The Theory of Gravitational Radiationbcc.impan.pl/13Gravitational/uploads/lecture3.pdf ·...

Doppler shift between freely falling observersLong-wavelength approximation

Responses of the solar-system-based detectors

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 Doppler shift between freely falling observers

2 Long-wavelength approximation

3 Responses of the solar-system-based detectorsLISA-type detector: time-delay interferometryGround-based laser interferometric detectorGround-based resonant bar detector




A weak plane gravitational wave in the TT coordinate system(with coordinates x0 = c t, x1 = x , x2 = y , x3 = z).

The wave is traveling in the +z direction.

The line element of spacetime

ds2 =− c2dt2 +

(1 + h+

(t − z

c

))dx2 +

(1− h+

(t − z

c

))dy 2

+ 2 h×(

t − z

c

)dx dy + dz2, (1)

where h+ and h× are the two polarizations of the wave.




Let us consider a photon which travels along a null geodesic of the metric(1). The photon’s world line is described by equations

xα = xα(λ), (2)

where λ is some affine parameter along the world line.

The 4-vector dxα/dλ tangent to the geodesic (2) is null,

gαβdxα

dλ

dxβ

dλ= 0. (3)




The metric (1) admits three Killing vectors:

Kα1 = (0, 1, 0, 0), (4a)

Kα2 = (0, 0, 1, 0), (4b)

Kα3 = (1, 0, 0, 1). (4c)

Because with any Killing vector Kα one can relate a constant of motionalong the null geodesic (2):

gαβKα dxβ

dλ= const, (5)

with the three Killing vectors (4) we can relate three such constants:

gαβKαndxβ

dλ= cn, n = 1, 2, 3. (6)




Three Eqs. (6) together with Eq. (3) form a system of four algebraicequations which can be solved with respect to the four componentsdxα/dλ of the tangent vector.

This unique solution, accurate to the terms linear in h, reads

cdt

dλ= −c2

1 + c22 + c2

3

2c3+

c21 − c2

2

2c3h+ +

c1c2c3

h× +O(h2), (7a)

dx

dλ= c1 − c1 h+ − c2 h× +O

(h2), (7b)

dy

dλ= c2 + c2 h+ − c1 h× +O

(h2), (7c)

dz

dλ=

c23 − c2

1 − c22

2c3+

c21 − c2

2

2c3h+ +

c1c2c3

h× +O(h2). (7d)




Let us consider two observersfreely falling in the field of the gravitational wave.Let us also assume that their spatial coordinates remain constant.

Their world lines are described by equations

t(τa) = τa, x(τa) = xa, y(τa) = ya, z(τa) = za, a = 1, 2, (8)

where (xa, ya, za) are constant spatial coordinates of the ath observer andτa is the proper time of this observer.

The 4-velocity Uαa := dxα/dτa of the ath observer has components

Uαa = (1, 0, 0, 0), a = 1, 2. (9)




Let us imagine that two observers measure,in their proper reference frames,the frequency of the same photon traveling along the world line (2).

Let us call the events, at which the photon’s frequency is measured, by E1and E2 (for the observer 1 and 2, respectively).These events have the following spacetime coordinates:

E1 : (t1, x1, y1, z1), E2 : (t2, x2, y2, z2). (10)

We assume that t2 > t1.




The photon’s frequency ν is proportional to the energy of the photonmeasured by the observer,

ν ∝ −gαβUα dxβ

dλ. (11)

One can rewrite Eq. (11), making use of Eqs. (1) and (9),in the form

ν ∝ dx0

dλ. (12)




Let us consider the ratio ν2/ν1 of the photon’s frequenciesmeasured by the two observers at the events E2 and E1.Let us denote

y12 :=ν2ν1− 1. (13)

By virtue of Eqs. (12) and (7a) we obtain:

y12 = c+(h+(t1 − z1/c)− h+(t2 − z2/c)

)+ c×

(h×(t1 − z1/c)− h×(t2 − z2/c)

)+O

(h2), (14)

where we have collected the three constants cn into two new constantparameters c+ and c×:

c+ :=c21 − c2

2

c21 + c2

2 + c23

, c× :=2c1c2

c21 + c2

2 + c23

. (15)




Let us introduce the 3-vector v with components

v :=(dx

dλ,dy

dλ,dz

dλ

), (16)

together with its Euclidean length:

|v| :=

√(dx

dλ

)2

+

(dy

dλ

)2

+

(dz

dλ

)2

. (17)

