Ph 236a: General Relativity 17 October 2007WEEK 3: CONSERVATION LAWS, STRESS-ENERGY TENSOR,ACCELERATED OBSERVERS, CONNECTION AND GEODESICSRevision 1 - With Supplementary Reading addedRecommended Reading:1. MTW: Chapter 5; Also Sec. 4.7.2. Blandford and Thorne chap. 0601.2.K: Secs. 1.11 and 1.12. Be sure to use version0601.2.K of chap ter 1 of Blandford an d Thorne athttp://www.pma.caltech.edu/Courses/ph136/yr20 06/text.html3. MTW Chapter 6 (especially Sec. 6.6)4. MTW Sections 8.1–8.55. MTW Section 9.6 [ on commutators]6. Blandford and Thorne Sec. 23.3 (of version 0623.1.K of Chap. 23).Possible Suppl ementary Reading:Conservation Laws and Stress-Energy Tensor7. Schutz (A First Course in General Relativity): Chapter 4. This discusses the stress-energy tensor from a more elementary viewpoint than MTW or Blandford and T horne,with the central focus being on a perfect fluid and on analyses in an inertial referenceframe rather than frame-independent.8. Wald (General Relativity): Section 4.2. This discusses the stress-energy tensor andthe flow of 4-momentum through spacetime in much the same language as mine.9. Wald: Appendix B. This discusses integration over spacetime (and other manifolds)and over 3-surfaces in spacetime (and submanifolds embedded in other manifolds),using the language of exterior calculus.10. Ca rroll (Spacetime and Geometry): Sections 1.9 and 1.10. This discusses the stress-energy tensor, including its connection to field theory (not treated in our course).11. Hartle (Gravity: Sections 22.1, 22.2. This is a clear, elementary discussion of thestress-energy tensor and conservation of 4 -momentum.Accelerated Observers[The flat-spacetime metric in the coordinates of a family of uniformly acceleratedobservers i s cal led the Rindler Metric . I should have told you that in class.]12. Wald, pp. 149–152. This is a moderately sophisticated treatment of the Rindler metric13. Ca rroll, pp. 403–405. So is this.Connection Coefficients and geodesics14. Schutz (A First Course in General Relativity): Chapter 5. This covers essentiallythe same material as the recommended sections of MTW Chapter 8, but i n a moreelementary and leisurely way.15. MTW, chapters10, This covers parallel transport and connection coefficients in greatdetail, in manifolds where you might not have a metric.16. Wald (General Relativity): Section 3.1. This covers connection coefficients in a quick,mathematically sophisticated way.117. Ca rroll: Sections 3.1–3.5. This is a very nice treatment of connection coefficients andgeodesics.18. Hartle: Sections 8.1–8.3. So is this.Problems. Each problem is worth 10 points. The to tal problem set is worth a maximumof 50 points. Your solutions must be turned in at the beginning of class next Wednesday,25 October.1. Global conservation of 4-momentu m in a Lorentz frame. Exercise 1.26 ofBlandford and Thorne.2. Rest-mass-flux 4-vector, Lorentz contraction of rest-mass density, and rest-mass conservation for a fluid. Exercise 1.24 o f Blandford and Thorne.3. Stress-energy tensor and energy-momentum conservation for a perfectfluid. Exercise 1.27 of Blandford and Thorne: Parts (b) and (c). Do not do part(a) unless you feel a l ittle shaky a bout the solution to this that I gave in my Mondaylecture. Note that the notation here differs from my lecture: Pαβis the Blandford-Thorne not ation for the tensor that projects orthogonal to an observer’s world line(the observer’s spatial metric); in my lecture I used the notation γαβ.4. Ine rtial mass per unit volume. Exercise 5.4 of MTW.5. Entropy and the Se c ond Law of ThermodynamicsIt turns out, as we shall see later in this course, that the entropy of a system that hasmany microscopic degrees of freedome (e.g. a gas) can be described macroscopicallyby an entropy density-flux 4-vector ~s.a. Give a definition of this ~s anal ogous to our definition of the charge-current 4-vector~J and the stress-energy tensor T.b. Relying on that definition, formulate the second law of thermodynamics in ageometric, frame-independent manner, and discuss the justification for your for-mulation.6. Stress-Energy Tensor for the Electromagnetic Fi eld. Exercise 5 .1 of MTW.Note: I encourage you to read carefully the discussion of energy-momentum conser-vation for the electromagnetic field interacting with a block of electrically chargedrubber in Sec. 5.1 0 of MTW.7. Charge-Current 4-Ve c tor for a Point ParticleI claim that the charge-current 4-vector for a classical point particle with charge q,proper time τ, world line P(τ), and 4-velocity ~u(τ) = dP/dτ is~J(P′) = qZ~u(τ)δ4[P(τ), P′]dτ , (2)where δ4[P, P′] is the 4-dimensional Dirac delta function whose spacetime integralover P′,Rδ4[P, P′]d4Ω′, is unity if the region of integration includes the point P andzero otherwise.2Verify that expression (2) is, indeed, the particle’s charge-current 4-vector by showingthat it satisfies the defining property of~J:RSJαd3Σα= q if the particle passesthrough the 3-volume S in the positive directio n, andRSJαd3Σα= 0 if the particledoes not pass through S. Note: In your calculation you may wish to introduce a localLorentz reference frame i n which, at the event where the particle passes t hrough S,S lies in a coordinate pla ne—the plane t = 0 if S is spacelike; the plane z = 0 if S istimelike.8. Stress-Energy Tensor for a point particle.a. Consider a point particle with rest mass m, proper time τ, world line P(τ), and4-velocity ~u = dP/dτ. Write down a frame-independent equation for the stress-energy tensor of this particle T(P) as an integral along the particle’ s world lineanalogous to Eq. (2) for the charge-current 4-vector of a point particle.b. Verify that this is i ndeed the particle’s stress-energy tensor by performing a sur-face integral analogous to that used for the charge-current 4-vector in the previousproblem.c. Your expression in part (a) only works for a particle with finite rest mass.Construct an alternative expression for the stress-energy tensor involving an“affine parameter” along the particle’s world li ne P (ζ), such that the particle’s4-momentum is ~p = dP/dζ. More specifically, write T(P) as an integral along theworld line that involves ~p(ζ), P(ζ), and not in any way m(ζ). Your expressionshould work for particles with finite rest mass and also for particles with zero restmass.9. 4-Accelera tion in uniformly accelerated reference frameExercise 6.7 of MTW. It is the ma gnitude of