- •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
263 |
|
|
10.5 The interior structure of the star |
|
|||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
These have the solutions |
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
m(r) = M = const., |
|
|
(10.34) |
|||||||
|
|
|
|
|
|
|
e2 = 1 − |
2M |
, |
|
|
|
|
(10.35) |
|||
|
|
|
|
|
|
|
r |
|
|
|
|
|
|||||
|
|
where the requirement that → 0 as r → ∞ has been applied. We therefore see that the |
|||||||||||||||
|
|
exterior metric has the following form, called the Schwarzschild metric: |
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
ds2 = − 1 − |
|
r |
dt2 + 1 − 2r |
− |
dr2 + r2 d 2. |
(10.36) |
|||||||||
|
|
|
|
|
2M |
|
|
|
|
|
M |
|
1 |
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
For large r this becomes |
|
|
|
|
|
dt2 |
+ 1 + |
|
|
dr2 + r2 d 2. |
|
||||
|
|
ds2 ≈ − 1 − 2r |
|
r |
(10.37) |
||||||||||||
|
|
|
|
|
|
|
M |
|
|
|
|
|
2M |
|
|
|
|
|
|
We can find coordinates (x, y, z) such that this becomes |
|
|
|
||||||||||||
|
|
ds2 ≈ − |
1 − 2R |
dt2 + |
1 + |
|
R |
(dx2 + dy2 + dz2), |
(10.38) |
||||||||
|
|
|
|
|
M |
|
|
|
|
2M |
|
|
|
|
where R := (x2 + y2 + z2)1/2. We see that this is the far-field metric of a star of total mass M (see Eq. (8.60)). This justifies the definition, Eq. (10.28), and the choice of the symbol M.
Generality of the metric
A more general treatment, as in Misner et al. (1973), establishes Birkhoff’s theorem, that the Schwarzschild solution, Eq. (10.36), is the only spherically symmetric, asymptotically flat solution to Einstein’s vacuum field equations, even if we drop our initial assumptions that the metric is static, i.e. if we start with Eq. (10.5). This means that even a radially pulsating or collapsing star will have a static exterior metric of constant mass M. One conclusion we can draw from this is that there are no gravitational waves from pulsating spherical systems. (This has an analogy in electromagnetism: there is no ‘monopole’ electromagnetic radiation either.) We found this result from linearized theory in Exer. 40, § 9.7.
10.5 T h e i n t e r i o r s t r u c t u re o f t h e s t a r
Inside the star, we have ρ =0, p =0, and so we can divide Eq. (10.27) by (ρ + p) and use it to eliminate from Eq. (10.31). The result is called the Tolman–Oppenheimer–Volkov (T–O–V) equation:
264 |
|
Spherical solutions for stars |
|
||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
dp |
|
(ρ + p)(m + 4π r3 p) |
. |
(10.39) |
||
|
|
|
dr = − |
|
|||||
|
|
|
r(r |
− |
2m) |
|
|||
|
|
|
|
|
Combined with Eq. (10.30) for dm/dr and an equation of state of the form of Eq. (10.25), this gives three equations for m, ρ, and p. We have reduced to a subsidiary position; it can be found from Eq. (10.27) once the others have been solved.
General rules for integrating t he equations
Since there are two first-order differential equations, Eqs. (10.30) and (10.39), they require two constants of integration, one being m(r = 0) and the other p(r = 0). We now argue that m(r = 0) = 0. A tiny sphere of radius r = ε has circumference 2π ε, and proper radius |grr|1/2ε (from the line element). Thus a small circle about r = 0 has ratio of circumference to radius of 2π |grr|−1/2. But if spacetime is locally flat at r = 0, as it must be at any point of the manifold, then a small circle about r = 0 must have ratio of circumference to radius of 2π . Therefore grr(r = 0) = 1, and so as r goes to zero, m(r) must also go to zero, in fact faster than r. The other constant of integration, p(r = 0) := pc or, equivalently, ρc, from the equation of state, simply defines the stellar model. For a given equation of state p = p(ρ), the set of all spherically symmetric static stellar models forms a one-parameter sequence, the parameter being the central density. This result follows only from the standard uniqueness theorems for first-order ordinary differential equations.
