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Gravitational Radiation
reading course on “Gravitational Waves”
Marcelo Ponce:
CCRG, RIT
Rochester, NY – October, 2008
mponce
«Gravitational Radiation»
Outline
1
2
3
4
5
Introduction to GW
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
Non-linear wave solution
Detection of GW
Resonant detector
Generation of GW
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitudes estimates
Exact solution of the wave equation
Energy carried by Gravitational Waves
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: A Binary Pulsar
Summary
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Linearized Einstein equations
Basic properties of GW
Einstein field eqs.
Gµν
Rµν 12 gµν R 8πTµν
10 linearly independent
eqs.
Gravitational Wave as solution for
Einstein eqs. is valid under idealized assumption.
vacuum and asymptotically flat
space-time, and
a linearized regime.
search for wave-like solutions to Einstein eqs. in a
space-time with very modest curvature and with a metric
line element which is that of flat space-time but for small
derivations on nonzero curvature,
gµν
ηµν
mponce
hµν
O prhµν sq
«Gravitational Radiation»
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations (cont.)
Basic properties of GW
therefore linearized expressions for Christoffel symbols and
Riemann tensor are required
Γµαβ
12 gµν pgνα,β
Rµν
12
gβν,α gαβ,ν q hµα,να
1 µ
h α,β
2
hνα,µα hµν,αα h,µν
R gµν Rµν
hνα,α µ hµν, αα h,µν
mponce
ηµν Rµν
ù Einstein equations –linearized form–
hµα,αν
hµ β,α hαβ,µ
ηµν
hαβ,αβ
h,αα 16πTµν
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
asympt. flat, “Lorentz gauge”, ...
Basic properties of GW – Linearized Einstein equations (cont.)
introduce «trace-free» tensors, defined as
h̄µν
hµν 21 ηµν h
ñ
R̄µν
Gµν ,
¯
h̄µν
hµν ,
h̄ h
the linearized Einstein eqs. take a more compact form,
h̄µν,αα ηµν h̄αβ,αβ
h̄να,α µ
16πTµν
D’Alambertian (or wave) operator ù h̄µν,αα
gauge freedom inherent to GR, h̄µα,α
lh̄µν 16πTµν
devoid
GGGGGGGGGGGA
of matter
mponce
lh̄µν
0 (“Lorentz” gauge)
«Gravitational Radiation»
lh̄µν 0
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
a wave solution to Einstein equations
Basic properties of GW – Linearized Einstein equations
Ÿ grav. field weak but not stationary,
vacuum – linearized eqs.: 3d-wave eq.
plane wave, (solution)
h̄µν
R
BBt
2
2
Aµν eiκα x
∇2 h̄µν
α
0
(
η µν κµ κν κν κν 0 Ð condition for having a solution
ñ κµ null 4-vector, i.e. tang. to the world-line of a photon,
ç κµ Ñ pω, ~κq wave vector (direction of the wave).
µ
x pλq κµ λ lµ , photon’s path
ñ κµ xµ κ0 t ~κ ~x const κµ lµ
the photon travels with the GW, staying forever at the same
phase
6 the wave itself travels at the speed of light
dispersion relation for the wave, ω 2 |~κ|2
ñ wave’s phase velocity = group velocity = 1 ù light
speed.
mponce
«Gravitational Radiation»
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
TT gauge
Linearized Einstein equations – gauge condition
gauge condition for Einstein linearized eqs,
ñ
Aµν must be orthogonal to ~κ,
r
Aµν κν
h̄µα,α
0 s
0
using the gauge freedom, it’s possible to change the gauge
while remaining within the Lorentz class of gauges,
Aµµ
0
and
r
transverse-traceless (TT) gauge
Aµν U ν
Aµν κν 0, orthogonality cond.
Aµµ 0, infintesimal gauge transf.
Aµν U ν 0, global Lorentz frame.
mponce
0 s.
example: Lorentz frame,
Minkowski spacetime
background – plane waves in
z-direction
TT
ãÑ only 2 dof, ATT
xx and Axy .
«Gravitational Radiation»
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
TT gauge
Linearized Einstein equations – gauge condition
gauge condition for Einstein linearized eqs,
ñ
Aµν must be orthogonal to ~κ,
r
Aµν κν
h̄µα,α
0 s
0
using the gauge freedom, it’s possible to change the gauge
while remaining within the Lorentz class of gauges,
Aµµ
0
and
r
transverse-traceless (TT) gauge
Aµν U ν
Aµν κν 0, orthogonality cond.
Aµµ 0, infintesimal gauge transf.
