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Phase-Plane Analysis of Perihelion Precession and Schwarzschild Orbital Dynamics Bruce Dean

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Phase-Plane Analysis of Perihelion Precession and Schwarzschild Orbital Dynamics Bruce Dean Faculty Scholarship 1999 Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics Bruce Dean Follow this and additional works at: https://researchrepository.wvu.edu/faculty_publications Digital Commons Citation Dean, Bruce, "Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics" (1999). Faculty Scholarship. 284. https://researchrepository.wvu.edu/faculty_publications/284 This Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarship by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics Bruce Deana) Physics Department, West Virginia University, Morgantown, West Virginia 26506-6315 ~Received 22 December 1997; accepted 11 June 1998! A calculation of perihelion precession is presented that utilizes a phase-plane analysis of the general relativistic equations of motion. The equations of motion are reviewed in addition to the phase-plane analysis required for the calculation. ‘‘Exact’’ phase planes for orbital dynamics in the Schwarzschild geometry are discussed, and bifurcations are identified as a dimensionless parameter involving the angular momentum is varied. © 1999 American Association of Physics Teachers. I. INTRODUCTION frames, the phase diagrams in each case are not identical. This is due to the existence of an additional phase-plane The perihelion precession of planetary orbits has provided fixed point that appears in the coordinate reference frame at one of the earliest experimental tests of Einstein’s general the event horizon. This fixed point is obviously coordinate theory of relativity. In the standard textbook presentation of 1 dependent, but must exist to explain the apparent ‘‘slowing this calculation there are essentially two approaches taken to down’’ of objects ~and redshift of signals! approaching the calculate its value from the nonlinear equations of motion: horizon as seen by an observer in the coordinate reference ~a! approximate an elliptic integral, frame. ~b! find a perturbative solution to the general relativistic For comparison with the relativistic case, the correspond- equations. ing Newtonian phase-plane results are discussed in an Ap- pendix. Not only does this analysis complement the dynam- Although ~a! and ~b! are the most common methods appear- ics considered in Sec. VI, but it is shown that an analysis of ing in the literature, other approximation methods do exist. Newtonian orbits using time as an independent variable is Wald,1 for instance, considers small oscillations about an 1 just as instructive and no more complicated in principle than elliptical orbit; Misner, Thorne, and Wheeler ~MTW! con- using the equatorial angle as the independent variable ~how- sider nearly circular orbits and then later use the PPN ~‘‘Pa- ever, the opposite is true when using the standard methods of rametrized Post-Newtonian’’! formalism. The purpose of this analysis, e.g., Ref. 4!. Furthermore, by considering t as the paper is to illustrate how the perihelion calculation may be independent variable rather than w, an additional fixed point performed and to present an analysis of the Schwarzschild appears at infinity. But more importantly, the emphasis of the orbital dynamics based on a standard technique of nonlinear analysis is shifted from trying to find an explicit closed form analysis: the phase-plane approach ~see also Refs. 2 and 3!. solution ~i.e., the standard approach! to a more intuitive and Not only is the calculation simpler to perform in the modern qualitative description based on the energy method. setting of phase-plane analysis, but there is more physics to Finally, the phase-plane analysis is applied to the kinemat- be learned with less algebra compared with the standard pro- ics of light rays in the Schwarzschild black hole spacetime. cedures. The standard results are discussed and then compared with The contents of this paper are organized as follows. In the timelike phase-plane results. The added significance of Sec. II the general relativistic equations of motion are de- the photon orbits ~in the phase-plane context! is that the rived. The goal here is to not only make the presentation as equilibrium points of the differential equations exhibit a tran- self-contained as possible, but to ‘‘tailor’’ the derivation to- scritical bifurcation ~i.e., a change in stability! at these pa- ward a discussion emphasizing the phase-plane analysis, and rameter values. for easy comparison with the corresponding Newtonian cal- culation. In Sec. III, the phase-plane analysis is