Gravitational Waves

PHYS 471: Introduction to Relativity and Cosmology

This is an exciting time for general relativity! The revolutionary detections of gravita-tional waves by LIGO and friends confirms one of the earliest predictions of Einstein’stheory. It also gives us pretty definite proof that black holes are real, and exist inrather copious numbers out in the farthest depths of space. But how do gravitationalwaves come about? And what is it we detect that convinces us they are real? Inorder to understand this, you’ll need to know Einstein’s equations... luckily you do!

1 A Brief History of Gravitational Waves

Before actually describing how gravitational waves arise, let’s spend a some timedescribing their history. Shortly after Einstein derived his field equations in 1915,gravitational waves emerged as one of its very first predictions. Their effect on theworld was so incredibly small, however, it was immediately deemed impossible thatthey could be detected.

Fast forward to the 1970s, when a pair of astronomers at the University of Mas-sachusetts, Ahmerst were observing the orbital behavior of a binary pulsar systemby the name of PSR B1913+16. They noticed something rather peculiar about thisbinary system: the orbit was decaying. This was particularly odd, since they didn’texpect it. The two astronomers, Russell Hulse and Joseph Taylor, Jr., wereconfounded. The pulsar binary had to be losing energy – but how? The only knownexplanation was entirely theoretical, originally proposed by Einstein 60 years earlier:the energy was being radiated away as gravitational radiation! They matchedtheir rotation rate data with Einstein’s predictions, and found an exact match! Fortheir work – the first indirect evidence of gravitational waves – the two were laterawarded the 1993 Nobel Prize in Physics.

Hulse and Taylor published their groundbreaking data in 1979 as almost incontro-vertible proof that gravitational waves existed. There was nothing to say that thepulsars were losing energy through some other means, though. In order to say thatgravitational waves DID exist, we’d actually have to detect them! How would we doso? By measuring the tiny stretching and contracting of spacetime ithat they induce.

Attempts to build a gravitational wave detector to measure these spacetime deforma-tions actually started decades before the Hulse-Taylor discovery. This shouldn’t bea surprise, considering gravitational waves were predicted by Einstein shortly afterhe came up with General Relativity. In 1969 by Joseph Weber, called “Weber bars”.These were essentially a series of concentric aluminum cylinders that would in theory

Figure 1: The orbital period of the binary pulsar system PSR B1913+16 is slowly de-caying year after year! [Figure borrowed from J. M. Wesiberg and J. H. Taylor, “TheRelativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis”,Binary Radio Pulsars ASP Conference Series, Vol. 328, 2005].

deform as a gravitational wave passed through them. In 1987, Weber claimed to haveobserved exactly this behavior following the explosion of Supernova SN1987A. Thedata was weak and controversial, however, and it was generally disregarded.

The concept for the Laser Interferometer Gravitational Wave Observatory – LIGO– was also spawned in the 1960s, and is attributed to be the brainchild of threeindividuals: Kip Thorne (of Interstellar fame!... and also LIGO fame...), RonaldDrever, and Rainer Weiss. By the 1980s, prototype detectors were being built as aproof-of-concept in various laboratories. By the 1990s, the Hanford, Washington andLivingston, Louisiana locations were secured and construction began on the 4 km longinterferometers. LIGO started its detections in the early 2000s, and went throughvarious incarnations and technological upgrades, always with the promise but neverwith a positive detection. That all changed when Advanced LIGO went online inSeptember 2015, when the first crystal-clear detection of gravitational waves camethrough loud and clear. Amusingly, the detection occurred on September 14th, threedays before the official scientific run began!

At the time this was written, LIGO has observed the gravitational waves from fourbinary black hole mergers, with a fifth, weaker detection still inconclusive. Theyinclude:

1. GW170914: M1 = 36M� , M2 = 29M� , Mfinal = 62M� ; ∆M = 3M�

2. GW151226: M1 = 14.2M� , M2 = 7.5M� , Mfinal = 20.8M� ; ∆M = 1M�

3. GW170104: M1 = 31.2M� , M2 = 19.4M� , Mfinal = 48.7M� ; ∆M = 2M�

4. GW170814: M1 = 30.5M� , M2 = 25.3M� , Mfinal = 53.2M� ; ∆M = 1.7M�

The naming convention refers to the event (GW - gravitational wave!), and the year-month-day of detection.

