- •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
125 |
5.3 Tensor calculus in polar coordinates |
5.3 Te n s o r c a l c u l u s i n p o l a r co o rd i n a t e s
The fact that the basis vectors of polar coordinates are not constant everywhere, leads to some problems when we try to differentiate vectors. For instance, consider the simple vector ex, which is a constant vector field, the same at any point. In polar coordinates it has components ex → ( rx, θ x) = (cos θ , −r−1 sin θ ). These are clearly not constant, even though ex is. The reason is that they are components on a nonconstant basis. If we were to differentiate them with respect to, say, θ , we would most certainly not get ∂ex/∂θ , which must be identically zero. So, from this example, we see that differentiating the components of a vector does not necessarily give the derivative of the vector. We must also differentiate the nonconstant basis vectors. This is the key to the understanding of curved coordinates and, indeed, of curved spaces. We shall now make these ideas systematic.
Figure 5.6
Derivatives of basis vectors
Since ex and ey are constant vector fields, we easily find that |
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er = |
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(5.37a) |
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∂r |
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er = |
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∂θ |
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= − sin θ ex + cos θ ey = |
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eθ . |
(5.37b) |
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These have a simple geometrical picture, shown in Fig. 5.6. At two nearby points, A and B, er must point directly away from the origin, and so in slightly different directions. The derivative of er with respect to θ is just the difference between er at A and B divided by θ .
y
B
A
θ
x
Change in er, when θ changes by θ .
126 |
Preface to curvature |
The difference in this case is clearly a vector parallel to eθ , which then makes Eq. (5.37b) reasonable.
Similarly,
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eθ = |
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ey) |
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eθ = −r cos θ ex − r sin θ ey = −r er. |
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The student is encouraged to draw a picture similar to Fig. 5.6 to explain these formulas.
Derivatives of general vectors
Let us go back to the derivative of ex. Since |
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ex = cos θ er − |
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∂ |
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(cos θ ) er + cos θ |
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(er) |
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To get this we used Eqs. (5.37) and (5.38). Simplifying gives |
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just as we should have. Now, in Eq. (5.40) the first and third terms come from differentiating the components of ex on the polar coordinate basis; the other two terms are the derivatives of the polar basis vectors themselves, and are necessary for cancelling out the derivatives of the components.
r θ
A general vector V has components (V , V ) on the polar basis. Its derivative, by analogy
with Eq. (5.40), is |
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∂V |
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eθ ) |
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(V |
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Vθ |
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and similarly for ∂V/∂θ . Written in index notation, this becomes
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(Here α runs of course over r and θ .)
127 |
5.3 Tensor calculus in polar coordinates |
This shows explicitly that the derivative of V is more than just the derivative of its components Vα . Now, since r is just one coordinate, we can generalize the above equation to
∂V |
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(5.43) |
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where, now, xβ can be either r or θ for β = 1 or 2.
The C hristoffel symbols
The final term in Eq. (5.43) is obviously of great importance. Since ∂eα /∂xβ is itself a vector, it can be written as a linear combination of the basis vectors; we introduce the symbol μαβ to denote the coefficients in this combination:
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The interpretation of μαβ is that it is the μth component of ∂eα /∂xβ . It needs three indices: one (α) gives the basis vector being differentiated; the second (β) gives the coordinate with respect to which it is being differentiated; and the third (μ) denotes the component of the resulting derivative vector. These things, μαβ , are so useful that they have been given a name: the Christoffel symbols. The question of whether or not they are components of tensors we postpone until much later.
We have of course already calculated them for polar coordinates. From Eqs. (5.37) and (5.38) we find
(1) |
∂er/∂r = 0 μrr = 0 |
for all μ, |
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∂er/∂θ |
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∂eθ /∂θ = −r er rθ θ = −r, θ θ θ = 0. |
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In the definition, Eq. (5.44), all indices must refer to the same coordinate system. Thus, although we computed the derivatives of er and eθ by using the constancy of ex and ey, the Cartesian bases do not in the end make any appearance in Eq. (5.45). The Christoffel symbols’ importance is that they enable us to express these derivatives without using any other coordinates than polar.
The covariant derivative
Using the definition of the Christoffel symbols, Eq. (5.44), the derivative in Eq. (5.43) becomes
∂V |
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αβ eμ. |
(5.46) |
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In the last term there are two sums, on α and μ. Relabeling the dummy indices will help here: we change μ to α and α to μ and get
128 |
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Preface to curvature |
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∂V |
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eα + V |
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The reason for the relabeling was that, now, eα can be factored out of both terms:
∂xβ |
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So the vector field ∂V/∂x has components
∂Vα /∂xβ + Vμ α μβ .
Recall our original notation for the partial derivative, notation and define a new symbol:
eα . (5.48)
(5.49)
∂Vα /∂xβ = Vα,β . We keep this
Vα ;β := Vα ,β + Vμ α μβ . |
(5.50) |
Then, with this shorthand semicolon notation, we have
∂V |
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a very compact way of writing Eq. (5.48).
β
Now ∂V/∂x is a vector field if we regard β as a given fixed number. But there are two
values that β can have, and so we can also regard ∂V/∂xβ as being associated with a |
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tensor field which maps the vector eβ into the vector ∂V |
/∂x , as in Exer. 17, § 3.10. |
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tensorfield is called the covariant derivative of V, denoted, naturally enough, as V. Then
its components are
( V)α β = ( β V)α = Vα ;β . |
(5.52) |
On a Cartesian basis the components are just Vα ,β . On the curvilinear basis, however, the derivatives of the basis vectors must be taken into account, and we get that Vα ;β are the
components of V in whatever coordinate system the Christoffel symbols in Eq. (5.50) refer to. The significance of this statement should not be underrated, as it is the foundation
of all our later work. There is a single |
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tensor called V. In Cartesian coordinates its |
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coordinates |
{ |
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its components are called Vα |
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and can be obtained in either of two equivalent ways: (i) compute them directly in {xμ } using Eq. (5.50) and a knowledge of what the α μ β coefficients are in these coordinates; or (ii) obtain them by the usual tensor transformation laws from Cartesian to {xμ }.
What is the covariant derivative of a scalar? The covariant derivative differs from the partial derivative with respect to the coordinates only because the basis vectors change. But a scalar does not depend on the basis vectors, so its covariant derivative is the same as
its partial derivative, which is its gradient: |
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α f = |
∂f /∂xα ; |
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df . |