Aµν U ν 0, global Lorentz frame.
mponce
0 s.
example: Lorentz frame,
Minkowski spacetime
background – plane waves in
z-direction
TT
ãÑ only 2 dof, ATT
xx and Axy .
«Gravitational Radiation»
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
TT gauge
Linearized Einstein equations – gauge condition
gauge condition for Einstein linearized eqs,
ñ
Aµν must be orthogonal to ~κ,
r
Aµν κν
h̄µα,α
0 s
0
using the gauge freedom, it’s possible to change the gauge
while remaining within the Lorentz class of gauges,
Aµµ
0
and
r
transverse-traceless (TT) gauge
Aµν U ν
Aµν κν 0, orthogonality cond.
Aµµ 0, infintesimal gauge transf.
Aµν U ν 0, global Lorentz frame.
mponce
0 s.
example: Lorentz frame,
Minkowski spacetime
background – plane waves in
z-direction
TT
ãÑ only 2 dof, ATT
xx and Axy .
«Gravitational Radiation»
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
TT gauge
Linearized Einstein equations – gauge condition
gauge condition for Einstein linearized eqs,
ñ
Aµν must be orthogonal to ~κ,
r
Aµν κν
h̄µα,α
0 s
0
using the gauge freedom, it’s possible to change the gauge
while remaining within the Lorentz class of gauges,
Aµµ
0
and
r
transverse-traceless (TT) gauge
Aµν U ν
Aµν κν 0, orthogonality cond.
Aµµ 0, infintesimal gauge transf.
Aµν U ν 0, global Lorentz frame.
mponce
0 s.
example: Lorentz frame,
Minkowski spacetime
background – plane waves in
z-direction
TT
ãÑ only 2 dof, ATT
xx and Axy .
«Gravitational Radiation»
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
TT gauge
Linearized Einstein equations – gauge condition
gauge condition for Einstein linearized eqs,
ñ
Aµν must be orthogonal to ~κ,
r
Aµν κν
h̄µα,α
0 s
0
using the gauge freedom, it’s possible to change the gauge
while remaining within the Lorentz class of gauges,
Aµµ
0
and
r
transverse-traceless (TT) gauge
Aµν U ν
Aµν κν 0, orthogonality cond.
Aµµ 0, infintesimal gauge transf.
Aµν U ν 0, global Lorentz frame.
mponce
0 s.
example: Lorentz frame,
Minkowski spacetime
background – plane waves in
z-direction
TT
ãÑ only 2 dof, ATT
xx and Axy .
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Effect on free particles
Making sense of the TT gauge
Lorentz frame, particle initially at rest, TT-gauge.
particle init. in a wave-free region encounters a GW, free
d
α
particle obeys the geodesic eq.,
Γαµν U µ U ν 0
dτ U
ñ particle remains at rest forever regardless of the GW
6 TT-gauge ÞÑ coord.system attached to individual
particles.
nearby particles, beginning at rest, proper distance
between
³ them
³
1
∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT
xx px 0q ε
6 proper distance does change with time (gral. hTT
xx 0).
obs: difference between computing a coord-dependent nbr.
(position) and a coord-indep. nbr. (proper distance);
the effect of the wave is unambiguously seen in the
mponce
«Gravitational Radiation»
coord-indep. nbr.
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Effect on free particles
Making sense of the TT gauge
Lorentz frame, particle initially at rest, TT-gauge.
particle init. in a wave-free region encounters a GW, free
d
α
particle obeys the geodesic eq.,
Γαµν U µ U ν 0
dτ U
ñ particle remains at rest forever regardless of the GW
6 TT-gauge ÞÑ coord.system attached to individual
particles.
nearby particles, beginning at rest, proper distance
between
³ them
³
1
∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT
xx px 0q ε
6 proper distance does change with time (gral. hTT
xx 0).
obs: difference between computing a coord-dependent nbr.
(position) and a coord-indep. nbr. (proper distance);
the effect of the wave is unambiguously seen in the
mponce
«Gravitational Radiation»
coord-indep. nbr.
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Effect on free particles
Making sense of the TT gauge
Lorentz frame, particle initially at rest, TT-gauge.
particle init. in a wave-free region encounters a GW, free
d
α
particle obeys the geodesic eq.,
Γαµν U µ U ν 0
dτ U
ñ particle remains at rest forever regardless of the GW
6 TT-gauge ÞÑ coord.system attached to individual
particles.
nearby particles, beginning at rest, proper distance
between
³ them
³
1
∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT
xx px 0q ε
6 proper distance does change with time (gral. hTT
xx 0).
obs: difference between computing a coord-dependent nbr.
