Gravitational Waves - Ripples in the Spacetime

Akash JainDepartment of Physics, IISER Bhopal

April 27, 2013

1 Introduction

Nothing travel faster than the speed of light - a postulate which has great consequences.The postulate has nearly abandoned1 the word ’instantaneous’ from Physics. Whateveryou do, the information takes some finite amount of time to propagate to a distantlocation. Throw a pebble to a quiet pond. When the pebble touches the surface, afeather floating at some distance have no news of it. Neither will it have until the ripplecreated by the pebble reaches the feather. A similar analogy goes in nearly every branchof Physics including Mechanics, Electrodynamics or Gravitation. Whatever form theinformation may take, it is bound by an upper limit of propagation speed, which resultsin nice phenomenon including the subject of this Term Paper - Gravitational Waves.

Einstein proposed his ’General Theory of Relativity’ in 1915. Along with the unparalleledexcellence of imagination and intellect, GTR predicted many Gravitational Phenomenonwhich could otherwise never arise in the Newtonian Theory. The precession of the peri-helia of Mercury, bending of light near massive objects and gravitational red-shift beingsome of the consequences of the proposed space-time curvature, were successfully ver-ified in the days to come. Among others GTR predicted the existence of GravitationWaves, ripples in the curved space-time as a consequence of abrupt disturbances in themass configuration. Though the theory present a bold claim of their existence, there has

1In the context of action at a distance.

1

been no direct detection till date. In 1993, Nobel Prize was awarded for first success-ful indirect evidence of Gravitational Waves in the orbital decay of Hulse-Taylor BinaryPulsar. Many Gravitational Wave Detectors are functioning worldwide in internationalcollaborations, but yet remain unsuccessful to register any claimable detection.

Failure to detect Gravitational Waves is primarily due to the weakly interacting nature ofGravitational Forces. Though we are surrounded all around with plausible cosmic sourcesof GW2, expected measurements involved are of the order of 10−21m. This makes themvery sensitive to measurement errors and background noise, and thus hard to detect.But this weakly coupling nature of GW is what makes them more interesting to study.Successful understanding of these can shift the horizon of detectable universe farther indistance and backward in time. We might be able to look closer to the Big Bang thenwhat is currently provided by Cosmic Microwave Background, studying which is one ofthe prime goals of modern physics.

2 Derivation of the Gravitational Waves

2.1 Gravitational Wave Emission by an Accelerating Particle

An accelerating charge produces Electromagnetic Waves. Same is true for gravitation -an accelerating mass produces Gravitation Waves. I will not go through the mathematicsbehind the process, interested readers might look it up in any standard text on GeneralRelativity. But it’s worth going through the physical process involved.

Consider a massive particle far away from everything moving uniformly with velocity v.While it is moving, it is continuing to disturb the space-time canvas, which disturbanceis propagated to the entire canvas at speed c. But since the particle is moving withuniform velocity, the disturbance it creates moves with uniform velocity along with it,hence resulting in no overall influence. But if the particle is accelerating, at every instantit will create a disturbance, and very next instant it will accelerate and detach itself fromthe disturbance. The detached disturbance thus propagate freely on the canvas. Thislost energy is is what we understand as Gravitational Radiation. Same as EM Waves,their radial dependence dies out far away from the source. Thus, we will not be botheredmuch about the exact form these radiations take, but what is of interest is their limit inthe asymptotically flat space-time far away from the source.

Gravitational Waves, though sharing much similarity with the EM Waves, have an im-portant and rather complicated difference. Unlike EM Waves, GW carry their own sourcewith them. In other words, GW can effect their own propagation, as they carry with themvarying energy-momentum, which again acts as a source for GW. This makes the wavesolutions to the Einstein’s Equations highly non-linear and complicated to solve. Com-plete analytical solutions for the non-linear GW equations are extremely intractable, andone has to find refuge to numerical methods to solve them. However nice and tractablesolutions to the Field Equations can be achieved in Weak Field Linear Approximations.Though an approximation, Linear Solutions yield much information and insight of thephenomenon. Due to weak interacting nature of the Gravitational Forces, significantly

2Shorthand for ’Gravitational Waves’

2

high intensities of GW are only produced at sites of frequently changing highly curvedspace-time. Far (astronomical scales) from their sources, GW do behave linearly as theyare not strong enough to effect their own propagation significantly.

