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Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
The Theory of Gravitational Radiation
Piotr Jaranowski
Faculty of Physcis, University of Bia lystok, Poland
01.07.2013
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Notation and conventions
General relativity
We adopt notation and conventions of the textbook by Misner, Thorne,and Wheeler.
Greek indices α, β, . . . run from 0 to 3and Latin indices i , j , . . . run from 1 to 3.
We employ the Einstein summation convention on repeated indices.
We use spacetime metric of signature (−1, 1, 1, 1), so the line elementof the Minkowski spacetime in Cartesian (inertial) coordinates(x0 = c t, x1 = x , x2 = y , x3 = z) reads
ds2 = ηµν dxµdxν = −c2dt2 + dx2 + dy 2 + dz2.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Notation and conventions
3-vectors
For any 3-vectors a = (a1, a2, a3) and b = (b1, b2, b3)we define their usual Euclidean scalar product a · b,and |a| denotes Euclidean length of a 3-vector a:
a · b :=3∑
i=1
aibi , |a| :=√a · a =
√√√√ 3∑i=1
(ai )2.
Derivatives
The partial differentiation with respect to xµ:∂φ
∂xµ≡ ∂µφ ≡ φ,µ.
The partial differentiation with respect to time t:∂φ
∂t≡ ∂tφ ≡ φ.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
General relativity describes gravitation as geometry.
General-relativistic spacetime is curved but locally flat,that is locally Minkowskian.The local Minkowski frame is the frame of a freely falling observer.
In a general coordinate system the spacetime interval is given by
ds2 = gµν(xα) dxµdxν ,
where position-dependent functions gµν(xα)are components of the metric tensor,which measures the proper time and proper distance.
A freely falling particle follows a geodesic of the metric,defined as a locally straight worldline.
The metric tensor has 10 different components (gµν = gνµ),but there are 4 degrees of freedom to choose coordinates,therefore only 6 metric components is independent.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
The tensorial description of the geometry is throughthe Riemann curvature tensor, which can be expressedin terms of the Christoffel symbols and their first derivatives.
The Christoffel symbols Γµαβ dependon the metric components and their first derivatives:
Γµαβ =1
2gµν
(∂αgβν + ∂βgνα − ∂νgαβ
). (1)
The Riemann curvature tensor has components
Rµνρσ = gρλ(∂µΓλνσ − ∂νΓλµσ + ΓλµηΓηνσ − ΓλνηΓηµσ
). (2)
Next one introduces the symmetric Ricci tensor,
Rµν = gρσRρµσν , (3)
and its trace known as the Ricci scalar,
R = gµνRµν . (4)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Finally, the Einstein tensor is defined as
Gµν := Rµν −1
2gµνR. (5)
The Einstein field equations,
Gµν =8πG
c4Tµν , (6)
relates spacetime geometry expressed by the Einstein tensor Gµνwith sources of a gravitational field representedby an energy-momentum tensor Tµν .
Components of Tµν contain the energy density, momentum density,and stresses inside the source of the field, so in general relativity not onlyenergy density, but also momentum and stress create gravity.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
General relativity is a nonlinear theory, therefore in general there isno clear distinction between waves and the rest of the metric.
One can use the notion of a wave in certain limiting situations:—in linearized theory,—as small perturbations of a smooth background metric,—in post-Newtonian theory.
We will concentrate on linearized theory.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
If the gravitational field is weak, then it is possible to finda coordinate system (xα) such that the components gµνof the metric tensor in this system will be small perturbations hµνof the flat Minkowski metric components ηµν :
gµν = ηµν + hµν , |hµν | � 1. (7)
The coordinate system satisfying the condition (7) is sometimescalled an almost Lorentzian coordinate system.
For the almost Lorentzian coordinates (xα) we will also use anothernotation:
x0 ≡ c t, x1 ≡ x , x2 ≡ y , x3 ≡ z . (8)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
We assume that the indeces of the hµν will be raised and lowered bymeans of the ηµν and ηµν (and not by gµν and gµν).
Therefore we have, e.g.,
h βα = ηβµhαµ, hαβ = ηαµhµ
β = ηαµηβνhµν . (9)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
If the condition (7) is satisfied,one can linearize the Einstein field equationswith respect to the small perturbation hµν .