(position) and a coord-indep. nbr. (proper distance);
the effect of the wave is unambiguously seen in the
mponce
«Gravitational Radiation»
coord-indep. nbr.
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Effect on free particles
Making sense of the TT gauge
Lorentz frame, particle initially at rest, TT-gauge.
particle init. in a wave-free region encounters a GW, free
d
α
particle obeys the geodesic eq.,
Γαµν U µ U ν 0
dτ U
ñ particle remains at rest forever regardless of the GW
6 TT-gauge ÞÑ coord.system attached to individual
particles.
nearby particles, beginning at rest, proper distance
between
³ them
³
1
∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT
xx px 0q ε
6 proper distance does change with time (gral. hTT
xx 0).
obs: difference between computing a coord-dependent nbr.
(position) and a coord-indep. nbr. (proper distance);
the effect of the wave is unambiguously seen in the
mponce
«Gravitational Radiation»
coord-indep. nbr.
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Effect on free particles (cont.)
Making sense of the TT gauge, «geodesic deviation»
eq. of geodesic deviation,
ï connecting vector ξ for 2 parts., dτd
2
Uµ
ξµ
2
ξα
dxµ {dτ ; to the lowest order, ñ Uµ
p0, ε, 0, 0q.
two particles initially separeted
in the x-direction,
$
'
&
B2 x
B t2 ξ
2
1
TT
2 ε t2 hxx
'
%
B2 y
B t2 ξ
2
1
TT
2 ε t2 hxy
Rαµνβ UµUν ξβ
p1, 0, 0, 0q,
two particles initially separeted
in the y-direction,
B
B
$
'
&
B2 y
B t2 ξ
2
1
TT
2 ε t2 hyy
B
B
'
%
B2 x
B t2 ξ
2
1
TT
2 ε t2 hxy
mponce
«Gravitational Radiation»
B
B
B
B
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Polarization of Gravitational Waves
Linear polarization
ring of free particles initially at rest in the x y plane, GW in
the z-direction.
« » polarization
hTT
xx
e
0,
hTT
xy
«» polarization
0
hTT
xy
e~x b e~x e~y b e~y
e
mponce
TT
0, hTT
xx hyy 0
e~x b e~y
«Gravitational Radiation»
e~y b e~x
Linearized Einstein equations
Wave solution to Einstein equations
Transverse-Traceless gauge
Polarization of GW
non-linear plane wave
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Non-linear wave solution
any wave can be written as superposition of plane waves
linearized theory, describes the kind of waves arrive to
Earth
linear plane wave ÞÑ solution of the non-linear equation
null coordinates:
ñ ds2 dudv
utz
dx2
vt
dy2
vacuum field equation,
z
dudv
g2 {g 0
ds2
f 2 {f
f 2 puqdx2
g2 puqdy2
prescribe an arbitrary function gpuq, and solve for f puq.
same freedom as in the linear case
g 1 εpuq, near the linear case Ñ f 1 εpaq ñ linear
wave eq.
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Resonant detector
Detection of Gravitational Waves
General Considerations – resonant detector
gravitational wave spectrum, unexplored
GR carries information that no electromag. rad. can
provide.
«antenna» monitoring electromag. signals traveling
between 2 freely falling particles
technical difficulties (small perturb, noise,...)
oldest type of detector proposed: resonant oscillator
pioneered by J.Weber (1961)
ultimate propossal, use of entangled photons through their
polarization state... (0810.3911)
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Resonant detector
Resonant detector
2 point massive (m) particles, spring (k) and damping (ν)
"
mx1,00
mx2,00
kpx1 x2
kpx2 x1
l0 q ν px1 x2 q,0
l0 q ν px2 x1 q,0
Damped Harm. Osc.
GW passing...
1
ξ,00
2γξ,0
ω02 ξ
0
! )
1
free particle remains at rest in TT coordinates xα .
assuming motions only due to GW, i.e. ξ Opl0 |hµν |q ! l0
TT-coords.
j
ñ mx,0j1 101 Fj1
mx,00
Fj Op|hµν |2q
x, x,0 , x,00 Ophµν q
GGGGGGGGGGGGGGGGGGGA
2
ñ
only nongravit. force spring 9 lptq ξ,00
2γξ,0
ω02 ξ
12 l0hTT
xx,00
mponce
³
r1
1{2
hTT
xx ptqs dt
forced, damped harm. osc.
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Resonant detector
Resonant detector
2 point massive (m) particles, spring (k) and damping (ν)
"
mx1,00
mx2,00
kpx1 x2
kpx2 x1
l0 q ν px1 x2 q,0
l0 q ν px2 x1 q,0
Damped Harm. Osc.
GW passing...