Once m(r), p(r), and ρ(r) are known, the surface of the star is defined as the place where p = 0. (Notice that, by Eq. (10.39), the pressure decreases monotonically outwards from the center.) The reason p = 0 marks the surface is that p must be continuous everywhere, for otherwise there would be an infinite pressure gradient and infinite forces on fluid elements. Since p = 0 in the vacuum outside the star, the surface must also have p = 0. Therefore we stop integrating the interior solution there and require that the exterior metric should be the Schwarzschild metric. Let the radius of the surface be R. Then in order to have a smooth geometry, the metric functions must be continuous at r = R. Inside the star we have
grr = 1 − 2m(r) −1 r
and outside we have
grr = 1 − 2M −1 . r
Continuity clearly defines the constant M to be
M := m(R). |
(10.40) |
265 |
|
10.5 The interior structure of the star |
|
||
|
|
|
|
|
|
|
|
Thus the total mass of the star as determined by distant orbits is found to be the integral |
|||
|
|
||||
|
|
|
|
|
|
|
|
M = '0R |
4π r2ρ dr, |
(10.41) |
|
|
just as in Newtonian theory. This analogy is rather deceptive, however, since the integral |
||||
|
is over the volume element 4π r2dr, which is not the element of proper volume. Proper |
||||
|
|
volume in the hypersurface t = const. is given by |
|
||
|
|
|g|1/2 d3x = e r2 sin θ dr dθ dφ, |
(10.42) |
||
|
|
which, after doing the (θ , φ) integration, is just 4π r2 e + dr. Thus M is not in any sense |
|||
|
|
just the sum of all the proper energies of the fluid elements. The difference between the |
|||
|
|
proper and coordinate volume elements is where the ‘gravitational potential energy’ con- |
|||
|
|
tribution to the total mass is placed in these coordinates. We need not look in more detail |
|||
|
|
at this; it only illustrates the care we must take in applying Newtonian interpretations to |
|||
|
|
relativistic equations. |
|
|
|
|
|
Having obtained M, this determines g00 outside the star, and hence g00 at its surface: |
|||
|
|
g00(r = R) = − 1 − 2R . |
(10.43) |
||
|
|
|
|
M |
|
This serves as the integration constant for the final differential equation, the one which determines inside the star: Eq. (10.27). We thereby obtain the complete solution.
Notice that solving for the structure of the star is the first place where we have actually assumed that the point r = 0 is contained in the spacetime. We had earlier argued that it need not be, and the discussion before the interior solution made no such assumptions. We make the assumption here because we want to talk about ‘ordinary’ stars, which we feel must have the same global topology as Euclidean space, differing from it only by being curved here and there. However, the exterior Schwarzschild solution is independent of assumptions about r = 0, and when we discuss black holes we shall see how different r = 0 can be.
Notice also that for our ordinary stars we always have 2m(r) < r. This is certainly true near r = 0, since we have seen that we need m(r)/r → 0 at r = 0. If it ever happened that near some radius r1 we had r − 2m(r) = ε, with ε small and decreasing with r, then by the T–O–V equation, Eq. (10.39), the pressure gradient would be of order 1/ε and negative. This would cause the pressure to drop so rapidly from any finite value that it would pass through zero before ε reached zero. But as soon as p vanishes, we have reached the surface of the star. Outside that point, m is constant and r increases. So nowhere in the spacetime of an ordinary star can m(r) reach 12 r.
The structure of Newtonian stars
Before looking for solutions, we shall briefly look at the Newtonian limit of these equations. In Newtonian situations we have p ρ, so we also have 4π r3p m. Moreover, the
266 |
|
Spherical solutions for stars |
|||||
|
|
|
|
|
|
|
|
|
metric must be nearly flat, so in Eq. (10.29) we require m |
r. These inequalities simplify |
|||||
|
|||||||
|
Eq. (10.39) to |
|
|
|
|
||
|
|
dp |
= − |
ρm |
|
||
|
|
|
|
|
. |
(10.44) |
|
|
|
|
dr |
r2 |
This is exactly the same as the equation of hydrostatic equilibrium for Newtonian stars (see Chandrasekhar 1939), a fact which should not surprise us in view of our earlier interpretation of m and of the trivial fact that the Newtonian limit of ρ is just the mass density. Comparing Eq. (10.44) with its progenitor, Eq. (10.39), shows that all the relativistic corrections tend to steepen the pressure gradient relative to the Newtonian one. In other words, for a fluid to remain static it must have stronger internal forces in GR than in Newtonian gravity. This can be interpreted loosely as indicating a stronger field. An extreme instance of this is gravitational collapse: a field so strong that the fluid’s pressure cannot resist it. We shall discuss this more fully in § 10.7 below.
10.6 E x a c t i n t e r i o r s o l u t i o n s
In Newtonian theory, Eqs. (10.30) and (10.44) are very hard to solve analytically for a given equation of state. Their relativistic counterparts are worse.1 We shall discuss two interesting exact solutions to the relativistic equations, one due to Schwarzschild and one by Buchdahl (1981).
The Schwarzschild constantdensity interior solution
To simplify the task of solving Eqs. (10.30) and (10.39), we make the assumption
ρ = const. |
(10.45) |
This replaces the question of state. There is no physical justification for it, of course. In fact, the speed of sound, which is proportional to (dp/dρ)1/2, is infinite! Nevertheless, the interiors of dense neutron stars are of nearly uniform density, so this solution has some interest for us in addition to its pedagogic value as an example of the method we use to solve the system.
We can integrate Eq. (10.30) immediately:
m(r) = 4πρr3/3, r R, |
(10.46) |
where R is the star’s as yet undetermined radius. Outside R the density vanishes, so m(r) is constant. By demanding continuity of grr, we find that m(r) must be continuous at R. This implies
1If we do not restrict the equation of state, then Eqs. (10.44) and (10.39) are easier to solve. For example,
we can arbitrarily assume a function m(r), deduce ρ(r) from it via Eq. (10.30), and hope to be able to solve Eq. (10.44) or Eq. (10.39) for p. The result, two functions p(r) and ρ(r), implies an ‘equation of state’ p = p(ρ)
by eliminating r. This is unlikely to be physically realistic, so most exact solutions obtained in this way do not interest the astrophysicist.