We start from linearizing the Christoffel symbols Γµαβ ,they take the form
Γµαβ =1
2ηµν(∂αhβν + ∂βhνα − ∂νhαβ
)+O
(h2)
=1
2
(∂αhµβ + ∂βhµα − ηµν∂νhαβ
)+O
(h2). (10)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Christoffel symbols are first-order quantities, therefore the onlycontribution to the linearized Riemann tensor comes from thederivatives of the Christoffel symbols.
Making use of Eqs. (2) and (10) we get
Rµνρσ =1
2
(∂ρ∂νhµσ+∂σ∂µhνρ−∂σ∂νhµρ−∂ρ∂µhνσ
)+O
(h2). (11)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Next we linearize the Ricci tensor. Making use of (11) we obtain
Rµν =1
2
(∂α∂µhαν + ∂α∂νhαµ − ∂µ∂νh −�hµν
)+O
(h2), (12)
where h is the trace of the metric perturbation hµν ,
h := ηαβhαβ , (13)
and where we have introduced the d’Alembertian operator �in the flat Minkowski spacetime:
� := ηµν∂µ∂ν = − 1
c2∂2t + ∂2
x + ∂2z + ∂2
z . (14)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Finally we linearize the Ricci scalar,
R = ηµνRµν +O(h2)
= ∂µ∂νhµν −�h +O(h2). (15)
We are now ready to linearize the Einstein tensor. We get
Gµν =1
2
(∂µ∂αhαν + ∂ν∂αhαµ − ∂µ∂νh −�hµν
+ ηµν(�h − ∂α∂βhαβ
))+O
(h2). (16)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
It is possible to simplify a bit the right-hand side of Eq. (16)by introducing the trace-reversed metric perturbation
hµν := hµν −1
2ηµνh. (17)
From the definition (17) follows that
hµν = hµν −1
2ηµν h, (18)
where h := ηαβ hαβ (let us also observe that h = −h ).
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Substituting the relation (18) into (16), one obtains
Gµν =1
2
(∂µ∂αhαν+∂ν∂αhαµ−�hµν−ηµν∂α∂β hαβ
)+O
(h2). (19)
Making use of Eq. (16) or Eq. (19) one can write downthe linearized form of the Einstein field equations.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
If spacetime (or its part) admits one almost Lorentzian coordinatesystem, then there exists in spacetime (or in its part)infinitely many almost Lorentzian coordinates.
We describe below two kinds of coordinate transformationsleading from one almost Lorentzian coordinate systemto another such system:—global Poincare transformations,—gauge transformations.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
The global Poincare transformation,
x ′α(xβ) = Λαβ xβ + aα. (20)
The constant (i.e. independent on the spacetime coordinates xµ)numbers Λαβ are the components of the matrix representing thespecial-relativistic Lorentz transformation, and the constantquantities aα represent some translation in spacetime.
The matrix (Λαβ) fulfills the condition
Λαµ Λβν ηαβ = ηµν . (21)
This condition means that the Lorentz transformation does notchange the Minkowski metric.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
If the new coordinates are related to an observer which moves withrespect to another observer related to the old coordinates along its xaxis with velocity v , then the matrix built up from the coefficientsΛαβ is the following
(Λαβ) =
γ −vγ/c 0 0
−vγ/c γ 0 00 0 1 00 0 0 1
, γ :=
(1− v 2
c2
)−1/2
. (22)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
The transformation inverse to that given in Eq. (20) leads from new(x ′α) to old (xα) coordinates and is given by the relations
xα(x ′β) = (Λ−1)αβ(x ′β − aβ), (23)
where the numbers (Λ−1)αβ form the matrix inverse to thatconstructed from Λαβ :
(Λ−1)αβ Λβγ = δαγ , Λαβ (Λ−1)βγ = δαγ . (24)
The matrix ((Λ−1)αβ) also fulfills the requirement (21),
(Λ−1)αµ (Λ−1)βν ηαβ = ηµν . (25)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Let us now assume that the old coordinates (xα) are almostLorentzian, so the decomposition (7) of the metric holds in thesecoordinates.