Equations (7) imply that

|v| =c21 + c2

2 + c23

2 |c3|+O

(h). (18)




Let us also introduce three angles α, β, γ ∈ [0, π] such that their cosinesare equal to

cosα = − 2c1c3c21 + c2

2 + c23

, (19a)

cosβ = − 2c2c3c21 + c2

2 + c23

, (19b)

cos γ =c21 + c2

2 − c23

c21 + c2

2 + c23

. (19c)

It is easy to check that

cos2 α + cos2 β + cos2 γ = 1. (20)




Making use of Eqs. (7), one can show that

1

|v|dx

dλ= cosα +O

(h), (21a)

1

|v|dy

dλ= cosβ +O

(h), (21b)

1

|v|dz

dλ= cos γ +O

(h). (21c)

Equations (21) deliver simple geometrical interpretation of the angles α,β, and γ: if one neglects the spacetime curvature caused by thegravitational wave, then α, β, and γ are the angles between the path ofthe photon in the 3-space and the coordinate axis x , y , or z , respectively.




Making use of Eqs. (19), one can express the constants c+ and c× fromEq. (15) by cosines of the angles α, β, and γ:

c+ =cos2 α− cos2 β

2(1− cos γ), c× =

cosα cosβ

1− cos γ. (22)




y12 = c+(h+(t1 − z1/c)− h+(t2 − z2/c)

)+ c×

(h×(t1 − z1/c)− h×(t2 − z2/c)

)+O

(h2) (14)

Because the events E1 and E2 lie on the same null geodesic (2),the arguments t1 − z1/c and t2 − z2/c of the functions h+ and h× fromEq. (14) can be related to each other.

Let us introduce the coordinate time duration t12 of the photon’s tripfrom the event E1 to E2:

t12 := t2 − t1, (23)

and the Euclidean coordinate distance L12 between the observers:

L12 :=√

(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2. (24)




Direct integration of Eqs. (7) gives(we assume that λ = 0 corresponds to the event E1):

c t(λ) = c t1 −c21 + c2

2 + c23

2c3λ+O

(h), (25a)

x(λ) = x1 + c1λ+O(h), (25b)

y(λ) = y1 + c2λ+O(h), (25c)

z(λ) = z1 +c23 − c2

1 − c22

2c3λ+O

(h). (25d)




From Eqs. (25) follows that

c t12 = L12 +O(h). (26)

Another useful relation one gets from Eqs. (25) and (19c),

z2 − z1c

= t12 cos γ +O(h), (27)

so we can write

t2 −z2c

= t1 −z1c

+(

t12 −z2 − z1

c

)= t1 −

z1c− (1− cos γ)

L12

c+O

(h). (28)




Making use of Eqs. (22) and (28), the ratio (14) of the photon’sfrequencies can be written in the form

y12 =cos2 α− cos2 β

2(1− cos γ)

(h+

(t1 −

z1c

)− h+

(t1 −

z1c

+ (1− cos γ)L12

c

))

+cosα cosβ

1− cos γ

(h×(

t1 −z1c

)− h×

(t1 −

z1c

+ (1− cos γ)L12

c

))+O

(h2). (29)




Let us introduce the three-dimensional matrix Hof the spatial metric perturbationproduced by the gravitational wave:

H(t) :=

h+(t) h×(t) 0h×(t) −h+(t) 0

0 0 0

. (30)

Let us note that the form of the line element (1) impliesthat the functions h+ = h+(t) and h× = h×(t)describe the components of the wave-induced metric perturbationat the origin of the TT coordinate system (where z = 0).




Let us introduce the 3-vector n of unit Euclidean length directed along theline connecting the two observers.

The components of this vector we arrange into the column matrix n [wethus distinguish here the 3-vector n from its components being theelements of the matrix n; we remember that the same 3-vector can bedecomposed into components in different spatial coordinate systems]:

n := (cosα, cosβ, cos γ)T =

cosα

cosβ

cos γ

, (31)

where the subscript T denotes matrix transposition.




The frequency ratio y12 from Eq. (29) can be more compactly written asfollows:

y12 =1

2(1− cos γ)nT ·

(H(

t1 −z1c

)− H

(t1 −

z1c

+ (1− cos γ)L12

c

))· n

+O(h2), (32)

where the dot means matrix multiplication.




