Gravitational wave generation, propagation and detection

The science enabled by gravitational wave observations

1Thursday, February 5, 2009

2Thursday, February 5, 2009

3Thursday, February 5, 2009

1 ∼(v

c

)2∼ GM

c2L∼ φG

VirialThm

Energetic

4Thursday, February 5, 2009

No (negligible) reddening

No (negligible) extinction

Undistorted view of central engine dynamics

Charge? Mass

Current? Momentum

Radiation? Time varying mass, current distributions

No gravitational wave zone of avoidance!

5Thursday, February 5, 2009

6Thursday, February 5, 2009

∂2

∂t2

(zj1 − zj2

) ( ∂2φG∂xj∂xk

) (zk1 − zk2

)

Free test-body trajectories depend only on initial position, velocity

Can’t distinguish gravity from coordinate system acceleration on basis of single trajectory

“Tidal force” acting on separation between nearby trajectories

7Thursday, February 5, 2009

gμν = ημν + hμν

ḧjk ∼ ∂2φG

∂xj∂xk

ds2 = gμνdxμdxν

∂2

∂t2

(zj1 − zj2

) ( ∂2φG∂xj∂xk

) (zk1 − zk2

)

Space-time interval

Metric

Infinitesimal coordinate separation between space-time points

Minkowski metric

Metric perturbation

8Thursday, February 5, 2009

9Thursday, February 5, 2009

�h̄μν = −16πGc2

Tμν

∂

∂xμh̄μν = 0

(h̄μν = hμν − 12ημνh

h = ηαβhαβ)

�Aμ = −4πJμ∂Aμ

∂xμ= 0

gμν = ημν + hμν

hμνAμ

10Thursday, February 5, 2009

�Aμ = −4πJμ

Aμ(t,x) = −4π∫

d3x′Jμ(t − |x−x′|c ,x′)

|x − x′|

1|x|

[Jμ(x′) +

x′j

cJ̇μ(x′) + . . .

]ret

11Thursday, February 5, 2009

Ijk =∫

d3x

[xjxk − x

2

3δjk

]ρ(�x)

hTTjk =2Gc2r

[Ïjk

]TTret

“Transverse-Traceless” gauge specialization. Project transverse to direction of wave propagation and remove trace

Trace-free quadrupole moment of matter distribution

12Thursday, February 5, 2009

L =1

32Gcπ

∫d2Ωr2ḣTTjk ḣ

TTjk

=15

G

c5...I jk

...I jk

∼ Gc5

(Mv2

T

)2∼ K.E./T

c5/G

K.E.T

h ∼ 10−39 1 kmr

M

1000 kg

(v

300 m/s

)

c5

G= 3.6 × 1059 erg

s

forb~125 Hz

h ∼ 10−23 100 Mpcr

M

3M�150 km

R

(!)

13Thursday, February 5, 2009

14Thursday, February 5, 2009

For radiation propagating along z axis...

Ixx =Mr2

4cos2 ωt

Ixy =Mr2

4cos ωt sinωt

Iyy =Mr2

4sin2 ωt

Ixx =Mr2

4

[cos2 ωt − 1

3

]

Ixy =Mr2

4cos ωt sinωt

Iyy =Mr2

4

[sin2 ωt − 1

3

]

Izz = −Mr2

12

hxx = −hyy = −GM (ωr)2

c2Rcos 2ωt

hxy = hyx = −GM (ωr)2

c2Rsin 2ωt

Ïxx = −M2 (ωr)2 cos 2ωt

Ïxy = −M2 (ωr)2 sin 2ωt

Ïyy =M

2(ωr)2 cos 2ωt

L =2G5c5

(ωr)4 (ωM)2

15Thursday, February 5, 2009

S. L. Shapiro, S. A. Teukolsky. 1983. Black Holes, White Dwarfs and Neutron Stars (Wiley: New York)

P. C. Peters, J. Mathews. 1963. Gravitational radiation from point masses in a Keplerian orbit. Phys. Rev. 131:435

T. X. Thuan, J. P. Ostriker. 1974. Gravitational radiation from stellar collapse. Ap. J. Lett. 191:L105 – L107.

C. Van Den Broeck. 2005. The gravitational wave spectrum of non-axisymmetric, freely precessing neutron stars. Class. Quant. Grav. 22:1825 – 1839.

What is the radiation from a rapidly rotating, non-axisymmetric, non-precessing neutron star?

What is the braking index of for a pulsar whose spin-down is dominated by gravitational waves?

Consider a circular orbit binary system, losing energy adiabatically to gravitational waves. How does the binary period change with time?

16Thursday, February 5, 2009

17Thursday, February 5, 2009