In the following few sections I will go through the mathematics behind the Weak-fieldLinear Metric for Gravitational Waves which will require some basic understanding ofthe General Theory of Relativity and Einstein’s Field Equations. Unequipped (or unin-terested) readers might skip directly to equation (14).

2.2 Linearised Field Equations

In the weak-field approximation, metric of the space-time can be considered as smallperturbation to the flat Minkowski metric given by:

gµν = ηµν + εhµν (1)

where g is the metric of our space-time and η is the flat Minkowski space-time metricgiven by:

η =

−1 0 0 00 1 0 00 0 1 00 0 0 1

ε is a small dimensionless parameter, whose 2nd or higher powers will be neglected in fur-ther calculation. hµν is the perturbation to the Minkowski metric, which we are supposedto find as a solution to the field equations. Additionally we assume that our space-timeis asymptotically flat, i.e. perturbation due to the existence of GW is not carried all theway to infinity.

limr→∞

hµν = 0

In the weak field limit raising and lowering of indices is done by η. Therefore wehave:

hαβ = ηαµηβνhµν

Note that:(ηαβ − εhαβ)(ηαν + εhαν) = δαν +O(ε2)

⇒ gαβ = ηαβ − εhαβ

Once metric is known, it is not hard to find its higher derivatives.

Christoffel Symbols:

Γλµν =1

2gλα (gµα,ν + gνα,µ − gµν,α)

=1

2ε(ηλα − εhλα) (hµα,ν + hνα,µ − hµν,α)

Γλµν =1

2ε(hλµ,ν + hλν,µ − h ,λ

µν

)(2)

3

Riemann Christoffel Curvature Tensor:

Rλµνσ =1

2(gλν,µσ − gλσ,µν − gµν,λσ + gµσ,λν) + gητ

(ΓηλνΓ

τµσ − ΓηλσΓτµν

)Rλµνσ =

1

2ε (hλν,µσ − hλσ,µν − hµν,λσ + hµσ,λν) (3)

Ricci Tensor and Ricci Scalar

Rµσ = ηλνRλµνσ

=1

2ε(hλλ,µσ − hλσ,µλ − h λ

µ ,λσ + h λµσ,λ

)+O(ε2)

Rµσ =1

2ε(h,µσ − hλσ,µλ − h λ

µ ,λσ +�hµσ)

(4)

R = ηµσRµσ

R = ε(�h− hλµ,λµ

)(5)

Here h is defined to be hµµ, i.e. the trace of hµν tensor. Also check that the covariantderivatives of Rλµνσ, Rµσ and R reduces to their normal derivatives. I will do it here forRicci Tensor, all others follow similarly:

Rµσ;ρ = Rµσ,ρ − ΓαρµRασ − ΓαρσRµα

= Rµσ,ρ +O(ε2)

Hence we have linear Einstein’s Field Equation:

Gµσ =1

2ε(h,µσ − hλσ,µλ − h λ

µ ,λσ +�hµσ − ηµσ�h+ ηµσhλα,λα

)= −8πGTµσ (6)

2.3 Gauge Freedom

As we know that the Einstein’s Equations are generally covariant, we can do an arbitraryinfinitesimal coordinate transformation and yet preserving the form of the (linear) fieldequation.

xµ → x′µ = xµ + εξµ (7)

g′µν =∂xα

∂x′µ∂xβ

∂x′νgαβ

=(δαµ − εξα,µ

) (δβν − εξβ,ν

)gαβ

= gµν − ε(ξµ,ν + ξν,µ)

η′µν + εh′µν = ηµν + εhµν − ε(ξµ,ν + ξν,µ)

h′µν = hµν − (ξµ,ν + ξν,µ)

4

hµν → h′µν = hµν − (ξµ,ν + ξν,µ) (8)

This is called the equation of Gauge Transformation of hµν . It can be checked thatRiemann Tensor and its contracted forms are gauge invariant. If we choose:

�ξν = hµν,µ −1

2h,ν

h′µν,µ −1

2h′,ν = hµν,µ − (ξµ,νµ + ξ µ

ν,µ )− 1

2hµµ,ν +

1

2(ξµ,µν + ξ µ

µ,ν )

