- •Contents
- •Preface to the second edition
- •Preface to the first edition
- •1 Special relativity
- •1.2 Definition of an inertial observer in SR
- •1.4 Spacetime diagrams
- •1.6 Invariance of the interval
- •1.8 Particularly important results
- •Time dilation
- •Lorentz contraction
- •Conventions
- •Failure of relativity?
- •1.9 The Lorentz transformation
- •1.11 Paradoxes and physical intuition
- •The problem
- •Brief solution
- •2 Vector analysis in special relativity
- •Transformation of basis vectors
- •Inverse transformations
- •2.3 The four-velocity
- •2.4 The four-momentum
- •Conservation of four-momentum
- •Scalar product of two vectors
- •Four-velocity and acceleration as derivatives
- •Energy and momentum
- •2.7 Photons
- •No four-velocity
- •Four-momentum
- •Zero rest-mass particles
- •3 Tensor analysis in special relativity
- •Components of a tensor
- •General properties
- •Notation for derivatives
- •Components
- •Symmetries
- •Circular reasoning?
- •Mixed components of metric
- •Metric and nonmetric vector algebras
- •3.10 Exercises
- •4 Perfect fluids in special relativity
- •The number density n
- •The flux across a surface
- •Number density as a timelike flux
- •The flux across the surface
- •4.4 Dust again: the stress–energy tensor
- •Energy density
- •4.5 General fluids
- •Definition of macroscopic quantities
- •First law of thermodynamics
- •The general stress–energy tensor
- •The spatial components of T, T ij
- •Conservation of energy–momentum
- •Conservation of particles
- •No heat conduction
- •No viscosity
- •Form of T
- •The conservation laws
- •4.8 Gauss’ law
- •4.10 Exercises
- •5 Preface to curvature
- •The gravitational redshift experiment
- •Nonexistence of a Lorentz frame at rest on Earth
- •The principle of equivalence
- •The redshift experiment again
- •Local inertial frames
- •Tidal forces
- •The role of curvature
- •Metric tensor
- •5.3 Tensor calculus in polar coordinates
- •Derivatives of basis vectors
- •Derivatives of general vectors
- •The covariant derivative
- •Divergence and Laplacian
- •5.4 Christoffel symbols and the metric
- •Calculating the Christoffel symbols from the metric
- •5.5 Noncoordinate bases
- •Polar coordinate basis
- •Polar unit basis
- •General remarks on noncoordinate bases
- •Noncoordinate bases in this book
- •5.8 Exercises
- •6 Curved manifolds
- •Differential structure
- •Proof of the local-flatness theorem
- •Geodesics
- •6.5 The curvature tensor
- •Geodesic deviation
- •The Ricci tensor
- •The Einstein tensor
- •6.7 Curvature in perspective
- •7 Physics in a curved spacetime
- •7.2 Physics in slightly curved spacetimes
- •7.3 Curved intuition
- •7.6 Exercises
- •8 The Einstein field equations
- •Geometrized units
- •8.2 Einstein’s equations
- •8.3 Einstein’s equations for weak gravitational fields
- •Nearly Lorentz coordinate systems
- •Gauge transformations
- •Riemann tensor
- •Weak-field Einstein equations
- •Newtonian limit
- •The far field of stationary relativistic sources
- •Definition of the mass of a relativistic body
- •8.5 Further reading
- •9 Gravitational radiation
- •The effect of waves on free particles
- •Measuring the stretching of space
- •Polarization of gravitational waves
- •An exact plane wave
- •9.2 The detection of gravitational waves
- •General considerations
- •Measuring distances with light
- •Beam detectors
- •Interferometer observations
- •9.3 The generation of gravitational waves
- •Simple estimates
- •Slow motion wave generation
- •Exact solution of the wave equation
- •Preview
- •Energy lost by a radiating system
- •Overview
- •Binary systems
- •Spinning neutron stars
- •9.6 Further reading
- •10 Spherical solutions for stars
- •The metric
- •Physical interpretation of metric terms
- •The Einstein tensor
- •Equation of state
- •Equations of motion
- •Einstein equations
- •Schwarzschild metric
- •Generality of the metric
- •10.5 The interior structure of the star
- •The structure of Newtonian stars
- •Buchdahl’s interior solution
- •10.7 Realistic stars and gravitational collapse
- •Buchdahl’s theorem
- •Quantum mechanical pressure
- •White dwarfs
- •Neutron stars
- •10.9 Exercises
- •11 Schwarzschild geometry and black holes
- •Black holes in Newtonian gravity
- •Conserved quantities
- •Perihelion shift
- •Post-Newtonian gravity
- •Gravitational deflection of light
- •Gravitational lensing
- •Coordinate singularities
- •Inside r = 2M
- •Coordinate systems
- •Kruskal–Szekeres coordinates
- •Formation of black holes in general
- •General properties of black holes
- •Kerr black hole
- •Dragging of inertial frames
- •Ergoregion
- •The Kerr horizon
- •Equatorial photon motion in the Kerr metric
- •The Penrose process
- •Supermassive black holes
- •Dynamical black holes
- •11.6 Further reading
- •12 Cosmology
- •The universe in the large
- •The cosmological arena
- •12.2 Cosmological kinematics: observing the expanding universe
- •Homogeneity and isotropy of the universe
- •Models of the universe: the cosmological principle
- •Cosmological metrics
- •Cosmological redshift as a distance measure
- •The universe is accelerating!
