Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematical formulation of the Standard Model wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Kaluza–Klein theory wikipedia , lookup
Photon polarization wikipedia , lookup
Wave packet wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Gravitational Radiation: 1. The Physics of Gravitational Waves and their Generation Kip S. Thorne Lorentz Lectures, University of Leiden, September 2009 PDFs of lecture slides are available at http://www.cco.caltech.edu/~kip/LorentzLectures/ each Thursday night before the Friday lecture Introduction • The Gravitational-Wave window onto the universe is likely to be opened - in the next decade in four widely different frequency bands, spanning 22 decades: 10-5 10-10 h CMB Anisotropy Pulsar Timing Arrays ELF VLF 10-15 10-20 10-25 10-16 10-8 LISA LF 1 Frequency, Hz LIGO/Virgo HF 10+8 2 Radical New Windows ➠ Great Surprises • • Radio Window: 1940s & 50s - - 104 x lower frequency than optical radio galaxies, quasars, pulsars, cosmic microwave background, ... X-Ray Window: 1960s & 70s 103 x higher frequency than optical black holes, accreting neutron stars, hot intergalactic gas, ... Radical New Windows ➠ Great Surprises • Gravitational Waves are far more radical than Radio or X-rays Completely new form of radiation! Frequencies to be opened span 22 decades fHF / fELF ~ 1022 • ‣ ‣ ‣ • What will we learn from Gravitational Waves? “Warped side of the universe” our first glimpses, then in-depth studies The nonlinear dynamics of curved spacetime Answers to astrophysical & cosmological puzzles: How are supernovae powered? How are gamma-ray bursts powered? What was the energy scale of inflation? ... Surprises - Growth of GW Community • • 1994: LIGO Approved for Construction: ~ 30 scientists Today: ~ 1500 scientists - influx from other fields needed for success drawn by expected science payoffs These Lectures 1. The Physics of Gravitational Waves and their Generation Today 2. Astrophysical and Cosmological Sources of Gravitational Waves, and the Information they Carry Next Friday, Sept 25 3. Gravitational Wave Detection: Methods, Status, and Plans Following Friday, Oct 2 These Lectures • Prerequisites for these lectures: - Knowledge of physics at advanced undergraduate level - Especially special relativity and Newtonian gravity - Helpful to have been exposed to General Relativity; not necessary • Goals of these lectures: - Overview of gravitational-wave science - Focus on physical insight, viewpoints that are powerful • Pedagogical form of these lectures: - Present key ideas, key results, without derivations - Give references where derivations can be found These Lectures • Pedagogical references that cover this lectureʼs material: - LH82: K.S. Thorne, “Gravitational Radiation: An Introductory Review” in Gravitational Radiation, proceedings of the 1982 Les Houches summer school, eds. N. Deruelle and T. Piran (North Holland, 1983) requires some knowledge of general relativity - NW89: K.S. Thorne, Gravitational Waves: A New Window onto the Universe (unpublished book, 1989), available on Web at http:// www.its.caltech.edu/~kip/stuff/Kip-NewWindow89.pdf - does not require prior knowledge of general relativity, except in Chaps. 