Analytic models for compact binaries with spin Jan Steinhoff Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Potsdam-Golm, Germany Weekly ACR group seminar at AEI, Golm, Germany 1 / 21
Outline 1 Introduction Experiments Neutron stars and black holes Models for multipoles 2 Dipole/Spin Two Facts on Spin in Relativity Spin gauge symmetry Point Particle Action in General Relativity Spin and Gravitomagnetism 3 Quadrupole Quadrupole Deformation due to Spin Dynamic tides: External field and response Dynamic tides: Results 4 Universal relations Universal relation: I Love Q! Overview Universal relations for fast rotation Combination of relations 5 Conclusions 2 / 21
Experiments Pulsars and radio astronomy: � pics/doublepulsar pics/ska Double Pulsar (MPI for Radio Astronomy) Square Kilometre Array (SKA) γ -rays, X-rays, . . . Gravitational wave detectors: pics/ligo pics/lisa pics/xraybin Advanced LIGO eLISA space mission e.g. large BH spins in X-ray binaries 3 / 21
Neutron stars and black holes Neutron star picture by D. Page www.astroscu.unam.mx/neutrones/ ”Lab“ for various areas in physics magnetic field, plasma crust (solid state) superfluidity superconductivity pics/neutronstar unknown matter in core condensate of quarks, hyperons, kaons, pions, . . . ? accumulation of dark matter ? Black holes are simpler, but: strong gravity horizon analytic models? 4 / 21
Neutron stars and black holes Neutron star picture by D. Page www.astroscu.unam.mx/neutrones/ ”Lab“ for various areas in physics magnetic field, plasma crust (solid state) superfluidity superconductivity pics/neutronstar unknown matter in core condensate of quarks, hyperons, kaons, pions, . . . ? accumulation of dark matter ? Black holes are simpler, but: strong gravity horizon analytic models? 4 / 21
Neutron stars and black holes Neutron star picture by D. Page www.astroscu.unam.mx/neutrones/ ”Lab“ for various areas in physics magnetic field, plasma crust (solid state) superfluidity superconductivity pics/neutronstar unknown matter in core condensate of quarks, hyperons, kaons, pions, . . . ? accumulation of dark matter ? Black holes are simpler, but: strong gravity horizon analytic models? 4 / 21
Models for multipoles of compact objects Starting point: single object, e.g., neutron star M ← → S Q ← → state variables ( p , V , T ) multipoles ( m , S , Q ) thermodynamic potential ← → dynamical mass M correlation ← → response Idea Multipoles describe compact object on macroscopic scale Higher multipole order → smaller scales → more (internal) structure Multipoles describe the gravitational field and interaction Multipoles of neutron stars fulfill universal (EOS independent) relations 5 / 21
Models for multipoles of compact objects Starting point: single object, e.g., neutron star M ← → S Q ← → state variables ( p , V , T ) multipoles ( m , S , Q ) thermodynamic potential ← → dynamical mass M correlation ← → response Idea Multipoles describe compact object on macroscopic scale Higher multipole order → smaller scales → more (internal) structure Multipoles describe the gravitational field and interaction Multipoles of neutron stars fulfill universal (EOS independent) relations 5 / 21
Outline 1 Introduction Experiments Neutron stars and black holes Models for multipoles 2 Dipole/Spin Two Facts on Spin in Relativity Spin gauge symmetry Point Particle Action in General Relativity Spin and Gravitomagnetism 3 Quadrupole Quadrupole Deformation due to Spin Dynamic tides: External field and response Dynamic tides: Results 4 Universal relations Universal relation: I Love Q! Overview Universal relations for fast rotation Combination of relations 5 Conclusions 6 / 21
Two Facts on Spin in Relativity 2. Center-of-mass 1. Minimal Extension fast & heavy v ∆ z R spin slow & light V ring of radius R and mass M now moving with velocity v spin: S = R M V relativistic mass changes inhom. maximal velocity: V ≤ c frame-dependent center-of-mass ⇒ minimal extension: need spin supplementary condition: MV ≥ S S S µν p ν = 0 R = e.g., Mc 7 / 21
Two Facts on Spin in Relativity 2. Center-of-mass 1. Minimal Extension fast & heavy v ∆ z R spin slow & light V ring of radius R and mass M now moving with velocity v spin: S = R M V relativistic mass changes inhom. maximal velocity: V ≤ c frame-dependent center-of-mass ⇒ minimal extension: need spin supplementary condition: MV ≥ S S S µν p ν = 0 R = e.g., Mc 7 / 21
