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arXiv:physics/990

8041v1

[physics.ed-ph]21Aug1999

Gravitational Waves: An Introduction

Indrajit Chakrabarty

Abstract

In this article, I present an elementary introduction to the theory

of gravitational waves. This article is meant for students who havehad an exposure to general relativity, but, results from general rela-tivity used in the main discussion has been derived and discussed inthe appendices. The weak gravitational field approximation is firstconsidered and the linearized Einsteins equations are obtained. Wediscuss the plane wave solutions to these equations and consider thetransverse-traceless (TT) gauge. We then discuss the motion of testparticles in the presence of a gravitational wave and their polarization.The method of Greens functions is applied to obtain the solutions tothe linearized field equations in presence of a nonrelativistic, isolatedsource.

Mehta Research Institute, Chhatnag Road, Jhusi. Allahabad. 211 019 INDIAE-mail: indrajit@mri.ernet.in

1

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1 Introduction

In the past few years, research on the detection of gravitational waves hasassumed new directions. Efforts are now underway to detect gravitationalradiation from astrophysical sources, thereby enabling researchers to pos-sess an additional tool to study the universe (See [6] for a recent review).According to Newtons theory of gravitation, the binary period of two pointmasses (e.g., two stars) moving in a bound orbit is strictly a constant quan-tity. However, Einsteins general theory of relativity predicts that two starsrevolving around each other in a bound orbit suffer accelerations, and, as aresult, gravitational radiation is generated. Gravitational waves carry awayenergy and momentum at the expense of the orbital decay of two stars,

thereby causing the stars to gradually spiral towards each other and givingrise to shorter and shorter periods. This anticipated decrease of the orbitalperiod of a binary pulsar was first observed in PSR 1913+16 by Taylor andWeisberg ([4]). The observation supported the idea of gravitational radia-tion first propounded in 1916 by Einstein in the Proceedings of the RoyalPrussian Academy of Knowledge. Einstein showed that the first order con-tributon to the gravitational radiation must be quadrupolar in a particularcoordinate system. Two years later, he extended his theory to all coordinatesystems.

The weak nature of gravitational radiation makes it very difficult todesign a sensitive detector. Filtering out the noisy background to isolatethe useful signal requires great technical expertise. itself a field of research.Various gravitational wave detectors are fully/partially operational and weexpect a certain result to appear from the observations in the near future.

This article gives an elementary introduction to the theory of gravita-tional waves. Important topics in general relativity including a brief intro-duction to tensors and a derivation of Einsteins field equations are discussedin the appendices. We first discuss the weak gravitational field approxima-tion and obtain the linearized Einsteins field equations. We then discussthe plane wave solutions to these equations in vacuum and the restriction onthem due to the transverse-traceless (TT) gauge. The motion of particles inthe presence of gravitational waves and their polarization is then discussedin brief. We conclude by applying the method of Greens functions to show

that gravitational radiation from matter at large distances is predominantlyquadrupolar in nature.

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2 The weak gravitational field approximation

Einsteins theory of general relativity leads to Newtonian gravity in the limitwhen the gravitational field is weak & static and the particles in the gravita-tional field move slowly. We now consider a less restrictive situation wherethe gravitational field is weak but not static, and there are no restrictionson the motion of particles in the gravitational field. In the absence of grav-ity, space-time is flat and is characterised by the Minkowski metric, . Aweak gravitational field can be considered as a small perturbation on theflat Minkowski metric[3],

g = + h, |h| 1 (1)

Such coordinate systems are often called Lorentz coordinate systems. Indicesof any tensor can be raised or lowered using or respectively asthe corrections would be of higher order in the perturbation, h. We cantherefore write,

g = h (2)Under a background Lorentz transformation ([3]), the perturbation trans-forms as a second-rank tensor:

h =

h (3)

The equations obeyed by the perturbation, h, are obtained by writing the

Einsteins equations to first order. To the first order, the affine connection(See App. A) is,