5 & 6. - BT09: R.D. Blandford and K.S. Thorne, Applications of Classical Physics (near ready for publication, 2009), available on Web at http:// www.pma.caltech.edu/Courses/ph136/yr2008/ - contains an introduction to general relativity. Resources • The best introductory textbook on general relativity: - • James B. Hartle, Gravity an Introduction to General Relativity (Addison Wesley, 2003) The best course-length introduction to gravitational-wave science: - Gravitational Waves, a Web-Based Course (including videos of lectures, readings, problem sets, problem solutions): http://elmer.caltech.edu/ph237/ Outline of This Lecture 1. Gravitational waves (GWs) in the language of tidal gravity 2. GWs in the linearized approximation to general relativity 3. GW generation a. Linearized sources b. Slow-motion sources c. Nonlinear, highly dynamical sources: Numerical relativity 4. GWs in curved spacetime; geometric optics; GW energy 5. Interaction of GWs with matter and EM fields 1. Gravitational Waves in the Language of Tidal Gravity Relative Motion of Inertial Frames time GW-induced motion of Local Inertial Frames Non-meshing of local inertial frames ⇒ Curvature of Spacetime 1 GW GW field like ẍj = ∂gj xk ẍj = ḧjk xk ∂xk 2 Riemann Tensor −Rjtkt −∂ 2 Φ xk ∂xj ∂xk 1 GW hjk xk 2 analogous to Ej = −ȦT j (transverse Lorenz gauge) δxj = A B δx proper distance (C) Out[18]= Tidal Gravity B proper distance proper time Local Inertial Frame of A A The GW field GW hjk • The gravitational-wave field, hGW jk • Symmetric, transverse, traceless (TT); two polarizations: +, x + Polarization hGW xx = h+ (t − z/c) = h+ (t − z) hGW yy = −h+ (t − z) Lines of force 1 GW ẍj = ḧjk xk 2 ẍ = ḧ+ x ÿ = −ḧ+ y • x Polarization GW hGW xy = hyx = h× (t − z) x z y Gravitons • Quantum spin and rest mass: imprint on classical waves 180o spin = return angle photon graviton y E return angle = 360o spin=1 x return angle = 180o spin=2 propagation speed = c ≡1 ⇒ rest mass = 0 Behavior of z boosts in z direction GW Field z’ x’ GW hjk Under v hGW jk , h+ , h× y’ EM waves in Transverse Lorenz gauge transform as scalar fields � � h�GW (t − z ) jk y x t − z = D(t� − z � ) D= � 1+v 1−v = hGW jk (t − z) � � = hGW (D(t − z )) jk Riemann amplified 1 � GW = ḧjk 2 = D2 Rjtkt � Rjtkt boost weight 2 AT j is transverse: T AT x (t − z), Ay (t − z) transform as scalar fields � � A�T (t − z ) j � � = AT (D(t − z )) j Electric field amplified Ej� = −ȦT j = −DEj boost weight 1 GWs as Seen in Laboratory on Earth • Proper Reference Frame: analog of local Lorentz frame • GWs unaffected by earthʼs gravity ds2 = −(1 + 2g · x)dt2 + dx2 + dy 2 + dz 2 • except for a very tiny, unimportant gravitational blue shift Total gravitational force 1 GW ẍj = ḧjk xk + gj 2 L L -6 Laser L+6L Photodetector 2. GWs in Linearized Approximation to General Relativity Metric Perturbation, Lorenz Gauge, Einstein Field Equation • Metric: gµν = ηµν + hµν Grav’l Field Flat • • Field theory in flat spacetime hµν analogous to Aµ µ µ µ Gauge freedom (ripple coordinates) xnew = xold − ξ old hnew = h µν µν + • 1 Lorenz gauge h̄µν ≡ hµν − hα α ηµν 2 ∂ h̄µν ∂Aν ∂xν • • ∂ξµ ∂ξν ∂φ new old + analogous to A = A + µ µ ∂xν ∂xµ ∂xµ = 0 analogous to ∂xν =0 Einstein field equation in Lorenz gauge �h̄µν ≡ η αβ r wave zone ϕ θ near zone ∂ ∂ µν µν µ µ h̄ = −16πGT analogous to �A = −4πJ ∂xα ∂xβ Gravitational-wave field In wave zone, gauge change with �ξα = 0 (analogous to �φ = 0) → new old TT Project out TT piece; get GW field: hnew = hnew = hGW tt jt = 0, hjk = (hjk ) jk 1 1 TT old TT where (hold = hold = hold θϕ ) θϕ = h× , (hθθ ) θθ − (hθθ + hϕϕ ) = (hθθ − hϕϕ ) = h+ , 2 2 old T analogous to obtaining Transverse Lorenz gauge by projecting: AT j = (Aj ) 3. Gravitational Wave Generation Example: Linearized, Point Particles in Lorenz Gauge • • In wave zone Ej = −(Ȧj ) pα kα (Liénard-Wiechart) Gravitational