And hot off the press for 2017!! On October 3rd, the Nobel Prize for Physics wasawarded to Kip Thorne, Barry C. Barish, and Raiier Weiss for this outstanding andparadigm shifting piece of physics!

2 Sources of Gravitational Waves

Gravitational waves will result from either a strong gravity system and/or a violentstrong gravity event. These can be classified in four basic categories:

• Continuous wave sources: Any sufficiently large dynamic strong gravity sys-tem can emit gravitational waves on a continuous basis. These include pulsarsor other rotating compact object (e.g. spinning black holes). Although veryregular and distinct, these waves are difficult to isolate from the backgroundbecause the systems themselves aren’t usually strong enough to stand out.

• Single burst sources: Large energy events such as supernovae can result ina short gravitational wave burst. Again, these are difficult to isolate on theirown due to their extremely short duration. It is possible, however, to locatesuch a gravitational wave source by targeting the timestamp and location ofcorresponding electromagnetic signals. That is, if we see a supernova, we cancheck LIGO to see if it coincidentally detected anything in the direction of theevent. The supposed gravitational wave signal detected by Weber from SN1987would have been such a source.

• Compact binary coalescence: Colloquially referred to as CBC events (notto be confused with Canadian television shows), these involve the inspiral andmerger of two compact mass sources – neutron stars and/or black holes. Theyare the strongest sources of gravitational waves, so it’s no surprise that theyhave been the first to be detected by LIGO.

• Stochastic waves: High energy fluctuations of spacetime in the early universe,i.e. shortly after the Big Bang, could have produced gravitational waves aswell. Although these in principle could have been even stronger events thatCBCs, they have since been cosmologically redshifted to background “noise” of

extremely low frequency (less than a few mHz). It’s virtually impossible for anyEarth-based gravitational wave experiment to detect these, since their frequencyfalls well below the seismic cutoff. Future space-based detectors, however, mighthave a chance!

3 Linearizing Einstein’s Equations

But enough about history and stuff. Let’s explore the physics of gravitational waves!When a gravitational field is weak (e.g. far from a big source, or close to a smallsource), we can approximate the spacetime to be essentially flat, but with a smallperturbation. Mathematically, we can write this as

gµν ≈ ηµν + hµν

where ηµν = diag(+1,−1,−1,−1) is the flat spacetime metric, and hµν represents theperturbation. Its inverse is

gµν ≈ ηµν − hµν

because if h is small, then it’s like finding the approximation of 11+x

for x near 0 (thinkTaylor series!). This looks like a locally inertial frame (it sort of is!), so as you sawon your Problem Set, the Riemann tensor can be simplified to read

Rαβµν ≈1

2[∂α∂νhβµ + ∂β∂µhαν − ∂α∂µhβν − ∂β∂νhαµ]

Remember that all higher-order terms in h vanish.

From the Riemann tensor as written above, we can obtain the linearized Ricci tensorand Ricci scalar in the usual fashion,

Rµν = gαβRαµβν ≈ ηαβRαµβν

R = gµνRµν ≈ ηµνRµν

The metric gµν is replaced by ηµν because remember, all higher-order terms in h van-ish! (I just said that...)

