The Theory of Gravitational Radiation - IM PANbcc.impan.pl/13Gravitational/uploads/lecture1.pdf ·...

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




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





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.





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.





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.






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)






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.






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.






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)






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)






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)






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)






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)






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)






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 ).






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.






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.

















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.






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)






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)






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)






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.

















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)






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)






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)

















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)






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)






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)















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)).





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)





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).





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 .





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.















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).





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.





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.





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)





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’.





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)





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.





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)





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.






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)).






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].






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.






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.






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)






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

.






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.






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.

















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.






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.






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.






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.






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.

















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)






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)






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.






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.















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)





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


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




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


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.





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)






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.






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.






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)






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)






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)

















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)

















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)






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)


Date post:	27-Feb-2018
Category:	Documents
Upload:	vuduong
View:	216 times
Download:	3 times

The Theory of Gravitational Radiation - IM PANbcc.impan.pl/13Gravitational/uploads/lecture1.pdf ·...

Documents