Undergraduate Project in Physics Gitit Feingold Advisor: Prof. Eduardo Guendelman Department of Physics

Undergraduate Project in Physics
Gitit Feingold
Advisor: Prof. Eduardo Guendelman
Department of Physics
Ben Gurion University
Gravitational Radiation
Abstract:
On 1905, Albert Einstein predicted the existence of gravitational waves to be
emitted macroscopic (as in the merging of two large masses) scales.
However, to the best of my knowledge it should also be produced microscopic
(for example – in atomic transitions between levels) scales.
Gravitational waves have not been detected as yet, but a great degree of
effort is undertaken worldwide by researchers interested in this field, since it
connects the theory of general relativity and microscopic physics.
The aim of this project is to present an overview of the phenomenon and the
nature of gravitational radiation.
I will try to show a similarity between Maxwell’s equations and Einstein’s
general relativity equations, which will allow finding a radiative solution for
Einstein’s equations as is known to be possible for Maxwell’s.
I will conclude by describing state-of-the-art of the detection of gravitational
radiation emitted from large masses.
Contents:
1. Introduction
3
2. The gravitational and electromagnetic waves
3
3. The week filed approximation
6
4. Plane waves
9
5. Energy and momentum of plane waves
12
6. Quadrupole radiation
14
7. Scattering and absorption of gravitational radiation
17
8. Detection of gravitational radiation
20
9. References
23
2
Introduction
Looking at gravitational equations one can find many similarities to the theory
of electromagnetism, therfore can anticipate, both Einstein and Maxwell's
equations to have a radiative solution.
Gravitational radiation is an amount of energy being produced from a large
mass colliding or rotating, which spreads in spacetime in the form of waves.
The movement of mass is predicted to be making ripples propagating in
spacetime, binary-stars for example have its very special "sound".
In curved spacetime, the distance between two points isn't stable; it can
change as a result of the gravitational wave propagation.
If proved to exist, gravitational radiation might be the key to completion of the
filed theorem for providing a crucial link between general relativity and the
microscopic frontier.
The nonlinearity of Einstein's equations causes the theory of gravitational
radiation to be very complicated, as a result, no general radiative solution had
been found. Two approaches to this difficulty are: one is to study the weekfiled solution of Einstein equations, which describe waves carrying not enough
energy and momentum to affect the propagation of themselves. The other is
to look for special solutions.
This project will be dealing with the first approach; one motivation to do so is
that any real detection of gravitational radiation is likely to be in very low
intensity.
The gravitational and electromagnetic waves
Electrodynamics:
3
With no influence of gravitational field, the Maxwell equations can be written
as:
∂ αβ
F = −J β
α
∂x
(2.1)
(2.2)
∂
∂
∂
Fβγ + β Fγα + γ Fαβ = 0
α
∂x
∂x
∂x
Where:
J β is the current four-vector {J , ε }
F αβ is the field strength tensor
Using J µ and F µν in general coordinates and requiring that they reduce to
F αβ and J β demanded to behave as tensors under general coordinate