This gives us, respectively

Rνβ ≈1

2[∂µ∂νhβµ + ∂β∂

αhαν − ∂µ∂µhβν − ∂β∂νhαα]

and

R ≈ 1

2

[∂µ∂βhβµ + ∂β∂

αhβα − ∂µ∂µhββ − ∂β∂νh

αµ

](1)

Once we have these, we can plug them into Einstein’s equations for a given sourceTµν . Since a weak gravitational field is caused by weak curvature, we can assume thatthe source is nothing! That is, Tµν = 0:

Rµν −1

2gµνR = 0

We can pretty much ignore the cosmological constant term (which isn’t there), sinceit’s so small. If you look closely at the expressions for Rµν and R given above, youwill note some common terms between them. In particular, terms like ∂α∂αhµν from

the Ricci tensor and −∂α∂αhββ would combine nicely if we defined a new tensor ofthe form

Hµν = hµν −1

2ηµνh

αα

So let’s do it! It turns out that when this redefinition is made, the Einstein equationssimplify considerably to the following:

∂α∂αHµν = 0

This looks familiar – it’s the D’Alembertian, a.k.a.

∂α∂α = ∂2t −∇2

Rewriting the above equation for Hµν , we see that it is(∂2t −∇2

)Hµν = 0

which is a wave equation! The solutions are

Hµν = Aµνeikαxα

which are appropriately termed gravitational waves! The wave vector kα tells usabout the wavelength of the wave, its speed, as well as the direction of travel andmomentum. The tensor Aµν is a measure of how much the metric itself is deformedby the wave, and is called the polarization tensor.

4 Characteristics of Gravitational Waves

4.1 Propagation of Gravitational Waves

We can uncover additional characteristics of Aµν and kµ by placing constraints onthe wave itself. First, we expect gravitational waves to be transverse – that is, theyoscillate perpendicular to their direction of travel. This means that Aµν and kµ mustbe orthogonal, and so we can say

Transverse wave : =⇒ Aµνkν = 0

Gravitational waves propagate in a similar fashion to water waves or electromagneticwaves – the are transverse, not longitudinal.

4.2 Speed of Gravitational Waves

Next, let’s plug the solution into the wave equation to obtain information about kµ.Since the derivatives affect only the exponential, we get(

∂2t −∇2

)Aµνeikαx

α

= (k20 − k2

1 − k22 − k2

3)eikαxα

= 0

=⇒ k20 − k2

1 − k22 − k2

3 = kµkµ = 0

If you remember from special relativity (or even if you don’t), a four-vector whoselength is zero is called a null vector, and null vectors represent things that travelat the speed of light! So, we have a second property of gravitational waves:

Wave speed : =⇒ kµkµ = 0 means velocity = 1

Wave frequency : =⇒ ω = k0

Gravitational waves therefore propagate through the universe at the speed of light!

4.3 Polarization of Gravitational Waves

We still need to know something about the amplitude and polarization of the wave,so we’ll add some constraints to Aµν . We know that gµν is symmetric, and thereforeso is hµν , and therefore so is Hµν , and therefore so is Aµν . Usually, we also say thatthe metric is diagonal, and we’d be tempted to impose that constraint on hµν , Hµν ,and Aµν . Since these are small perturbations to flat spacetime, however, we can relaxthis property and allow for non-diagonal terms. Why? Because this represents theamplitude of a wave, which has to oscillate in different directions perpendicular tothe direction of motion.

Let’s say our gravitational wave is traveliing purely in the z-direction. That meansthe wave vector is kµ = (k0, 0, 0, kz). But since it is a null vector, kµkµ = 0, thismeans

kµ = (k0, 0, 0, k0) =⇒ kµkµ = (k0)2 − (k0)2 = 0

The orthogonality condition with Aµν tells us, therefore, that this tensor cannot havecomponents in the t(0) or z(3) indices. Also, it means that

Aµνkν = Aµ0k0 − Aµ3k0 = 0 =⇒ Aµ0 = Aµ3 = 0

That is, the 0th and 3rd columns are zeroes. Since Aµν is symmetric, the same goesfor the 0th and 3rd rows:

Aµν =

0 0 0 00 A11 A12 00 A21 A22 00 0 0 0

We already know A12 = A21, so now we only need constraints on the diagonal elementsA11 and A22. In the spirit of the null vector condition kµkµ = 0, we’re going to imposethe similar demand on Aµν (and by proxy Hµν) that they be traceless. That is, werequire