- •12.3 Cosmological dynamics: understanding the expanding universe
- •Critical density and the parameters of our universe
- •12.4 Physical cosmology: the evolution of the universe we observe
- •Dark matter and galaxy formation: the universe after decoupling
- •The early universe: fundamental physics meets cosmology
- •12.5 Further reading
- •Appendix A Summary of linear algebra
- •Vector space
- •References
- •Index
239 |
9.4 The energy carried away by gravitational waves |
may be turned around, however, to argue that the flux we have constructed is the only acceptable definition of energy for the waves, since our calculation shows it is conserved, when added to other energies, to lowest order in hμν . The qualification ‘to lowest order’ is important, since it is precisely because we are almost in flat spacetime that, at lowest order, we can construct conserved quantities. At higher order, away from linearized theory, local energy cannot be so easily defined, because the time dependence of the true metric becomes important. These questions are among the most fundamental in relativity, and are discussed in detail in any of the advanced texts. Our equations should be used only in linearized theory.
Energy lost by a radiating system
Consider a general isolated system, radiating according to Eqs. (9.82)–(9.87). By integrating Eq. (9.122) over a sphere surrounding the system, we can calculate its net energy loss rate. For example, at a distance r along the z axis, Eq. (9.122) is
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Use of the identity |
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(which follows from Eq. (9.87)) gives, after some algebra, |
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16π r2 |
Now, the index z appears here only because it is the direction from the center of the coordinates, where the radiation comes from. It is the only part of F which depends on the location on the sphere of radius r about the source, since all the components of –I ij depend on time but not position. Therefore we can generalize Eq. (9.125) to arbitrary locations on the sphere by using the unit vector normal to the sphere,
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nj = xj/r. |
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We get for F |
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+ n n n n I ijI kl . |
(9.127) |
16π r2 |
The total luminosity of the source is the integral of this over the sphere of radius r. In Exer. 45, § 9.7 we prove the following integrals over the entire sphere
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njnk sin θ dθ dφ = |
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Gravitational radiation |
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It then follows that the luminosity L of a source of gravitational waves is |
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The generalization to cases where I ij has a more general time dependence is |
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where dots denote time derivatives.
It must be stressed that Eqs. (9.121) and (9.132) are acccurate only for weak gravitational fields and slow velocities. They can at best give only order-of-magnitude results for highly relativistic sources of gravitational waves. But in the spirit of our derivation and discussion of the order-of-magnitude estimate of hij in Eq. (9.70), we can still learn something about strong sources from Eq. (9.121). Since Ijk is of order MR2, Eq. (9.121) tells us that L M2R4 6 (M/R)2(R )6 φ02v6. The luminosity is a very sensitive function of the velocity. The largest velocities we should expect are of the order of the velocity of free fall, v2 φ0, so we should expect
L (φ0)5. |
(9.133) |
Since φ0 1, the luminosity in geometrized units should never substantially exceed one. In ordinary units this is
L 1 = c5/G ≈ 3.6 × 1052 W. |
(9.134) |
We can understand why this particular luminosity is an upper limit by the following simple argument. The radiation field inside a source of size R and luminosity L has energy density L/R2 (because |T0i| |vi|T00 = cT00 = T00), which is the flux across its surface. The total energy in radiation is therefore LR. The Newtonian potential of the radiation alone is therefore L. We shall see in the next chapter that anything where the Newtonian potential substantially exceeds one must form a black hole: its gravitational field will be so strong that no radiation will escape at all. Therefore L 1 is the largest luminosity any source can have. This argument applies equally well to all forms of radiation, electromagnetic as well as gravitational. The brightest quasars and gamma-ray bursts, which are the most luminous classes of object so far observed, have a (geometrized) luminosity 10−10. By contrast, black hole mergers (Ch. 11) have been shown by numerical simulations to reach peak luminosities 10−3, all of the energy of course emitted in gravitational waves.
An example. The Hulse –Taylor binary puls ar
In § 9.3 we calculated –I ij for a binary system consisting of two stars of equal mass M in circular orbits a distance l0 apart. If we use the real part of Eq. (9.97) in Eq. (9.132), we get
241 |
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9.4 The energy carried away by gravitational waves |
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This expression illustrates two things: first, that L is dimensionless in geometrized units and, second, that it is almost always easier to compute in geometrized units, and then convert back at the end. The conversion is
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So for the binary pulsar system described in § 9.3, if its orbit were circular, we would have ω = 2π/P = 7.5049 × 10−13 m−1 and
L = 1.71 × 10−29 |
(9.138) |
in geometrized units. We can, of course, convert this to watts, but a more meaningful procedure is to compare this with the Newtonian energy of the system, which is (defining the orbital radius r = 12 l0),
E = 21 Mω2r2 |
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The physical question is: How long does it take to change this? Put differently, the energy radiated in waves must change the orbit by decreasing its energy, which makes |E| larger and hence ω larger and the period smaller. What change in the period do we expect in, say, one year?