=1

2[h + h h] + O(h2) (4)

Therefore, the Riemann curvature tensor reduces to

R = (5)

The Ricci tensor is obtained to the first order as

R R(1) =1

2 h + h

nu h2h (6)

where, 2 is the DAlembertian in flat space-time. Contractingagain with , the Ricci scalar is obtained as

R = h 2h (7)

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The Einstein tensor, G, in the limit of weak gravitational field is

G = R 12

R =1

2[h

+ h

h + 2h2h]

(8)The linearised Einstein field equations are then

G = 8GT (9)

We cant expect the field equations (9) to have unique solutions as any solu-tion to these equations will not remain invariant under a gauge transforma-tion. As a result, equations (9) will have infinitely many solutions. In otherwords, the decomposition (1) of g in the weak gravitational field approx-

imation does not completely specify the coordinate system in space-time.When we have a system that is invariant under a gauge transformation, we

fix the gauge and work in a selected coordinate system. One such coordinatesystem is the harmonic coordinate system ([5]). The gauge condition is

g = 0 (10)

In the weak field limit, this condition reduces to

h =

1

2h (11)

This condition is often called the Lorentz gauge. In this selected gauge, the

linearized Einstein equations simplify to,

2h 12

2h = 16GT (12)

The trace-reversed perturbation, h, is defined as ([3]),

h = h 12

h (13)

The harmonic gauge condition further reduces to

h = 0 (14)

The Einstein equations are then

2h = 16GT (15)

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3 Plane-wave solutions and the transverse-traceless(TT) gauge

From the field equations in the weak-field limit, eqns.(15), we obtain thelinearised field equations in vacuum,

2h = 0 (16)

The vacuum equations for h are similar to the wave equations in electro-magnetism. These equations admit the plane-wave solutions,

h = Aexp(kx) (17)

where, A is a constant, symmetric, rank-2 tensor and k is a constantfour-vector known as the wave vector. Plugging in the solution (17) into theequation (16), we obtain the condition

kk = 0 (18)

This implies that equation (17) gives a solution to the wave equation (16)if k is null; that is, tangent to the world line of a photon. This showsthat gravitational waves propagate at the speed of light. The time-likecomponent of the wave vector is often referred to as the frequency of thewave. The four-vector, k is usually written as k (, k). Since k is null,it means that,

2

= |k|2

(19)This is often referred to as the dispersion relation for the gravitational wave.We can specify the plane wave with a number of independent parameters;10 from the coefficients, A and three from the null vector, k. Using theharmonic gauge condition (14), we obtain,

kA = 0 (20)

This imposes a restriction on A : it is orthogonal (transverse) to k. Thenumber of independent components of A is thus reduced to six. We haveto impose a gauge condition too as any coordinate transformation of theform

x = x + (x) (21)

will leave the harmonic coordinate condition

2x = 0 (22)

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satisfied as long as2 = 0 (23)

Let us choose a solution

= Cexp(kx) (24)

to the wave equation (23) for . C are constant coefficients. We claimthat this remaining freedom allows us to convert from the old constants,

A(old) , to a set of new constants, A

(new) , such that

A(new) = 0 (traceless) (25)

and

AU

= 0 (26)where, U is some fixed four-velocity, that is, any constant time-like unitvector we wish to choose. The equations (20), (25) and (26) together arecalled the transverse traceless (TT) gauge conditions ([3]). Thus, we haveused up all the available gauge freedom and the remaining components ofA must be physically important. The trace condition (25) implies that

hTT = hTT (27)

Let us now consider a background Lorentz transformation in which thevector U is the time basis vector U = 0. Then eqn.(26) implies thatA0 = 0 for all . Let us orient the coordinate axes so that the wave

is travelling along the z-direction, k (, 0, 0, ). Then with eqn.(26),eqn.(20) implies that Az = 0 for all . Thus, A

TT in matrix form is

ATT =

0 0 0 00 Axx Axy 00 Axy Axx 00 0 0 0

(28)

4 Polarization of gravitational waves

In this section, we consider the effect of gravitational waves on free particles.