Aµµ = Hµµ = 0

=⇒ A00 − A11 − A22 − A33 = 0

=⇒ A11 = −A22

Furthermore, the traceless nature of Hµν means that

Hµµ = hµµ −

1

2ηµµh

αα = 0

The contraction of the Minkowski metric is ηµµ = ηµνηµν = 1 + 1 + 1 + 1 = 4, and sowe can write

0 = hµµ − 2hµµ0 = −hµµ

which obviously tells us that hµµ = 0, and so in this case the two perturbation tensorsare equal, Hµν = hµν . This is, appropriately enough, called the transverse-tracelesscondition (gauge), and the metric perturbation associated with it is traditionallywritten hµνTT. The tensor hµν is called the strain, and it’s what we ultimately mea-sure in detectors such as LIGO and Virgo.

The complete version of the polarization amplitude tensor is therefore

Aµν =

0 0 0 00 h+ h× 00 h× −h+ 00 0 0 0

with the quantities h+ and h× representing the two polarization modes of thewaves. This is sometimes written

Aµν = h+εµν+ + h×ε

µν×

where

εµν+ =

0 0 0 00 1 0 00 0 −1 00 0 0 0

, εµν× =

0 0 0 00 0 1 00 1 0 00 0 0 0

are called the polarization tensors.

This is (one of the places) where gravitational waves are interesting! Like the polar-ization vectors of the waves you know and love, these modes are orthogonal; there isno way to construct εµν+ as a linear superposition of εµν× . But the physical manifesta-tion of the + and × polarizations are not orthogonal! One is a mix of the xx andyy directions, the other is a mix of the xy and yx direction. We can visualize this asfollows:

Figure 2: The two orthogonal polarization modes of gravitational waves representedby εµν+ and εµν× are physically oriented at 45◦ to one another.

The effect of these modes are to stretch spacetime in one direction, and contract itin the perpendicular direction. Sounds like the kind of thing an interferometer wouldbe useful for measuring...

5 Detecting Gravitational Waves: LIGO and the

Strain!

The basic set-up of gravitational wave detection is as follows:

• Gravitational waves of amplitude h0 are produced somewhere, out there, withintensity (luminosity) L0.

• The waves travel for a time t and distance ct.

• The waves eventually pass through our interferometer detectors with decreasedamplitude h(t).

• The spacetime around the detectors is stretched/compressed according to thepolarization modes.

• One arm of the interferometer becomes shorter than the other by a factor ofh(t), and the interference pattern of the two laser beams shift.

• We work backwards to figure out the initial conditions that created these waves(source type, masses, distance).

The design of LIGO’s detectors is that of a basic Michaelson interferometer: a laserbeam is split into two, and each is sent down a perpendicular arm and back. Thebeams recombine as an interference pattern. If something happens along one armand not the other (e.g. a gravitational wave passes through), the interference patternshifts, as shown below.

What can we make of the values of h+ and h×? They represent the amplitude ofthe gravitational waves, and by proxy the relative degree to which the spacetimeis stretched and compressed by each mode. In other words, when the wave passesthrough an object of length `0, the object’s size will change by an amount

h =∆`

`0

=⇒ ∆` = h`0

In the derivation of the wave equation, we assumed that the stress energy tensorwas that of vacuum, i.e. Tµν = 0. This isn’t really the case for gravitational waves,though, since they are carrying energy through spacetiime! The actual tensor for awave is

Tµν =1

32π〈(∂µhTTij )(∂νh

TTij )〉

Figure 3: A simple schematic of the LIGO interferometer, showing normal spacetimein which the two arms are of equal length (top), and the passage of a gravitationalwave (bottom), where the polarization changes each arm’s length. The interferencepattern of the split beams shifts accordingly.

where hTTij are the spatial components of the transverse-traceless tensor, and the 〈..〉means we average the value over a small region of space. Since we set the gravita-tional waves traveling in the z-direction, the Einstein equations will only have termscorresponding to t and z coordinates, and you can show the stress energy componentsare

Ttt = Tzz = −Ttz =1

32π

ω2c2

G

(h2

+ + h2×)

(2)