Making use of the rule relating the componets of the metric tensorin two coordinate systems,
g ′αβ(x ′) =∂xµ
∂x ′α∂xν
∂x ′βgµν(x), (26)
by virtue of Eqs. (7), (23), and (21) one easily gets
g ′αβ = ηαβ + h′αβ , (27)
where we have defined
h′αβ := (Λ−1)µα (Λ−1)νβ hµν . (28)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
This last equation means that the metric perturbations hµνtransform under the Poincare transformationas the components of a (0, 2) rank tensor.
The result of Eqs. (27)–(28) also means that the new coordinatesystem
(x ′α)
will be almost Lorentzian, provided the numericalvalues of the matrix elements (Λ−1)αβ are not too large, becausethen the condition |hµν | � 1 implies that also |h′αβ | � 1.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Infinitesimal coordinate transformations known as gaugetransformations are of the form
x ′α = xα + ξα(xβ), (29)
where the functions ξα are small in this sense, that
|∂βξα| � 1. (30)
Equations (29)–(30) imply that
∂x ′α
∂xβ= δαβ + ∂βξ
α, (31a)
∂xα
∂x ′β= δαβ − ∂βξα +O
((∂ξ)2
). (31b)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Let us now assume that the coordinates (xα) are almost Lorentzian.
Making use of Eqs. (7), (26), and (31), we compute the componentsof the metric in the (x ′α) coordinates:
g ′αβ = ηαβ + hαβ − ∂αξβ − ∂βξα +O(h ∂ξ, (∂ξ)2
), (32)
where we have introduced
ξα := ηαβξβ . (33)
The metric components g ′αβ can thus be written in the form
g ′αβ = ηαβ + h′αβ +O(h ∂ξ, (∂ξ)2
), (34)
where we have defined
h′αβ := hαβ − ∂αξβ − ∂βξα. (35)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Because the condition (30) is fulfilled,the new metric perturbation h′αβ is small,
|h′αβ | � 1,
and the coordinates (x ′α) are almost Lorentzian.
From Eq. (35), making use of the definition (17),one gets the rule of how the metric perturbation hαβchanges under the gauge transformation,
h′αβ = hαβ − ∂αξβ − ∂βξα + ηαβ ∂µξµ. (36)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Among all almost Lorentzian coordinates one can choosecoordinates for which the following additionalharmonic gauge conditions are fulfilled:
∂β hβα = 0. (37)
The conditions (37) can equivalently be written as
ηβγ∂β hγα = 0. (38)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
If these conditions are satisfied, the linearized Einstein tensor Gµνfrom Eq. (19) reduces to
Gµν = −1
2�hµν +O
(h2), (39)
and the linearized Einstein field equations take the simple form ofthe wave equations in the flat Minkowski spacetime:
�hµν +O(h2)
= −16πG
c4Tµν . (40)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Global Poincare transformationsGauge transformationsHarmonic coordinates
Harmonic coordinates are not uniquely defined:
— the harmonic gauge conditions are preserved by thePoincare transformations,
— they are also preserved by the infinitesimal gaugetransformations of the form (29), provided all thefunctions ξα satisfy homogeneous wave equations:
�ξα = 0. (41)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Linearized Einstein field equationsin harmonic coordinates in vacuum (Tµν = 0),
�hµν = 0. (42)
Time-dependent solutions of these equations can be interpretedas weak gravitational waves.
The simplest solution of Eq. (42) is a monochromatic plane wave,which is of the form
hµν(xα) = Aµν cos(kαxα − α(µ)(ν)
). (43)
Aµν and α(µ)(ν) is the constant amplitude and the constant initialphase, of the µν component of the wave, and kα are another fourreal constants (we have encircled the indices µ and ν of the initialphases by parentheses to indicate that there is no summation overthese indices on the right-hand side of Eq. (43)).
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
One checks that the functions (43) are solutions of Eqs. (42)if and only if
ηαβkαkβ = 0, (44)
what means that if we define
kα := ηαβkβ ,
then kα are the components of a null (with respect to theMinkowski metric) 4-vector.
Einstein field equations take the simple form (42) only if theharmonic gauge conditions (37) are satisfied. This leads to therequirement that the plane wave solution hµν is orthogonal to kµ:
hµνkν = 0. (45)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
The contraction kαxα can be written as
kαxα = k0 x0+3∑
i=1
ki x i = −k0 x0+3∑
i=1
k i x i = −c k0 t + k · x, (46)
where we have introduced the two 3-vectors:k with components (k1, k2, k3) and x with components (x1, x2, x3).