Consider some particles described by a single velocity field, U

and a sepa-ration vector, . Then, the separation vector obeys the geodesic equation(See App. A)

d2

d2= RU

U (29)

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where, U = dx/d is the four-velocity of the two particles. We consider the

lowest-order (flat-space) components ofU only since any corrections to U

that depend on h will give rise to terms second order in the perturbation inthe above equation. Therefore, U = (1, 0, 0, 0) and initially = (0, , 0, 0).Then to first order in h, eqn. (29) reduces to

d2

d2=

2

t2= R00x = R0x0 (30)

Using the definition of the Riemann tensor, we can show that in the TTgauge,

Rx

0x0

= Rx0x0 =

1

2hTT

xx,00

Ry0x0 = Ry0x0 = 1

2hTTxy,00

Ry

0y0 = Ry0y0 = 1

2hTTyy,00 = Rx0x0 (31)

All other independent components vanish. This means that two particlesinitially separated in the x-direction have a separation vector which obeysthe equation

2x

t2=

1

2

2

t2hTTxx ,

2y

t2=

1

2

2

t2hTTxy (32)

Similarly, two particles initially separated by in the y-direction obey the

equations

2y

t2=

1

2

2

t2hTTyy =

1

2

2

t2hTTxx

2x

t2=

1

2

2

t2hTTxy (33)

We can now use these equations to describe the polarization of a gravita-tional wave. Let us consider a ring of particles initially at rest as in Fig.1(a). Suppose a wave with hTTxx = 0, hTTxy = 0 hits them. The particlesrespond to the wave as shown in Fig. 1(b). First the particles along thex-direction come towards each other and then move away from each other

as hTTxx reverses sign. This is often called + p olarization. If the wave hadhTTxy = 0, but, hTTxx = hTTyy = 0, then the particles respond as shown in Fig.1(c). This is known as polarization. Since hTTxy and hTTxx are independent,the figures 1(b) and 1(c) demonstrate the existence of two different states

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of polarisation. The two states of polarisation are oriented at an angle of

45o to each other unlike in electromagnetic waves were the two states ofpolarization.

5 Generation of gravitational waves

In section III, we obtained the plane wave solutions to the linearized Ein-steins equations in vacuum, eqns.(16). To obtain the solution of the lin-earised equations (15), we will use the Greens function method. The Greensfunction, G(x y), of the DAlembertian operator 2, is the solution ofthe wave equation in the presence of a delta function source:

2 G(x y) = (4)(x y) (34)

where (4) is the four-dimensional Dirac delta function. The general solutionto the linearized Einsteins equations (15) can be written using the Greensfunction as

h(x) = 16G

d4y G(x y)T(y) (35)

The solutions to the eqn.(34) are called advanced or retarded according asthey represent waves travelling backward or forward in time, respectively.We are interested in the retarded Greens function as it represents the neteffect of signals from the past of the point under consideration. It is givenby

G(x y) = 14|x y|

|x y| (x0 y0)

(x0 y0) (36)

where, x = (x1, x2, x3) and y = (y1, y2, y3) and |x y| = [ij(xi yi)(xj yj)]1/2. (x0 y0) is the Heaviside unit step function, it equals 1 whenx0 > y0, and equals 0 otherwise. Using the relation (36) in (35), we canperform the integral over y0 with the help of the delta function,

h(t,x) = 4G d3y 1|x y|T(t |x y|,y) (37)where t = x0. The quantity

tR = t |x y| (38)

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is called the retarded time. From the expression (37) for h, we observe that

the disturbance in the gravitational field at (t,x) is a sum of the influencesfrom the energy and momentum sources at the point (tR,y) on the pastlight cone.

