- •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
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1.9 The Lorentz transformation |
Any single observer can make consistent observations using concepts that are valid for him but that may not transfer to other observers. All the so-called paradoxes of relativity involve, not the inconsistency of a single observer’s deductions, but the inconsistency of assuming that certain concepts are independent of the observer when they are in fact very observer dependent.
Two more points should be made before we turn to the calculation of the Lorentz transformation. The first is that we have not had to define a ‘clock’, so our statements apply to any good timepiece: atomic clocks, wrist watches, circadian rhythm, or the half-life of the decay of an elementary particle. Truly, all time is ‘slowed’ by these effects. Put more properly, since time dilation is a consequence of the failure of simultaneity, it has nothing to do with the physical construction of the clock and it is certainly not noticeable to an observer who looks only at his own clocks. Observer O¯ sees all his clocks running at the same rate as each other and as his psychological awareness of time, so all these processes run more slowly as measured by O. This leads to the twin ‘paradox’, which we discuss later.
The second point is that these effects are not optical illusions, since our observers exercise as much care as possible in performing their experiments. Beginning students often convince themselves that the problem arises in the finite transmission speed of signals, but this is incorrect. Observers define ‘now’ as described in § 1.5 for observer O¯ , and this is the most reasonable way to do it. The problem is that two different observers each define ‘now’ in the most reasonable way, but they don’t agree. This is an inescapable consequence of the universality of the speed of light.
1.9 T h e L o re n t z t ra n s f o r m a t i o n
We shall now make our reasoning less dependent on geometrical logic by studying the algebra of SR: the Lorentz transformation, which expresses the coordinates of O¯ in terms of those of O. Without losing generality, we orient our axes so that O¯ moves with speed v on the positive x axis relative to O. We know that lengths perpendicular to the x axis are the same when measured by O or O¯ . The most general linear transformation we need consider, then, is
¯t = αt + βx |
y¯ = y, |
x¯ = γ t + σ x |
z¯ = z, |
where α, β, γ , and σ depend only on v.
From our construction in § 1.5 (Fig. 1.4) it is clear that the ¯t and x¯ axes have the equations:
¯t axis (x¯ = 0) : vt − x = 0, x¯ axis (¯t = 0) : vx − t = 0.
The equations of the axes imply, respectively:
γ /σ = −v, β/α = −v,
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Special relativity |
which gives the transformation
¯t = α(t − vx),
x¯ = σ (x − vt).
Fig. 1.4 gives us one other bit of information: events (¯t = 0, x¯ = a) and (¯t = a, x¯ = 0) are connected by a light ray. This can easily be shown to imply that α = σ . Therefore we have, just from the geometry:
¯t = α(t − vx),
x¯ = α(x − vt).
Now we use the invariance of the interval:
−( ¯t)2 + ( x¯)2 = −( t)2 + ( x)2.
This gives, after some straightforward algebra,
α = ±1/√(1 − v2).
We must select the + sign so that when v = 0 we get an identity rather than an inversion of the coordinates. The complete Lorentz transformation is, therefore,
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the corresponding ones in O. Spatial rotations of the axes relative to one another produce more complicated sets of equations than Eq. (1.12), but we will be able to get away with Eq. (1.12).
1.10 T h e v e l o c i t y - co m p o s i t i o n l a w
The Lorentz transformation contains all the information we need to derive the standard formulae, such as those for time dilation and Lorentz contraction. As an example of its use we will generalize the Galilean law of addition of velocities (§ 1.1).
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1.11 Paradoxes and physical intuition |
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This is the Einstein law of composition of velocities. The important point is that |W | never exceeds 1 if |W| and |v| are both smaller than one. To see this, set W = 1. Then Eq. (1.13) implies
(1 − v)(1 − W) = 0,
that is that either v or W must also equal 1. Therefore, two ‘subluminal’ velocities produce another subluminal one. Moreover, if W = 1, then W = 1 independently of v: this is the universality of the speed of light. What is more, if |W| 1 and |v| 1, then, to first order, Eq. (1.13) gives
W = W + v.
This is the Galilean law of velocity addition, which we know to be valid for small velocities. This was true for our previous formulae in § 1.8: the relativistic ‘corrections’ to the Galilean expressions are of order v2, and so are negligible for small v.
1.11 Pa ra d oxe s a n d p h y s i c a l i n t u i t i o n
Elementary introductions to SR often try to illustrate the physical differences between Galilean relativity and SR by posing certain problems called ‘paradoxes’. The commonest ones include the ‘twin paradox’, the ‘pole-in-the-barn paradox’, and the ‘space-war paradox’. The idea is to pose these problems in language that makes predictions of SR seem inconsistent or paradoxical, and then to resolve them by showing that a careful application of the fundamental principles of SR leads to no inconsistencies at all: the paradoxes are apparent, not real, and result invariably from mixing Galilean concepts with modern ones. Unfortunately, the careless student (or the attentive student of a careless teacher) often comes away with the idea that SR does in fact lead to paradoxes. This is pure nonsense. Students should realize that all ‘paradoxes’ are really mathematically ill-posed problems, that SR is a perfectly consistent picture of spacetime, which has been experimentally verified countless times in situations where gravitational effects can be neglected, and that SR
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Special relativity |
‘Its top speed is 186 mph – that’s 1/3 600 000 the speed of light.’
Figure 1.15 The speed of light is rather far from our usual experience! (With kind permission of S. Harris.)
forms the framework in which every modern physicist must construct his theories. (For the student who really wants to study a paradox in depth, see ‘The twin “paradox” dissected’ in this chapter.)
Psychologically, the reason that newcomers to SR have trouble and perhaps give ‘paradoxes’ more weight than they deserve is that we have so little direct experience with velocities comparable to that of light (see Fig. 1.15). The only remedy is to solve problems in SR and to study carefully its ‘counter-intuitive’ predictions. One of the best methods for developing a modern intuition is to be completely familiar with the geometrical picture of SR: Minkowski space, the effect of Lorentz transformations on axes, and the ‘pictures’ of such things as time dilation and Lorentz contraction. This geometrical picture should be in the back of your mind as we go on from here to study vector and tensor calculus; we shall bring it to the front again when we study GR.
1.12 F u r t h e r re a d i n g
There are many good introductions to SR, but a very readable one which has guided our own treatment and is far more detailed is Taylor and Wheeler (1966). Another widely admired elementary treatment is Mermin (1989). Another classic is French (1968). For treatments that take a more thoughtful look at the fundamentals of the theory, consult Arzeliès (1966), Bohm (2008), Dixon (1978), or Geroch (1978). Paradoxes are discussed