw and qSchw to sub-leading order in ω.

Beyond the leading order

2 µ Now focus on the case ξ = 0, when both ∆ φ and ∆ hTν i vanish. Search for a solution to the radial equation of the form

∞ 2Mω 2Mω X n qSchw = ω (r − 2M) (2Mω) Yn (ωr) n=0

This is the method of Poisson and Sasaki (1994).

D E Jake Percival φ2 far from stars and black holes December 18, 2018 17 / 22 The equations up to Y1 can be solved for ` = 0, giving us pSchw and qSchw to sub-leading order in ω.

Beyond the leading order

2 µ Now focus on the case ξ = 0, when both ∆ φ and ∆ hTν i vanish. Search for a solution to the radial equation of the form

∞ 2Mω 2Mω X n qSchw = ω (r − 2M) (2Mω) Yn (ωr) n=0

This is the method of Poisson and Sasaki (1994).

Substituting this in gives a set of 2nd order ODE’s, one for each Yn.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 17 / 22 Beyond the leading order

2 µ Now focus on the case ξ = 0, when both ∆ φ and ∆ hTν i vanish. Search for a solution to the radial equation of the form

∞ 2Mω 2Mω X n qSchw = ω (r − 2M) (2Mω) Yn (ωr) n=0

This is the method of Poisson and Sasaki (1994).

Substituting this in gives a set of 2nd order ODE’s, one for each Yn.

The equations up to Y1 can be solved for ` = 0, giving us pSchw and qSchw to sub-leading order in ω.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 17 / 22 M2 ∆ φ2 = . 4π2r 4 3M2 ∆ hT µi = − diag [1, −1, 2, 2] ν 8π2r 6

In general, universality does not hold beyond the leading order in r −1.

Beyond the leading order

For ξ = 0 at sub-leading order in r −1 we now have

D E Jake Percival φ2 far from stars and black holes December 18, 2018 18 / 22 In general, universality does not hold beyond the leading order in r −1.

Beyond the leading order

For ξ = 0 at sub-leading order in r −1 we now have

M2 ∆ φ2 = . 4π2r 4 3M2 ∆ hT µi = − diag [1, −1, 2, 2] ν 8π2r 6

D E Jake Percival φ2 far from stars and black holes December 18, 2018 18 / 22 Beyond the leading order

For ξ = 0 at sub-leading order in r −1 we now have

M2 ∆ φ2 = . 4π2r 4 3M2 ∆ hT µi = − diag [1, −1, 2, 2] ν 8π2r 6

In general, universality does not hold beyond the leading order in r −1.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 18 / 22 ∆ φ2 , ξ = 0

D E Jake Percival φ2 far from stars and black holes December 18, 2018 19 / 22 t ∆ hTt i, ξ = 0

D E Jake Percival φ2 far from stars and black holes December 18, 2018 20 / 22 This global behaviour depends on the structure of the gravitational source. Universality for ξ = 0 does not hold beyond leading order in r −1. Work for ξ 6= 0 is ongoing. Generalisations to larger spin fields and different structures of gravitational source are in principal possible, although Kerr spacetime isn’t obvious due to the loss of Birkhoff’s theorem.

Conclusion

2 µ The global behaviour of ∆ φ and ∆ hTν i can be found via taking diﬀerences in the same quantum state on diﬀerent spacetimes.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 21 / 22 Universality for ξ = 0 does not hold beyond leading order in r −1. Work for ξ 6= 0 is ongoing. Generalisations to larger spin fields and different structures of gravitational source are in principal possible, although Kerr spacetime isn’t obvious due to the loss of Birkhoff’s theorem.

Conclusion

D E Jake Percival φ2 far from stars and black holes December 18, 2018 21 / 22 Generalisations to larger spin fields and different structures of gravitational source are in principal possible, although Kerr spacetime isn’t obvious due to the loss of Birkhoff’s theorem.

Conclusion

D E Jake Percival φ2 far from stars and black holes December 18, 2018 21 / 22 Conclusion

2 µ The global behaviour of ∆ φ and ∆ hTν i can be found via taking differences in the same quantum state on different spacetimes. This global behaviour depends on the structure of the gravitational source. Universality for ξ = 0 does not hold beyond leading order in r −1. Work for ξ 6= 0 is ongoing. Generalisations to larger spin fields and different structures of gravitational source are in principal possible, although Kerr spacetime isn’t obvious due to the loss of Birkhoff’s theorem.

D E Jake Percival φ2 far from stars and black holes December 18, 2018 21 / 22 Thanks for listening!

D E Jake Percival φ2 far from stars and black holes December 18, 2018 22 / 22