Gravitational Waves. Basic theory and applications for core-collapse supernovae. Moritz Greif. 1. Nov Stockholm University 1 / 21

Transcription

1 Gravitational Waves Basic theory and applications for core-collapse supernovae Moritz Greif Stockholm University 1. Nov / 21

2 General Relativity Outline 1 General Relativity Basic GR Gravitational waves from eld equations Examples and eects of Gravitational waves 2 Observation and measurement Nobel prize 1993 Measurements 3 Astrophysical Sources Core collapse supernovae 4 References 2 / 21

3 General Relativity Basic GR Metric Flat space-time: η µν = (1) Dierential Geometry: g µν = F ( x) x µ, F ( x) x ν, for surface F ( x). Christoel symbol (tangential part of 2 F/ x ν x ρ ): Γ µ νρ = 1 2 σ g µσ [ gσν x ρ + g σρ x ν g ] νρ x σ (2) 3 / 21

4 General Relativity Basic GR Geodesics and Curvature Free particles move on geodesics, 2 x λ τ 2 + x µ x ν Γλ µν τ τ = 0 Exp.: Great circle is the geodesic on a sphere Shift a vector v i at point x to a point x + dx, then it has the components: Riemannian Curvature Tensor: v i + dv i = v i Γ i mjdx m v j R α µβγ = Γα βµ x γ Γα γµ x β + Γα γνγ ν βµ Γα βν Γν γµ Flat space: R = 0, Γ = 0, g = diag(1, 1, 1, 1),parallel shift independent on path 4 / 21

5 General Relativity Basic GR Einstein Field Equations Some denitions: Ricci tensor: R µγ = g β αr α µβγ = Rα µαγ Riemann scalar: R = g µγ R µγ Einstein tensor: G µν = R µν 1 2 g µνr Energy-momentum tensor: T µν = (ρ + p c 2 ) u µ u ν pg µν (3) (ρ: mass density, p: pressure, u µ : 4-velocity) Einstein eld equations G c 4 = cm 2 cm3 erg G µν = 8πG c 4 T µν 5 / 21

6 General Relativity Gravitational waves from eld equations Gravitational waves I: Linearise and Gauge Weak gravity: g µν = η µν + h µν, h 1, µ h 1 Linearise equations: Γ µ νρ 1, R αβ... R = µ ν h µν h G αβ = 1 2 ( α µ h µβ + β µ h µα α β h µ µ h αβ + η αβ h η αβ µ ν h νµ ) Notation: h αβ h αβ 1 2 η αβh, h h µ µ (trace-reversing: h = h) Gauge-Freedom: h αβ h αβ + α χ β + β χ α, equivalent to coordinate change: x µ x µ χ µ Leaves R α βγδ invariant! (require µχ α 1) (See: Literature 1,2,3) 6 / 21

7 General Relativity Gravitational waves from eld equations Gravitational waves II: Gauge and formal solution Lorentz-Gauge: µ hµν = 0 (cf. E-dynamic: µ A µ = 0) Einstein tensor now: G αβ = 1 2 h αβ Linearised Field equations: General solution: h αβ = 16πG c 4 T αβ h αβ = 4G c 4 cf. E-Dynamic: 1 2 ϕ c 2 + ϕ = 4πρ t2 Not all components of h µν in Riemann tensor! ) Tαβ ( r, t r r c r r d 3 r 7 / 21

8 General Relativity Gravitational waves from eld equations Gravitational waves III: transverse solutions Example: wave in x-direction: h µν = h 22 h h 32 h 33 In general: spatial, transverse and traceless components gauge-inv. radiation! traceless: h µ µ = 0 spatial: h 0i = h i0 = h 00 = 0 transverse: i h ij = 0 (h i ) k for propagation k Dene: P ij = δ ij n i n j, n = k/ k Make every h kl transverse-traceless (TT): h T T ij = h kl ( P ki P lj 1 2 P klp ij ) 8 / 21

9 General Relativity Gravitational waves from eld equations Gravitational waves IV: energy conservation Energy-conservation in curved space: 0 = µ T µν = µ T µν Γ λ µµ T λν = µ T µν }{{} 0 ν = 0 0 T 00 + j T j0 = 0 j T j0 = t T 00 ν = j 0 T 0j + i T ij = 0 i T ij = t T 0j...combined: i j T ij = 2 T t 2 00 multiply with x r x k, and integrate 2 times by part: x i x j r k T rk d 3 r = x i x j k T rk df r S x i k T ik d 3 r = S T ij ( r, t)d 3 r = 2 t 2 x i T ik df k + x i k T ik d 3 r T ij d 3 r x i x j T 00 ( r, t)d 3 r 9 / 21

