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 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 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 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 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 µνr r − ξR (x)] G x; x0 = − p 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 µνr r − ξR (x)] G x; x0 = − p 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 µνr r − ξR (x)] G x; x0 = − p 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 !` !` φ 1 = φ 2 + d! 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 = d! F !` − 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 = d! F !` − counterterms ` A difference in this between two scenarios can be written Z 2 ren 2 ren X !` !` φ 1 = φ 2 + d! 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 = d! F !` − counterterms ` A difference in this between two scenarios can be written Z 2 ren 2 ren X !` !` φ 1 = φ 2 + d! 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.