Ph 236a 5 December 2007Kip ThorneWEEK 10: Geometric Optics, GW Stress-Energy Tensor, andGeneral Relativity a s a N onlinear Field Theory in Flat SpacetimeRecommended Reading:1. Stress-energy tensor for gravitational wavesa. For a quick derivation, omitting lots of details, along the lines of what I did inclass, read Sec. 26.3.4 of Chapter 26 of Blandford and Thorne, version 0626.1.K.pdf;available a t http://www.pma.caltech.edu/Courses/ph136/yr2006/ .b. For a more detailed treatment, read Sections 35. 13–35.15 of MTW.c. For the energy and angular momentum carried away from a slow-motion source bygravita tional waves, computed as a surface integral of the GW energy flux T0rGW, readpage 992 of Sec. 36.7 of MTW.2. Geometric optics for wave p ropa ga tion in cu rved spacetime.a. I suggest beginning with Section 22 .5 of MTW. This is a leisurely treatment ofgeometric optics for quasi-monochromatic electromagnetic waves, giving a lot moredetail than I did in my lectures.b. Then read (but do not do) MTW Exercise 35.15, which is the analogous geometric-optics treatment of q uasi-monochromatic gravitational waves.c. Then read Sections 26. 3.5 and 26.3.6 of Blandford and Thorne, Chapter 26, version0626.1.K.pdf . This writes down the geometric optics solution in the time domain inthe form I gave in class, and verifies that it satisfies the Einstein field equations.d. Final ly read Section 5.D (pages 134–138 ) of an unpublished 198 9 book that I havewritten on gravitational waves (on our website), which derives the geometri c opticsequations in the manner that I did in class.3. General relat ivity as a nonlinear field theory i n flat sp acetime, and its use toderive the evolution laws for the mass, momentum, and angular momentumof a semi-isolated body.a. I suggest beginning by reading MTW chapter 19 (“Mass and Angular Momentum ofa Gravitating System”) and Secs. 20.1–20.5 (nonlinear field theory and derivation ofevolution laws).b. Having read MTW, I suggest reading pages 341–344 of Sec. 100 (“The energy-momentum pseudotensor”) of Landau and Lifschitz, The Classical Theory of Fields(revised second edition, 1962; in a later edition this might be section 10 1 and differ-ent page numbers). This sketches a derivation of the nonliner field theory eq uationsthat I presented in class and that are discussed in MTW — though in a somewhatdifferent notation than I used in class and than MTW.c. The nonlinear field theory equations and evolution equations are writ ten in a handoutthat I passed out in class, titled “Landau-Lifshitz Formulation of the Einstein FieldEquations”. That handout is on our website immediately after this Assignment 10 .1Supplementary Reading on GW Stress-Energy Tensor4. For computation of the energy, linear momentum, and angular momentum carried bygravita tional waves, as integrals over a sphere surrounding the source, wit h the answersexpressed a s sums over the source’s multip ole moments, see Thorne, Reviews of ModernPhysics , 52, 299 (1980), Sections 4B,C,D (pages 317 –319).5. For a beautiful computation of the energy flux in a gravitational wave based on ananalysis of the energy extracted from the wave by a dense collection of mechanicaloscillators, see Sec. 9.4 of Schutz, A First Course in General RelativitySupplementary Reading on Ge omet ric Optics:6. Sections 6.2 and 6.3 o f Version 0406.3.K.pdf of Chapter 6 of Blandford and Thorne,avai lable at http://www.pma. caltech.edu/Courses/ph136/y r2004/ . This treats geo-metric optics for most any type of wave (electromagnetic waves in dielectric media,sound waves in solids such as the interior of the earth, sound waves in fluids, ...).Supplementary Reading on Applications of the Nonlinear Field Theory For-mulation of General Relativity:7. MTW Sec. 20.6 on “Equations of Motion Derived from Field Equations”. This material[due to John Wheeler] describes the conceptual foundations for the derivation of the lawsof motion and precession of a self-graviting body (e.g., a black hole) moving throughcurved spacetime; and it uses those foundations to show that [if one i gnores couplingof the body’s multipol e moments to the external spacetime curvature], the body movesalong a geodesic.8. Kip S. T horne and James B. Hartle, “Laws of Moti on and Precession for Black Holesand Other Bodies”, Physical Review D, 31, 1 815 (1985). This paper combines theLandau-Lifshitz formalism (general relativity as a field theory in flat spacetime; Refs.3.b and 3.c a bove) with the conceptual foundations for laws of moti on given in MTW(Ref. 5 above), to derive the actual laws and equations of motion and precession forcompact bodies such as neutron stars and black holes.9. Kip S. Thorne and Yekta G¨ursel, “The Free Precession of Slowly Rotating NeutronStars: Rigid-Body Motion in General Relativi ty”, Monthly Notices of the Royal Astro-nomical Society, 205, 809 (1983). This paper uses the Landau-Lifshitz formalism toprove that the equations of free precession for a slowly and rigidly rotati ng body areindependent of the strength of the body’s internal gravity.Problems [Each problem is worth 10 points unless otherwise indicated. The maximumnumber of points you can get from this set is 50 points plus whatever you earn for problem9.]1. Stress-Energy Te nsor for a Plane Gravit ational Wave in terms of h+and h×a. In MTW it is shown that, in any Lorenz gauge (¯hαβ|β= 0) in which¯hαβhas beenmade traceless by a further gauge specialization (e.g., in TT gauge), the wave’sstress-energy tensor has the formTGWµν=132πh¯hαβ|µ¯hαβ|νi . (6)2a. Consider a plane gravitational wave propagating in the z direction as seen in somelocal Lorentz reference frame. Show that the wave’s stress-energy tensor in this casehas as its only nonzero componentsT00GW= T0zGW= Tz0GW= TzzGW=116πh˙h2++˙h2×i , (7)where the dots mean ∂/∂t.b. Show that, in the language of problem 6 of Assignment 8, these waves’ energy densityand flux have boost weight two (i.e. under a boost a long the propagation direction,their magnitudes are changed by the square of the doppler shift factor). Deduce thisfrom the vanishing boost weight of h+and h×(problem 7.b of Assignment 8 , with atypo corrected: the words “spin weight” should be “boost weight”). Also deduce itfrom the Lorentz transformatio n law for any stress-energy tensor.2. Cross S ectional Are a of a Bundle of R ays [15 Points]a. MTW Exercise 22.13.b. Show that the square root of