Lecture Notes for PHYS 4147

Deirdre Shoemaker

Tuesday: Clay How could we extract energy from a black hole?

1 An Introduction to Gravitational Waves (Chapter 16)

”So far, gravity has been a silent engine. Scientists have never yet directly measured gravityfrom systems outside the Solar System. When gravitational wave detection becomes partof astronomy, astronomers will record in their laboratories the changing gravitational fieldsproduced by some very distant bodies. Gravity will no longer be silent. It will tell us itsstory directly. Gravity will speak to us.” Chapter 22, Schutz.

1.1 Gravitational waves are inevitable

• Gravitational waves arise from the restriction that no influence can travel faster thanlight.

• This is not unusual! Newtonian gravity was unusual that it did not allow for waves.

• Action at a Distance: In Newtonian theory, when two stars in a binary system movearound, their gravitational fields change instantaneously everywhere. So even if anexperimenter is millions of light-years away, she could in principle feel the effect ofthe changing positions immediately, without any delay. This was called action at adistance.

• Educated Guess on properties of real gravitational waves: in ordinary materials, thestiffer the material, the faster the wave speed. Since gravitational waves will travelwith the fastest possible speed the speed of light it follows that space itself iseffectively the stiffest possible material. In stiff materials, it takes a lot of force andenergy to make a small disturbance, so we can expect that gravitational waves willhave small amplitudes even when created by major events, like supernova explosions,and that they will carry large energies in their small amplitudes.

1.2 Transverse waves of tidal acceleration

• Tidal accelerations is a fundamental manifestation of gravity.

1

• Consider: two stones failling to the earth, one above the other. What are theirspeeds some time later in their fall? (Non-uniform gravitational field). Their verticalseparation increases!

• Consider: two stones side by side. How does their initial separation change?

• The gravity of moon on earth is detectable through its time-dependent tidal forces.The effect if the moon on the earth is to deform it from a sphere into an ellipse.

• Remember, the equivalence principle is local!

• Gravitational waves act in a similar manner, but carry time-dependent spatial cur-vature.

– transverse: produce tidal accelerations only in directions perpendicular to di-rection traveling

– whatever action a gravitational wave has on matter is also the motion by whichmatter produces gravitational waves

– Electromagnetic waves are also transverse, EM waves carry oscillating electricfields that make electrons move and forth along a line - the direction of that lineis the direction of the polarization of the wave.

– In contrast, GW produce deformations in the transverse plan that turn circlesinto ellipses, but!! this is not quite the same as the deformation produced bythe moon which is partially longitundinal in direction.

– any spherical source produces a time- independent gravitational field outside it- no gravitational waves!

– Gravitational waves are derived from the Einstein Equation when solved fornon-spherical masses in nonuniform motion.

– GWs have 2 polarizations, plus and cross

– Strength of GWs is measured by the relative deformation whose amplitude iscalled strain

h = 2δ`

`

We can approximate the upper bound of the strain

h ≤ 2GM

rc2

A neutron star of 1.4M� at 4.6× 1023m away will give us h ≤ 6× 10−21

2

– GWs are produced by the mass-equivalent of kinectic energy and are approxi-mated by

h =8G

rc2

(K

c2

)called the quadrupole formula and is a good approximation for systems withv < c.

– When gravitational waves move through a region they do not induce differencesbetween the rates of nearby clocks. Instead, they deform proper distances.

– They carry a large amount of energy. If we consider a plane wave (the source isfar away and the wave passes us with a flat wave front). We will assume that theGW is a simple sine-wave oscillation with frequency f . The energy flux (energycarried by the wave through a unit of are per unit of time) is about

F =πc3

4Gf2h2

Each of the polarizations contribute to the energy with amplitude squared. Theenergy is large even if the amplitude is small because c3/G is a big number.

1.3 Sources

• Binary star systems (white dwarfs, neutron stars and black holes being the strongests)

• supernova explosions

• gamma ray brusts

• non-spherical collapse to a black hole

• big bang

• more exotic like cosmic strings

• The frequency of a gravitational wave is determined by the typical time-scale forthings to happen in its source. If the masses radiating the waves move in and out in1 s, then the waves will have periods near one second and frequencies near 1 Hz.

• The upper bound on expected frequencies is about 104 Hz, because it is difficult toget large astronomical bodies, with masses comparable to the Sun or larger, to doanything on time-scales shorter than a tenth of millisecond or so.

• no lower bound on frequency

3

1.4 Hulse-Taylor Pulsar

• Many possible sources of gravitational waves in the Universe

• Weakness of the gravitational interaction means they are difficult to detect.

• They have not been directly detected.

• There existence has been indirectly detected or inferred from the Huse-Taylor pulsar.This binary star system has an increasing orbital period as the stars approach eachother from a loss of gravitational radiation (waves).

