Ph 236a: General Relativity 3 October 2007Kip Thorne CaltechWEEK 1: FOUNDATIONS OF SPECIAL RELATIVITYRecommended Reading related to my first week lectures: This looks like a lotof reading. However, most students will already be familiar with a la rge portion of thismaterial. Fo r those students, I urge that you try to understand the material from thegeometric point of view that I am developing, which may be new to you. That v iewp ointis central to general relativity.1. Blandford and Thorne, Applications of Classical Physics, Sections 1.1–1.4 (pages 1–20) and Section 1.7.3 (pages 34–36) of Chapter 1, version 0601.2.K. Note: This bookis not yet published. It is on line athttp://www.pma.caltech.edu/Courses/ph136/yr2006/text.html2. MTW (Misner, Thorne and Wheeler, Gravitation): Chapter 1. A pdf file of t hischapter will be available sometime Thursday in the password-protected part of ourwebsite. Because of copyright law, please do not give this file to anyone except peoplein our class.3. MTW, Section 2.1–2.6. A pdf file of this chapter will be availa ble in the password-protected part of our website sometime T hursday. Because of copyright law, pleasedo not give this file to anyone except people in our class.Possible Supplementary Reading:4. Bernard F. Schutz (A First Course in General Relativity): Chapter 1.5. Robert M. Wald (General Relativity): Chapter 1.6. Sean Caroll (An Introduction to Spacetime and Geometry): Sections 1 .1–1.3.7. James B. Hartle (Gravity): Chapters 1–4. This is especially good and detailed if youhave little or no previous exposure to special relativity.ProblemsEach problem is worth 10 points. The maximum number of poi nts that will be given forthis problem set is 50 points. The number of points you will get wi ll be the maximumof 50, and the sum of the points you a ctual ly achieve on all the problems you at tempt.Please wo rk those problems that are most useful for you, given your previous preparationand experience. If you have never done much with spacetime diagra ms, please give specialattent ion to problems 4 and 5. See the document Course procedures on our class websitefor a discussion of homework and grading in this course.1. Geometrized Units.“Geometrized units” are units in which t he speed of lightc ≡ 2.99792458000000000000000 000000000000000000000000cm/secis set to unity, and thereby we regard one second as equal to 2.99792458 centimeters.(Later we will also set Newton’s gravitation constant to unity.)1a. How can experimental physicists have actually measured the speed of light to theenormous accuracy indicated by the above string of digits, or did they? Whatare the next 17 digits in the above value of c? How do you know?b. What is one year, ex pressed in centimeters?c. What is your age, in centimeters?d. What is your height, in seconds?e. The following equations are written in geometrized units. Restore the factors ofc so they are in cgs units. [This can b e done by simply inserting what ever factorsof c are required to make the equat ions dimensionally consistent, i n cgs units.](Planck length)=√G¯h, where ¯h is Planck’s constant and G is Newton’s gravita-tion constant.(Energy density in a n electromagnetic wave)= (E2+ B2)/(8π).(Lorentz force on a proton)= e(E + v × B).2. Proof of Invariance of a Timelike Interval.Note: The solution is sketched briefly in Sec. 1.2.2 and Exercise 1.2 of Blandford andThorne, and a very leisurely version of the solutio n is given in Secs. 3.6 and 3.7 ofTaylor and Wheeler, Spacetime Physics (1992).Consider two events P and Q in spacetime, with a timelike separation vector~A.Examine these events in two different reference frames S and¯S which move with speedv relative to each other. Choose the origins of the two frames’ spacetime coordinatesto coincide, and to be at the event P, and orient the spati al axes o f the two frames sotheir relative motion is in the x direction and Q lies in t he x −y plane. The followingdiagram depicts this in a spacetime diagram drawn from the viewpoint of frame S.xtyQPa. Convi nce yourself that, wherever may be t he events P and Q, the origins andaxes of the two frames can be adjusted as described above.The foll owing experiment is a foundation for proving the invariance of the intervalbetween P and Q. The experiment is sketched below in two purely spatial diagrams(time not shown) from the frames’ two different viewp oints.2xyQPααSxyQPααS’A photon (light pulse) is emitted from P, and travels alo ng a straight line in the x-yplane until it hits a mirror that reflects it; the photon then travels again in a straightline in the x-y plane, arriving at the event Q. The position of the reflecting mirror isadjusted so the photon reaches the spatial location of Q precisely at the time of Q.The state of motion of the mirror is not important; the key thing is that, as seen inframe S, the photon’s direction of motion makes an angle α with the x axis that isthe same before and after reflection (“angle of incidence equals angle of reflection”),as shown in the diagram. In the following do not use the Lorentz tra nsformatio nequations. Assume that t hey have not yet been derived; we are working our waytoward them, and our first step is to derive the invariance of the interval from thePrinciple of Relativi ty.b. Use the Principle of Relati vity to show that the heights of the reflection pointare the same in the two frames, yrefl= ¯yrefl, and that the angles o f incidence a ndreflection a re equal in frame¯S, ¯α = ¯α′, just as t hey a re equal in frame S.c. Use the Principle of Relativity, the constancy of the speed of light, and simplegeometric considerations to show that the interval between P and Q is the same,as computed in the two reference frames:−(∆t)2+ (∆x)2+ (∆y)2+ (∆z)2= −(∆¯t)2+ (∆¯x)2+ (∆¯y)2+ (∆¯z)2.3. L orentz TransformationFo r the two frames of Exercise 2, assume that the relationship between the coordinatesis linear.a. Use arguments from the Principle of Relativity and symmetry arguments to showthat for any event (e.g. Q), y = ¯y and similarly z = ¯z; and thus it is only x andt that get mixed up among each other.b. Write the transformation for x and t asx = A¯x + B¯t , t = C¯t + D¯x .Use the invariance of the interval, and the fact that frame¯S moves at speed vin the x direction as seen from frame S, to show that A, B, C, and D have thestandard values for a Lorentz boost in the x direction, A = C =