1
ξ,00
2γξ,0
ω02 ξ
0
! )
1
free particle remains at rest in TT coordinates xα .
assuming motions only due to GW, i.e. ξ Opl0 |hµν |q ! l0
TT-coords.
j
ñ mx,0j1 101 Fj1
mx,00
Fj Op|hµν |2q
x, x,0 , x,00 Ophµν q
GGGGGGGGGGGGGGGGGGGA
2
ñ
only nongravit. force spring 9 lptq ξ,00
2γξ,0
ω02 ξ
12 l0hTT
xx,00
mponce
³
r1
1{2
hTT
xx ptqs dt
forced, damped harm. osc.
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Resonant detector
resonant detector for sources of GW of fixed freq. (e.g.
pulsars or close binary stars or even bursts)
assuming hTT
xx A cos Ωt, the
steady solution
ξ R cos pΩt φq
osc., avg. energy, resonant,...
21 mpx1,0q2 12 mpx2,0q2 21 kξ2
xEy 81 mR2pω2 Ω2q
A
Rreson 41 l0 ApΩ{γ q
R 12 rpω Ωlq Ω 4Ω
γ s{
1
Ereson 64
ml02 Ω2 A2 pΩ{γ q2
tan φ 2γΩ{pω02 Ω2 q
Q ω0 {2γ
massive cylindrical bars, ‘spring’ Ñ elasticity of the bar
0
0
2
E
2
2 2 1 2
-stretched along its axis-; bar hited broadside, excite its
long. modes
noise sources: random, thermal, external vibrations
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Resonant detector
resonant detector for sources of GW of fixed freq. (e.g.
pulsars or close binary stars or even bursts)
assuming hTT
xx A cos Ωt, the
steady solution
ξ R cos pΩt φq
osc., avg. energy, resonant,...
21 mpx1,0q2 12 mpx2,0q2 21 kξ2
xEy 81 mR2pω2 Ω2q
A
Rreson 41 l0 ApΩ{γ q
R 12 rpω Ωlq Ω 4Ω
γ s{
1
Ereson 64
ml02 Ω2 A2 pΩ{γ q2
tan φ 2γΩ{pω02 Ω2 q
Q ω0 {2γ
massive cylindrical bars, ‘spring’ Ñ elasticity of the bar
0
0
2
E
2
2 2 1 2
-stretched along its axis-; bar hited broadside, excite its
long. modes
noise sources: random, thermal, external vibrations
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Resonant detector
resonant detector for sources of GW of fixed freq. (e.g.
pulsars or close binary stars or even bursts)
assuming hTT
xx A cos Ωt, the
steady solution
ξ R cos pΩt φq
osc., avg. energy, resonant,...
21 mpx1,0q2 12 mpx2,0q2 21 kξ2
xEy 81 mR2pω2 Ω2q
A
Rreson 41 l0 ApΩ{γ q
R 12 rpω Ωlq Ω 4Ω
γ s{
1
Ereson 64
ml02 Ω2 A2 pΩ{γ q2
tan φ 2γΩ{pω02 Ω2 q
Q ω0 {2γ
massive cylindrical bars, ‘spring’ Ñ elasticity of the bar
0
0
2
E
2
2 2 1 2
-stretched along its axis-; bar hited broadside, excite its
long. modes
noise sources: random, thermal, external vibrations
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
Generation of Gravitational Waves
General considerations
amplitude of any GW incident on Earth SMALL, hµν
amplitude of GW falls off as r1 far from the source
Op1q.