= hµν,µ −1

2h,ν −�ξν

= 0

Dropping the primes, we get the Einstein’s Gauge:

hµν,µ =1

2h,ν (9)

Under Einstein’s Gauge, field equations (Eqn. 6) transform to:

1

2ε�

(hµν −

1

2ηµνh

)= −8πGTµν (10)

In vacuum Tµν = 0, and hence we have:

�hµν =1

2ηµν�h

ηνµ�hµν = ηνµ1

2ηµν�h

�h = 2�h

�hµν = 0 (11)

It should be noted that the Einstein’s Gauge does not kill the complete gauge freedom.We can still make transformations like:

hµν → h′µν = hµν − (ζµ,ν + ζν,µ) s.t. �ζν = 0 (12)

2.4 Plane Wave Solutions

Once the Field Equation in hand, all what is left is to solve it for hµν. First we noticefrom (Eqn. 11) that:

�hµν = 0

⇒ ~∇2hµν =∂2hµν∂t2

5

Therefore, the speed of wave is 1 (speed of light, c). Lets assume, without loss of general-ity, that the wave is propagating in x direction. As a consequence metric only depends onx and t, and partials with respect to y and z vanishes. Also we impose that our solutionshould move in x direction with speed 1, giving us the form:

hµν = hµν(t− x) (13)

Einstein Gauge (Eqn. 9) then on lowering with η gives:

−h00,0 + h10,1 =1

2h,0

−h01,0 + h11,1 =1

2h,1

−h02,0 + h12,1 = 0

−h03,0 + h13,1 = 0

We denote ∂/∂(t− x) as (˙). Since, ∂/∂(t− x) = ∂/∂t = −∂/∂x

−h00 − h10 =1

2h ⇒ −h00 − h10 −

1

2h = f1

−h01 − h11 = −1

2h ⇒ −h01 − h11 +

1

2h = f2

−h02 − h12 = 0 ⇒ −h02 − h12 = f3

−h03 − h13 = 0 ⇒ −h03 − h13 = f4

But, h has to vanish at ∞, therefore fi’s vanishes. Expanding h we get:

1

2(h00 + h11) = −h10 −

1

2(h22 + h33)

1

2(h00 + h11) = −h01 +

1

2(h22 + h33)

h02 = −h12h03 = −h13

hµν =

h00

12(h00 + h11) h02 h03

12(h00 + h11) h11 −h02 −h03

h02 −h02 h22 h23h03 −h03 h23 −h22

We have additional freedom in our gauge (Eqn. 12), which we can use to make the aboveresult even simpler. Choose ζ such that:

ζµ = ζµ(t− x)

ζ0 =1

2h00; ζ1 = −1

2h11; ζ2 = h02; ζ3 = h03

Thus it satisfies the condition �ζµ = 0. The profit we gain is:

h00 → h00 − 2ζ0,0 = h00 − 2ζ0 = 0

h11 → h11 − 2ζ1,1 = h11 + 2ζ1 = 0

h02 → h02 − ζ0,2 − ζ2,0 = h02 − ζ2 = 0

h03 → h03 − ζ0,3 − ζ3,0 = h03 − ζ3 = 0

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while h22 and h23 remain unaltered. Finally we have our metric for weak GravitationalWave ripples propagating in x direction:

gµν = ηµν +

0 0 0 00 0 0 00 0 h22(t− x) h23(t− x)0 0 h23(t− x) −h22(t− x)

(14)

3 Properties of Gravitational Waves

Now when we have with us our metric, we would like to extract from it some importantproperties of GW. Especially in weak field limit, GW share much similarity with EMWaves, including speed, transverse nature, polarization etc. We shall discuss them insome detail here.

3.1 Transverse Nature and Polarization of Gravitational Waves

Notice that the metric for GW (Eqn. 14) has two independent uncoupled parameters,namely h22 and h23. One can safely separate the two and write the metric in form:

hµν = h(22)µν + h(23)µν

h(22)µν =

0 0 0 00 0 0 00 0 h22 00 0 0 −h22

; h(23)µν =

0 0 0 00 0 0 00 0 0 h230 0 h23 0

(15)

If h23 is zero, metric of a GW moving in x direction is given by:

dτ 2 = −gµνxµxν

= dt2 − dx2 − (1 + εh22)dy2 − (1− εh22)dz2

Now, consider six test particles infinitesimally close to origin, each sitting on either sideof a particular axis.