From Eq. (9.139), by taking logarithms and differentiating, we get
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Since dE/dt is just −L, we can solve for dP/dt:
dP/dt = (3 PL)/(2E) ≈ −15 PM−1(M )8/3
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which is dimensionless in any system of units. It can be reexpressed in seconds per year:
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242 |
Gravitational radiation |
This estimate needs to be revised to allow for the eccentricity of the orbit, which is considerable: e = 0.617. The correct formula is derived in Exer. 49, § 9.7. The result is that the true rate of energy loss is some 12 times our estimate, Eq. (9.138). This is such a large factor because the stars’ maximum angular velocity (when they are closest) is larger than the mean value we have used for , and since L depends on the angular velocity to a high power, a small change in the angular velocity accounts for this rather large factor of 12. So the relativistic prediction is:
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The observed value as of 2004 is (Weisberg and Taylor 2005)
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The effect has been observed in other binaries as well (Lorimer 2008).
9.5 A s t ro p h y s i c a l s o u rce s o f g ra v i t a t i o n a l wa v e s
Overview
Physicists and astronomers have made great efforts to understand what kinds of sources of gravitational waves the current detectors might be able to see. This has been motivated partly by the need to decide whether the large investment in these detectors is justified, and partly because the sensitivity of the detectors depends on the accuracy with which waveforms can be predicted, so that they can be recognized against detector noise. While the astrophysics of potential sources is beyond the scope of this book, it is useful to review the basic classes of sources, learn what general relativity says about them, and understand why they might be interesting to observe. We shall consider four groups of sources: binary systems, spinning neutron stars, gravitational collapse, and the Big Bang.
Binary systems
We have seen how to compute the expected wave amplitude from a binary system in Eqs. (9.98) and (9.99), as well as in Exers. 29 and 39 in § 9.7. There are a number of known binary systems in our Galaxy which ought, by these equations, to be radiating gravitational waves in the frequency band observable by LISA, and with amplitudes well above LISA’s expected instrumental noise. When LISA begins its observations, therefore, scientists will be looking for these signals as proof that general relativity is correct at this basic level, as well as that the spacecraft is operating properly.
The ground-based detectors will not, however, be looking for signals from long-lived binary systems. The reason is evident if we combine some of our previous computations. We have calculated what the gravitational wave luminosity of an equal-mass binary is
243 |
9.5 Astrophysical sources of gravitational waves |
(Eq. (9.136)) and what its binding energy is (Eq. (9.139)). We used the last two equations to derive the lifetime of the Hulse–Taylor binary pulsar. We can generalize this to find the lifetime of any equal-mass circular binary system, expressing it in terms of the masses M of the stars and the frequency of the orbit f = ω/2π :
τgw = − |
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−8/3, |
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What the second equation tells us is that, if LIGO observes a binary system composed of stars of solar mass, it has only a few seconds to make the observation. During this time, the signal increases in frequency, something the gravitational wave scientists call a ‘chirp’. There are no realistic systems in the frequency range of ground-based detectors that are long-lived. Instead, these detectors look for inspiral events ending in the merger of the two objects, which itself might produce a burst of radiation. For neutron stars, the merger happens when the signal frequency reaches about 2 kHz, a number that is sensitive to the neutron star equation of state (see the next chapter). If the objects in the binary are black holes of mass 10 M , the final frequency is similar, and it scales inversely with the masses. Advanced LIGO and VIRGO should see several neutron-star mergers per year, and while the event rate for black holes is harder to predict, it is likely to be similar.
It is particularly interesting from the point of view of general relativity to observe the merger of two black holes. This can be simulated numerically, and by comparing the observed and predicted waveforms we have a unique test of general relativity in the strongest possible gravitational fields. This is also the only direct way to observe a black hole: after a merger of black holes or neutron stars has led to a single black hole, that hole will oscillate for a short time until is radiates away all its deformities and settles down as a smooth Kerr black hole (see Ch. 11). This ‘ringdown radiation’ carries a distinctive signature that will distinguish the black hole from any neutron star or other material system.
LISA, observing between 0.1 mHz and 10 mHz, will follow the coalescence and merger of black holes around 106M . Astronomers know that such black holes exist in the centers of most galaxies, including our own Milky Way (Merritt and Milosavljevic 2005), as we discuss in Ch. 11. LISA will have sufficient sensitivity to see such mergers anywhere in the universe, even back to the time of the formation of the first stars and galaxies. Its observations may be very informative about the way galaxies themselves formed and merged in the early universe.
Notice that, if we observe a chirp signal well enough to measure its inspiral timescale τ from the rate of change of the signal’s frequency, then we can infer the mass M of the system from Eq. (9.147). LISA by itself, or a network of ground-based detectors working together, can measure the degree of circular polarization of the waves. This contains the information about the inclination of the orbit to the line of sight, and allows us to compute from the observed amplitude of the waves some standard amplitude, such as the amplitude the same system would be radiating if it were oriented face-on. Then we can go back