a distance R from a source of mass M, largest amplitude
waves expected are of order M {R (e.g. formation of a
10Md BH in a supernova explosion in a nearby galaxy
1023 m away, is about 1017 )
Goal: to solve wave eq. with source
2
BBt2
∇2 h̄µν
16πTµν
mponce
Ÿ approx. soln.: simplified,
but realistic assumptions
Ÿ exact solution
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
approximate calculation of wave generation
Generation of GW
Sµν pxiqeiΩt
2nd assumption, region (Sµν 0) is small compared with
2π {Ω λGW – «slow-motion assumption»
ãÑ soln, h̄µν Bµν pxiqeiΩt ñ ∇2 Ω2 Bµν 16πSµν
“pB2{Bt2 ∇2qf g”
outside of the source (Sµν 0) ù outgoing radiation
Ÿ determine Aµν ÞÑ source is
solution outside the source
nonzero only inside a sphere of
Z
A
iΩr
Bµν r eiΩr
r e
radius ε ! 2π {Ω
Ÿ no ingoing wave Ñ Zµν 0
1st assumption, Tµν
µν
µν
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
approximate calculation of wave generation
Generation of GW, determining the outside solution
∇2 Ω2 Bµν
16πSµν h̄µν Bµν eiΩt Bµν Ar eiΩr
³ 2
Ω Bµν d3 x ¤ Ω2 |Bµν |max 4πε3 {3, ù negligible term
³ 2
¶
d
∇ Bµν d3 x n ∇Bµν dS ÝÑ 4πε2 dr
Bµν rε 4πAµν
µν
* expressions for GW generated by the sources, neglecting
terms r2 and r1 that are higher order in εΩ
def. Jµν
³
Sµν d3 x; Aµν
εÑ0
ÝÑ
4Jµν
ñ
h̄µν
h̄µν are componentes of a single tensor
4Jµν eiΩprtq{r
conservation law for T µν , T µν,ν 0 6 T|µν
r
if Ω 0, J µ0 0 ñ h̄µ0 0
mponce
0
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
approximate calculation of wave generation
Generation of GW, quadrupole approximation for gravitational radiation
expression for Jij ,
for a source in slow motion, T 00 ρ, Newtonian mass
density
Quadrupole moment tensor of mass distribution,
I lm
h̄jk
³
T 00 xl xm d3 x Dlm eiΩt
2Ω2Djk eiΩprtq{r
2Ω2Ijk e r
Jlk
21 Djk Ω2eiΩprtq{r
}Quadrupole approximation for
gravitational radiation~
iΩr
terms neglected of order r2 but also r1, not dominant in the
slow-motion approximation
h̄jk,k is of higher order, and this guarantees the gauge
condition h̄µν,ν 0 is satisfied at the lowest order in r1 and Ω
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
approximate calculation of wave generation
Generation of GW, quadrupole approximation for gravitational radiation
expression for Jij ,
for a source in slow motion, T 00 ρ, Newtonian mass
density
Quadrupole moment tensor of mass distribution,
I lm
h̄jk
³
T 00 xl xm d3 x Dlm eiΩt
2Ω2Djk eiΩprtq{r
2Ω2Ijk e r
Jlk
21 Djk Ω2eiΩprtq{r
}Quadrupole approximation for
gravitational radiation~
iΩr
terms neglected of order r2 but also r1, not dominant in the
slow-motion approximation
h̄jk,k is of higher order, and this guarantees the gauge
condition h̄µν,ν 0 is satisfied at the lowest order in r1 and Ω
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
approximate calculation of wave generation
Generation of GW, quadrupole approximation for gravitational radiation
expression for Jij ,
for a source in slow motion, T 00 ρ, Newtonian mass
density
Quadrupole moment tensor of mass distribution,
I lm
h̄jk
³
T 00 xl xm d3 x Dlm eiΩt
2Ω2Djk eiΩprtq{r
2Ω2Ijk e r
Jlk
21 Djk Ω2eiΩprtq{r
}Quadrupole approximation for
gravitational radiation~
iΩr
terms neglected of order r2 but also r1, not dominant in the
slow-motion approximation
h̄jk,k is of higher order, and this guarantees the gauge
condition h̄µν,ν 0 is satisfied at the lowest order in r1 and Ω
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
approximate solution for GW generation – TT gauge
choosing axes so that the point where the wave is
measured, travels in the z-direction,
$ TT
& h̄zi
0
TT
2
iΩr {r
h̄ h̄TT
yy Ω p-Ixx -Iyy qe
% xx
TT
2
iΩr
h̄xy 2Ω -Ixy e {r
mponce
-Ijk
Ijk 31 δjkIll
Trace-free reduced
quadrupole
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
GW generation, approximate solution I
examples
Examples:
1
waves emitted by a simple oscillator «laboratory oscillator»,
both masses osc. with ω à only one non-zero component:
Ixx mrpx1 q2 px2 q2 s mrp 21 l0 A cos ωtq2 p 12 l0 A cos ωtq2 s
const. mA2 cosp2ω tq 2ml0A cosploωoo n tq
å ω-term:
å 2ω-term:
"
4
iωt
ß same radiation pattern,
-Ixx 3 ml0 Ae
TT h̄TT 4mω 2 l Ae2iω prtq {r
2
iωt
h̄
0
-Iyy -Izz 3 ml0 Ae
xx
yy
TT h̄TT 0,
h̄
no
rad.
in x dir.
ß radiation in z-direction,
xy
ij
TT
TT
2
iω
p
r
t
q
h̄xx h̄yy 2mω l0 Ae
{r ë Total radiation:
TT
TT
h̄xy h̄ij 0,
no rad. in x dir.