When GW passes this system, distance between particles on a particular axis will be:

X : dτ 2 = −dx2 (16)

Y : dτ 2 = −(1 + εh22)dy2 (17)

Z : dτ 2 = −(1− εh22)dz2 (18)

Notice that the distance between particles on x axis is unchanged. This illustrates thefact that GW are Transverse in nature; i.e. motion of the particles is perpendicular tothe direction of propagation of the wave. Assume h22 to have some oscillatory form (likesinusoidal) such that its amplitude is both positive and negative depending on phase.Then, distance between particles on y and z axes oscillates out of phase. This state ofwave is said to be (+) Polarized.

7

Figure 1: Effect of a (+) polarized Gravitational Wave on system of particles. Dark spheresare the initial position of particles, and white ones the final.

Instead if we set h22 to be zero, metric takes the form:

dτ 2 = −gµνxµxν

= dt2 − dx2 − dy2 − dz2 − 2εh23dydz

= dt2 − dx2 − dy2 − dz2 − dydz + dydz − εh23(

2dydz +dy2

2+dz2

2− dy2

2− dz2

2

)= dt2 − dx2 − 1

2(dy + dz)2 − 1

2(dz − dy)2 − εh23

(1

2(dy + dz)2 − 1

2(dz − dy)2

)

If we perform a spatial rotation about x axis 45o counterclockwise, new coordinates aregiven by:

t′ = t; x′ = x; y′ =1√2

(y + z); z′ =1√2

(z − y)

Therefore the line element modifies exactly to what we had in (+) Polarization:

dτ 2 = dt′2 − dx′2 − (1 + εh22)dy′2 − (1− εh22)dz′2

This state of wave is called (×) Polarized. The (×) and (+) polarizations differ onlyby an angle of 45o. Thus it suffices to study properties of GW in (+) polarization.Any general oblique polarization can be generated from linear combinations of the abovetwo.

An important point to note is the distinction here from EM Waves, whose polarizationsare separated by 90o rather than 45o. This is a consequence of the fact that EM WaveEquations are vector equations, however GW Equations are symmetric 2nd rank tensorequations.

8

3.2 Motion of Particles under effect of Gravitational Waves

3.2.1 Effect of GW on Position Coordinates

We have seen that GW only effects particles transverse to their direction of motion. Butwe will show that positions of the particles are not actually altered due to GW. It is easyto see from the Geodesic Equation. Every particle will always follow its own geodesicirrespective of the structure of the space-time.

d2xi

dτ 2= −Γiµν

dxµ

dτ

dxν

dτ

Small variation to it introduced due to the GW ripple will be given by:

d2δxi

dτ 2= −δΓiµν

dxµ

dτ

dxν

dτ− 2Γiµν

dδxµ

dτ

dxν

dτ

In absence of GW, space-time was flat and thus Γ vanishes. Also if we assume our particleto be initially at rest, four velocities are given by (1, 0, 0, 0).

d2δxi

dτ 2= −δΓiµν

dxµ

dτ

dxν

dτ= −δΓi00 (19)

For our metric (Eqn. 14) Γi00 is given by the following using (Eqn. 2):

Γi00 =1

2ε(hi0,0 + hi0,0 − h

,i00

)= 0

⇒ d2δxi

dτ 2= 0

Since in absence of GW, particle was sitting silently at rest, δx(0) = 0 anddδx(0)

dτ=

0δx(τ) = 0 (20)

Thus we see that the coordinates of particles are not effected under influence of GW.Just the space between them is curved, resulting in straight lines no longer being thegeodesics.