– A neutron star ≈ 20km in diameter, mass of about 1.4M� (note: on Earth, oneteaspoonful on a neutron star would weigh a billion tons)

– neutron stars are an end-state of stars with masses about 4-8 times the Sun. Af-ter a supernova explosion, the central region collapses so much that the protonsand electrons combine to form neutrons.

– Pulsars are rotating neutron stars that have jets of particles moving almost atc that stream out above their magnetic poles.

– Binary pulsar PSR B1913+16 has a pulsar and unseen companion with an orbitalperiod of 7.75hours. GR predicts the decrease in the orbital period due to theemission of gravitational radiation on order of 10µ seconds per year.

– GR is at 13% accuracy right now.

• The energy emitted in gravitational waves is not small! It is the weak coupling ofgravity to matter that makes it difficult to detect.

• A world-wide network of gravitational wave detectors exist. In the USA, we have twodetectors that make up the LASER Interferometric Gravitational wave Observatory(LIGO) - one in LA and one in WA.

• We may have a space-based detector called LISA in the next decade.

• Science:

– Fundamental detection of gravitational waves

– Test of GR in fully non-linear regime (black hole mergers)

– Weakness that they penetrate deeply into regimes that are obscured for electro-magnetic radiation.

– Gravitational wave offer information from the earliest epcohs of our Universe,into the interior of galaxies and near the black hole horizon.

4

Figure 1: Model of a pulsar.

5

Figure 2: Evidence of gravitational waves from the Nobel prize winning work of Hulse andtaylor.

6

1.5 Linearized Gravitational Waves

• Properties:

– propagate with the speed of light

– are transverse

– have two polarizations

– detected by their effect on the relative motion of test masses

– carry energy

• Simplest example:

– We will study a gravitational wave spacetime that is a small ripple of curvaturethat is propogating in one direction (longitudinal) and is independent of theother two spatial directions (transverse): plane wave.

– Metrics that are close to flat are written:

gαβ(x) = ηαβ + hαβ(x)

– hαβ is an amplitude of a small perturbation of flat spaceime. It describes thegravitational wave.

– Here is one example of a plane wave in the z-direction

hαβ =

0 0 0 00 1 0 00 0 −1 00 0 0 0

f(t− z).

– The function f(t−z) is any function of t−z as long as it is small, |f(t−z)| << 1.

– The line element for the entire spacetime is given by

ds2 = −dt2 + [1 + f(t− z)]dx2 + [1− f(t− z)]dy2 + dz2 .

– This gives a wave with speed c = 1 moving in the z-direction. hαβ is dimension-less number, as is the amplitude of the wave.

– There are many choices for the function, for example, it may be f(t − z) =aexp[−(t−z)2/σ2] representing a Gaussian wave packet, or f(t−z) = asin[ω(t−z)] which is a gravitational wave with amplitude a, frequency ω and wavelengthλ = 2π/ω.

• This plane gravitational wave spacetime is not a solution to the Einstein equation.It is a solution to the linearized Einstein equation and becomes closer to the truesolution, the smaller its amplitude.

7

1.5.1 Detecting Gravitational Waves

• How? the motion of test bodies moving along geodesics.

• One test mass will remain at rest within its freely falling frame with a wave or not.

• Equivalence Principle tells us we need at least 2 bodies or we will not detect the GW

• Simple Example using 2 test masses labeled A and B

– GW propagating in z-direction

ds2 = −dt2 + [1 + f(t− z)]dx2 + [1− f(t− z)]dy2 + dz2

– Location of masses before wave passes xiA(τ) = (0, 0, 0) and xiB(τ) = (xB, yB, zB)

– Stationary so uαA = uαB = (1, 0, 0, 0)

– What is the location of the masses after the wave passes? Solve geodesic foreach particle, A and B

– Since the amplitude is small, we will only solve to first order in δxiA and δxiB.

d2x

dτ2= −Γiαβ

dxα

dτ

dxβ

dτ

–

d2δxi

dτ2= −Γiαβ

dδxα

dτ

dxβ

dτ− Γiαβ

dxα

dτ

dδxβ

dτ− δΓ1

αβ

dxα

dτ

dxβ

dτ(1)

= −δΓiαβdxα

dτ

dxβ

dτ(2)

= −δΓiαβuαuβ (3)

– The A and B test masses are stationary so

d2δxi

dτ2= −δΓitt

– but since Γitt = 0 for our metric so Γitt = 0; and, therefore d2xi

dτ2= 0 the two test

masses do not move in coordinate space.

– Because d2δxi

dτ2= 0 then δxiA = δxiB = 0 since δxi(τ) = 0 initially.

8

• But δL(t), the distance between 2 test masses in plane orthogonal to direction ofpropagation?