transformations
F µν ≡
(2.3)
Jµ ≡
∂x µ ∂xν αβ
F
∂ξ α ∂ξ β
∂x µ α
J
∂ξ α
Replacing all derivatives by covariant derivatives, we get:
(2.4)
F µν ; µ ≡ − J ν
Fµν ;λ + Fλµ ;ν + Fνλ ; µ = 0
Lowering or raising indices will now be possible using:
(2.5)
Fλ k ≡ gλµ gkν F µν
Keeping in mined that F µν and Fµν are anti-symmetric, we shall write Maxwell
equations as:
(2.6)
∂
g F µν ≡ − g J ν
∂x µ
∂
∂
∂
Fµν + ν Fλµ + µ Fνλ = 0
λ
∂x
∂x
∂x
4
Maxwell equations written in that form are true when there is no influence of
gravitation and they are generally covariant. We can use the Principal of
General Covariance to write the electromagnetic force on a charge e:
f α = eF α β
(2.7)
dx β
dτ
In general coordinates the electromagnetic force in an arbitrary gravitational
field is:
f µ = eF µν
(2.8)
dxν
dτ
Using the Principal of General Covariance once more, we can estimate the
current vector (integrating along the n’th particle trajectory):
(2.9)
J α = ∑ en ∫ δ 4 ( x − xn )dxnα (In special relativity)
n
In general relativity the conservation law appears as:
(2.10)
J µ ; µ = 0 or
∂
g 1/ 2 J µ ) = 0
µ (
∂x
Notice that g 1/ 2 guards the constancy of en .
Energy and momentum:
The density and current of energy and momentum can be represented as a
symmetric tensor T αβ satisfying the following conservation equation:
(2.11)
∂T αβ
= Gβ
α
∂x
G β is the density of an external force acting on the system.
We will define T µν , Gν as contravariant tensors so the generally covariant
equation in the locally inertial system:
5
(2.12) Or using:
T µν ; µ = Gν
∂
µ
g ∂x
1
(
)
gT µν = Gν − Γνµλ T µν
The second term on the right side represents a gravitational force density; this
force depends on the system itself through the energy-momentum tensor
alone.
For a system consisting of both particles and radiation, considering the
moment to the purely material energy-momentum tensor we can write:
(2.13)
µ 0 1/ 2 3
∫ T g d x = ∑ mn
n
dxn µ
dτ
The summation is over all the particles in the volume of the integration. This
suggests that T µ 0 g1/ 2 is to be regarded in general as spatial density of energy
and momentum. We are interested in defining the energy, momentum and
angular momentum for arbitrary system as:
(2.14)
P µ ≡ ∫ T µ 0 g1/ 2 d 3 x
J µν ≡ ∫ x µ T ν 0 − xν T µ 0 g1 / 2 d 3 x
Since these quantities are not contravariant tensors and are not conserved,
we are forced to take into consideration the fact that energy and momentum
are being exchanged between matter and gravitation.
The weak field approximation
We have already discussed the complications in finding a general solution to
Einstein's equations. To overcome these deficulties it is comfortable to look at
the radiation far from the source where we can work with pure plane waves
that have no affect on their own source.
6
Minkowski spacetime is the mathematical setting in which Einstein's theory of
special relativity is most conveniently formulated. In this setting the
dimensions of space are combined with a dimension of time to form a fourdimensional representation of spacetime.
We will use the next approximation to the Minkowski metric η µν :
g µν = η µν + hµν
(3.1)
Where: hµν << 1 . Deriving Ricci’s tensor to the first order of h.
(3.2)
Rµν ∂ λ
∂
Γ λµ − λ Γ λµν
ν
∂x
∂x
and the affine connection (to the first order of h):
Γ λµν (3.3)
η λρ  ∂

