- •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
91 |
4.4 Dust again: the stress–energy tensor |
Re presentation of a frame by a one -form
Before going on to discuss other properties of fluids, we should mention a useful fact. An inertial frame, which up to now has been defined by its four-velocity, can be defined also
by a one-form, namely that associated with its four-velocity g(U, ). This has components
Uα = ηαβ Uβ
or, in this frame,
U0 = −1, Ui = 0.
dt (since their components are equal). So we could equally |
|
This is clearly also equal to −˜¯ |
|
dt. This has a nice picture: dt is to be pictured as a set of |
|
well define a frame by giving ˜ |
˜ |
surfaces of constant t, the surfaces of simultaneity. These clearly do define the frame, up to
˜
spatial rotations, which we usually ignore. In fact, in some sense dt is a more natural way
to define the frame than U. For instance, the energy of a particle whose four-momentum
is p is |
= |
|
|
E = ˜ |
p0. |
(4.10) |
|
dt, p |
There is none of the awkward minus sign that we get in Eq. (2.35)
E |
= − |
· |
U. |
p |
|
4.4 D u s t a g a i n : t h e s t re s s– e n e rg y t e n s o r
So far we have only discussed how many dust particles there are. But they also have energy and momentum, and it will turn out that their energy and momentum are the source of the gravitational field in GR. So we must now ask how to represent them in a frame-invariant manner. We will assume for simplicity that all the dust particles have the same rest mass m.
Energy density
In the MCRF, the energy of each particle is just m, and the number per unit volume is n. Therefore the energy per unit volume is mn. We denote this in general by ρ:
ρ := energy density in the MCRF. |
(4.11) |
Thus ρ is a scalar just as n is (and m is). In our case of dust,
ρ = nm (dust). |
(4.12) |
In more general fluids, where there is random motion of particles and hence kinetic energy of motion, even in an average rest frame, Eq. (4.12) will not be valid.
92 |
|
|
|
Perfect fluids in special relativity |
|
|
||
|
|
|
|
|
|
|
|
|
|
|
O |
we again have that the number density is n/√(1 |
− |
v2), but now the |
|||
|
|
|||||||
|
|
In the frame ¯ |
|
|||||
|
energy of each particle is m/√(1 − v2), since it is moving. Therefore the energy density is |
|||||||
|
mn/√(1 − v2): |
|
ρ |
energy density in a frame in |
|
|
||
|
|
|
|
|
= |
which particles have velocity v . |
|
(4.13) |
|
|
|
|
1 − v2 |
|
This transformation involves two factors of (1 v )− ¯ 0, because both volume and
0
2 1/2 =
−
energy transform. It is impossible, therefore, to represent energy density as some compo-
nent of a vector. It is, in fact, a component of a 20 tensor. This is most easily seen from the point of view of our definition of a tensor. To define energy requires a one-form, in order to select the zero component of the four-vector of energy and momentum; to define a density also requires a one-form, since density is a flux across a constant-time surface. Similarly, an energy flux also requires two one-forms: one to define ‘energy’ and the other to define the surface. We can also speak of momentum density: again a one-form defines which component of momentum, and another one-form defines density. By analogy there is also momentum flux: the rate at which momentum crosses some surface. All these things require two one-forms. So there is a tensor T, called the stress–energy tensor, which has all these numbers as values when supplied with the appropriate one-forms as arguments.
Stress–energy te nsor
The most convenient definition of the stress–energy tensor is in terms of its components in some (arbitrary) frame:
(dxα , dxβ ) |
|
Tαβ : |
flux of α momentum across |
. |
|
|
= |
= a surface of constant xβ |
(4.14) |
||||
T ˜ ˜ |
|
(By α momentum we mean, of course, the α component of four-momentum: pα :=
˜ α
dx , p .) That this is truly a tensor is proved in Exer. 5, § 4.10.
Let us see how this definition fits in with our discussion above. Consider T00. This is defined as the flux of zero momentum (energy) across a surface t = constant. This is just the energy density:
T00 = energy density. |
(4.15) |
Similarly, T0i is the flux of energy across a surface xi = const:
T0i = energy flux across xi surface. |
(4.16) |
Then Ti0 is the flux of i momentum across a surface t = const: the density of i momentum,
Ti0 = i momentum density. |
(4.17) |
Finally, Tij is the j flux of i momentum:
Tij = flux of i momentum across j surface. |
(4.18) |