3.2.2 Effect of GW on Extended Bodies

Consider a circular ring of particles centred at origin in y − z plane. We would like tostudy its behaviour when GW passes through it. Note that it gives nearly all informationwhat we need to know about how 3D systems behave in the influence of GW, as xdirection is remain always unaffected. Considering the GW to be (+) polarized we writethe expression for distance of particles on ring from origin.

dA2 = gijdxidxj | i, j = 1, 2, 3

9

gij =

1 0 00 1 + εh22 00 0 1− εh22

Let’s switch to cylindrical polar coordinates given by:

x = ρ; y = rcosθ; z = rsinθ

Then our metric transforms to:

g(P )ij =

∂xl

∂xi(P )

∂xm

∂xj(P )

glm; xi(P ) = {ρ, r, θ}

=

1 0 00 1 + εh22cos2θ 00 0 r2(1− εh22cos2θ)

Thus we have:

dA2 = dρ2 + (1 + εh22cos2θ) dr2 + r2 (1− εh22cos2θ) dθ2

Since we are interested in distance from the origin and ring is in y− z plane, dθ2 = dρ2 =0

A =

∫ R

0

√1 + εh22cos2θ dr

= R√

1 + εh22cos2θ

R2 =A2

1 + εh22cos2θ

≈ A2(1− εh22cos2θ)

A bit of algebra from here can bring it to nicer form:

A2cos2θ

R2(1 + εh22)+

A2sin2θ

R2(1− εh22)= 1 (21)

You will easily recognize this form as an ellipse in A − θ plane. If we turn off h22 itwill return to a circle of radius R as we would expect. If we consider a nice sinusoidalwave h22 = hosin(ω(t − x)), this will give a varying ellipse whose major (minor) axis isoscillating with frequency ω. Small ε will make sure that the oscillations are not too large(or even significant3). Under (×) polarization the effect will be no different, just rotatedwith 45o.

In r − θ plane our ring is still circular irrespective of h22. It is also expected becausepositions of particles should not be altered under influence of GW, only distance betweenthem should be. It can be visualized as if the space between particles is itself shrunk(and expanded) while GW passes through them.

3Expected amplitude of GW on Earth due to distant violent sources is expected to be of order of10−21m

10

Figure 2: Effect of GW on a ring of particles. First figure shows the expansion and contractionof space between particles in r−θ coordinate system. Second figure shows the elliptical distortionof the ring in A− θ coordinate system.

Figure 3: Effect of Gravitational Waves on extended 3D objects.

While we have done it for 2D ring, it shouldn’t be hard to guess effect of GW on 3Dobjects. In (Fig. 3) I have provided Mathematica models for effects of GW on twobasic objects - Sphere and Cylinder, though amplitudes are highly exaggerated obviously.Notice that there is a phase difference in the distortion in the direction of propagation ofwave, which is the result of (t− x) dependence of h22. These ripples propagate throughthese objects once we allow the time to flow.

3.3 Energy of Gravitational Waves

Waves are of any physical significance only if they carry Energy. To our relief, as in EMWaves, GW also carry energy. Recall that the energy density of a gravitational field inNewtonian Limit is given by (See ref.[3]):

ρ =(∇φ)2

8πG

11

But in Einstein’s Theory, there is no Gravitational ’Force’, so is not any concept ofGravitational Energy. This is due to the Principle of Equivalence which states that noeffects of gravitation can be felt locally in a freely falling frame. Though when talking ofasymptotically flat space-time, one might associate an energy to curvature with referenceto the flat space at∞. I will not go into details, but will mention the result for the energyflux of EM Waves.

F =1

16πG

((h22)

2 + (h23)2)

(22)

Notice that the energy flux is independent for different polarizations, which should beexpected.

F+ =1

16πG(h22)

2; F× =1

16πG(h23)

2

4 Detection of Gravitational Waves

At the end of the day Physics is an experimental science. Until we make claimableobservations of a predicted phenomenon, their mathematical or even conceptual eleganceis of no use. Same is true for Gravitational Waves too. First passive confirmation of GWwas done in 1993 by studying the decay of the orbit of Hulse-Taylor Binary Pulsar, whichwas credited by a Nobel Prize. The observations matched exactly with the predictionsof GTR, giving GW a firm base. Many passive observations have been done since then,but we still lack to detect any direct evidence of GW. Many GW detectors are workingworldwide trying to enhance their sensitivity and observation range to get a successfuldetection.

4.1 Detectors

GW detectors are based on the principle of Michelson Interferometer, with obviously alot sophistications. Some of them which are currently functional are:

1. LIGO (Laser-Interferometer Gravitational Wave Observatory) - Two in Hanford,Washington (4km and 2km), and one in Livingston, Louisiana (4km).