– Let L∗ be the distance between A and B in the unperturbed flat spacetime

L(t) =

∫ L∗

0dx[1 + hxx(t, 0)]1/2

– soL(t) ≈ L∗[1 + 1/2hxx(t, 0)]

and the test masses on the x-axis at t=0

L(t)− L∗L∗

=δL(t)

L∗=

1

2hxx(t, 0)

– The distance between test masses change in distance along x-axis oscillates withthe wave.

– If f(t− z) = asin[ω(t− z) + δ) then

δL(t)

L∗=

1

2asin(ωt+ δ)

– Generalize this equation off of the x-axis. Place A at the origin and B at anarbitrary location in plane transverse to direction of propagation. B is L∗ awayfrom A in direction of ~n in z=0 plane.

δL(t)

L∗=

1

2hij(t, 0)ninj

is the fractional strain produced by wave.

– GW does not change coordinate location of 2 test masses, however, the path isno longer a geodesic.

– Optimal way of detecting a GW is not from separation, rather it is better to seeif laser light deviates from straight lines. The light follows straight lines in flatspacetime but will deviated by δxi if there is a GW.

1.5.2 GW Polarization

• Introduce new coordinates (X,Y ) for the x-y plane still in z = 0 assuming z-directionof gw propagation

X =

(1 +

1

2asin(wt)

)x

Y =

(1− 1

2asin(wt)

)y

9

The line element in the X-Y plane of these new coordinates is

dS2 = dX2 + dY 2

to order in a2

• The (x,y) do not vary in time but (X,Y) do.

• Now calculate distance between masses in x-y plane from X(t) and Y(t) using Eu-clidean plane geometry to first order in wave amplitude using our plane wave with

f(t− z) = asin[ω(t− z)]

• You will note that this corresponds to the plus polarization.

• Note Beyond: radiation field of any spin field has 2 orthogonal states of linearizedpolarization EM s=1 θ = 90◦, GR s=2 θ = 45◦, Neutrino s=1/2 θ = 180◦

• The metric we have been using is missing this second polarization. To generalize:

– introduce new coordinates by rotating x-y axes by φ = 45◦ such that x =1√2(x′ + y′) and y = 1√

2(x′ − y′)

10

– The original line element and metric are given as

ds2 = −dt2 + [1 + f(t− z)]dx2 + [1− f(t− z)]dy2 + dz2

hαβ =

0 0 0 00 1 0 00 0 −1 00 0 0 0

f(t− z).

– We will use our coordinate transformation to move to metric in new coordinates

gα′β′ = gαβ∂xα

∂xα′∂xβ

∂xβ′

– so

hx′x′ = hxx

(∂x

∂x′

)2

+ 2hxy

(∂x

∂x′

)(∂y

∂x′

)+ hyy

(∂y

∂x′

)2

(4)

= f(t− z)1

2+ 0− f(t− z)1

2(5)

= 0 (6)

hx′y′ = hxx

(∂x

∂x′

)(∂x

∂y′

)+ hxy

(∂x

∂x′

)(∂y

∂y′

)+ hyx

(∂y

∂x′

)(∂x

∂y′

)+ hyy

(∂y

∂x′

)(∂y

∂y′

)(7)

= f(t− z)1

2+ 0 + 0− f(t− z) 1√

2

(− 1√

2

)(8)

= f(t− z) (9)

= hy′x′ (10)

hy′y′ = 0 (11)

– So the new metric addition is rotated by φ = 45◦

hαβ =

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

f(t− z).

• This is a linearly independent 2nd polarization orthogonal to the first

11

• The general linearized GW in z-direction is given by

hαβ(t, z) =

0 0 0 00 f+(t− z) f×(t− z) 00 f×(t− z) −f+(t− z) 00 0 0 0

.

1.6 No Local Gravitational Energy in General Relativity

• Energy density in a Newtonian gravitational field

εNewt(~x) = − 1

8πG[~∇Φ(~x)]2 = − 1

8πG[~g(~x)]2

• These are analogous to the energy density in the electric field but (-) because gravityis attractive.

• What is the expression in GR? There is none.

• Why? Well following the analogy between Newtonian and Electromagnetism, wecould compute first derivatives of the metric but these all vanish in the local inertialframe.

• This is consistent with the principle of equivalence

• More deeply, the lack of a local gravitational energy density is due to the connectionbetween conserved quantities and symmetries. We did calculate the energy and an-gular momentum conserved in Schwarzschild orbits - but these followed directly fromthe symmetries of the spacetime. For Maxwell’s equation you have the symmetriesof flat spacetime.

• In GR, we do not have a fixed spacetime geometry! There are no symmetries thatcharacterize all spacetimes.

• The absence of a local energy density is expected - we shifted from gravity as a forcefield operating in spacetime to gravity as curved spacetime.

12