It is convenient to introduce the 3-vector kpointing to the gravitational-wave source.In our coordinate system the wave is traveling in the +z direction,therefore the components of the 3-vector k, arranged into the columnmatrix k, are

k = (0, 0,−1)T. (33)

Let us finally introduce the 3-vectors xa (a = 1, 2)describing the positions of the observers with respect to the origin of thecoordinate system. Again the components of these 3-vectors we put intothe column matrices xa:

xa = (xa, ya, za)T, a = 1, 2. (34)




Making use of Eqs. (33)–(34) we rewrite the basic formula (32) in thefollowing form [where we have also employed Eqs. (26) and (27)]

y12 =

nT ·(

H(

t1 +kT · x1

c

)− H

(t1 +

L12

c+

kT · x2c

))· n

2(1 + kT · n)+O

(h2).

(35)




1

2

3

O

n1

n2

n3 L1

L2

L3

x1

x2

x3

Fig. 1. Configuration of three freely falling particles as a detector ofgravitational waves. The particles are labeled 1, 2, and 3. We denote by O theorigin of the TT coordinate system and by xa (a = 1, 2, 3) be the 3-vectorjoining O and the ath particle. The Euclidean coordinate distances between theparticles are denoted by La, where the index a corresponds to the oppositeparticle. The 3-vectors na of unit Euclidean lengths point between pairs ofparticles, with the orientation indicated.




The configuration from Fig. 1 is general enough to obtain responsesfor all currently working and planned detectors.

Two particles model Doppler tracking experiment where one particle is theEarth and the other one is a distant spacecraft.

Three particles model a ground-based laser interferometer where theparticles (mirrors) are suspended from seismically isolated supportsor a space-borne interferometer where the particles are shielded insatellites driven by drag-free control systems.




Let us denote by ν0 the frequency of the coherent beam used in thedetector (laser light in the case of an interferometer and radio waves inthe case of Doppler tracking).

Let the particle 1 emits the photonwith frequency ν0 at the moment t0 towards the particle 2,which registers the photon with frequency ν′ at the momentt′ = t0 + L3/c +O(h).

The photon is immediately transponded (without change of frequency)back to the particle 1,which registers the photon with frequency ν at the momentt = t0 + 2L3/c +O(h).

We express the relative changes of the photon’s frequency

y12 :=ν′ − ν0ν0

and y21 :=ν − ν′

ν′(36)

as functions of the instant of time t.




Making use of Eq. (35) we obtain

y12(t) =1

2(1− kT · n3)

× nT3 ·(

H(

t − 2L3

c+

kT · x1c

)− H

(t − L3

c+

kT · x2c

))· n3

+O(h2), (37a)

y21(t) =1

2(1 + kT · n3)

× nT3 ·(

H(

t − L3

c+

kT · x2c

)− H

(t +

kT · x1c

))· n3

+O(h2). (37b)




The total frequency shift y121 of the photon during its round trip:

y121 :=ν

ν0− 1

=ν

ν′ν′

ν0− 1

= (y21 + 1)(y12 + 1)− 1 = y12 + y21 +O(h2). (38)




The reduced wavelength of the gravitational wave:

λ :=λ

2π. (39)

In the long-wavelength approximation

λ � L, (40)

where L is the size of the detector.

Then the time delays across the detector are much shorter than the periodof the gravitational wave and can be neglected,

ω∆t ∼ L

λ� 1. (41)

It means that with a good accuracy the gravitational-wave field can betreated as being uniform (but time-dependent) in the space region thatcover the entire detector.




Let us consider the wave with some dominant angular frequency ω.

To detect such wave one must collect data during time interval longer(sometimes much longer) than the gravitational-wave period. It impliesthat in Eq. (32) the typical value of the quantity t := t1 − z1/c will bemuch larger than the retardation time ∆t := L12/c, t � ∆t.

Let us take any element hij of the matrix H, Eq. (30),and expand it with respect to ∆t:

hij(t + ∆t) = hij(t) + hij(t) ∆t +1

2hij(t) ∆t2 + · · ·

= hij(t)

(1 +

hij(t) ∆t

hij(t)+

hij(t) ∆t2

2 hij(t)+ · · ·

), (42)

where overdot denotes differentiation with respect to time t.




The time derivatives of hij can be estimated by

hij ∼ ωhij , hij ∼ ω2hij , and so on.