If we additionally introduce the quantity ω := c k0,then the plane wave solution (43) becomes
hµν(t, x) = Aµν cos(ωt − k · x + α(µ)(ν)
). (47)
We can assume, without loss of generality, that ω ≥ 0.Then ω is angular frequency of the wave(measured in radians per second).
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
We will also use frequency f of the wave (measured in hertz, i.e. incycles per second), related with ω by the equation
ω = 2πf . (48)
The 3-vector k is known as a wave vector, it points to the directionin which the wave is propagating and its Euclidean length is relatedto the wavelength λ,
λ|k| = 2π. (49)
Equation (44) written in terms of ω and k takes the form
ω = c |k|. (50)
This is the dispersion relation for gravitational waves. It implies thatboth the phase and the group velocity of the waves are equal to c .
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Summing up: the solution
hµν(t, x) = Aµν cos(ωt − k · x + α(µ)(ν)
)represents a plane gravitational wave with frequency f = ω/(2π)and wavelength λ = 2π/|k|, which propagates through the 3-spacein the direction of the 3-vector k with the speed of light.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
At each event in the spacetime region covered by some almostLorentzian and harmonic coordinates (xα) let us choose timelikeunit vector Uµ,
gµνUµUν = −1.
Let us consider a gauge transformation generated by the functionsξα of the form
ξα(t, x) = Bα cos(ωt − k · x + β(α)
), (51)
with ω = c |k| and k the same as in the plane-wave solution (47).
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
It is possible to choose the quantities Bα and β(α) in such a way,that in the new almost Lorentzian and harmonic coordinatesx ′α = xα + ξα the following conditions are fulfilled
h′µνU ′ν = 0, ηµν h′µν = 0. (52a)
The gauge transformation based on the functions (51) preserves thecondition (45):
h′µνk ′ν = 0. (52b)
Let us also note that, as a consequence of Eqs. (52a),
h′µν = h′µν . (53)
Equations (52) define transverse and traceless (TT in short)coordinate system related to the 4-vector field U ′µ.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Equations (52) comprise eight independent constraintson the components of the plane-wave solution h′µν :any plane monochromatic gravitational wavepossesses two independent degrees of freedom,often also called wave’s polarizations.
The simplest way to describe these two polarizations is by makingmore coordinate changes:We have used all freedom related to the gauge transformations,but we are still able to perform global Lorentz transformations,which preserve equations (52) defining TT gauge.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
One can first move to coordinates in which the vector Uµ
(from now we omit the primes in the coordinate names)has components Uµ = (1, 0, 0, 0).Then the first equation (52a) implies
hµ0 = 0. (54)
Further, one can orient spatial coordinate axes such that the wavepropagates in, say, +z direction. Then k = (0, 0, ω/c),kµ = (ω/c , 0, 0, ω/c), and Eqs. (52b) together with (54) give
hµ3 = 0. (55)
The last constraint provides the second equation (52a)supplemented by (54) and (55). It reads
h11 + h22 = 0. (56)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
It is common to use the following notation:
h+ := h11 = −h22, h× := h12 = h21. (57)
The functions h+ and h× are called plus and cross polarization ofthe wave, respectively.
We will label all quantities computed in the TT coordinate systemby super- or subscript ‘TT’.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Equations (54)–(56) allow us to rewrite the plane-wave solution (47)in the TT coordinates in the following matrix form:
hTTµν (t, x) =
0 0 0 00 h+(t, x) h×(t, x) 00 h×(t, x) −h+(t, x) 00 0 0 0
, (58)
where the plus h+ and the cross h× polarizations of the plane wavewith angular frequency ω travelling in the +z direction are given by
h+(t, x) = A+ cos(ω(
t − z
c
)+ α+
), (59a)
h×(t, x) = A× cos(ω(
t − z
c
)+ α×
). (59b)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Any gravitational wave can be represented as a superposition of planemonochromatic waves.
Because the equations describing TT gauge,
∂ν hµν = 0, hµνU
ν = 0, ηµν hµν = 0, (60)
are all linear in hµν , it is possible to find TT gauge for any gravitationalwave.