We can simplify this down a bit by assuming that the size of h+ and h× is about thesame, so

Ttt =1

16π

ω2c2h2

G=π

4

f 2c2h2

G

The substitution ω = 2πf has been made in the latter equality, because f is theactual frequency we measure. Since π ≈ 4, we can further simplify this to

Ttt ∼f 2c2h2

G

Just how big is h for typical gravitational waves that pass through the Earth? A littlebit of dimensional analysis goes a long way! First, let’s re-write the expression aboveto the nearest order of magnitude, Since T00 is a measure of energy density, we cansay

Ttt ∼EGWV

where EGW is the energy of the gravitational waves, and V is the volume of spacethrough which is has travelled. What kind of values can we expect for EGW and r?The processes that create gravitational waves are extremely violent and high-energy,and for typical sources, they can be on the order of a few solar masses,

EGW ∼M�c2

Distance-wise, these events should be very far away – otherwise we’re hosed, eh! Wewant them to be at least a few megaparsecs (Mpc) distant (the visible universe isabout 1000 Mpc in size). A parsec (pc) is about 3 lightyears in size, and a lightyearis the distance light travels in a year (or π × 107 s), so r is on the order

r ∼ 1 Mpc ∼ 3 · (c · π × 107) · 106 ∼ 1022 metres

The volume is therefore V ∼ r3, where r is the distance from the source to thedetector, so the energy density is on the order of

Ttt ∼M�c

2

r3

Equating this to the actual value T00 given in Equation 2, we find

f 2c2h2

G∼ M�c

2

r3

Solving for h, we get the expression

h ∼(M�G

f 2r3

) 12

Looks pretty simple! Not bad for some handy guesswork. Plugging in values: thelowest sensitivity of gravitational wave we can hope to detect on the Earth is of theorder f = 102 Hz. Noting also that M� ∼ 1030 kg and G ∼ 10−10 N·m2

kg2 , we get

h ∼(

1030 · 10−10

104 · 1066

) 12

=(10−50

) 12 = 10−25

YIKES! For a typical gravitational wave source we’re likely to detect, h is prettysmall. What does this mean for LIGO? It’s interferometer arms are 4 km in size, sothis amount of strain will change the length by

∆` = h`0 ∼ (10−25)(103) = 10−22 m

WOW!!! That’s small. That’s REALLY small! In fact, it’s so small that wereally can’t detect it!

The secret to getting around this is: instead of sending the laser beam down theinterferometer and back once, we do it hundreds of times! This means the effectivelength isn’t 4 km, but 400 km! That bumps ∆` up by a few orders of magnitude,bringing it into the realm of measurability.

The above estimate involved cutting a few corners, but in reality it isn’t that far fromthe actual optimal strain for LIGO (about 10−23), as shown in the following figure(borrowed from Hild, S. Class.Quant.Grav. 29 (2012) 124006):

In fact, we can improve the estimate of the strain by taking into account that theenergy density of the wave isn’t really smeared out over the entire volume of spacearound the source. Instead, it’s contained in a spherical shell of radius r and widthL:

ρE ∼M�c

2

r2L

This changes the expression for h slightly to

h ∼(M�G

f 2r2L

) 12

=

(M�G

L

) 12

· f−1r−1

This shows a bit more cleanly how the strain depends on the parameters involved.We need a value for L that makes sense in terms of a gravitational wave detection.It can’t just be L = λGW, since the energy is not contained in a single wavelength.But it makes sense that it should be some multiple of λGW. Since gravitational waves

travel at the speed of light, the “thickness” of the wavetrain is

L = c∆t

and so the expression for the strain becomes

h ∼(M�G

c∆t

) 12

· f−1r−1

6 Gravitational Waves and Quantum Gravity

The orthogonal polarization modes of the gravitational waves actually tell us some-thing very interesting about the nature of the nature of any particle that might carrythe gravitational force. If you’ve studied quantum mechanics and/or particle physics,you’ll remember that force carriers are bosons, which means they have integer spin.So, if there is such a thing as a graviton (we don’t know yet), it will be such aparticle. But what spin will it have?