Spin gauge symmetry in an action principle choice of center should be physically irrelevant ⇒ gauge symmetry? Construction of an action principle (flat spacetime): introduce orthonormal corotating frame Λ 1 µ , Λ 2 µ , Λ 3 µ complete it by a time direction Λ 0 µ such that η AB Λ A µ Λ B ν = η µν realize that Λ 0 µ is redundant/gauge since one can boost Λ A µ such that Boost (Λ 0 ) ∝ p ( p µ : linear momentum) find symmetry of the kinematic terms in the action: z µ + 1 µ ˙ Λ A ν p µ ˙ 2 S µν Λ A z µ → z µ + ∆ z µ S µν → S µν + p µ ∆ z ν − ∆ z µ p ν Λ → Boost p → Λ 0 + ǫ Boost Λ 0 → p Λ find invariant quantities, minimal coupling 8 / 21
Spin gauge symmetry in an action principle choice of center should be physically irrelevant ⇒ gauge symmetry? Construction of an action principle (flat spacetime): introduce orthonormal corotating frame Λ 1 µ , Λ 2 µ , Λ 3 µ complete it by a time direction Λ 0 µ such that η AB Λ A µ Λ B ν = η µν realize that Λ 0 µ is redundant/gauge since one can boost Λ A µ such that Boost (Λ 0 ) ∝ p ( p µ : linear momentum) find symmetry of the kinematic terms in the action: z µ + 1 µ ˙ Λ A ν p µ ˙ 2 S µν Λ A z µ → z µ + ∆ z µ S µν → S µν + p µ ∆ z ν − ∆ z µ p ν Λ → Boost p → Λ 0 + ǫ Boost Λ 0 → p Λ find invariant quantities, minimal coupling 8 / 21
Spin gauge symmetry in an action principle choice of center should be physically irrelevant ⇒ gauge symmetry? Construction of an action principle (flat spacetime): introduce orthonormal corotating frame Λ 1 µ , Λ 2 µ , Λ 3 µ complete it by a time direction Λ 0 µ such that η AB Λ A µ Λ B ν = η µν realize that Λ 0 µ is redundant/gauge since one can boost Λ A µ such that Boost (Λ 0 ) ∝ p ( p µ : linear momentum) find symmetry of the kinematic terms in the action: z µ + 1 µ ˙ Λ A ν p µ ˙ 2 S µν Λ A z µ → z µ + ∆ z µ S µν → S µν + p µ ∆ z ν − ∆ z µ p ν Λ → Boost p → Λ 0 + ǫ Boost Λ 0 → p Λ find invariant quantities, minimal coupling 8 / 21
Spin gauge symmetry in an action principle choice of center should be physically irrelevant ⇒ gauge symmetry? Construction of an action principle (flat spacetime): introduce orthonormal corotating frame Λ 1 µ , Λ 2 µ , Λ 3 µ complete it by a time direction Λ 0 µ such that η AB Λ A µ Λ B ν = η µν realize that Λ 0 µ is redundant/gauge since one can boost Λ A µ such that Boost (Λ 0 ) ∝ p ( p µ : linear momentum) find symmetry of the kinematic terms in the action: z µ + 1 µ ˙ Λ A ν p µ ˙ 2 S µν Λ A z µ → z µ + ∆ z µ S µν → S µν + p µ ∆ z ν − ∆ z µ p ν Λ → Boost p → Λ 0 + ǫ Boost Λ 0 → p Λ find invariant quantities, minimal coupling 8 / 21
Spin gauge symmetry in an action principle choice of center should be physically irrelevant ⇒ gauge symmetry? Construction of an action principle (flat spacetime): introduce orthonormal corotating frame Λ 1 µ , Λ 2 µ , Λ 3 µ complete it by a time direction Λ 0 µ such that η AB Λ A µ Λ B ν = η µν realize that Λ 0 µ is redundant/gauge since one can boost Λ A µ such that Boost (Λ 0 ) ∝ p ( p µ : linear momentum) find symmetry of the kinematic terms in the action: z µ + 1 µ ˙ Λ A ν p µ ˙ 2 S µν Λ A z µ → z µ + ∆ z µ S µν → S µν + p µ ∆ z ν − ∆ z µ p ν Λ → Boost p → Λ 0 + ǫ Boost Λ 0 → p Λ find invariant quantities, minimal coupling 8 / 21
Spin gauge symmetry in an action principle choice of center should be physically irrelevant ⇒ gauge symmetry? Construction of an action principle (flat spacetime): introduce orthonormal corotating frame Λ 1 µ , Λ 2 µ , Λ 3 µ complete it by a time direction Λ 0 µ such that η AB Λ A µ Λ B ν = η µν realize that Λ 0 µ is redundant/gauge since one can boost Λ A µ such that Boost (Λ 0 ) ∝ p ( p µ : linear momentum) find symmetry of the kinematic terms in the action: z µ + 1 µ ˙ Λ A ν p µ ˙ 2 S µν Λ A z µ → z µ + ∆ z µ S µν → S µν + p µ ∆ z ν − ∆ z µ p ν Λ → Boost p → Λ 0 + ǫ Boost Λ 0 → p Λ find invariant quantities, minimal coupling 8 / 21
Point Particle Action in General Relativity Westpfahl (1969); Bailey, Israel (1975); Porto (2006); Levi & Steinhoff (2014) Minimal coupling to gravity, in terms of invariant position: Dz µ d σ − p µ S µν µ D Λ A ν � Dp ν d σ + 1 − λ � � 2 H − χ µ C µ S PP = d σ p µ 2 S µν Λ A p ρ p ρ d σ H := p µ p µ + M 2 = 0 , C µ := S µν ( p ν + p Λ 0 µ ) constraints: Dynamical mass M includes multipole interactions Application: post-Newtonian approximation for bound orbits one expansion parameter, ǫ PN ∼ v 2 c 2 ∼ GM c 2 r ≪ 1 (weak field & slow motion) 9 / 21
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