It means that

hij∆t/hij ∼ ω∆t, hij∆t2/hij ∼ (ω∆t)2, and so on.

We thus see that in the right-hand side of Eq. (42) the first term added to1 is a small correction, and all the next terms are corrections of evenhigher order.




We can therefore expand the equation

y12 =1

2(1− cos γ)nT ·

(H(t)− H

(t + (1− cos γ)∆t

))· n

+O(h2),

with respect to ∆t and keep terms only linear in ∆t.

After doing this one obtains the following formula for the relativefrequency shift in the long-wavelength approximation:

y12(t) = −L12

2cnT · H (t) · n +O

(h2). (43)




For the configuration of particles shown in Fig. 1, the relative frequencyshifts y12 and y21 given by Eqs. (37) can be written,by virtue of the formula (43), in the form

y12(t) = y21(t) = − L3

2cnT3 · H (t) · n3 +O

(h2), (44)

so they are equal to each other up to terms O(h2).

The total round-trip frequency shift y121 [cf. Eq. (38)] is thus equal

y121(t) = −L3

cnT3 · H (t) · n3 +O

(h2). (45)

There are important cases where the long-wavelength approximation isnot valid: These include satellite Doppler tracking measurements and thespace-borne LISA-type detectors for gravitational-wave frequencies largerthan a few mHz.




LISA-type detector: time-delay interferometryGround-based laser interferometric detectorGround-based resonant bar detector








Real gravitational-wave detectors do not move along geodesics of thespacetime metric related to the passing gravitational wave, because theyalso move in the gravitational field of the solar system bodies, as in thecase of the LISA-type space-borne detectors, or are fixed to the surface ofEarth, as in the case of Earth-based laser interferometers or resonant bardetectors.

The motion of the detector with respect to the solar system barycenter(SSB) will modulate the gravitational-wave signal registered by thedetector.





The detector moves in spacetime with the metricwhich approximately can be written as

ds2 = −(1 + 2φ)c2dt2 + (1− 2φ)(dx2 + dy 2 + dz2) + hTTij dx idx j , (46)

where φ is the Newtonian potential(divided by c2 to make it dimensionless)produced by different bodies of the solar system.

The Newtonian gravitational field is weak, i.e.

|φ| � 1, but |φ| � |h|.

Both effects are small and can therefore be treated independently,i.e. all the effects of the order O(φh) can be neglected.





The Newtonian-field-induced effects have to be “subtracted”before the gravitational-wave data analysis begins.This subtraction requires accurate modelling of the motion of the detectorwith respect to the SSB using the solar system ephemeris.

In the rest of this lecture we will assumethat such subtraction was already done.Therefore we will put φ = 0 into the line element (46) and reduce ouranalysis to consideration of the motion of particles and photons in thefield of gravitational wave only.

After doing this it is possible to use in the computation of the responsesof the real detectors the results we already discussed.We have to drop the assumption that the particles modelling thedetector’s parts are at rest with respect to the TT coordinatesystem—now their spatial coordinates can change in time, so the particlesmove along some world lines, which in general are not geodesics of thegravitational-wave-induced spacetime metric.





Let the origin of the TT coordinate system coincides with the SSB.

The detailed computations show that as far as the velocities of thedetector’s parts (particles) with respect to the SSB are nonrelativistic(which is the case for all existing or planned detectors),the formulae discussed above can still be used,provided the quantities na and xa (a = 1, 2, 3) will be interpreted as madeof the time-dependent components of the 3-vectors na and xa computedin the SSB coordinate system.





Equation (37b) takes now the form

y21(t) =1

2(1 + kT · n3(t))

× nT3 (t) ·

(H(

t − L3(t)

c+

kT · x2(t)

c

)− H

(t +

kT · x1(t)

c

))· n3(t)

+O(h2). (47)

In this equation all time-dependent quantities are computed at the samemoment of time t: corrections coming from the retardation effects wouldlead to additional terms of the order of O(vh), but such terms we neglect.





Let us consider the proper reference frame of the detectorwith coordinates (xα).

Because the motion of this frame with respect to the SSBis nonrelativistic, we can assume that the transformation between the TTcoordinates (xα) and the proper-reference-frame coordinates (xα) has theform

t = t, x i (t, xk) = x iO(t) + Oi

j(t) x j , (48)

where the functions x iO

(t) describe the motion of the origin O of thedetector’s proper reference frame with respect to the SSB, and thefunctions Oi

j(t) account for the different and changing in time relativeorientations of the spatial coordinate axes of the two reference frames.

