by analogy with the development here. What we have to add to all this is a discussion of parallelism, of how to measure the extent to which the Euclidean parallelism axiom fails. This measure is the famous Riemann tensor.

5.7 F u r t h e r re a d i n g

The Eötvös and Pound–Rebka–Snider experiments, and other experimental fundamentals underpinning GR, are discussed by Dicke (1964), Misner et al. (1973), Shapiro (1980), and Will (1993, 2006). See Hoffmann (1983) for a less mathematical discussion of the motivation for introducing curvature. For an up-to-date review of the GPS system’s use of relativity, see Ashby (2003).

The mathematics of curvilinear coordinates is developed from a variety of points of view in: Abraham and Marsden (1978), Lovelock and Rund (1990), and Schutz (1980b).

5.8E xe rc i s e s

1Repeat the argument that led to Eq. (5.1) under more realistic assumptions: suppose a fraction ε of the kinetic energy of the mass at the bottom can be converted into a photon and sent back up, the remaining energy staying at ground level in a useful form. Devise a perpetual motion engine if Eq. (5.1) is violated.

2Explain why a uniform external gravitational field would raise no tides on Earth.

3(a) Show that the coordinate transformation (x, y) → (ξ , η) with ξ = x and η = 1 violates Eq. (5.6).

(b)Are the following coordinate transformations good ones? Compute the Jacobian and list any points at which the transformations fail.

(i)ξ = (x2 + y2)1/2, η = arctan(y/x);

(ii)ξ = ln x, η = y;

(iii)ξ = arctan(y/x), η = (x2 + y2)−1/2.

4A curve is defined by {x = f (λ), y = g(λ), 0 λ 1}. Show that the tangent vector (dx/dλ, dy/dλ) does actually lie tangent to the curve.

5Sketch the following curves. Which have the same paths? Find also their tangent vectors where the parameter equals zero.

(a) x = sin λ, y = cos λ; (b) x = cos(2π t2), y = sin(2π t2 + π ); (c) x = s, y = s + 4;

(d) x = s2, y = −(s − 2)(s + 2); (e) x = μ, y = 1.

6Justify the pictures in Fig. 5.5.

7Calculate all elements of the transformation matrices α β and μν for the transformation from Cartesian (x, y) – the unprimed indices – to polar (r, θ ) – the primed indices.

(a)Show that the spacelike line described by Eq. (5.96) with a as the variable parameter and λ fixed is orthogonal to his world line where they intersect. Changing λ in Eq. (5.96) then generates a family of such lines.

(b)Show that Eq. (5.96) defines a transformation from coordinates (t, x) to coordi-

nates (λ, a), which form an orthogonal coordinate system. Draw these coordinates and show that they cover only one half of the original t − x plane. Show that the coordinates are bad on the lines |x| = |t|, so they really cover two disjoint quadrants.

(c)Find the metric tensor and all the Christoffel symbols in this coordinate system. This observer will do a perfectly good job, provided that he always uses Christoffel symbols appropriately and sticks to events in his quadrant. In this sense, SR admits

accelerated observers. The right-hand quadrant in these coordinates is sometimes called Rindler space, and the boundary lines x = ±t bear some resemblance to the

black-hole horizons we will study later.

22 Show that if Uα α Vβ = Wβ , then Uα α Vβ = Wβ .