If we restrict to plane (but not necessarily monochromatic) waves andorient spatial axes of coordinate system such that the wave propagatesin the +z direction, then equations
hµ0 = 0, hµ3 = 0, h11 + h22 = 0,
are still valid.
Moreover, because all monochromatic components of the wave dependon space-time coordinates only through the combination t − z/c, thesame dependence will be valid also for the general wave.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Therefore any weak plane gravitational wave propagating in the +zdirection is described in the TT gauge by the metric perturbation of thefollowing matrix form:
hTTµν (t, x) =
0 0 0 00 h+(t − z/c) h×(t − z/c) 00 h×(t − z/c) −h+(t − z/c) 00 0 0 0
. (61)
If we introduce the polarization tensors e+ and e× by means of equations
e+xx = −e+
yy = 1, e×xy = e×yx = 1, all other components zero, (62)
then one can reconstruct the full gravitational-wave field from its plus andcross polarizations as
hTTµν (t, x) = h+(t, x) e+
µν + h×(t, x) e×µν . (63)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
Let us consider two nearby observers freely falling in the field ofa weak and plane gravitational wave.
The wave produces tiny variations in the proper distance betweenthe observers. We describe now these variations from the point ofview of one of the observers, which we call the basic observer.
We endow this observer with his proper reference frame, whichconsists of a small Cartesian latticework of measuring rodsand synchronized clocks.The time coordinate t of that frame measures proper time along theworldline of the observer whereas the spatial coordinate x i
(i = 1, 2, 3; we will also use the notation: x1 ≡ x , x2 ≡ y , x3 ≡ z)measures proper distance along his ith axis.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
Spacetime coordinates (x0 := c t, x1, x2, x3) are locally Lorentzianalong the whole geodesic of the basic observer, i.e. the line elementof the metric in these coordinates has the form
ds2 = −c2dt2 + δijdx idx j +O((x i )2
)dxαdxβ , (64)
so it deviates from the line element of the flat Minkowski spacetimeby terms which are at least quadratic in the values of the spatialcoordinates x i .
Let the basic observer A be located at the origin of his properreference frame, so his coordinates are xαA(t) = (c t, 0, 0, 0).A neighbouring observer B moves along the nearby geodesic and itpossesses the coordinates xαB (t) = (c t, x1(t), x2(t), x3(t)).
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
We define the deviation vector ξα describing the instantaneousrelative position of the observer B with respect to the observer A:
ξα(t) := xαB (t)− xαA(t) =(0, x1(t), x2(t), x3(t)
). (65)
The relative acceleration D2ξα/dt2 of the observers is related to thespacetime curvature through the equation of geodesic deviation,
D2ξα
dt2= −c2 Rα
βγδuβ ξγ uδ, (66)
where uβ := dxα/(c dt) is the 4-velocity of the basic observer[all quantities in Eq. (66) are evaluated on the basic geodesic,so they are functions of the time coordinate t only].
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
If one chooses the TT coordinates in such a way that the 4-velocityfield needed to define TT coordinates coincides with the 4-velocityof our basic observer, then the TT coordinates (t, x i ) and theproper-reference-frame coordinates (t, x i ) differ from each other inthe vicinity of the basic observer’s geodesic by quantities linear in h.
It means that, up to the terms quadratic in h, the components ofthe Riemann tensor in both coordinate systems coincide,
Ri0j0 = RTTi0j0 +O(h2). (67)
Making use of the above relation, after neglecting some termsO(h2), one gets
d2x i
dt2=
1
2
∂2hTTij
∂ t2x j , (68)
where the second time derivative ∂2hTTij /∂ t2 is to be evaluated
along the basic geodesic x = y = z = 0.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
In the derivation of the above relation it is not needed to assumethat the wave is propagating in the +z direction, thus this relationis valid for the wave propagating in any direction.
Let us now imagine that for times t ≤ 0 there were no waves(hTT
ij = 0) in the vicinity of the two observers,
x i (t) = x i0 = const, for t ≤ 0. (69)
At t = 0 some wave arrives.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
We expect that x i (t) = x i0 +O(h) for t > 0, therefore, because we
neglect terms O(h2), one can make the replacement
d2x i
dt2=
1
2
∂2hTTij
∂ t2x j −→ d2x i
dt2=
1
2
∂2hTTij
∂ t2x i
0 . (70)
We can immediately integrate (70),
x i(t)
=(δij +
1
2hTTij
(t))
x i0, t > 0. (71)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
Let us orient the spatial axes of the proper reference frame suchthat the wave is propagating in the +z direction.