The transformation inverse to that of Eq. (48) reads

t = t, x i (t, xk) = (O−1)ij(t)(x j − x j

O(t)). (49)





The functions Oij and (O−1)ij are elements of two mutually inverse

matrices, which we denote by O and O−1, respectively.The following relations are thus fulfilled:

Oij(t) (O−1)jk(t) = δik , (O−1)ij(t) Oj

k(t) = δik , (50)

which in matrix notation read

O(t) · O(t)−1 = I, O(t)−1 · O(t) = I, (51)

where I is 3× 3 identity matrix.





The following scalar quantity is the building block for responses ofdifferent gravitational-wave detectors

Φa(t) :=1

2na(t)T · H

(t − tr(t)

)· na(t)

=1

2hij

(t − tr(t)

)nia(t) nj

a(t), a = 1, 2, 3, (52)

where tr(t) is some retardation [see Eq. (47)],hij are elements of the 3× 3 matrix H,and ni

a are elements of the 3× 1 column matrix na.

The value of Φa is invariant under the transformation (48),because ni

a transform as components of a contravariant 3-vectorand hij transform as components of a (0,2) tensor:

nia = Oi

j(t) nja(t), hij(t) = (O−1)ki (t) (O−1)lj(t) hkl(t), (53)

where we have indicated that the proper-reference-frame components nia

of the unit vector na usually can be treated as constant(at least to a good approximation).





Let us introduce the 3× 1 column matrix na with elements nia

and the 3× 3 matrix H with elements hij .Then the transformation formulae (53) in matrix notation read

na = O(t) · na(t), H(t) = (O(t)−1)T · H(t) · O(t)−1. (54)

If the transformation matrix O is orthogonal, then O−1 = OT,and the relation between the matrices H and H reads

H(t) = O(t) · H(t) · O(t)T. (55)





By virtue of Eqs. (54) the quantity Φa defined in Eq. (52) takes in theproper reference frame the following form:

Φa(t) =1

2nTa · H

(t − tr(t)

)· na. (56)

To get the above equality we have had to replace in some placesO(t − tr(t)) by O(t), what means that we again neglectall terms of the order of O(vh).





The space-borne LISA-type detectors will have multiple readoutscorresponding to the laser Doppler shifts measured between differentspacecraft and also to the intra-spacecraft Doppler shifts(measured between optical benches located at the same spacecraft).

It is possible to combine, with suitable time delays,the time series of the different Doppler shiftsto cancel both the frequency fluctuations of the lasersand the noise due to the mechanical vibrations of the optical benches.

The technique used to devise such combinations is knownas time-delay interferometry (TDI).





The TDI responses were first derived under the assumption that thespacecraft array is stationary (such responses are called sometimesfirst-generation TDI).

But the rotational motion of the spacecraft array around the Sunand the relative motion of the spacecraft prevent the cancellation of thelaser frequency fluctuations in the first-generation TDI responses.Therefore new combinations was devised that are capable of suppressingthe laser frequency fluctuations for a rotating spacecraft array.They are called second-generation TDI.





The second-generation TDI responses are rather complicatedcombinations of delayed time series of the relative laser frequencyfluctuations measured between the spacecraft.

We present here the so called second-generation Michelson observablesXa, a = 1, 2, 3. We adopt the notation for the delay of the time seriessuch that, for instance,

y12,3(t) := y12(t − L3(t)).

Using this notation the Michelson observable X1 reads (the observables X2

and X3 can be obtained by cyclical permutations of spacecraft labels):

X1 = (y31 + y13,2) + (y21 + y12,3),22 − (y21 + y12,3)− (y31 + y13,2),33

−((y31 + y13,2) + (y21 + y12,3),22

− (y21 + y12,3)− (y31 + y13,2),33),2233

. (57)





The responses Xa can be constructed from expressions of the type (52),each of them can be written as a linear combination of the wavepolarization functions h+ and h×:

Φa(t) = Fa+(t) h+

(t − tr(t)

)+ Fa×(t) h×

(t − tr(t)

), (58)

where tr is some retardation and the functions Fa+ and Fa× are called thebeam-pattern functions.