Using the expression (37), we now consider the gravitational radiationemitted by an isolated far away source consisting of very slowly movingparticles (the spatial extent of the source is negligible compared to the dis-tance between the source and the observer). The Fourier transform of theperturbation h is

h(, x) =12

dt exp(t) h(t,x) (39)

Using the expression (37) for h(t,x), we get

h = 4G

d3y exp(|x y|) T

(, y)

|x y| (40)

Under the assumption that the spatial extent of the source is negligible com-pared to the distance between the source and the observer, we can replacethe term exp(|x y|)/|x y| in (40) by exp(R)/R. Therefore,

h(, x) = 4Gexp(R)

R

d3y T(, y) (41)

The harmonic gauge condition (14) in Fourier space is

h(t,x) =

d h

exp(t) = 0 (42)

Separating out the space and time components,

0

d h

0(, x)exp(t) + i

d h

i(, x)exp(t) = 0 (43)

Or,

h0

= ihi

(44)

Thus, in eqn.(41), we need to consider the space-like components ofh(, y).Consider,

d3y k

yiTkj

=

d3y

ky

i

Tkj +

d3y yi

kT

kj

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On using Gauss theorem, we obtain,d3y Tij(, y) =

d3y yi

kT

kj

(45)

Consider the Fourier space version of the conservation equation for T, viz.,T

(t,x) = 0. Separating the time and space components of the Fouriertransform equation, we have,

iTi = T0 (46)

Therefore,

d

3

yTij

(, y) =

d3

y yi

T0j

=

2

d3

y

yi

T0j

+ yj

T0i

(47)

Considerd3y l

yi yjT

0l

=

d3y

ly

i

yj +

lyj

yi

T0l+

d3y y i yj

lT

0l

(48)Simplifying the equation above, we obtain for the left hand side

d3y

yi T0j + yjT0i

+

d3y yi yj

lT

0l

Since the term on the left hand side of eqn.(47) is a surface term, it vanishes

and we obtaind3y

yiT0j + yjT0i

=

d3y yi yj

lT

0l

(49)

Using the equations (46) and (48), we can write,d3y Tij(, y) =

2

d3y l

yiyj T0l

(50)

Using the eqn(45), we can write,

d3y Tij(, y) =

2

2d

3y yiyj T00 (51)

We define the quadrupole moment tensor of the energy density of the sourceas

qij() = 3

d3y yiyj T00(, y) (52)

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In terms of the quadrupole moment tensor, we have

d3y Tij(, y) =

2

6qij() (53)

Therefore, the solution (41) becomes

hij(, x) = 4Gexp(R)

R

2

6qij()

(54)

Simplifying further,

hij(, x) =

2

3

G2

Rexp(R) qij() (55)

Taking the Fourier transform of eqn.(54), and simplifying, we finally obtainfor the perturbation

hij(t,x) =2G

3R

d

dt2qij(tR) (56)

where, tR = t |x y| is the retarded time. In the expression (54), we seethat the gravitational wave produced by an isolated, monochromatic andnon-relativistic source is therefore proportional to the second derivative ofthe quadrupole moment of the energy density at the point where the pastlight cone of the observer intersects the cone. The quadrupolar nature of

the wave shows itself by the production of shear in the particle distribution,and there is zero average translational motion. The leading contribution toelectromagnetic radiation comes from the changing dipole moment of thecharge density. This remarkable difference in the nature of gravitationaland electromagnetic radiation arises from the fact that the centre of massof an isolated system cant oscillate freely but the centre of charge of acharge distribution can. The quadrupole momnet of a system is generallysmaller than the dipole moment and hence gravitational waves are weakerthan electromagnetic waves.

6 EpilogueThis lecture note on gravitational waves leaves several topics untouched.There are a number of good references on gravitation where the inquisitivereader can find more about gravitational waves and their detection. I have

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freely drawn from various sources and I dont claim any originality in this

work. I hope I have been able to incite some interest in the reader about atopic on which there is a dearth of literature.