2
h̄TT
xx r2mω l0 A cos ω pr tq
ë off-diagonal vanished,
4mω 2 A2 cos 2ω pr tqs{r 1034 {r
ë radiation linearly polarized
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
GW generation, approximate solution II
examples
2
gravitational source: Binary System, two stars of mass m,
idealized as points in circular orbit
è orbit equation: m2"
{l02 mω2 pl0{2q ñ ω p2m{l03q1{2
x1 ptq 21 l0 cos ωt y1 ptq 12 l0 sin ωt
î curves of motion:
x2 ptq x1 ptq
y2 ptq y1 ptq
å Quadrupole tensor:
$
& Ixx
%
Iyy
Ixy
14 ml02 cosp2ωtq const.
14 ml02 cosp2ωtq const.
14 ml02 sinp2ωtq
å Reduced Quadrupole tensor:
"
I-xx
-Ixy
-Iyy 14 ml02 ω 2 e 2iωt
1
2 2 2iωt
4 iml0 ω e
mponce
î all radiation comes out Ω 2ω,
ß along z-direction (perp. to the
plane of the orbit) circularly polarized radiation,
h̄TT
xx
h̄TT
xy
2 2 2iω prtq {r
h̄TT
yy 2ml0 ω e
2
2
2iω
p
2iml0ω e rtq{r
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
GW generation, approximate solution III
examples
ß Linear polarization radiation on x-direction:
h̄TT
yy
1 2 2 2iωprtq
h̄TT
{r
zz ml0 ω e
2
aligned with the orbital plane
e.g. pulsar PSR 1913+16,
two very compact stars orbiting each other closely,
orbital period (Doppler shift) 7h 45min 7s
both stars have masses approx. equal to 1.4Md
5 kpc away from the Earth ß its radiation amplitude 1020 at
Earth
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
Order-of-magnitudes Estimates
MR2, for a system of mass M and size R
radiation amplitude about M pΩRq2 {r v2 pM {Rq
Djk
a collapsing mass moving under its own gravitational
forces: v2 φ0 (typical Newtonian potential in the source),
M {r φr (Newtonian potential of the source at distance r)
ñ h φ0φr 6 wave amplitude À Newtonian potential
BUT we detect h not φ
spherically symmetric motions do not radiate
if T 00 is spherically symmetric ñ I lm 9δ lm and I- lm vanishes.
conservation of energy eliminates ‘monopole’ radiation in
linearized theory ( charge conservation eliminates
monopole radiation in EM)
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
approximate calculation of wave generation
Examples, GW generation–approx. soln.
order-of-magnitude estimates
Exact Solution of the Wave Equation
Exact Solution of the Wave Equation
Arbitrary Tµν , outgoing-wave
outgoing-wave for arbitrary source
h̄µν pt, xi q 4
³
p q d3 y
Tµν t R,yi
R
|xi| r " |yi| y, ñ
using conservation of
T µν,ν 0 (total enegy
and momentum
conserved),
h̄jk pt, xi q 2r Ijk,00 pt rq
,
R |xi yi |
h̄µν pt, xi q 4
r
³
Tµν pt R, yi qd3 y
adopting TT gauge
h̄TT
xx
h̄TT
xy
mponce
1r r-Ixx,00pt rq -Iyy,00pt rqs
2r -Ixy,00pt rq
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: a binary pulsar
Energy carried away by Gravitational Waves
GW can put energy into things they pass through (detector)
carry energy away from their sources
effects of the loss of energy by GW can be observed
although the GW itself not
á harmonic oscillator as a
detector of waves: ‘test body’
without influence on the GW è
inconsistent!
à detector extract energy from
the wave è waves must be
weaker (‘downstream’ ‘upstream’) ß detector (oscillator)
in motion will radiate itself
mponce
ß waves of two frequencies will
be emitted
ß waves with same freq. as
incident GW (Ω):
total downstream wave field
Ý amplitude, sum of two
waves
interfere desctructively,
producing a net decrease in
«Gravitational
Radiation»
the dowstream
amplitude
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: a binary pulsar
The energy flux of a Gravitational Wave I
energy flux: energy carried by the GW across a surface per unit area per unit time
á array of oscillators, filling the plane z 0 ( continuous
distribution, σ oscillators per unit area)
à incident wave (TT gauge),
$ TT
ç net energy flux (decreasing), δF & h̄xx A cosrΩpz tqs
TT
TT
21 σmγΩ2R2
h̄yy h̄xx
%
ç each oscillator, Ixx ml0 R cospΩt
0, other components
ç each oscillator
steady state φq
ç producing a wave amplitude,
oscillation,
ξ R cospΩt φq
2
ç energy given to each osc., δ h̄xx 2Ω ml0 R cosrΩpr tq φs{r
ç total radiated field due to all the
dE{dt ν pdξ {dtq2 mγ pdξ {dtq2
total
Ù averaging,
steady energy loss, oscs., δ h̄xx 4πσmΩl0 R sinrΩpz ³
tq φs
x dEdt y 2π1 p dEdt qdt 12 mγΩ2R2
Ω
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: a binary pulsar
The energy flux of a Gravitational Wave II
energy flux: energy carried by the GW across a surface per unit area per unit time
Total Flux
F
32π1 Ω2A2 ñ
F
TTµν y
32π1 Ω2xh̄TT
µν h̄
invariant under background Lorentz transformations,
but not under gauge changes
applies to all polarizations (Lorentz inv.)