2  ∂x µ
hρν −

∂2
∂2
h
+
h
ν ρµ
ρ µν 
∂x
∂x

Using η µν to lower indices we will have Ricci’s tensor appear as:
(3.4)

1
∂
∂2
∂2
Rµν (1) ≡  2 hµν − λ µ h λν − λ ν h λ µ + µ ν h λ λ 
2
∂x ∂x
∂x ∂x
∂x ∂x

Einstein field equation:
∂
∂2
∂2
λ
λ
h
−
h
+
hλ λ = −16π GSµν
ν
µ
∂xλ ∂x µ
∂xλ ∂xν
∂x µ ∂xν
1
where
Sµν ≡ Tµν − ηµν T λ λ
2
∂ µ
satistifying conservation condition µ T v = 0 since T µ v is independentof hµν .
∂x
2 hµν −
(3.5)
T µν - is the stress energy tensor.
if the gravitational force strongly affect the structure of the radiating system
then we replace T µν with the generic tensor τ µν of the system.
linearized Ricci tensor satisfies Bianchi identities of the form:
(3.6)
∂ µ
1 ∂
Rν =
µ
∂x
2 ∂xν
 2 λ
 1 ∂ λ
∂2
h
−
h λν  =
R λ
λ

λ
ν
ν
∂x ∂x

 2 ∂x
derived to the first order.
7
Coordinate transformations perform many solutions to the field equation (3.5),
therefore, there is no unique solution.
Supporting the week field approximation the coordinate transformation we
shall use:
x µ → x 'µ = x µ + ε µ ( x )
(3.7)
∂ε µ
is at most of the order of magnitude as hµν .
∂xν
From the new metric, we shall have:
h 'µν = h µν −
(3.8)
∂ε µ λν ∂ε ν ρµ
η − ρη
∂x λ
∂x
and drive the solution for the field equation:
h 'µν = hµν −
(3.9)
∂ε µ
ν
∂x
−
∂εν
∂x µ
ε µ ≡ ε νη µν are four small arbitrary functions of x µ giving us the gauge
invariance of the field equation.
Solution becomes simpler using particular gauge: harmonic coordinate
system.
g µν Γ λµν = 0
(3.10)
⇒
∂ µ
1 ∂ µ
hν =
h µ ( on the first order ) .
µ
2 ∂xν
∂x
We now have to solve the d’Alembert equation of the form:
(3.11)
2 hµν = −16π GS µν
The solution is the retarded potential:
(3.12)
hµν ( x, t ) = 4G ∫ d 3 x '
S µν ( x ', t −
x − x')
x − x'
8
Notice that the term
t − x− x'
shows that gravitational effects propagate with
the velocity c – the speed of light!
We found solution (3.12) for the gravitational radiation produced by the source
S µν , staying aware of the fact that any additional term satisfying
2 hµν = 0
(3.13)
∂ µ
1 ∂ µ
hν =
h µ
µ
∂x
2 ∂xν
represents a gravitational radiation coming in from infinity can also solve
(3.11). The solutions will be dependent of the form of the waves.
Plane waves
We expect the waves produced for example by a rotating large mass, to
extend in the form of spheres from the source out, additionally we assume
that if we will be able to detect these waves it would be at a very large
distance from the source causing the form of the wave to be approximately a
plane wave.
The general solution for:
2 hµν = 0
1 ∂ µ
∂ µ
hν =
h µ
µ
∂x
2 ∂xν
when r → ∞ is linear superposition of the form:
(4.1)
λ
hµν ( x ) = eµν eikλ x + eµν * e − ikλ x
λ
This satisfies the equations if:
(4.2)
kµ k µ = 0
and
(4.3)
k µ e µν =
1
kν e µ µ
2
The polarization tensor will be defined as:
9
eµν = eνµ
(4.4)
There are six independent components in the representing 4x4 matrix of eµν
but only two of them have physically significant degree of freedom
independent of the coordinates.
However, in order to emphasize the difference between the components of
the polarization tensor we subject the coordinate system to a rotation about
the axis of the propagation of the wave z - this is just a Lorentz transformation
of the form:
R µν
(4.5)
0
1
 0 cos θ
→
 0 sin θ