Acknowledgements

This expository article grew out of a seminar presented at the end of theGravitation and Cosmology course given by Prof. Ashoke Sen. I am gratefulto all my colleagues who helped me during the preparation of the lecture.

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Appendix A: Some topics in general theory of relativity

An event in relativity is characterised by a set of coordinates (t,x,y,z)in a definite coordinate system. Transformations between the coordinatesof an event observed in two different reference frames are called Lorentztransformations. These transformations mix up space and time and hencethe coordinates are redefined so that all of them have dimensions of length.We write x0 ct,x1 x, x2 y, x3 z and a general component of a fourvector (x0, x1, x2, x3) as x. A Lorentz transformation is then written as

x = x (57)

where,

=

0 0 0 0

0 0 1 00 0 0 1

(58)

At this point, it is useful to note the Einstein summation convention: when-ever an index appears as a subscript and as a superscript in an expression,we sum over all values taken by the index. Under a Lorentz transformation,the spacetime interval (ct)2 + x2 + y2 + z2 remains invariant. The lengthof a four-vector is given by

|x| = (x0)2 + (x1)2 + (x2)2 + (x3)2 (59)

We never extract a square root of the expression (59) since |x| can be nega-tive. Four-vectors that have negative length are called time-like, while thosewith positive lengths are called space-like. Four-vectors with zero length arecalled null. The notion of norm of a four-vector is introduced with thehelp of the Minkowski metric:

=

1 0 0 00 1 0 00 0 1 0

0 0 0 1

(60)

Then, we have,|x| = xx (61)

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There are two kinds of vectors that are classified in the way they transform

under a Lorentz transformation:

Contravariant :x = x

Covariant :x =

x (62)

Vectors are tensors of rank one. () is called the metric tensor; itis a tensor of rank two. There are other higher rank tensors which wewill encounter later. If two coordinate systems are linked by a Lorentztransformation as:

x = x (63)

then, multiplying both sides of the equation above by and differentiat-

ing, we get,x

x = (64)

Therefore, we see that

x =

x(65)

Thus,

/x =

1

c

t,

x,

y,

z

(66)

transforms as a covariant vector. The differential operates on tensors toyield higher-rank tensors. A scalar s can be constructed using the Minkowski

metric and two four-vectors u and v as:

s = uv (67)

A scalar is an invariant quantity under Lorentz transformations. Using thechain rule,

dx =x

xdx (68)

we have,

s =

x

xx

x

uv (69)

If we defineg x

xx

x(70)

then,s = g u

v (71)

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g is called the metric tensor; it is a symmetric, second-rank tensor.

To follow the motion of a freely falling particle, an inertial coordinatesystem is sufficient. In an inertial frame, a particle at rest will remainso if no forces act on it. There is a frame where particles move with auniform velocity. This is the frame which falls freely in a gravitational field.Since this frame accelerates at the same rate as the free particles do, itfollows that all such particles will maintain a uniform velocity relative to thisframe. Uniform gravitational fields are equivalent to frames that accelerateuniformly relative to inertial frames. This is the Principle of Equivalencebetween gravity and acceleration. The principle just stated is known as theWeak Equivalence Principle because it only refers to gravity.

In treating the motion of a particle in the presence of gravity, we define

the Christoffel symbol or the affine connection as

=1

2g

gx

+gx

gx

(72)

plays the same role for the gravitational field as the field strength tensordoes for the electromagnetic field. Using the definition of affine connection,we can obtain the expression for the covariant derivative of a tensor:

DA A

x+ A

(73)

It is straightforward to conclude that the covariant derivative of the metric

tensor vanishes. The concept of parallel transport of a vector has im-portant implications. We cant define globally parallel vector fields. Wecan define local parallelism. In the Euclidean space, a straight line is theonly curve that parallel transports its own tangent vector. In curved space,we can draw nearly straight lines by demanding parallel transport of thetangent vector. These lines are called geodesics. A geodesic is a curve ofextremal length between any two points. The equation of a geodesic is

d2x

d2+

dx

d

dx

d= 0 (74)