10 General Relativity Gravitational waves from eld equations Gravitational waves IV: quadrupole approximation Dene Quadrupole-moment: T ij ( r, t)d 3 r = 2 t 2 2 x i x j T 00 ( r, t)d 3 r 1 2 t 2 I ij The lowest order approximation r r R is a quadrupole-approximation! Quadrupole formula: h T ij T = 2 G d 2 I kl (P R c 4 dt 2 ki P lj 12 ) P klp ij 10 / 21

11 General Relativity Examples and eects of Gravitational waves No dispalcement of particles For plane wave in x 1 -direction (see Lit.1): ẍ 1 O(β 2 ) ẍ 2 O(β) ẍ 3 O(β) No coordinate change! But particles move! Consider wave with h 23 = h 32 = 0: ds 2 = c 2 dt 2 + dx 2 + g 22 dy 2 + g 33 dz 2 Remember: g µν = η µν + h µν g 22 = (1 h 22 ), g 33 = (1 + h 22 ) L. Ju, D. G. Blair and C. Zhao, Rept. Prog. Phys. 63 (2000) / 21

12 Observation and measurement Outline 1 General Relativity Basic GR Gravitational waves from eld equations Examples and eects of Gravitational waves 2 Observation and measurement Nobel prize 1993 Measurements 3 Astrophysical Sources Core collapse supernovae 4 References 12 / 21

13 Observation and measurement Nobel prize 1993 Nobel prize 1993 for 50 years only theoretically predicted Binary pulsar PSR , Period: 7,75h Russell Hulse and Joseph Taylor: Nobel prize for measurements of decreasing orbit period 13 / 21

14 Observation and measurement Measurements Measurement via Interferometry: LIGO 14 / 21

15 Astrophysical Sources Outline 1 General Relativity Basic GR Gravitational waves from eld equations Examples and eects of Gravitational waves 2 Observation and measurement Nobel prize 1993 Measurements 3 Astrophysical Sources Core collapse supernovae 4 References 15 / 21

16 Astrophysical Sources Core collapse supernovae Expected Amplitude for Core-collapse 16 / 21

17 Astrophysical Sources Core collapse supernovae Modelling h(t) for supernovae multi-scale multi-physics problem: t comp 10 6 s, t SN = 1 2 s, 3D, GR,Hydro,EOS,... Three models for collapse and bounce and discussed: Type I Nuclear EOS such, that waveform strong at bounce, with quick 'ring down' Type II Core bounce aected by rotation (centrif. forces),low densities, multiple bounces like damped harmonic oscillator Type III Fast collapse because of very ecient electron capture and small masses of the homologously collapsing inner core, low signal (see Dimmelmeier et al.) GW-burst signal depends mostly on: pre-collapse central angular velocity progenitor mass (iron core mass important at bounce) 17 / 21

18 Astrophysical Sources Core collapse supernovae Type I signal for core collapse SN 18 / 21

19 References Outline 1 General Relativity Basic GR Gravitational waves from eld equations Examples and eects of Gravitational waves 2 Observation and measurement Nobel prize 1993 Measurements 3 Astrophysical Sources Core collapse supernovae 4 References 19 / 21

20 References References Adler,Bazin Schier, Introduction to general relativity, McGraw Hill 1975 S. A. Hughes, Gravitational waves from merging compact binaries, Ann. Rev. Astron. Astrophys. 47 (2009) 107 [arxiv: [astro-ph.co]]. L. Ju, D. G. Blair and C. Zhao, Detection of gravitational waves, Rept. Prog. Phys. 63 (2000) SeaM. Carroll, An Introduction to General Relativity Spacetime and Geometry, Book Addison Weasley (2004) B. P. Abbott et al. [LIGO Scientic Collaboration], LIGO: The Laser interferometer gravitational-wave observatory, Rept. Prog. Phys. 72 (2009) [arxiv: [gr-qc]]. C. D. Ott, The Gravitational Wave Signature of Core-Collapse Supernovae, Class. Quant. Grav. 26 (2009) [arxiv: / 21