0
0
0
− sin θ
cos θ
0
0
0 
0

1
we can see that Rνµ kµ = k µ therefore the only effect is to transform
eµν → e 'µν = Rµρ Rνσ eρσ
(4.6)
Using the relations:
(4.7)
e '± = e±2iθ e±
; e± ≡ e11 ∓ ie12 = −e22 ∓ ie12
f '± = e± iθ f ±
; f ± ≡ e31 ∓ ie32 = −e01 ∓ ie02
e '33 = e33 , e '00 = e00
Generally, any plane wave Ψ , which is transformed by an angle θ about the
direction of the propagation into Ψ ' = eihθ Ψ is said to have helicity h . We have
seen that a gravitational plane wave can be decomposed into parts:
(4.8)
e± ;
h = ±2
f± ;
h = ±1
e33 , e00 ; h = 0
10
However, one can reveal that the parts containing helicity h = −1, 0,1 vanishes
by a right choice of coordinates, and the only physically significant part
contain h = ±2 .
In both gravitation and electrodynamics we can choose to work now in an
inertial coordinate system 2 ≡ η αβ
∂2
, in that case we can find a plane
∂xα ∂x β
wave solution for electrodynamics waves of the form:
Aα = eα e
(4.9)
ik β x β
+ eα * e
− ik β x β
kα k α = 0
kα eα = 0
eα has only three independent components satisfying the formal conditions.
Furthermore, leaving the electric and the magnetic fields unchanged, without
Lorentz gauge we can use gauge transformation to write:
(4.10)
Aα → A 'α = Aα +
Φ ( x) = iε e
ik β x β
∂Φ
∂xα
− iε * e
− ikβ x β
then the new potential can be written as:
(4.11)
A 'α = e 'α e
ik β x β
− e 'α * e
− ik β x β
e 'α = eα − ε kα
ε is arbitrary parameter meaning we have three independent components of
eα like we had before only two of them have physical significant. We would
like to identify these two components, considering a gravitational wave
traveling along z-axis with:
(4.12)
k 
0
kα =   ; k > 0
0
 
k 
11
Since, kα eα = 0 we can determine e0 = −e3 , using gauge transformation
changes e3 → e '3 = e3 − ε k , but we already said ε to be arbitrary able us to
choose ε =
e3
and make e '3 = 0 leaving the physical significant to e2 and e1 .
k
Finally, we reveal the physical meaning of the two components by subjecting
the plane electromagnetic wave to the rotation matrix. The polarization vector
will change into:
e 'α = Rαβ eβ ⇒
(4.13)
e '± = e ± iθ e±
e '3 = e3
e± ≡ e1 ∓ ie2
The electromagnetic wave can be decomposed to parts with different values
of helicity h = −1, 0,1 , parts with h = ±1 have physically significant, equivalent to
gravitational waves that have h = ±2 . This is the meaning (classically
speaking) that both electromagnetic and gravitational waves are carried by
waves of spin 1 and spin 2 respectively.
Energy and momentum of plane waves
The solution of plane wave leads to calculating the energy and momentum it
carries. The energy momentum tensor of gravitation given to the order of h 2
by:
(5.1)
t µν 1  1
1
1

− hµν η λρ Rλρ (1) + η µν h λρ Rλρ (1) + Rµν (2) − η µν η λρ Rλρ (2) 

8π G  2
2
2

Rµν ( N ) is the term in the Ricci tensor of order N in hµν . The metric
gµν = ηµν + hµν satisfies the first order Einstein equations Rµν (1) = 0 so we get:
(5.2)
t µν 1  (2) 1

Rµν − η µν η λρ Rλρ (2) 
8π G 
2

We would like to calculate Rµν (2) in order to do so we would add the solution
we found for plane wave into:
12
(5.3)
1
Rµ k (2) = − h λν
2
 ∂ 2 hλν
∂ 2 hµν
∂ 2 hµ k  1  ∂hν σ ∂hν ν
∂ 2 hλ k
 k µ − k λ − ν µ + ν λ  + 2 ν − σ
∂x ∂x
∂x ∂x
∂x ∂x  4  ∂x
∂x
 ∂x ∂x
∂h
∂h
1  ∂h
−  σλk + σλk − λσk
4  ∂x
∂x
∂x
σ
  ∂h µ ∂hσ k ∂hµ k 
2
+ µ −

−

k
∂xσ 
∂x
  ∂x
σ
λ
  ∂h µ ∂hσλ ∂h µ 
+
−


 ∂x
∂xσ 
∂x µ
  λ
Simplifying the complicated result can be doable if taking an average over
region of space larger than
1
.
k
and we get:
(5.4)
1