The parameter is called an affineparameter. A curve having the same path

as a geodesic but parametrised by a non-affine parameter is not a geodesiccurve. The Riemannian curvature tensor is defined as

R =

x

x+

(75)

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In a flat space,

R = 0 (76)

Geodesics in a flat space maintain their separation; those in curved spacesdont. The equation obeyed by the separation vector in a vector field Vis

DV DV = RV V (77)If we differentiate the Riemannian curvature tensor and permute the indices,we obtain the Bianchi identity:

R + R + R = 0 (78)

Since in an inertial coordinate system the affine connection vanishes, theequation above is equivalent to one with partials replaced by covariantderivatives. The Ricci tensor is defined as

R R = R (79)

It is a symmetric second rank tensor. The Ricci scalar (also known as scalarcurvature) is obtained by a further contraction,

R R (80)

The stress-energy tensor(also called energy-momentum tensor) is defined as

the flux of the -momentum across a surface of constant x

. In componentform, we have:

1. T00 = Energy density =

2. T0i = Energy flux (Energy may be transmitted by heat cinduction)

3. Ti0 = Momentum density (Even if the particles dont carry momen-tum, if heat is being conducted, then the energy will carry momentum)

4. Tij = Momentum flux (Also called stress)

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Appendix B: The Einstein field equation

The curvature of space-time is necessary and sufficient to describe grav-ity. The latter can be shown by considering the Newtonian limit of thegeodesic equation. We require that

the particles are moving slowly with respect to the speed of light, the gravitational field is weak so that it may be considered as a per-

turbation of flat space, and,

the gravitational field is static.In this limit, the geodesic equation changes to,

d2x

d2+ 00(

dt

d)2 = 0 (81)

The affine connection also simplifies to

00 = 1

2gg00 (82)

In the weak gravitational field limit, we can lower or raise the indices of atensor using the Minkowskian flat metric, e.g.,

h

= h

(83)

Then, the affine connection is written as

00 = 1

2h00 (84)

The geodesic equation then reduces to

d2x

d2=

1

2

dt

d

2h00 (85)

The space components of the above equation are

d2xi

d2 =1

2 (dt

d)2

ih00 (86)

Or,d2xi

dt2=

1

2ih00 (87)

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The concept of an inertial mass arises from Newtons second law:

f = mia (88)

According to the the law of gravitation, the gravitational force exerted on anabject is proportional to the gradient of a scalar field , known as the scalargravitational potential. The constant of proportionality is the gravitationalmass, mg:

fg = mg (89)According to the Weak Equivalence Principle, the inertial and gravitationalmasses are the same,

mi = mg (90)

And, hence,a = (91)

Comparing equations (86) and (91), we find that they are the same if weidentify,

h00 = 2 (92)Thus,

g00 = (1 + 2) (93)The curvature of space-time is sufficient to describe gravity in the Newto-nian limit as along as the metric takes the form (93). All the basic laws of

Physics, beyond those governing freely-falling particles adapt to the curva-ture of space-time (that is, to the presence of gravity) when we are work-ing in Riemannian normal coordinates. The tensorial form of any law iscoordinate-independent and hence, translating a law into the language oftensors (that is, to replace the partial derivatives by the covariant deriva-tives), we will have an universal law which holds in all coordinate systems.This procedure is sometimes called the Principle of Equivalence. For exam-ple, the conservation equation for the energy-momentum tensor T in flatspace-time, viz.,

T = 0 (94)

is immediately adapted to the curved space-time as,

DT = 0 (95)

This equation expresses the conservation of energy in the presence of agravitational field. We can now introduce Einsteins field equations which

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governs how the metric responds to energy and momentum. We would like

to derive an equation which will supercede the Poisson equation for theNewtonian potential:

2 = 4G (96)where, 2 = ijij is the Laplacian in space and is the mass density.A relativistic generalisation of this equation must have a tensorial form sothat the law is valid in all coordinate systems. The tensorial counterpart ofthe mass density is the energy-momentum tensor, T. The gravitationalpotential should get replaced by the metric. Thus, we guess that our newequation will have T set proportional to some tensor which is second-orderin the derivatives of the metric,

T = A (97)

where, A is the tensor to be found. The requirements on the equationabove are:-

By definition, the R.H.S must be a second-rank tensor. It must contain terms linear in the second derivatives or quadratic in

the first derivative of the metric.

