1. 18/04/2004New Windows on the Universe Jan Kuijpers Part 1: Gravitation & relativityPart 1: Gravitation & relativity J.A. Peacock, Cosmological Physics, Chs. 1 & 2 Part 2: Classical CosmologyPart 2: Classical Cosmology Peacock, Chs 3 & 4

2. 18/04/2004New Windows on the Universe Part 1: Gravitation & relativity GR Recap (1)GR Recap (1) Gravitational radiation (2.2-2.4)Gravitational radiation (2.2-2.4) Black holes (2.5-2.6)Black holes (2.5-2.6)

3. 18/04/2004New Windows on the Universe GR recap Inertial frames and Mach’s principle; Equivalence principle (1.1)Inertial frames and Mach’s principle; Equivalence principle (1.1) Equation of motion (1.2)Equation of motion (1.2) Energy-momentum tensor (1.4)Energy-momentum tensor (1.4) Field equations (1.5)Field equations (1.5)

4. 18/04/2004New Windows on the Universe Equation of motion (1.2) Freely falling observers: special relativity: General transformation:

5. 18/04/2004New Windows on the Universe Tensor transformations and derivativesTensor transformations and derivatives Vector components in non-orthogonal basis: Contravariant and covariant components: Covariant derivative:

6. 18/04/2004New Windows on the Universe Field equations Affine connection Affine connection Riemann Riemann curvature tensor curvature tensor Ricci tensor Einstein tensor Einstein tensor Einstein’s gravitational field equations

7. 18/04/2004New Windows on the Universe Relativistic fluid mechanics (1.4)Relativistic fluid mechanics (1.4) Energy-momentum tensor:

8. 18/04/2004New Windows on the Universe Gravitational radiation (2.2-2.4) Weak fieldsWeak fields Affine connections Riemann curvature tensor Ricci tensor Einstein tensor Field equation

9. 18/04/2004New Windows on the Universe In absence of matter In absence of matter: simple gravitational waves moving at speed of light, transverse In presence of matter In presence of matter: source of GWs

10. 18/04/2004New Windows on the Universe Relative acceleration of 2 test masses separated by X,Relative acceleration of 2 test masses separated by X, each satisfying eqn. of motion: each satisfying eqn. of motion: Equation of geodesic deviation to 1-order: Equation of geodesic deviation to 1-order: In rest frame: Effect of gravitational waves on test masses

11. 18/04/2004New Windows on the Universe

12. 18/04/2004New Windows on the Universe Energy of GWsEnergy of GWs If GWs carry energy where is the energy-momentum tensor? Landau-Lifshitz pseudotensor:

13. 18/04/2004New Windows on the Universe Excitation of GWs:Excitation of GWs: Excitation by changing mass distributions occurs in lowest order by quadrupolar changes Excitation by changing mass distributions occurs in lowest order by quadrupolar changes

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16. 18/04/2004New Windows on the Universe EstimatesEstimates Rotating rod of length L:Rotating rod of length L: Stellar binaries: Stellar binaries: Type II supernovae: once per century per galaxy Type II supernovae: once per century per galaxy Characteristic wave frequency: Characteristic wave frequency:

17. 18/04/2004New Windows on the Universe

18. 18/04/2004New Windows on the Universe GW detectors:GW detectors: - resonant bars/balls - resonant bars/balls - laser-based interferometers - laser-based interferometers Measure fractional length distortion: gravitational strainMeasure fractional length distortion: gravitational strain Energy flux density is:

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22. 18/04/2004New Windows on the Universe LIGOs, optimal performanceLIGOs, optimal performance Principle: Michelson interferometer with two long perpendicular arms, and reflecting mirror at each end; Laserlight is recombined, and movement of fringes measures the relative change in length of the two arms. Length: depends on frequency (light travel time along arm has to be less than wave period, < few 100 km. Practice: much smaller length but multiple reflections. Limits: Seismic noise Seismic noise Thermal noise Thermal noise Uncertainty principle Uncertainty principle Optimum laser power: 3 MW at 600 nm and n=100 (100 km) Optimum laser power: 3 MW at 600 nm and n=100 (100 km)

23. 18/04/2004New Windows on the Universe Uncertainty principle Optimum laser power Minimum strain detectable: 10 -23 for 1 km 1 tonne, 1 ms Too few photons: reduced fringe shift sensitivity Too many photons: radiation presure fluctuations too large Optimum power:

24. 18/04/2004New Windows on the Universe Hanford, Washington 4 km Livingston, Louisiana LISA

25. 18/04/2004New Windows on the Universe Binary pulsar(s) (2.4) Radio pulse period from one of the binary stars varies by Doppler shift:Radio pulse period from one of the binary stars varies by Doppler shift: Radial velocity curve determines 4 constraints on 6 orbit parameters (m1,m2,e,a,i,  ): - eccentricity e - position periastron  - projected semimajor axis - mass function

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27. 18/04/2004New Windows on the Universe Higher-order corrections to doppler and gravitational redshift

28. 18/04/2004New Windows on the Universe omegadot=4.2 degrees/yr!

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37. 18/04/2004New Windows on the Universe Black holes (2.5-2.6) Schwarzschild metricSchwarzschild metric

38. 18/04/2004New Windows on the Universe Orbits in Schwarzschild spacetimeOrbits in Schwarzschild spacetime Consider orbits in equatorial plane (  =  /2):

39. 18/04/2004New Windows on the Universe Precession of orbits (close to circular):Precession of orbits (close to circular): Differentiate radial eq. of motion:

40. 18/04/2004New Windows on the Universe Orbital stability:Orbital stability: Circular photon orbit:Circular photon orbit:

41. 18/04/2004New Windows on the Universe Radial orbit for particle falling in from infty:Radial orbit for particle falling in from infty: For external observer:

42. 18/04/2004New Windows on the Universe Accretion onto black hole:Accretion onto black hole: Energy per unit mass liberated by a particle orbiting a black hole down to marginally stable orbit: