Week 6: The Schwarzschild SolutionYanbei Chen(Dated: November 9, 2009; Due on Tuesday, November 17, 2009)I. READINGThis week, we discussed the Schwarzschild Geometry, which we now understand is the exterior solution of a sphericalstar; the Birkhoff Theorem states tha t a spherica lly symmetric va c uum space-time must be a Schwarzschild space-time. We discussed the structure of a spherical star, and started discussing the more interesting fea tur e s of theSchwarzschild space-time near the r = 2M sur face. Here are possible reading materials.• My lectures were roughly following Chapter 5 of Carroll, except Section 5.7, which I did not have time to discuss.• Secs. 24.1 – 24.3 of Blandford and Thorne covers similar mater ial, but in a somewhat differe nt style.• Sec. 9.5 of Shapiro and Te ukolsky, Black Holes, White Dwarfs, and Neutron Stars for a discussion on maximummass of stars.II. PROBLEMSEach problem is worth 10 points, unless otherwise noted. The maximum number of points you can get for thisassignment is 50, although you could choose to do problems that worth more than 50 points. You should choose todo those problems that you will learn the most from.1. Schwarzschild Geometry in Isotropic Coordinates. Show, either by hand or using a computer, that ifone changes radial coordinates from r to r′via the transformationr = r′(1 + M/2r′)2, (1)the Schwarzschild metric takes on the new formds2= −1 − M/2r′1 + M/2r′2dt2+1 +M2r′4[dr′2+ r′2(dθ2+ sin2θdφ2)] . (2)This new (t, r′, θ, φ) coordinate system is called “isotropic” because in it the spatial part of the metric is writtenas a function of r′times the flat, 3-dimensional Euclidean metric (in spherical coordinates), and this Euclideanmetric does not pick out any direction as special.2. Embedding Diagram for a “Wormhole”.Embedding Diagrams are s ometimes useful in depicting spa c e times, usually a particular slicing of some space-time is is ometrically embedded into a 3-Euclidean space. [See Blandford and Thorne, 24.3.5 for detailed discus-sions.](a) Show that in the isotropic coordinates of Eq. (2) the radial coordinate r′is e verywhere spatial—i.e.,~∂rpoints in a s pacelike direction. This contrasts with Schwarzschild’s original radial coordinate r, which is aspatial coordinate at r > 2M but is timelike at r < 2M.(b) Consider an equatorial two-dimensional surface t =c onstant, θ = π/2 =constant. Construct an embeddingdiagram for this surface. To do this construction, use the isotropic coordinates of Eq. (2); i.e., in a flatEuclidean space with coordinates (¯r, ¯z,¯φ) and ds2= d¯r2+ d¯z2+ ¯r2d¯φ2, construct a 2-surface labeled bysurface coordinates (r′, φ) with the same 2-metric a s you read off of Schwarzschild, Eq. (2). Your embeddingdiagram sho uld turn out to be a “wormhole” (also called an “Einstein-Rosen bridge”) connecting twoasymptotically flat spaces. [We will see how this “wormhole” fits into the whole picture of Schwarzschildgeometry during the next lecture.](c) Derive an eq uation ¯z = ¯z(¯r) for the shape of this surface.23. Periastron Precession of Nearly Circular Orbits. [15 Points] Even though Carroll has derived periastronprecession carefully in his textbook, here let us explore a simpler approach, that works quickly for cases withvery weak gravity and very nearly circular orbits.(a) Write down the radial equation of motion, draw a picture of the e ffective potential, and explain why theminimum of that p otential corresponds to the circular orbit.(b) Explain the connection of V′′(r) and the radial frequency, Ωr, namely 2π/Tr, where Tris the period (interms of proper time) of the radial motion. Show that this frequency is independent of the radial oscilla tionamplitude, when this amplitude is small. We know tha t for very large radius,Ωr=rMr3(3)Derive the leading GR corr ection to this formula (the correction is in terms of M/r).(c) Show that the angular frequency Ωϕ, to the limit of small oscillations, deviates from the circular-orbit valueonly to second order in amplitude of oscillation. In the limit of very large radius,Ωϕ= Ωr=rMr3(4)What is the leading GR correction to Ωϕ?(d) Periastron shift ∆ϕ is defined as the difference in ϕ of neighboring geodesics, subtra c ted by 2π. Deducethe relation between ∆ϕ and Ωrand Ωϕ, and express it for nearly circular orbits in terms of M/r.4. Kepler’s Law in Schwarzschild. [5 points] Show that for circular orbits in Schwarzschild, we havedϕdt=rMr3, (5)identical to Kepler’s law. (Note that dϕ/dt is different from the Ωφabove, which is written in terms of propertime, dϕ/dτ.)5. Bound orbits in Schwarzschild, in terms of Elliptic Functions. [15 Points]The effective potential in Schwarzschild is a third-order polynomial in 1/r. Integrals likeZ1pP (z)dz (6)are representable in terms of Elliptic Functions. It is therefore possible to obtain analytical representations ofgeodesics in Schwarzschild. Her e we define u ≡ M/r.(a) Consider orbits of a massive particle, in the Equa torial plane of a Schwarzschild black hole. The shapes ofsuch orbits, r(φ), are governed bydudφ=E2− (1 − 2u)(1 + (L/M)2u2)(L/M )2(7)This equation can be integrated in terms of Elliptic functions.(b) Show, in particula r, that for bound orbits that travel in and out b etween a maximum radius and minimumradius. The orbit is given byu = u1+ (u2− u1)sn2 φr(u3− u1)2u2− u1u3− u1!, (8)where u = M/r; the ujare related by u1+ u2+ u3= 1/2, and u1< u2< u3; and sn(θ|m) is the JacobiElliptic function with modulus m (sometimes denoted m = k2). Show, further, that the maximum andminimum radii are at r1= M/u1and r2= M/u2.3(c) Use Mathematica or Maple or some o ther graphics tool to plot these orbits in spherical polar c oordinates(r, φ) for various values of the inner and outer radii. As an ex ample, for r1= 100M and r2= 4.166 M ,your plot should look like this:100100100 10015151515(d) Consider a sequence of orbits with the form shown above, in which the conserved angular momentum perunit mass˜L is held fixed, w hile the conserved energy per unit mass˜E is increased slightly from one orbitto the next. Describe qualitatively how the orbits will change from the above. Justify your description interms of the properties of the effective potential˜V (r) for the particle’s radial motion.6. Observers and the Schwarzschild Horizon. [25 Points](a) So lve Problem 3 in C