developed and in Sec. IV applied to obtain the well-known value of II. GENERAL RELATIVISTIC ORBITS perihelion precession. In Sec. V, a discussion of the Schwarzschild orbital dynamics is given based upon an ‘‘ex- The general relativistic equations of motion for a point act’’ general relativistic phase plane. The standard results are mass with rest mass, m0 , orbiting a mass, M ~assuming for discussed, but also an alternative viewpoint for analyzing the simplicity that m0!M!, originate from the Schwarzschild orbital dynamics is presented based upon the separatrix line element: structure of the phase plane. In this approach, the critical 2 2 2 21 2 2 2 relationship that holds between energy and momentum at the ds 5c Ldt 2L dr 2r dV , unstable orbital radius ~i.e., the separatrix! summarizes the L512r /r, dV25du21sin2 udw2. ~1! range of physically possible orbits, and demonstrates a S saddle-center bifurcation as a dimensionless parameter in- Equation ~1! is expressed using spherical coordinates and rS volving the angular momentum is varied. is the Schwarzschild radius ~G is Newton’s gravitational In Sec. VI, a phase-plane analysis of the dynamical invari- constant and c is the speed of light!: ance between the coordinate and proper time reference r 52MG/c2. ~2! frames is given. Although the dynamical structure ~i.e., the S effective potential! is invariant between the two reference The Lagrangian is a constant of the motion: 78 Am. J. Phys. 67 ~1!, January 1999 © 1999 American Association of Physics Teachers 78 1 2 1 2 L5 2m0~ds/dt! 5 2m0c ; t[proper time, ~3! x85 f ~x,y!5y, 3 2 ~11! and if the orbit is confined to the equatorial plane, i.e., u y85g~x,y!5 2x 2x1s. 5p/2, L takes the explicit form (˙t5dt/dt, etc.!: To find the fixed points of ~11!~i.e., equilibrium points of the 1 2 1 2 ˙2 1 21 2 1 2 2 L5 2m0c 5 2m0c Lt 2 2m0L r˙ 2 2m0r w˙ . ~4! solution! we solve simultaneously: x85y850, for x and y. From the Euler–Lagrange equations there are two additional Therefore, the fixed points of ~11! are given by constants of motion @~4! is trivial; ~5! and ~6! are first inte- 11A126s 12A126s grals#: xW*5 ,0 , xW*5 ,0 . ~12! 1 S 3 D 2 S 3 D ]L/]t50 ]L/]˙t5E5m c2L˙t, ~5! ) 0 Alternatively, by expressing y in terms of x using ~8!: 2 ]L/]w50 ]L/]w˙ 5J5m0r w˙ . ~6! ) x85y56@2sEˆ 22~2s1x2!~12x!#1/250, ~13! Physically, E is the energy required for an observer at infin- ˆ 2 ity to place m0 in orbit about M @it is left as an exercise to and then solving simultaneously: x85y850, for E and x check this physical interpretation by considering radial mo- rather than x and y, the corresponding energies at each fixed tion in ~4! and then combining with ~5! in the Newtonian point are expressed solely in terms of s: limit#. J is the angular momentum of the system and since 2s@124s2~126s!1/2# this is constant, there will be no precession of the equatorial ˆ 2 E1215 1/2 3 , plane. @~126s! 21# Continuing with the equations of motion, using ~5! and ~6! ~14! 2s@2114s2~126s!1/2# to eliminate ˙t and ˙ from 4 and then rearranging algebra- ˆ 2 w ~ ! E2215 1/2 3 , ically gives the following result: @~126s! 21# 2 2 2 ˆ 2 2 2 2 2 respectively. Therefore, solving simultaneously for Eˆ 2 and x r˙ /c 5~dr/ds! 5E 2~11J /m0c r !L, ~7! gives additional information on the dynamics. Furthermore, ˆ 2 where E[E/m0c defines the total energy per unit rest en- the phase-plane equations analogous to ~11! that result from ergy. Noting the functional dependence of r on the equatorial the proper and coordinate time analysis considered in Sec. angle @i.e., r5r(w) r˙5(dr/dw)w˙ # allows ~7! to be further VI ~and also in the Newtonian case! give nonphysical roots expressed in terms of) the constant J. Furthermore, the degree when solving only for x and y ~i.e., they do not correspond to of this equation ~in r! is reduced by making the usual change the effective potential extrema!. However, these additional ˆ 2 of variable to u5rS /r. Simplifying algebraically gives the roots are eliminated by solving for E and x as illustrated following result: above and as discussed in Sec. VI. To give a general classification of the fixed points ~12! a du/d 252 Eˆ 22 2 1u2 , 8 ~ w! s ~ s !L ~ ! linear stability analysis must be performed. Essentially, this where s defines the dimensionless parameter: amounts to series expanding ~11! about an arbitrary fixed 1 1 point in the small parameters: dx5x2x* and dy5y2y*. s5 ~m cr /J!252~GMm /cJ!25 ~r /J!2. ~9! 2 0 s 0 2 S Dropping second-order terms, the resulting first-order linear Equation ~9! is expressed on the far right-hand side in the equations are expressed in matrix form: ‘‘geometrized’’ system of units ~i.e., G5c51, rS52M; 1 dx8 ]x f ]y f dx see, e.g., Shutz, p.
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