The other forces – electromagnetism, weak, and strong – are mediated by spin-1bosons: the photon, W± and Z0, and gluons. All these particles are represented byvectors. You’ve already learned about the vector potential Aµ in the electromag-netic field tensor (Fµν = ∂µAν − ∂νAµ). Although this is a classical theory, in thequantum version (QED – quantum electrodynamics) this vector ends up being thephoton itself. There are analogous vectors that represent the other force carriers.

Everything is fundamentally related to geometry (!), so let’s take a look at the po-larization vectors of a photon. For the sake of visualization, we’ll consider x and ypolariaztions, but the following argument works just as well for left- and right-circularpolarizations. If the polarization for a photon is initially in either one of those di-rections and we start rotating the vector, you will certainly agree that to get thevector back to it’s initial orientation, it must be rotated by 360◦ (or 2π for you radiantypes). Of course, any vector must be rotated through 360◦ to get back to its originalorientation.

What happens if we rotate a gravitational wave’s polarization modes? In order toget back to the original orientation, we only need to rotate each by 180◦! If youdon’t believe it, take another look at Figure 2 and see how much you need to rotatethe two ovals to get back to the original configuration. This sets gravity apart fromthe other fundamental forces. If there is a particle (boson) responsible for mediatinggravitational interactions, it cannot be spin-1. Rather, the graviton must be aspin-2 boson!

Figure 4: Photon polarization vectors must be rotated through 360◦ to get back totheir original orientation.

We can also understand this because gravitation is represented by a rank-2 tensor,while other bosons are represented by rank-1 tensors (i.e. vectors). For those of youwho have read up on particle physics, you will also remember that matter particles– electrons, quarks, neutrinos, etc... – are spin-1

2beasts, which must be rotated

through an angle 4π to recover the original. So, it’s all consistent!

7 Other Gravitational Waves Detectors

LIGO isn’t the only game in town! Although it gets the most press because, well,it actually detected gravitational waves, its two detectors in Hanford, WA and Liv-ingston, LA are only some of the other gravitational wave experiments currently inprogress. Because I’ve spent waaayy to long writing this document (you’d better ap-preciate it!!), I’ll only give them a brief mention here. There’s VIRGO in Italy, whichwas able to play a role in the August 2017 detection of a binary black hole mergery.A new LIGO detector under construction in India (called Ind-IGO, get it?), GEO,the Einstein Telescope (planned for 2030), and a couple others you can read abouton Wikipedia... But the most interesting telescopes aren’t the ones being constructedon Earth....

The biggest factor inhibiting ground-based gravitational wave telescopes like LIGOor VIRGO is seismic noise. No matter how hard you try, you can’t remove this back-cround – unless you remove the entire Earth from the equation! This is, in fact, thevery idea behind future interferometer experiments that will probe the low frequencyregime of gravitational waves. The most prominent of the planned experiments isthe Laser Interferometer Satellite Antenna, or LISA.

Figure 5: The Laser Interferometer Satellite Antenna consists of three interferometerarms 2.5 million km in length. [Image borrowed from http://lisa.nasa.gov/.]

LISA is an array of three satellites training the Earth’s orbit, which will have inter-ferometer arm lengths slightly longer than LIGO’s... 2,500,000 km as opposed to4 km. This will make them sensitive to gravitational waves of that wavelength, whichcorresponds to an optimal frequency threshold of about 0.1 Hz. Figure 7 shows thestrain sensitivity for LISA, as compared to Earth-based detectors.

Plagued by a lot of funding issues, LISA was originally supposed to be launchedin the 2020s, but is now projected to be in orbit by 2034.

Figure 6: LISA’s strain sensitivity will open the window to extremely low fre-quency gravitational waves, like those from burst sources, continuous waves, andultimately primordial waves. [Plot borrowed from From: Andersson, N., “The Roadto Gravitational-Wave Astronomy”, Prog. Part. Nucl. Phys. 66,239-248 (2011) .]