The R.H.S must be symmetric in and as T is symmetric.

Since T is conserved, the R.H.S must also be conserved.

The first two conditions require the right hand side to be of the form

R + Rg = T (98)

where R is the Ricci tensor, R is the scalar curvature and & areconstants. This choice is symmetric in and and hence satisfies the thirdcondition. From the last condition, we obtain

gD(R + Rg) = 0 (99)

This equation cant be satisfied for arbitrary values of and . This equa-

tion holds only if / is fixed. As a consequence of the Bianchi identity,viz.,

DR =1

2DR (100)

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we choose,

= 12

(101)

With this choice, the equation (42) becomes

(R 12

Rg) = T (102)

In the weak field limit,g00 2 (103)

the 00-component of the equation(42), viz.,

2g00 = T00

2

2 = (104)

Compare this result with Newtons equation (40), we obtain,

2a =1

4G(105)

Thus, we obtain the Einstein field equations in their final form as

R 12

Rg = 8GT (106)

References

[1] S. Caroll, Lecture notes on General Relativity, gr-qc/9712019

[2] L. D. Landau & E. M. Lifshitz, Classical Theory of Fields, PergamonPress (1980)

[3] B. F. Schutz, A first course in general relativity, Cambridge UniversityPress (1995)

[4] J. H. Taylor & J. M. Weisberg, ApJ, 253, 908 (1982)

[5] S. Weinberg, Gravitation and Cosmology, Princeton, N. Y.

[6] R. Weiss, Rev. Mod. Phys., 71, S187 (1999)

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Figures

000111 000111000111000111

000111 000111000111000111 000111

00000000000000000111111111111111110 0 0 0 0 0 0 01 1 1 1 1 1 1 1

x

y

Fig 1(a)

Figure 1: The initial configuration of test particles on a circle before a gravitational wave hitsthem.

0 00 01 11 10 00 01 11 1 0 00 01 11 1

0 00 01 11 1

0 00 01 11 10 00 00 01 11 11 1 000111

0 00 01 11 1 0011

0011 0 00 01 11 10 00 01 11 10 00 01 11 1 0 00 01 11 1

0 00 00 01 11 11 1 00110 00 01 11 10 00 00 01 11 11 1

00000001111111

00000000111111110 0 0 01 1 1 1 0 0 0 01 1 1 1

0 0 01 1 1 0 0 0 01 1 1 100000001111111

00000001111111

0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1Fig. 1(b)y

x

Figure 2: Displacement of test particles caused by the passage of a gravitational wave with the+ polarization. The two states are separated by a phase difference of.

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0 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 1

000111

000111

0 00 01 11 1000111

0 00 01 11 10 00 00 01 11 11 1

0 00 01 11 10 00 00 01 11 11 1 0 0

0 00 01 11 11 1000111 0 00 00 01 11 11 1

0 00 01 11 10 00 01 11 1

0 00 01 11 10 00 00 01 11 11 1

0 0 00 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 11 1 1

0 0 00 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 11 1 10 00 00 00 00 01 11 11 11 11 1

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 01 1 1 11 1 1 11 1 1 11 1 1 11 1 1 10 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1 0 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1

0 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1 0 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1

000000000000000000011111111111111111110 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1

x

y

Fig. 1(c)

Figure 3: Displacement of test particles caused by the passage of a gravitational wave with the polarization.

22