applies to any waveform (either plane waves or spherical...)
not proved energy conservation –GR– (assumption)
valid only in linearized theory (‘lowest order’)
mponce
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: a binary pulsar
Energy Lost by a Radiating System
general isolated system radiating according a
QUADRUPOLE
distribution
integrating the «total flux» over a sphere surrounding the system
net energy loss rate
A
Ω
2y
Iij -I ij 4-I zj-Iz j -Izz
at distance r along the z-axis, F 16πr
2 x2j
j
generalizing for arbitrary locations, n x {r (unit normal vector),
Ω6
F 16πr
Iij -I ij 4nj nk -Iji -Iki ni nj nk nl -Iij -Ikl y
2 x26
Luminosity L of a source of Gravitational Waves,
integral
the flux over the sphere of radius r,
³ of
2
L FR sin θdθdφ
ñ
L 15 Ω6 x-Iij -I ij y
1
Generalization to general time dependence L 5 x;-I ij;-I
only for weak gravitational fields and slow velocities. alt.deriv.
ij
mponce
«Gravitational Radiation»
y
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
strong sources
L is very sensitive to velocity
largest velocities, v2
ñ L À pφ0q5
φ0
φ0 À 1,
L À 1 c5 {G 3.6 1052 W
–luminosity upper limit –.
mponce
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: a binary pulsar
Ø anything whose Newtonian potential substancially exceeds 1 ì
must form a BH: its gravitational
field will be so strong that no radiation will escape at all
é L 1 is the largest luminosity
any source can have
ç Quasars, the most luminous class
of object known, geom.luminosity
À 107.
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
General considerations
The energy flux of a Gravitational Wave
Energy lost by a radiating system
Example: a binary pulsar
Example: a Binary Pulsar
ç -Iij (reduced quadrupole tensor) for a binary system consisting
of two stars of equal mass M in circular orbits a distance l0 ,
luminosity of the binary system,
4.0p
q
L
?
p q
{
8 2 4 6
32
Mω 10 3
5 M l0 ω
534
Mω 10 3 (geom.units)
{
Newtonian energy of the system
L(SI units) cG L(geometrized)
for the binary pulsar system,
circ.orb., ω 2π {P 7.5049 1013 m1 ñ L 1.71 1029
(geom.units)
5
L, dP{dt 3PL
2E 2 1013 6 106s yr1
eccentricity e 0.617 !!!
dE
dt
mponce
defining the orbital radius r 12 l0 ,
1
M2
2 2
E 21 Mω 2 r2
2 Mω r 2r 1.11 103m
energy radiates in waves change
the orbit by decreasing its
2 1 dω
2 1 dP
energy, E1 dE
dt 3 ω dt 3 P dt
dP{dt 7.2 105 s yr1 ,
p2.30 0.22q 1012 (obs.)
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
Summary
GW are fluctuations in
curvature that propagate
through the universe at speed
of light.
They are a natural
consequence of Einstein’s
theory of relativity.
GW are transverse and
traceless in nature and are
produced by changes in the
quadrupole moment of a mass
distribution.
Mass and momentum
conservation ensure that there
mponce
GW carry energy and
momentum, which create
background curvature. This is
equal to the energy and
momentum lost by radiation
reaction at the source.
GW can be lensed and
redshifted in the same way as
electromagnetic waves, but
not readily absorved or
dispersed.
«Gravitational Radiation»
Introduction to GW
Detection of GW
Generation of GW
Energy carried by Gravitational Waves
Summary of GW properties
References
Ÿ
Schutz, Bernard F.
{ A First Course in General Relativity | .
Ÿ
Rezzolla, Luciano
Gravitational Waves from Perturbed Black Holes and
Relativistic Stars.
gr-qc/0302025.
Ÿ
Gair, Jonathan R.