Rµν ( 2) = Re {e λρ *  k µ kν eλρ − k µ kλ eνρ − kν k ρ eµλ + kλ k ρ eµν  +  e λ ρ kλ − eλ λ k ρ  *  k µ e ρν + kν e ρ µ − k ρ eµν  −
2


1
−  kλ eρν + kν eρλ − k ρ eλν  *  k λ e ρ µ + k µ e ρλ − k ρ e λ µ 
2
}
It is comfortable to use harmonic coordinates for the solution, but we can start
by adding the next term into hµν ( x)
(5.5)
i ( qµ εν + qν ε µ ) eiqλ x − i ( qµ εν * + qν εν * ) e−iqλ x
λ
1
q−k
Averaging over a region greater than
λ
the destructive interference
between the two waves adds a term to Rµν ( 2) , with the right adjustments, the
second term vanishes and with the use of harmonic coordinates, we get:
(5.6)
Rµν ( 2) =
k µ kν  λρ *
1 λ 2
 e eλρ − e λ 
2 
2

and so we can easily calculate the energy-momentum tensor for plane
wave tµν (under the harmonic coordinate conditions):
(5.7)
t µν =
k µ kν  λρ *
1 λ 2
 e eλρ − e λ 
16π G 
2

we can also use gauge transformation to write:
(5.8)
t µν =
k µ kν
8π G
(e
2
11
+ e12
2
)
+ e−
2
)
or in term of helicity amplitudes:
(5.9)
t µν =
k µ kν
16π G
(e
+
2
13
Quadrupole radiation
In addition to the week field approximation, we use the approximation of "long
waves" 1 >> ω R where R is the source radius and is much smaller than the
wavelength.
Most of the radiation emitted at frequencies of order: v
R
where v is typical
velocity within the system.
Using the above approximation, we can write the energy-momentum tensor in
Fourier space:
Tij (k , ω ) ∫ d 3 xTij ( x, ω )
(6.1)
using the conservation law:
∂2
T ij ( x, ω ) = −ω 2T 00 ( x, ω )
∂xi ∂x j
(6.2)
Multiplying with xi x j and integrating over x, we have:
(6.3)
Tij (k , ω ) −
ω2
2
Dij (ω ) ; Dij (ω ) ≡ ∫ d 3 xx i x jT 00 ( x, ω )
we can now write:
(6.4)
dP Gω 6
=
Λ ij ,lm (k ) Dij* (ω ) Dlm (ω )
d Ω 4π
if the source is a Fourier series then the power is a sum of the above term, if
the source is a Fourier integral then it is clear that we can write the energy per
solid angle:
∞
(6.5)
dE G
= Λ ij ,lm (k ) ∫ ω 6 Dij* (ω ) Dlm (ω )d ω
dΩ 2
0
since Dij (ω ) independent of k direction we can calculate the integral over
solid angle using:
14
4π
δ ij
3
4π
∫ d Ωkˆi kˆ j kˆl kˆm = 15 (δ ijδ lm + δ ilδ jm + δ imδ jl )
∫ d Ωkˆ kˆ
i
(6.6)
j
=
The right side term should be invariant to symmetry and rotation contracting
indexes we can calculate:
∫ d ΩΛ
(6.7)
ij ,lm
(k ) =
2π
11δ ilδ jm − 4δ ijδ lm + δ imδ jl 
15 
Power emitted per discrete frequency:
P=
(6.8)
2Gω 6
5
2
1
 *
D
(
)
D
(
)
−
D
(
)
ω
ω
ω
ij
ij
ij