We can then employ Eq. (61),where we put z = 0 and replace t by t,
hTTµν (t) =
0 0 0 00 h+(t) h×(t) 00 h×(t) −h+(t) 00 0 0 0
.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
Equations (71) take the form
x(t)
= x0 +1
2
(h+
(t)x0 + h×
(t)y0
), (72a)
y(t)
= y0 +1
2
(h×(t)x0 − h+
(t)y0
), (72b)
z(t)
= z0. (72c)
Equations (72) shows that the gravitational wave is transverse:it produces relative displacements of the test particles only in theplane perpendicular to the direction of the wave propagation.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
Let us imagine that the basic observer is checking the presence ofthe wave by observing some neighbouring particles which form,before the wave arrives, a perfect ring in the (x , y) plane.
Let the radius of the ring be r0 and the center of the ring coincideswith the origin of the observer’s proper reference frame.Then the coordinates of any particle in the ring can be parametrizedby some angle φ ∈ [0, 2π] such that they are equal
x0 = r0 cosφ, y0 = r0 sinφ, z0 = 0. (73)
Equations (72) and (73) imply that in the field of gravitational wavethe z coordinates of all the ring’s particles remain equal to zero:
z(t)
= 0,
so only x and y coordinates of the particles should be analyzed.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
The gravitational wave is in the plus mode when h× = 0.Making use of Eqs. (72) and (73) one then gets
x(t)
= r0 cosφ(
1 +1
2h+
(t)), (74a)
y(t)
= r0 sinφ(
1− 1
2h+
(t)). (74b)
Initially, before the wave arrives, the ring of the particles is perfectlycircular. Does this shape change after the wave has arrived?
One can treat Eqs. (74) as parametric equations of a certain curve,with φ being the parameter. It is easy to combine the two equations(74) such that φ is eliminated.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
The resulting equation reads
x2(a+(t)
)2 +y 2(
b+(t))2 = 1, (75)
where
a+(t) := r0
(1 +
1
2h+
(t)), b+(t) := r0
(1− 1
2h+
(t)). (76)
Equations (75)–(76) describe an ellipse with its center at the originof the coordinate system.
The ellipse has semiaxes of the lengths a+(t) and b+(t), which areparallel to the x or y axis, respectively.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
a+(t) := r0
(1 +
1
2h+
(t)), b+(t) := r0
(1− 1
2h+
(t)).
If h+
(t)
is the oscillatory function, which changes its sign in time, thenthe deformation of the initial circle into the ellipse is the following:
in time intervals when h+
(t)> 0, the cirlce is stretched in the x
direction and squeezed in the y direction,
when h+
(t)< 0, the stretching is along the y axis and the
squeezing is along the x axis.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
The effect of a plane monochromatic gravitational wave with + polarizationon a circle of test particles placed in a plane perpendicular to the directionof the wave propagation.
The plots show deformation of the circle measured in the proper referenceframe of the central particle at the instants of time equal to
0,1
4T ,
1
2T ,
3
4T , T ,
where T is the period of the gravitational wave.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
Let us now fix some single particle in the ring. The motion of thisparticle with respect to the origin of the proper reference frame isgiven by Eqs. (74), for some fixed value of φ.What is the shape of the particle’s trajectory?
It is again easy to combine Eqs. (74) in such a way that the functionh+
(t)
is eliminated. The result is
x
r0 cosφ+
y
r0 sinφ− 2 = 0. (77)
Equation (77) means that any single particle in the ring is movingaround its initial position along some straight line.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
The gravitational wave is in the crosss mode when h+ = 0.From Eqs. (72) and (73) one then gets
x(t)
= r0
(cosφ+
1
2sinφ h×
(t)), (78a)
y(t)
= r0
(sinφ+
1
2cosφ h×
(t)). (78b)
Let us introduce in the (x , y) plane the new coordinates (x ′, y ′)related to the old ones by rotation around the z axisby the angle of α = 45◦ degrees,(
x ′
y ′
)=
(cosα sinα− sinα cosα
)(xy
)=
√2
2
(1 1−1 1
)(xy
). (79)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
We rewrite Eqs. (78) in terms of the coordinates (x ′, y ′),
x ′(t)
=
√2
2r0(sinφ+ cosφ)
(1 + h×
(t)), (80a)
y ′(t)
=
√2
2r0(sinφ− cosφ)
(1− h×
(t)). (80b)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
After eliminating from Eqs. (80) the parametr φ, one gets
x ′2(a×(t)
)2 +y ′2(
b×(t))2 = 1, (81)
where
a×(t) := r0
(1 +
1
2h×(t)), b×(t) := r0
(1− 1
2h×(t)). (82)
Equations (81)–(82) have exactly the form of Eqs. (75)–(76).