pα pβ �h̄ = −16πGT ⇒ at O, h̄ = G kµ pµ 4Gm tt in rest frame of particle, reduces to h̄ = r � j k �TT p p GW TT In wave zone hjk = (h̄jk ) = G kµ pµ αβ • O Electromagnetic q pα α α α �A = −4πJ ⇒ at O, A = kµ pµ q t in rest frame of particle, reduces to A = r T αβ αβ Gravitational-Wave Memory ∆hGW jk � =G ∆ � A 4 pjA pkA kA µ pµA �TT final h+ ∆h+ time initial Slow-Motion GW Sources Slow Motion: speeds << c = 1; wavelength = λ >> (source size) = L examples: me waving arms; pulsar (spinning neutron star); binary made r from two black holes ϕ Weak-field, near zone: Newtonian Potential θ m mass dipole mass quadrupole • Φ = −G • r &G r2 &G r3 & ... wave zone near zone 1 [by energy conservation] and dimensionless ⇒ r m ∂(mass dipole)/∂t ∂ 2 (mass quadrupole)/∂t2 ∼G &G &G & ... r r r Wave zone: hGW jk ∼ hGW jk momentum; cannot oscillate mass; cannot oscillate Mass canonical field theory radiation field carried by quadrupole quanta with spin s has multipoles confined to ℓ ≥ s dominates hGW jk = 2G � Ïjk r �TT for Newtonian source Ijk = � 1 2 3 ρ(x x − r )d x 3 j k Common Textbook Derivation 1. Linearized Approximation to General Relativity (set G=1) � jk � � T (t − |x − x |, x ) 3 � jk jk jk �h̄ = −16π h̄ ⇒ h̄ (t, x) = 4 d x � � jk |x − x� | 3 � 4 T (t − r, |x )d x jk slow motion ⇒ h̄ (t, x) = r 2. Conservation of 4-momentum T αβ ,β = 0 ⇒ 2T jk = (T tt xj xk ),tt − (T ab xj xk ),ab − 2(T aj xk + T ak xj ),a 3. Insert 2 into 1; integral of divergence vanishes � jk ¨ 2I (t − r) jk h̄ (t, x) = , where I jk = T 00 xj xk d3 x r 4. Take transverse � traceless � part h̄GW jk (t, x) = 2 Ïjk (t − r) r TT , where I jk = � T 00 (xj xk − r2 δ jk )d3 x PROBLEM: Derivation Not valid when self gravity influences source’s dynamics!! • Derivation via Propagation from Weak-Field Near Zone to Wave Zone Weak-gravity regions: Linearized approximation to GR �hαβ = 0 h̄αβ,β = 0 i.e. h̄tt h̄tj • ,t ,t = −h̄tj ,j = −h̄jk ,k In weak-field near zone (wfnz) Φ = − M 3Ijk n n , − 3 r 2r j k Quadrupolar solution in Induction Zone: � 1 6 j k � h̄tt = 2 Ijk (t − r) I n n in wfnz, jk 3 r r ,jk tiny � � −2 1 � 2 İjk nk in wfnz, h̄tj = 2 İjk (t − r) r r ,k � h̄jk = unit radial vector h̄tt = −4Φ pure gauge 2 � I¨jk nj nk in lwz r pure gauge � −2 Ïjk (t − r)nl in lwz 2 r � 2 Ïjk (t − r) Take TT part to get GW field in wfnz: h̄GW jk (t, x) = r �TT 2 Ïjk (t − r) r hGW jk Order of Magnitude • Source parameters: mass ~ M, size ~ L, rate of quadrupolar oscillations ~ ω, distance ~ r, internal kinetic energy of quadrupolar oscillations ~ Ekin ~ M(ωL)2 • GW strength: Ïjk ω 2 (M L2 ) Ekin /c2 = 2G ∼G ∼G r r r ∼ Φ produced by kinetic energy of shape changes. hGW jk hGW jk ∼ h+ ∼ h× ∼ 10 −21 � Ekin M⊙ c2 �� 100Mpc r � 1 100Mpc = 300 million light years ∼ (Hubble distance) 30 Slow-Motion Sources: Higher-Order Corrections • • Source Dynamics: Post-Newtonian Expansion � � 2 in v/c ∼ Φ/c ∼ P/ρc2 GW Field: Higher-Order Moments (octopole, ...) - computed in same manner as quadrupolar waves: by analyzing the transition from weak-field near zone, through induction zone, to local wave zone - actually Two families of moments (like electric and magnetic) - ‣ - moments of mass distribution, moments of angular-momentum distribution Use symmetric, trace-free (STF) tensors to describe the moments and the GW field ... (19th century approach; Great Computational Power) STF Tensors [an aside] • • Multipole moments of Newtonian gravitational potential +� � M�m Y�m (θφ) Φ∼ r�+1 - Spherical-harmonic description: - M�m has 2� + 1 components: m = −�, � + 1, ..., +� Ia1 a2 ...a� na1 na2 ...na� STF description: Φ∼ r�+1 Ia1 a2 ...a� has 2� + 1 independent components m=−� Multipolar Expansion of gravitational-wave field � � hGW jk = ∞ � 4 ∂ � Ijka1 ...a�−2 (t − r) a1 n ...na�−2 � �! ∂t r TT mass moments �=2 + �∞ � �=2 - 8� ∂ Sk)pa1 ...a�−1 (t − r) q a1 �pq(j � n n ...na�−1 (� + 1)! ∂t r � �TT angular momentum moments current moments Indices carry directional, multipolar and tensor information, all at once Strong-Gravity (GM/c2L ~1), Fast-Motion (v~c) Sources • • • • Most important examples [next week] - Black-hole binaries: late inspiral, collision, merger, ringdown - neutron star / neutron-star binaries: late inspiral, collision and merger - supernovae Black-hole / neutron-star binaries: late inspiral, tidal disruption and swallowing These are the strongest and most interesting of all sources Slow-motion approximation fails Only way to compute waves: Numerical Relativity 4. Gravitational Waves in curved spacetime; geometric optics; GW energy GWs Propagating Through Curved Spacetime (distant wave zone) • Definition of gravitational wave: the rapidly varying part of the metric and of the curvature λ̄ = λ/2π � L = (lengthscale on which background metric varies) � R gαβ = B gαβ + hαβ ≡ �gαβ � ≡ gαβ − �gαβ � Same definition used for waves in plasmas, fluids, solids • In local Lorentz frame of background: GW theory same as in flat spacetime (above) �hαβ = 0 in vacuum. Same propagation equation as for EM waves: �Aα = 0 GWs exhibit same geometric-optics behavior as EM waves • Geometric-Optics Propagation GWs and EM waves propagate along the same ray (tret , θ, φ) rays: null geodesics in the background spacetime - Label each ray by its direction (θ, ϕ) in sourceʼs local wave zone, and the retarded time it has in the local wave zone, tret ≡ (t − r)local wave zone Waveʼs amplitude dies out as 1/r in local wave zone. Along the the ray, in distant wave zone, define� A = (cross sectional area of a bundle of rays, and r ≡ ro A/Ao �eθ where ro and Ao are values at some location in local wave zone. Then amplitude continues to die out as 1/r in distant wave zone. - eθ and �eϕ parallel to Transport the unit basis vectors � themselves along the ray, from local wave zone into and through distant wave zone. Use them to define + and x - Then in distant wave zone: Aθ = h+ = Q+ (tret , θ, ϕ) , r h× = Qθ (tret , θ, ϕ) , r Q× (tret , θ, ϕ) r Aϕ = Qϕ (tret , θ, ϕ) r local wave zone Waveʼsamplitude amplitudedies diesout outasas1/r1/rininlocal localwave wavezone. zone. - - Waveʼs �eϕ distant wave zone Geometric-Optics Propagation ray (t • Form of waves: Aθ = Q+ (tret , θ, ϕ) h+ = , r • Qθ (tret , θ, ϕ) , r Aϕ = Qϕ (tret , θ, ϕ) r Q× (tret , θ, ϕ) h× = r GWs experience identically the same geometric-optics effects as EM waves: - ret , θ, φ) gravitational redshift, �eϕ �eθ distant wave zone cosmological redshift, gravitational lensing, ... at the focus of a gravitational lense: ‣ and the same diffraction effects as EM waves local wave zone Energy and Momentum in GWs • Einsteinʼs general relativity field equations say Gαβ = 8π G Tαβ 1 Energy-momentum-stress tensor Einstein’s curvature tensor • • • • • • B Break metric into background plus GW: gαβ = gαβ + hαβ Expand Einstein tensor in powers of hαβ (1) (2) Gαβ = GB + G + G αβ αβ αβ quadratic in hµν Background linear in hµν Einstein tensor Average over a few wavelengths to get quantities that vary on background scale L , not wavelength scale λ̄ (2) �Gαβ � = GB + �G (2) αβ αβ � = 8π�Tαβ � �G αβ � B GW GW Rearrange: Gαβ = 8π(�Tαβ � + Tαβ ) where Tαβ ≡ − 8π Evaluate the average: In Local Lorentz Frame tt tz zz TGW = TGW = TGW = GW Tαβ 1 = �h+,α h+,β + h×,α h×,β � 16π 1 �ḣ2+ + ḣ2× � . 