Download Gravitational Radiation:

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/