By Bernard Schutz
Readability, clarity and rigor mix within the moment variation of this widely-used textbook to supply step one into basic relativity for undergraduate scholars with a minimum heritage in arithmetic. themes inside relativity that fascinate astrophysical researchers and scholars alike are covered.
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Additional resources for A first course in general relativity
We know that lengths perpendicular to the x axis ¯ The most general linear transformation we need are the same when measured by O or O. 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. 5 (Fig. 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, t 22 Special relativity which gives the transformation ¯t = α(t − vx), x¯ = σ (x − vt).
Our interest in SR in this text is primarily because it is a simple special case of GR in which it is possible to develop the mathematics we shall later need. But SR is itself the underpinning of all the other fundamental theories of physics, such as electromagnetism and quantum theory, and as such it rewards much more study than we shall give it. See the classic discussions in Synge (1965), Schrödinger (1950), and Møller (1972), and more modern treatments in Rindler (1991), Schwarz and Schwarz (2004), and Woodhouse (2003).
On O’s (b) Calculate that ( s2 )AC = (1 − v 2 )( s2 )AB . 13 14 15 16 17 (c) Use (b) to show that O¯ regards O’s clocks to be running slowly, at just the ‘right’ rate. 5 × 10−8 s when the pion is at rest relative to the observer measuring its decay time. 6 × 10−7 s, as measured by an observer at rest. Suppose that the velocity v of O¯ relative to O is small, |v| 1. Show that the time dilation, Lorentz contraction, and velocity-addition formulae can be approximated by, respectively: (a) t ≈ (1 + 12 v 2 ) ¯t, (b) x ≈ (1 − 12 v 2 ) x¯ , 1 as well).