This means that the initial circle of particles is deformed into anellipse with its center at the origin of the coordinate system.
The ellipse has semiaxes of the lengths a×(t) and b×(t), which areinclined by the angle of 45◦ degrees to the x or y axis, respectively.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Plus polarizationCross polarization
The effect of a plane monochromatic gravitational wave with × polarizationon a circle of test particles placed in a plane perpendicular to the directionof the wave propagation.
The plots show deformation of the circle measured in the proper referenceframe of the central particle at the instants of time equal to
0,1
4T ,
1
2T ,
3
4T , T ,
where T is the period of the gravitational wave.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Energy-momentum tensor for gravitational wavesis obtained by averaging the squared gradient of the wave fieldover several wavelengths.
In the TT gauge it has components
T gwαβ =
c4
32πGηµνηρσ
⟨∂αhTT
µρ ∂βhTTνσ
⟩. (83)
If one additionally assumes that hTT0µ = 0, then Eq. (83) reduces to
T gwαβ =
c4
32πG
3∑i=1
3∑j=1
⟨∂αhTT
ij ∂βhTTij
⟩. (84)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
For the plane gravitational wave propagating in the +z direction,the tensor T gw
αβ takes the standard form for a bundle ofa zero-rest-mass particles moving in the speed of light in the +zdirection. It has components
T gw00 = −T gw
0z = −T gwz0 = T gw
zz =c2
16πG
⟨(∂h+
∂t
)2
+
(∂h×∂t
)2⟩
(85)(all other components are equal to zero).
The energy-momentum tensor for the monochromatic plane wavewith angular frequency ω has components
T gw00 = −T gw
0z = −T gwz0 = T gw
zz =c2ω2
16πG
×(
A2+
⟨sin2
[ω(t − z/c
)+ α+
]⟩+A2×⟨sin2
[ω(t − z/c
)+ α×
]⟩).
(86)P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Averaging the sine squared terms over one wavelength or one waveperiod gives 1/2. After substituting this to (86), and replacing ω bythe frequency f = ω/(2π) measured in hertz, one obtains
T gw00 = −T gw
0z = −T gwz0 = T gw
zz =πc2f 2
8G
(A2
+ + A2×
). (87)
Typical gravitational waves we might expect to observe at Earthhave frequencies between 10−4 and 104 Hz, and amplitudes of theorder of A+ ∼ A× ∼ 10−22. The energy flux in the +z direction forsuch waves can thus be estimated as
−T gwtz = −c T gw
0z = 1.6× 10−6
(f
1Hz
)2 A2+ + A2
×(10−22)2
erg
cm2 s.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
The simplest technique for computing gravitational-wave field hTTµν
is delivered by the famous quadrupole formalism.
This formalism is especially important because it is highly accuratefor many astrophysical sources of gravitational waves.
It does not require for high accuracy any constraint on the strengthof the source’s internal gravity, but requires that internal motionsinside the source are slow.
This requirement implies that
L� λ , (88)
where L is the source’s size and λ is the reduced wavelength of thegravitational waves it emits:
λ :=λ
2π. (89)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
Let us introduce some coordinate system (t, x is)
centered on a gravitational-wave sourceand let an observer at rest with respect to the sourcemeasures the gravitational-wave field hTT
µν generated by that source.