The long-wavelength approximation can be applied.

We consider a standard Michelson and equal-arm interferometricconfiguration (L2 = L3 = L in Fig. 1).

The observed relative frequency shift ∆ν(t)/ν0 is equal to the differenceof the round-trip frequency shifts in the detector’s arms:

∆ν(t)

ν0= y131(t)− y121(t)

=L

c

(nT2 · H

(t − zd

c

)· n2 − nT

3 · H(

t − zdc

)· n3

), (59)

zd is the 3rd spatial component of the 3-vector rd = (xd, yd, zd) whichdescribes the position of the characteristic point within the detector(the particle number 1 in Fig. 1) with respect of the origin of the TTcoordinates.





The difference ∆φ(t) of the phase fluctuations (measured by a photodetector), is related to the relative frequency fluctuations ∆ν(t) by

∆ν(t)

ν0=

1

2πν0

d∆φ(t)

dt. (60)

Making use of (59) one can integrate Eq. (60),

∆φ(t) = 4π ν0 L s(t), (61)

where the dimensionless function s,

s(t) =1

2

(nT2 · H

(t − zd

c

)· n2 − nT

3 · H(

t − zdc

)· n3

), (62)

is the response function of the interferometer to a gravitational wave.





Equations (61)–(62) were derived in some TT reference frame assumingthat the particles forming the triangle in Fig. 1 all have constant in timespatial coordinates with respect to this frame.We can choose the origin of the frame to coincide with the SSB.

The formula (62) can still be used in the case when the particles aremoving with respect to the SSB with nonrelativistic velocities,then the response of the Earth-based inteferometric detector reads

s(t) =1

2

(nT2 (t) · H

(t − zd(t)

c

)· n2(t)− nT

3 (t) · H(

t − zd(t)

c

)· n3(t)

),

(63)all quantities are computed in the TT coordinate systemconnected with the SSB.





The same response (63) can be rewrittenin terms of the proper-reference-frame quantities as

s(t) =1

2

(nT2 · H

(t − zd(t)

c

)· n2 − nT

3 · H(

t − zd(t)

c

)· n3

), (64)

where the matrix H is related to the matrix H by means of Eq. (55),the proper-reference-frame components n2 and n3 of the unit vectorsalong the inteferometer arms are treated as time-independent quantities(in fact they deviate from being constants by quantities of the order ofO(h), therefore these deviations contribute to the response function onlyat the order of O(h2)).

The response function (64) can directly be derived in the proper referenceframe of the interferometer by means of the equation of geodesicdeviation. In this derivation the response s is defined as the relativechange of the lengths of the two arms, i.e. s(t) := ∆L(t)/L.





Taking into account the form (30) of the matrix H one showsthat the response function s is a linear combinationof the two wave polarization functions h+ and h×:

s(t) = F+(t) h+

(t − zd(t)

c

)+ F×(t) h×

(t − zd(t)

c

), (65)

where F+ and F× are the interferometric beam-pattern functions.They depend on the location of the detector on Earth and on the positionof the gravitational-wave source in the sky.

Let the 3-vector n0 of unit Euclidean length be directed from the SSBtowards the gravitational-wave source. In the TT coordinate systemconsidered here it has components n0 = (0, 0,−1),so the z-component of the 3-vector rd connecting the SSBand the detector can be computed as

zd(t) = −n0 · rd(t). (66)





In the case of a resonant bar detectorthe long-wavelength approximation is very accurate.

If one additionally assumes that the frequency spectrumof the gravitational wave which hits the bar entirely lieswithin the sensitivity band of the detector, then the dimensionlessresponse function s can be computed from the formula

s(t) = nT · H(

t − zd(t)

c

)· n, (67)

where the column matrix n is made of the proper-reference-framecomponents of the unit vector n directed along thesymmetry axis of the bar.

The above formula can be derived in the proper reference frameof the detector using the equation of geodesic deviation,the response function s is then defined as s(t) := ∆L(t)/L,where ∆L(t) is the wave-induced change of the proper length L of the bar.





The response (67), like in the case of the interferometric detector, can bewritten as a linear combination of the wave polarization functions h+ andh×, i.e. it can be written in the form

s(t) = F+(t) h+

(t − zd(t)

c

)+ F×(t) h×

(t − zd(t)

c

), (68)

but with bar beam-pattern functions F+ and F×which are different from the interferometric ones.


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