Black Holes, Vortices And Thermodynamics

Black Holes, Vortices and ThermodynamicsLuke BarclayDurham, CPTluke.barclay@durham.ac.ukSupervisor: Ruth GregoryLuke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

OutlineBlack Hole ThermodynamicsVorticesBlack Holes with VorticesSolving the Hierarchy Problem?Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Black Hole ThermodynamicsHawking and Bekenstein in the early 70’s conjectured that blackholes have thermodynamic properties.Black holes have entropy S.Black holes have Hawking temperature TH , consistent withthermodynamic relation between energy, entropy andtemperature.ThermodynamicsS A4TH where A is the area of the event horizon.κ2πwhere κ in the surface gravity of the black hole.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Path Integral FormulationIn 1976 Hawking and Gibbons demonstrated that thesethermodynamic results could be attained via a path integralapproach to quantum gravity.In this approach one considers expressions of the formDhΦ2 , t2 Φ1 , t1 i Φ1 e iH(t2 t1 )EZ Φ2 D[Φ]e iS[Φ])if t2 t1 iβ and Φ1 Φ2 thenhΦ, t2 Φ, t1 i Tre βH Z ZD[Φ]e iS[Φ]Here Z is the partition function of an ensemble at temperatureT β 1 , D[Φ] is the measure of the space of all fields.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Euclideanisation and TemperatureIn quantum gravity this is the sameZZ d[g ]d[φ]e iS[g ][φ]For ease of calculation the metric is Euclideanised i.e. t iτand the metric becomes positive definite.Now, a periodicity can be associated with τ meaning β τ .Then, by including all metrics that are asymptotically flat andhave periodicity of the imaginary time coordinate β τ , thepath integral gives the partition function for a system attemperature T β 1 .Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Topology of Euclidean Black HolesEuclidean Schwarzschild black hole metricds 2 (1 rsrs)dτ 2 (1 ) 1 dr 2 r 2 (dθ2 sin2 θdφ2 )rrSingular at r rs .A change variables of ρ2 (2rs )2 (1 ds 2 ρ2 (τ 2) dρ2 rs2 dΩ2II2rsrsr ),givesasr rs . τ must be periodic with period β 4πrsSingularity is coordinate singularityMetric is only defined on rs r The metric has topology R2 S2 .Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Black Hole Partition functionFor black holes the key contributions to the path integral comefrom geometries that have topology such as this i.e. R2 S2 .For the Schwarzschild black hole including this geometry aloneresults in the partition function from which the famous resultscan be derived.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

OutlineBlack Hole ThermodynamicsVorticesBlack Holes with VorticesSolving the Hierarchy Problem?Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

VorticesA vortex is a non-perturbative, non-trivial solution of the fieldequations.This talk will consider only local Abelian Higgs vortices.They can be created during phase transitions.Abelian Higgs Lagrangianλ1L Fµν F µν Dµ φ(Dµ φ) (φφ η 2 )2 ,440φ(x) e i.e.Λ(x) φ(x),0Aµ (x) Aµ (x) µ Λ(x),Dµ φ µ φ ieAµ φ.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Vortex Formation1λL Fµν F µν Dµ φ(Dµ φ) (φφ η 2 )244If η 0 then it is the symmetry breaking scale, an energyscale below which φ(x) acquires a vev6 0, the symmetrybreaks and the theory undergoes a phase transition.It is likely that during a transition a non-trivial winding of thephase will appear about some point.For this winding to be reconciled at the origin, φ must rise upthe potential barrier to φ 0, thus a stable, localised,non-zero energy density appears which forms the vortex core.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Field ConfigurationFinite energy considerations imply that φ η as r (it’svacuum value) and Aµ must asymptotically be a pure gaugerotation.Simplest Field Configuration for Vorticesφ ηX (r )e ikθ ,Aµ 1(P(r ) k) µ θ,e(X (0) 0,X (r ) 1, r .(P(0) k,P(r ) 0, r .Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Field EquationsThis form simplifies the field equations for variables X (r ) andP(r ).Field Equations In Minkowski Background X 0 P 2 Xλη 2 2 2 (X 1)X ,rr2P0P 00 2e 2 η 2 X 2 P.rX 00 These coupled, second order, ordinary differential equationscan be solved numerically.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Field .50.2X(r)P(r)Energy DensityX(r)P(r)Energy Density000246radial distance r81002468radial distance rFigure: Field distribution for k 1 and k 2 vorticesLuke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics10

Vortices and GravityTo include gravity:The Minkowski metric must be replaced by gµν , the generalmetric.The field equations must now include components of themetric and be coupled to the Einstein equations. Giving moredifferential equations of more variables.These equations have been solved for a vortex in an otherwise flatspacetime and give an interesting result.The geometry of the spacetime outside the core is locallyidentical to Minkowski but not globally.The effect of the vortex is to introduce a ‘deficit angle’making the spacetime that of a snub-nosed cone. 8πG µwhere is the deficit angle, G is Newtons constant and µ is thevortex mass per unit length.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