16π Energy and Momentum Conservation • GW = 8π(�T � + T Einsteinʼs field equations GB αβ αβ αβ ) guarantee energy and momentum conservation, e.g. • Source loses mass (energy) at a rate � � dM 1 tr �ḣ2+ + ḣ2× �dA = − TGW dA = − dt 16π S S r S • Source loses linear momentum at a rate � � j dp 1 jr = − TGW dA = − (�ej · �er )�ḣ2+ + ḣ2× �dA dt 16π S S • Angular momentum is a little more delicate 5. Interaction of Gravitational Waves with matter and EM fields Plane GW Traveling Through Homogeneous Matter • Fluid: - 1 GW shears the fluid, (rate of shear) = σjk = ḣGW 2 jk no resistance to shear, so no action back on wave Viscosity η ∼ ρvs = (density)(mean speed of particles)(mean free path) GW NOTE: s must be < λ̄ produces stress Tjk = −2ησjk = −η ḣjk GW TT GW Linearized Einstein field equation: �hjk = −16π(Tjk ) = 16πη ḣjk 1 1 Wave attenuates: hGW where ∼ exp(−z/� ) � = = att att jk 8πη 8πρvs Fluidʼs density curves spacetime (background Einstein equations) 1 ∼ GB 00 = 8πρ 2 R Therefore �att R2 Rc Rc ∼ =R �R �R vs s v λ̄ v The viscous attenuation length is always far larger than the background radius of curvature. Attenuation is never significant! Plane GW Traveling Through Homogeneous Matter • Elastic Medium: - 1 GW 1 ḣjk , (shear)=Σjk = hGW 2 2 jk GW = −2µΣjk − 2ησjk = −µhGW jk − η ḣjk GW shears the medium, (rate of shear) = σjk = Medium resists with stress Tjk TT GW GW Einstein equation becomes �hGW = −16π(T ) = 16π(µh + η ḣ jk jk jk jk ) Insert hGW jk ∝ exp(−iωt + ikz) . Obtain dispersion relation ω 2 − k 2 = 16π(µ − iωη); i.e. ω = k(1 + 8πλ̄2 µ) − i8πη, where λ̄ = 1/k Same attenuation length as for fluid: �att = 1 �R 8πη Phase and group velocities (dispersion): ω dω vphase = = 1 + 8πλ̄2 µ, vgroup = = 1 − 8πλ̄2 µ k dk λ̄ Dispersion length (one radian phase slippage) � = δvphase 1 R2 = � �R 8πλ̄µ λ̄ The dispersion length is always far larger than the background radius of curvature. Dispersion is never significant! GW Scattering • Strongest scattering medium is a swarm of black holes: hole mass M, number density of holes n - Scattering cross section σ � M 2 Graviton mean free path for scattering 1 1 1 R2 �= � = ∼ �R nσ nM 2 ρM M The scattering mean free path is always far larger than the background radius of curvature. Scattering is never significant! Interaction with an Electric or Magnetic Field • • • Consider a plane EM wave propagating through a DC magnetic field Bwave = Bo sin[ω(t − z)]ey , BDC = BDC ey Beating produces a TT stress TT stress resonantly generates a GW h+ = • Txx = −Tyy = hGW xx = −hGW yy Bo BDC 4π TT �hGW jk = −16π(Tjk ) 2BDC Bo = z cos[ω(t − z)] The “Gertsenshtein effect” ω Ratio of GW energy to EM wave energy: tt TGW tt TEMwave �ḣ2+ �/16π z2 2 2 = = BDC z = 2 Bo2 /8π R The lengthscale for significant conversion of EM wave energy into GW energy is equal to the radius of curvature of spacetime produced by the catalyzing DC magnetic field. • The lengthscale for the inverse process is the same There can never be significant conversion in the astrophysical universe. Conclusion • Gravitational Waves propagate through the astrophysical universe without significant attenuation, scattering, dispersion, or conversion into EM waves • Next Friday: Astrophysical and Cosmological Sources of Gravitational Waves, and the Information they Carry - slides will be available Thursday night at http://www.cco.caltech.edu/~kip/LorentzLectures/