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A Student’s Manual for A First Course in General Relativity by Dr Robert B. Scott

By Dr Robert B. Scott

This complete scholar handbook has been designed to accompany the major textbook via Bernard Schutz, a primary direction usually Relativity, and makes use of distinct options, cross-referenced to numerous introductory and extra complicated textbooks, to let self-learners, undergraduates and postgraduates to grasp basic relativity via challenge fixing. the fitting accompaniment to Schutz's textbook, this guide publications the reader step by step via over 2 hundred workouts, with transparent easy-to-follow derivations. It presents exact suggestions to nearly half Schutz's workouts, and contains a hundred twenty five fresh supplementary difficulties that tackle the sophisticated issues of every bankruptcy. It contains a complete index and collects beneficial mathematical effects, corresponding to transformation matrices and Christoffel symbols for often studied spacetimes, in an appendix. Supported by means of a web desk categorising routines, a Maple worksheet and an teachers' guide, this article presents a useful source for all scholars and teachers utilizing Schutz's textbook.

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Example text

U)(1 − u cos(θ )) = √ ν 1 − u2 Schutz Eq. 23) Solution: In the reference frame O the photon has four-momentum p → (hν, hν cos(θ ), hν sin(θ ), 0). 24) This follows from the fact that the momentum of a photon must be a null vector. Transforming to the frame O moving at speed u along the x-axis, we apply the Lorentz transformation ⎛ ⎞ γ −uγ 0 0 ⎜−uγ 1 γ 0 0⎟ ⎟, , Λ(u) = ⎜ γ = √ ⎝ 0 ⎠ 0 1 0 1 − u2 0 0 0 1 to pα in eqn. 24) to obtain pα¯ : ⎛ γ hν − uγ hν cos(θ ) ⎜ −uγ hν + γ hν cos(θ ) p→⎜ hν sin(θ ) O ⎝ 0 ⎞ h¯ν ¯ ⎟ ⎜ p1 ⎟ ⎟ ⎟=⎜ .

Then W = 0 (= 0), the second equality added to remind you that in this case 0 is a four-vector. The magnitude of W would then be nil, so clearly not a four-vector. 8 Sometimes we write the total relativistic energy of a particle as the sum of its kinetic energy K and its rest mass energy: E = K + m. 46 Vector analysis in special relativity Use this to show that, in general, 3 K + 2Km = 2 (pi )2 . 9 A timelike or null vector is said to be future directed if it points into the future light cone. Show that the sum of a future directed timelike vector t α and a future directed null vector l α is timelike and future directed.

Pγ CM The non-zero pγ1 = h¯ ω violates conservation of momentum in the x-direction. It is possible to produce two γ -rays while conserving four-momentum. Suppose they travel in opposite directions with equal and opposite momentum in some frame of reference O. Then the final total four-momentum has zero spatial part. To satisfy spatial momentum conservation we only require that the positron and electron have equal and opposite momentum in O, so ve+ = − ve− with arbitrary ve+ , which can obviously be satisfied.

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