3
for small range of frequencies:
∞
E=
(6.9)
2
4π G
1

ω 6  Dij* (ω ) Dij (ω ) − Dij (ω )  d ω
∫
5 0 
3

We would like to calculate one special case but we need to clarify some
stipulations:
1. Usually the quadropole approximation is made for nonrelativistic
systems, for these systems the energy density T 00 ( x, ω ) is dominated by
the rest mass density. We don't need to take the potential and the
kinetic energy explicitly in T µν because it is not necessarily conserved
moreover, for systems of particles bounded by gravitational forces we
should in principal use τ µν (tensor that contains both matter and
gravitation), in the formal calculations we received terms including T 00
therefore we may use this approximation freely.
2. For rotating and\or vibrating systems it may be difficult to evaluate
Fourier transform T 00 ( x, ω ) and it may be easier to evaluate:
Dij (t ) ≡ ∫ d 3 xx i x jT 00 ( x, t )
Where we can evaluate Dij (ω ) using Fourier transform or sum of
Fourier components:
15
∞
Dij (t ) ≡ ∫ d ω Dij (ω )e − iωt + c.c. or Dij (t ) ≡ ∑ e − iωt Dij (ω ) + c.c.
ω
0
3. Choosing the origin of coordinates xi in the integral for Dij generally
does not matter because shifting the origin by a parameter, under
conservation of energy and momentum, causes an addition of linear
functions of time. Using the above approximation of T 00 ( x, ω ) it does not
change Dij so we can change the origin of coordinates freely.
Let us calculate the power of radiation emitting from a rotating body. We
consider a rotating rigid body about 3-axis with angular frequency Ω , the mass
density T 00 will take the form of:
T 00 ( x, t ) = ρ ( x ')
(6.10)
x' are fixed in the body coordinates defined as:
x1 ≡ x1' cos Ωt − x2' sin Ωt
x2 ≡ x1' sin Ωt + x2' cos Ωt
(6.11)
x3 ≡ x3'
By changing the coordinates, we can write Dij as:
I ij ≡ ∫ d 3 x ' xi' x 'j ρ ( x ')
(6.12)
Where I ij is the moment-of-inertia tensor in body fixed coordinates.
For simplicity, let us consider rotation around one of the principal axes of the
ellipsoid, so that I13 = I 23 = 0 .
We may choose x1' , x2' consolidate with the ellipsoid's axes here I12 = 0 then:
I11 + I 22 I11 − I 22
I −I
+
cos 2Ωt ; D12 (t ) = 11 22 sin 2Ωt ; D13 (t ) = 0
2
2
2
I +I
I −I
D22 (t ) = 11 22 − 11 22 cos 2Ωt ; D23 (t ) = 0
2
2
D33 (t ) = I 33
D11 (t ) =
(6.13)
For ω = 2Ω the remaining Fourier coefficients are:
(6.14)
D11 (2Ω) = D22 (2Ω) = iD12 (2Ω) =
I11 − I 22
4
The total power emitted at twice the rotation frequency is given by:
16
(6.15)
P (2Ω ) =
32GΩ 6 I 2 e2
I −I
; I ≡ I11 + I 22 , e ≡ 11 22
5
5c
I
I is the moment of inertia, e is the equatorial ellipticity.
Conclusions:
a) Circular symmetry body e = 0 will not emit gravitational radiation.
b) For a mass fixed at xi ' = (r , 0, 0) : I11 = mr 2 ⇒ I = mr 2 , e = 1 the power of
radiation:
P(2Ω) =
32GΩ6 m 2 r 4
5c 5
Scattering and absorption of gravitational radiation
Consider a plane gravitational wave impinging on a target at the origin. At
large distance, the wave will consist of a scattered wave and the incident
wave.
(7.1)

eiωr  − iωt
ikx
ˆ
hµν ( x, t ) 
→
e
e
+
f
(
x
)
e
µν
 µν
r →∞
r 

where: eµν is the polarization tensor of the plane wave.
We define: r ≡ x , xˆ ≡
x
, ω ≡ k and f µν is a scattering amplitude, which is
r
independent of r or t.
We would like to analyze the energy balance between the gravitational wave
and the target then we must decompose the wave written above into incoming
and outgoing parts. The plane wave has a Lagendre expression:
∞
(7.2)
eikx = ∑ ( 2l + 1) Pl (kˆ ⋅ xˆ )i l jl (ω r )
l =0
Pl is the usual Lagendre polynomial and jl is spherical Bessel function of
order l . Asymptotically we have:
(7.3)
i l jl (ω r ) →
1
eiω r − (−)l e −iω r 
2iω r
the sum over l become simply the Lagendre expression of delta function:
17
∑ ( 2l + 1) P (µ ) = 2δ (1 − µ )
l
l
(7.4)
∑ ( 2l + 1) (−) P (µ ) = 2δ (1 + µ )
l
l
l
The desirable outgoing and incoming waves:
eikx 
→
r →∞
(7.5)
eiωr
e −iω r
δ (1 − kˆ ⋅ xˆ ) −
δ (1 + kˆ ⋅ xˆ )
iω r
iω r
the gravitational wave then appears as:


hµν 
→  eµν eiωr + eµν e − iω r  e − iωt + c.c. ;
r →∞
in
 out

1 
eµν ( x) =
eµν δ (1 − kˆ ⋅ xˆ ) + iω f µν ( xˆ ) 

iω r 
out
1 
eµν ( x) = −
eµν δ (1 + kˆ ⋅ xˆ ) 

iω r 
in
(7.6)
The total power carried out of a large sphere (radius r) by the outgoing wave
will be:
oi
Pout = ∫ d Ω tout
xˆi r 2
(7.7)
oi
tout
is the mean energy flux (the average is over space-time with
dimensions>>
1
ω
and small compared to r).
we can write: Pout = Pscat + Pint + Pplane
Pscat =
(7.8)
2
1
ω2

d Ω  f λν * ( xˆ ) fλν ( xˆ ) − f λ λ ( xˆ ) 
∫
16π G
2


The interference term can be calculated as:
(7.9)
Pint =
ω2
 −1
1


Re  ∫ d Ωδ (1 − kˆ ⋅ xˆ ) eλν * f λν ( xˆ ) − eλ λ * f ν ν ( xˆ )  
8π G  iω
2


integrating over delta function:
(7.10)
Pint = −
ω
1


Im e λν * f λν (kˆ) − eλ λ * f ν ν (kˆ) 
4G 
2

This term is formally infinite in the limit where r goes to infinity. However, the
amount of power carried out by the plane wave is the same as the amount of
power brought into a sphere of radius r.
(7.11)
Pin = Pplane
18
Pplane cancels out of the equation of energy conservation, the power absorbed
by the target is in that case:
Pabsorb = Pin − Pout = − ( Pscat + Pint )
(7.12)
The energy flux in the impinged wave:
(7.13)
ω 2  λν *
1 ν 2
Φ ≡ t oi kˆi =
 e eλν − e ν 
16π G 
2

The effective cross-section for elastic scattering:

(7.14)
σ scat ≡
Pscat
=
Φ
∫ d Ω  f
λν *
1 λ 2
f λ 
2

1 ν 2
− eν 
2

( xˆ ) f λν ( xˆ ) −
 λν *
 e eλν

The total cross-section for scattering or absorption:
σ total ≡ −
(7.15)
Pscat + Pabsob
Φ
The total cross-section can be expressed in terms of the interference between
the incident and scattered waves:
σ total ≡ −
(7.16)
Pint
Φ
adding the formal terms for the flux and the interference:
(7.17)
σ total
1