2. GEO - Mitika, Japan (600m).

3. VIRGO - Cascina, Italy (3km).

4. TAMA - Sarstedt, Germany (300m).

All these detectors work under global collaboration as a single grid, which helps in mini-mizing the chance of any fake detection due to noise from surroundings. It will also helpto get information about the direction of the sources once a detection is made.

Michelson Interferometers were first devised to measure the speed of aether4 wind aroundthe earth, which turned to be an extremely important experiment in the development ofmodern science. Interferometer consists of two mutually perpendicular arms, each fitted

4A hypothetical fluid which was thought to be filled in the empty space between cosmic bodies inour universe. It was supposed to the absolute rest frame with respect to which the motion of all otherbodies are to be measured.

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Figure 4: Schematic diagram of a Michelson Interferometer

with a mirror at the end (See Fig 4). At the intersection of the arms rests a beam splitterthat splits the incoming laser beam to two halves and sends it to the either arms. Thereis a collector kept perpendicular to the source where the beams after reflections from therespective arms meet and interfere. As you can see in Figure 4, the resultant beams willhave a phase difference of π if the two arms are equal in length, and thus will undergo adestructive interference. If the lengths are not equal, phase difference is given by5:

∆φ = π +2π

λ(2∆L) (23)

where ∆L is the length difference of the two arms, and λ is the wavelength of the laserused.

We have seen in section 3.2 that a GW passing an object will expand it in one directionand squeeze perpendicular to it. So, lets consider a (+) polarized GW striking ourinterferometer kept in y − z plane with its equal length arms aligned along the axes. Asan influence of the GW, the arm lengths of the interferometer will differ, due to whichthe phase difference will no longer remain π, and we will see an interference patter in thecollector.

According to (Eqn. 21) ∆L is given by:

∆L = L√

1 + εh22 − L√

1− εh22

= L(1 +1

2εh22)− L(1− 1

2εh22)

= Lεh22

H =∆L

L= εh22 (24)

5Lookup any standard text or article on Michelson Interferometry for detailed derivation.

13

Figure 5: LIGO Gravitational Wave Interferometer at Livingston, Louisiana with arm-lengthof nearly 4km.

H is called the Dimensionless Strain, which is a measure of the strength of the GW. Notethat the factor is independent of the detector parameters, which should be expected.

∴ ∆φ = π +4πHL

λ

It is worth mentioning that H is of the order of 10−24 for the GW signals we expect onearth. Therefore ∆L goes as 10−21 for interferometers having arm-length of the orderof a km. More is the arm-length of the detector, better sensitivity can it provide to themeasurements, but it has its own drawbacks and limitations. If you are careful enoughyou will notice that change in ∆φ is not much appreciable.

∆φGW =4πHL

λ≈ 4π × 10−21

10−7≈ 4π × 10−14

Efforts are being made to rise this number so as to increase the sensitivity of the in-terferometers. One way it is done is using Fabry-Perot cavities at the ends of both thearms. Thus light undergoes multiple reflections inside the cavity before it interferes,which increase the path length and thus the phase difference significantly. But even thenew number does not make one very happy, where comes the role of technology whichgoes in making these interferometers precise to the scales we want. Currently LIGO in-terferometers are sensitive upto H value of around 10−20, which in itself is a jaw falling6

figure.

LISA (Laser Interferometer Space Antenna) is the next generation of GW Detectors. Itwill be a space based detector which will orbit around the sun in the Earth’s orbit itself.Two prime motivations to place a detector in space are:

1. Enormous length scales available, which can in turn increase the sensitivity of mea-surements.

6The length measurements involved are of the order of 10−18m, even below the nuclear scales by 3orders of magnitudes.

14

2. Low background noise compared to what faced by the ground based detectors.

Once in place, LISA is expected to be orders of magnitude more sensitive than theconventional ground based detectors.

4.2 Sources of Gravitational Waves

May it be a tiny atom or a super massive black-hole, all accelerating massive particles emitGravitational Waves. But due to the weekly interacting nature of gravity, they mightnot always leave significant detectable traces. Though we are surrounded all over bysources of GW, we do not experience them too often (not even with highly sophisticatedexperiments). Thus we have to rely on astronomical sources of GW for any significantchance of detection.