OutlineBlack Hole ThermodynamicsVorticesBlack Holes with VorticesSolving the Hierarchy Problem?Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Vortex on a Black HoleThe temperature of a black hole depends on the periodicity,β, of the imaginary temporal coordinate.The gravitational effect of a vortex on the surrounding spacetime is to reduce the period of the dimension in which itsphase lies.Therefore, one might expect that a vortex on a black holeconfigured such that its phase lies in the temporal directionmay effect the temperature of a black hole.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Set upWe now consider an Abelian Higgs Lagrangian with GeneralEuclidean Schwarzschild metricLagrangian and Metric1λL Fµν F µν Dµ φ(Dµ φ) (φφ η 2 )2 ,44ds 2 A2 dτ 2 A 2 dr 2 C 2 (dθ2 sin2 θdφ2 ).Field Configurationik 2πτφ ηX (r )e β ,2π2πAµ (P(r ) k) µ τ (Pµ k µ ).βeβeThis configuration ensures cylindrical symmetry about τ whichleads to A A(r ) and C C (r ).Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Field EquationsField EquationsVarying φ and Aµ gives212 0 02 2X P(CP) 2eηC2A221P X 4π 2 λη 22 2 0 0(CAX) X (X 2 1).C2A2 β 22Varying g µν gives the Einstein equations, which for this case are:C(T 0 Trr )A2 0((A2 )0 C 2 )0 8πGC 2 (2Tθθ Trr T00 )C 00 4πG1(A2 )0 C 0 2 (1 A2 C 02 ) 8πGTrrCCWhere Tii are components of the energy-momentum tensor.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Boundary conditionsThese coupled, ordinary differential equations must be solvedsimultaneously along with the boundary conditions specified byfinite energy constraints and regularity of the metric at the horizon.Boundary ConditionsC (rs ) rsX (rs ) 01rsX ( ) 1P(rs ) 1P( ) 0.A(rs ) 0A(rs )2 The problem complicated by the ‘mixed type’ boundaryconditions.The method used involves a Runge-Kutta algorithm on theequations for the gravity fields and successive under-relaxationon the matter fields, repeated on successive iterations.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

2X(r)P(r)A 2(r)C 2(r)/121(Energy Density)/5X(r)P(r)A 2(r)C 2(r)/121(Energy Density)/50.2001234567radial distance r8910111234567radial distance r891011Figure: Field distribution for G 0.0 and G 0.02 vorticesCaveat: There is a small numerical artefact in these solutions(not shown here) which needs some further investigation toascertaining its origin.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Results10.81Energy0.60.80.4Energy0.60.2X(f)P(f)A 2(f)C (-2)(f)*10,000(Energy Density)/30.400246f810120.2X(f)P(f)A 2(f)C (-2)(f)*10,000(Energy Density)/3002.5220406080100f120140160180200Figure: Field distribution forG 0.0 (close up), G 0.0 andG 0.02 vortices (coordinatesappear flat at the horizon).Energy1.510.5X(f)P(f)A 2(f)C (-2)(f)*10,000(Energy Density)/30020406080100f120140160180200Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

ConclusionsKey observations:Gravity fields are asymptotically Schwarzschild.A2 ’s asymptotic value is increased by the presence of thegravitating vortex.If we look at the asymptotic Schwarzschild where A2 has beenmultiplied by a constant λ.ds 2 λ2 A2 dτ 2 1 2 2A drλ2 1 2C (dθ2λ2 sin2 θdφ2 ).This factor can only be absorbed by a rescaling of dτ dr λdrdτλandβTherefore the period at infinity, β̃ τλ λ , is reduced andthe temperature of the black hole T H β̃1 is increased.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

SummaryVerified numerically that the presence of a vortex on aEuclidean Schwarzschild black hole increased the temperatureof the system.This supports previous work of my supervisor andcollaborators when looking analytically at the extreme case ofa thin weakly gravitating vortex on a black hole.These results apply to the more general case of thicker andstronger gravitating vortices.This may well have important implications on other currentwork in the field.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

OutlineBlack Hole ThermodynamicsVorticesBlack Holes with VorticesSolving the Hierarchy Problem?Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Discrete Local SymmetryIn the 70’s the idea of discrete local symmetry was used as away of endowing black holes with ‘hair’ despite the no hairtheorems.Consider a theory of two complex scalar fields φ and ϕ invariantunder a local U(1) gauge transformation.0φ(x) e ieΛ(x) φ0ϕ(x) eieΛ(x)Nϕ0Aµ (x) Aµ (x) µ Λ(x),for some integer N. Now consider the Lagrangian density1λL F 2 (Dφ)2 (Dϕ)2 (φ2 η 2 )2 V ( ϕ ).44Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics

Discrete Local SymmetrySuppose η is some energy scale at which φ condenses and ϕdoes not.φ will be invariant under additions of k 2πe to the phase whistϕ will not, being multiplied by the nth root of unity.The possible actions of these transformations generate a ZNsymmetry.Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor:BlackRuthHoles,GregoryVortices and Thermodynamics