From space-time to gravitation waves. Bubu 2008 Oct. 24

Transcription

1 From space-time to gravitation waves Bubu 008 Oct. 4

2 Do you know what the hardest thing in nature is? and that s not diamond. Space-time! Because it s almost impossible for you to change its structure.

3 Newton s Gravity = Einstein s space-time geometry Wheeler: matter tell space-time how to curve and space-time tell matter how to move

4 Gravitational radiation (wave) is the ripple _ space-time.

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7 Outline 1. Introduction. How motions of matter affect the space-time and the idea of gravitational radiation/wave 3. Order of magnitude estimation 4. Conclusion and Prospect

8 The idea of gravitational wave: Small-scale ripple propagate in a background of large-scale curvature (Short Wave Approximation)

9 a(dimensionless) D short wave limit a << 1 D / R s t << 1 :

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11 constraint on sources : R M source source / D << 1 / R source << 1 I m the source

12 How motions of matter affect the space-time? Matter with steady velocity Matter with acceleration Matter with rotation Two matters attract to each other via gravity Two matters in orbit with steady angular velocity

13 x e 1 Radiation from a electric dipole source : dipole = d ex < charge length > e Power d&& = e a m 1 and its gravitaional radiation analog : a 1 m a d&& r m a r ma = 0

14 Radiation from a magnetic dipole source : dipole = J r ρν Power J && v 1 m 1 and its gravitaional radiation analog : m a a 1 µ& & d ( ) r mivi dt i = 0 v

15 GW emitted from two masses in orbit - Quadrupole radiation v 1 y m 1 quadrupole mn = I = ρx m x n a 1 a x Power L GW &&& I m v Axial symmetry rotation constant quadrupole moment no gravitational waves

16 Order of magnitude estimation quadrupole = I ~ MR < mass length length > Power L GW &&& I MR ~ ( 3 T ) ( ) ~ ( energy / period)

17 The maximum power of GW Geometrized unit : c = G = 1 Virial theorem : (describe the relation between the inertial energy and the gravitational energy) M ~ υ R MR 4 6 M 6 M 6 M 5 LGW ~ ( ) ~ M R ω ~ ( ) ( Rω) ~ ( ) υ ( ) 3 T R R R The maximum power output occurs when the system is near its gravitational radius and the formula breaks down L GW 1 = 5 c G = ( erg / sec)

18 GW emitted form a Rotating Bar L GW MR ~ ( ) ( T 3 ~ ( Ml ω ) ( erg / sec) erg / sec) 59

19 GW emitted from the rotating of four masses at the corner of a square y y (0,a,0) (-a,0,0) (a,0,0) x x M (0,-a,0) I xx = ρ a I xx = ρ a No gravitational radiation M

20 Price Theorem: Whatever can be radiated is radiated

21 What s their shape after Collapsing and becoming a static black hole?

22 Linear Theory ds g h µν µν ( t = c = η = h µν µν + dt 1 η + h ) h + dx µν µν µν h + dy h α α µν << 1 + dz = 0 if in vacuum = η µν dx µ dx ν h = 0 (plus mode) for GW propagates along z axis : t x y z h + = 0 (cross mode) t 0 h+ h 0 x hµν ( t) = 0 h 0 h+ y z which changes the proper length between test particles

23 GW Polarization At any moment of time, a gravitational wave is invariant under a rotation of 180 degree Shear tensor spin= The two basic mode can combine to, e.x. circular polarization

24 de dt = LGW < 0 Photo credit:

25 Why not detect gravitational tidal force instead? Consider a spring with initial length r and characteristic frequency ω = k m : k ( r)h( ) = ( r) hω m I&& 1 MR where h ~ r r T M and ω R = R M r M ( r + r) M ~ r r r 1 = MR ω r (dimensionless) 1 4 hence the Gravitational Wave Force:( r) MR ω = r which is far larger than the Gravitational Tidal Force: r r M R 4 3

26 Other possible sources for GW Sources Compact object oscillation (periodic ) Rotation of NS (periodic ) Orbiting of compact binary (periodic ) Merge of compact stars (burst) Supernovae (burst) Early universe (stochastic) NS tidal disruption BH excitation Cosmic strings Associate physics Rich physics included Spin-spin, spin-orbital coupling Enable to observe really young universe for the first time NS e.o.s, Is it a soure of GRB? BH normal modes This is over my head

27 PSR B Discovered by Russell Alan Hulse and Joseph Hooton Taylor, Jr. in They were awarded the 1993 Nobel Prize in Physics. Mass of detected pulsar M Mass of companion M Orbital period hr Eccentricity

28 E = 1 1 E Mω r log E de dt = + 1 Mω r logω 3 1 dω = 3 ω dt M r 3 1 P dp dt ~ M 5 3 ω 3 r r C.M. dp dt = where 3 P E E L de dt ~ GW M = 3 M (r) ~ ( 3 T a more detail calculation shows and the observed value shows 5 3 ω 3 P E L GW ) ~ ~ ( M (r) dp dt dp dt 3 ω ) =.4 10 M ~ ( ) ω =.3± ~ (r) O(10 M 5 = Mω r sec/ yr)

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30 Conclusion Why GW is weak and why quadrupole radiation? What s the maximal power can GW carries? Why GW force larger than gravitational tidal force? The power of order of magnitude estimation.

31 Prospect Tools for observing our universe: 1. EM wave. Gravitational wave 3. Cosmic ray 4. Neutrino To know more than estimations we need numerical relativity to deal with the strong field near the sources (we have to solve the Einstein field equation and get the space-time structure)

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33 3+1 decomposition The EM analog: Constraints equations Evolution equations Two categories of problems in Numerical Relativity: Initial value problem Evolution problem