Depending on the nature of the sources we can expect different kinds of GW signals toreach us, which are classified in 4 vast categories as below:

1. Burst Sources (Supernovae Explosions, Gamma Ray Bursts etc.)

2. Pulsars - (Binary Pulsars, Asymmetric Neutron Stars, Blackhole Mergers etc.)

3. Stochastic Background

Figure 6: Supernovae Explosion. They act as Burst (Impulsive) Sources of GravitationalWaves

15

4.2.1 Burst Sources

At a stage near the end of a star’s life, due to enormous gravitational pull its dense coreundergoes a sudden violent collapse. This event is termed as Supernovae Collapse (orExplosion) of a star. Along with all other kinds of emissions, SN7 acts as a rich sourceof GW. Being explosive in nature, this is an impulsive source, and thus can reflect as asudden peak in our detectors.

4.2.2 Pulsars

Pulsars refer to oscillating sources of GW, where we can expect nice sinusoidal behaviourof signals. They generally contain binary pairs of astronomical bodies (Blackholes orNeutron Stars) rotating around each other, and loosing energy while they do so in the formof GW. Also asymmetric rotating Blackholes or Neuron Stars can give rise to pulsatingGW.

Figure 7: Blackhole Binaries. They act as strong pulsating sources of Gravitational Waves.

Depending on the acceleration and mass of the binaries, signals can be extremely shortlived (maybe a minute or so), or might be virtually constant. As more massive and moreaccelerating bodies will emit more and will sustain shorter, but will be more intense.

4.2.3 Stochastic Background

Stochastic Background refer to the signals that were originated at the time of Big Bang.Gravitational Forces came into existence way before the electromagnetic. So, if we areable to read the Gravitational Wave signals originated from that time scales, we mightbe able to look closer to the horizon of Big Bang. Also as we know, GW are weaklyinteracting, these signals are not quite effected or perturbed by the matter they haveencountered in their way since then, thus the information they contain is well preserved.All we need to do is hear.

7Supernovae Explosion

16

5 Discussion

In a presentation in 1913 in Vienna, Einstein was asked ’How quickly the action of Gravityis propagated in your theory? There must be very complicated connection between theseideas...’. As a reply to which he mentioned ’Yes, the mathematics will be...’.

From my point of view, once you start to talk about Gravitational Waves, the idea seemsso simple and obvious. All you are doing is propagating the information of change inmass distribution of your system to all over space. More or less the same way a pebble’sdisturbance propagate in the whole pond in form of ripples. Life will be simple if thepebble is spherical and you drop it quietly in the pond; you can write exact equations forthe emission and propagation of the ripples. Instead if the pebble is weird in shape (theone’s you generally encounter) and you throw it with an initial spin and orientation (whichis more likely to be the case), equations of the ripples will not be that easy to handle.But anyway, you will still understand the ripples in the pond, if not the mathematicsbehind it.

Obviously Gravitational Waves are much more complicated that the ripples in the pond,but a similar analogy goes there. Mathematics and tensor algebra that is involved (sim-plest of which I have given in Section 2) might be extremely cumbersome, the idea ofwaves is not a new one. The theory of Electromagnetic Waves is well established andplundered, which acts as our guide while we try understanding the waves of Gravity!

If detected, GW will provide another big confirmation of the General Theory of Relativity.Or if someone comes with a better theoretical understanding of why are we not able to seethem (similar to the explanation that goes in the quark confinement8), it will be anotherbreakthrough. Whatever may be the case, with the speed technology is getting betternowadays, I think the day is not far when we will have another news headline!

References

[1] D’ Inverno R., Introducing Einstein’s Relativity.

[2] James B. Hartle, Gravity.

[3] Jayant V. Narlikar, An Introduction to Relativity.

[4] Charles W. Misner, Kip S. Thorne, John A. Wheeler, Gravitation.

[5] Steven Weinberg, Gravitation and Cosmology.

[6] Lecture Notes from ISGWA (IndIGO School on Gravitational-Wave Astronomy) 2010.

8When the experimentalists were not able to find any fractional charged particles, as was proposedby the Quark Model, those who favoured the theory gave the concept of Quark Confinement - Quarksonly come as bound states in form of Baryons and Mesons, you cannot see a free quark!

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