4π Im eλν * f λν (kˆ) − eλ λ * f ν ν (kˆ) 
2


=
2
1


ω  eλν *eλν − eλ λ 
2


The cross-section is proportional to the imaginary part of the forward
scattering amplitude.
The aim of gravitational scattering theory is to calculate the scattering
amplitude and to determine the cross section.
19
Detection of gravitational radiation
Doing some great efforts to detect gravitational waves, what will be the sound
we'll listen to?
It is always a scientific challenge to discover something new- like a new type
of radiation, but more importantly to use the information to better
understanding of systems is universe. This information can not only give the
completion of knowledge collected today by telescopes, but also to open view
for new researches that dependent only on it.
It is easy to figure out that the intensity of the waves depends on the size,
density and the rate of the movement of the masses. Therefore, we can use
that knowledge to complete the information collected by telescopes and in
some cases, it will even be the only source of information.
Scientists expect to detect emitting of such waves from the following sources:
"Compact elements" - very massive small elements like white dwarfs,
neutron-stars, pulsars and black-holes.
We also expect to detect certain activities like collapse stars collision of two
objects, binary systems, supernova and even remains from the "Big Bang".
It was already mentioned before, that no one has yet been able to detect
gravitational wave but there is already indirect evidence for their existence.
In 1974 Russell A. Hulse and Joseph H. Taylor, Jr. searched the space
using radio telescope and discovered a radio signal that is typically to pulsars
(PSR 1913 + 16), but it was unique since it had a special periodic Doppler sift
every 7 hours and 45 minutes. Relaying on the information this discovery
provides, the two scientists understood that the system observed was a
binary- pulsar, a very energetic likely to emit gravitational waves system.
The binary system presented a reduction of the distance between the two
stars, meaning it has to emit energy of some kind, calculating the amount of
the effect we see perfect match to theory.
20
This phenomenon represent an indirect proof of the existence of gravitational
radiation awarded Hulse and Taylor a Nobel Prize in 1993.
The first kind of detectors were a bar detector invented by J. Weber of the
University of Maryland in College Park fashioned solid aluminum cylinders,
about 2 meters long and 1 meter in diameter, and suspended them on steel
wires. A passing gravitational wave would set one of these cylinders vibrating
at its resonant frequency, about 1660 hertz and piezoelectric crystals firmly
attached around the cylinder's waist would convert that ringing into an
electrical signal. Unfortunately, Weber findings were controversialÚ and turned
out to be misleading.
New type of detectors came as a replacement for the bar detector are
interferometer detectors.
The most famous one is the LIGO (Laser Interferometer Gravitational-Wave
Observatory)
The LIGO has two perpendicular “arms” 4 km long, kept in vacuum, each
consist of two floating mirrors. It is very sensitive to movements, the system is
able to detect waves with an amplitude of ~ 10 −21 .
Figure 8.1: The LIGO real picture on the left and an internal schema on the right.
near Hanford USA.
Additional detectors based on the same principle function were built around
the world: The VIRGO – in Italy with European cooperation, GEO – EnglishGerman interferometer built in Germany and TEMA in Japan.
21
Two more are plane to be build: the advanced LIGO who should have 10
times higher sensitivity, meaning it will have ~1000 times greater range than
the LIGO.
The other is the LISA (laser Interferometer Space Antenna), the LISA’s arms
are planed to be 5 ⋅ 106 km and it will have the form of an equilateral triangle
between three satellites.
The detection of gravitational waves is likely to open horizons to great
development is reserches, although no gravitational radiation had ever been
detected so far, scientists estimate the first disclosure in the early future.
22
References
[1]
Weinberg Steven, 1993 \ Gravitation and Cosmology; Principles and applications of the
general theory of relativity.
[2]
http://nobelprize.org/nobel_prizes/physics/laureates/1993/press.html
[3]
J. Weber. Gravitational-Waves Detector Events. Phys. Rev. Lett. 20, 1307-1308 (1968)
[4]
http://physicaplus.org.il/zope/home/1185176174/sounds_en?curr_issue=1185176174
Physica Plus/Issue No.9 Gravitational waves: Heavenly sounds/ Barak Kol
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