«Gravitational Waves Astronomy» online course notes @
Cavendish Lab., Cambridge.
http://www.ast.cam.ac.uk/jgair/GWAstronomy/
mponce
«Gravitational Radiation»
Appendixes
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Generation of GW I
alternative approach // EM analogy...
Ý classical electrodynamics, energy emitted put by an oscillating electric dipole d
Lelect.dip.
emitted)
(energy
23 q2a2 23 pd:q2;
(unit time)
d
qx
Ý Luminosity in GW produced by an oscillating mass-dipole, N
point-like particles of mass mA
~d
~d9
°NA1 mA~x9 A ~P
:
linear momentum conserv. ñ ~d ~p9 0 é Lmassdip. 0
therefore no mass-dipole radiation in GR ( no EM
°NA1 mA~xA,
radiation from an oscillating electric-monopole)
mponce
«Gravitational Radiation»
Appendixes
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Quadrupole radiation I
Generation of GW - alternative approach // EM analogy...
Ý electromagnetic energy emission produced by an oscillating
electric quadrupole
p ; q pQ;jk Q;jk q
where Qjk rpxA qj pxA qk 1
i s,
δ
p
x
q
p
x
q
is the electric
3 jk A i A
Lelect.quad.
1
2
20 Q
°N
A 1 qA
1
20
quadrupole for a distribution of N
charges
mponce
Ý in close analogy, energy
loss due to an oscillating mass
quadrupole
Lmass.quad.
51 cG5 x;I y2 1G ; ; 2
5 c5 x-Ijk -Ijk y
where -Ijk is the trace-less
mass
quadrupole
(or
“reduced”
mass
quadrupole)
I
jk °N
1
i
A³1 mA rpxA qj pxA qk 3 δjk pxA qi pxA q s
1
i
ρpxjxk 3 δjk xix qdV
«Gravitational Radiation»
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Appendixes
Quadrupole radiation II
Generation of GW - alternative approach // EM analogy...
A crude estimative of the third derivative of the mass
quadrupole
is given by,
;-I jk mass of the
system in motion
size of the
system
ptime scaleq
Lmass quad. cG Mv
τ
3
2
MRτ Mvτ
2
2
3
2
so that,
5
Although extremely simplified, this eqs. show that:
energy emission in GW is severely suppressed by the coef.
G{c5 1059 ÐÑ the conversion of any type of energy into
GW is, in general, not efficient.
mponce
«Gravitational Radiation»
Appendixes
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Quadrupole radiation III
Generation of GW - alternative approach // EM analogy...
time variation of the mass quadrupole, can become
considerable only for very large masses moving at
relativisitic speeds.
6 these conditions can not be reached by sources in terrestrial
laboratories, but can easily met by astrophisycal compact
objects, which therefore become the most promising sources of
gravitational radiation.
return
mponce
«Gravitational Radiation»
Appendixes
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Polarization of Gravitational Waves
Circular Polarization, Eliptical Polarization, General Polarization
Circularly polarized in the x y plane if hTT
yy
TT .
hTT
ih
xy
xx
eR
e ?2ie
eL
rotates clockwise
hTT
xx and
e ?2ie
rotates counter-clockwise
Elliptically polarized with principal axes x and y if
TT
TT
TT
hTT
xy iahxx , where a is a real number, and hxy hxx .
TT
If hTT
xy αhxx , where α is a complex number (general case
for a plane wave), new axes x1 and y1 can be found for
which the wave is elliptically polarized with these principal
axes.
circular and linear polarization are special cases of elliptical.
mponce
«Gravitational Radiation»
Appendixes
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Gravitational Wave spectrum
Frequency of GW in the Universe extends over many decades
Frequency of GWs from a system scales as 1/mass
mponce
«Gravitational Radiation»
Appendix i: alternative deduction of generation of GW
Appendix ii: «Polarization» – complement
Appendix iii: GW Spectrum
Appendix iv: }Formal complements~
Appendixes
Formulae
Generalization for an arbitrary binary system
average energy lost rate and angular momentum:
2
3
xdE{dty 325 aµ5pp1m e2Mq7q{2
1
2
5{2
xdL{dty 325 µa7p{2mp1 Meq2q2
average orbital elements:
xda{dty 645 aµpp1meMqq{ 1
µpm M q e
xde{dty 304
15 a p1e q { 1
µpm M q {
xdP{dty 192π
5 a { p1e q {
2
2 7 2
3
73 2
24 e
2
121 2
304 e
2 5 2
4
3 2
5 2
2 7 2
mponce
1
37 4
e
96
73 2
e
24
1
37 4
96 e
73 2
24 e
7 2
e
8
37 4
96 e
«Gravitational Radiation»