Let us further assume that the observer is situatedwithin the local wave zone of the source,where the background curvature both of the sourceand of the external universe can be neglected.It implies that the distance from the observer to the sourceis very large compared to the source’s size.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
Then the quadrupole formalism allows to writethe gravitational-wave field in the following form:
hTT0µ (t, x i
s) = 0, hTTij (t, x i
s) =2G
c4
1
R
d2J TTij
dt2
(t − R
c
), (90)
where R :=√δijx i
sx js is the distance from the point (x i
s) where the
gravitational-wave field is observed to the source’s center,t is proper time measured by the observer,and t − R/c is retarded time.The quantity Jij is the source’s reduced mass quadrupole moment(which we define below), and the superscript TT at Jij meansalgebraically project out and keep only the part that is transverseto the direction in which wave propagates and is traceless.
Equations (90) describe a spherical gravitational wave generatedby the source located at the origin of the spatial coordinates.
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
Let ni := x is/R be the unit vector in the direction of wave
propagation and let us define the projection operator P ij
which projects 3-vectors to a 2-plane orthogonal to ni ,
P ij := δij − ninj . (91)
Then the TT part of the reduced mass quadrupole momentcan be computed as
J TTij = Pk
i P lj Jkl −
1
2Pij
(PklJkl
). (92)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
For the wave propagating in the +z direction the unit vector ni hascomponents
nx = ny = 0, nz = 1. (93)
Making use of Eqs. (91)–(92) one can then easily compute the TTprojection of the reduced mass quadrupole moment:
J TTxx = −J TT
yy =1
2(Jxx − Jyy ), (94a)
J TTxy = J TT
yx = Jxy , (94b)
J TTzi = J TT
iz = 0 for i = x , y , z . (94c)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
Making use of these equations one can write the formulae for theplus and the cross polarizations of the wave progagating in the +zdirection of the coordinate system:
h+(t, x is) =
G
c4 R
(d2Jxxdt2
(t − R
c
)− d2Jyy
dt2
(t − R
c
)), (95a)
h×(t, x is) =
2G
c4 R
d2Jxydt2
(t − R
c
). (95b)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
We assume that the source has weak internal gravity and smallinternal stresses, so Newtonian gravity is a good approximation togeneral relativity inside and near the source.
Then Jij is the symmetric and trace-free (STF) part of the secondmoment of the source’s mass density ρ computed in a Cartesiancoordinate system centered on the source:
Jij(t) :=
(∫ρ(xk , t)x ix jd3x
)STF
=
∫ρ(xk , t)
(x ix j − 1
3r 2δij
)d3x .
(96)
Equivalently, Jij is the coefficient of the 1/r 3 term in the multipolarexpansion of the source’s Newtonian gravitational potential Φ,
Φ(t, xk) = −GM
r− 3G
2
Jij(t) x ix j
r 5− 5G
2
Jijk(t) x ix jxk
r 7+ · · · . (97)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
1 Linearized general relativityGlobal Poincare transformationsGauge transformationsHarmonic coordinates
2 Plane monochromatic gravitational waves
3 Description in the TT coordinate system
4 Description of in the observer’s proper reference framePlus polarizationCross polarization
5 Energy-momentum tensor for gravitational waves
6 Generation of gravitational wavesQuadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
From the quadrupole formula (90) and the formula (84)for the energy-momentum tensor of the gravitational waves,one can compute the fluxes of energy and angular momentumcarried by the waves.
After integrating these fluxes over a sphere surrounding the sourcein the local wave zone one obtains the rates Lgw
E and LgwJi
of emission respectively of energy and angular momentum:
LgwE =
G
5c5
3∑i=1
3∑j=1
⟨(d3Jijdt3
)2⟩, (98a)
LgwJi
=2G
5c5
3∑j=1
3∑k=1
3∑`=1
εijk
⟨d2Jj`dt2
d3Jk`dt3
⟩. (98b)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw
Linearized general relativityPlane monochromatic gravitational wavesDescription in the TT coordinate system
Description of in the observer’s proper reference frameEnergy-momentum tensor for gravitational waves
Generation of gravitational waves
Quadrupole formalismMass quadrupole moment of the sourceGravitational-wave luminosities
The laws of conservation of energy and angular momentum implythat radiation reaction should decrease the source’s energyand angular momentum at rates just equal to minus rates givenby Eqs. (98), leading thus to the following balance equations:
dE source
dt= −Lgw
E , (99a)
dJsourcei
dt= −Lgw
Ji. (99b)
P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw