Faculty Scholarship
1999 Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics Bruce Dean
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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 1998͒ 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 signals͒ approaching the calculate its value from the nonlinear equations of motion: horizon as seen by an observer in the coordinate reference ͑a͒ approximate an elliptic integral, frame. ͑b͒ 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 ͑a͒ and ͑b͒ 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 ͑MTW͒ 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. 4͒. Furthermore, by considering t as the paper is to illustrate how the perihelion calculation may be independent variable rather than , 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 3͒. solution ͑i.e., the standard approach͒ 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 context͒ 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 stability͒ 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 m0ӶM͒, 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 Ϫ1 2 2 2 relationship that holds between energy and momentum at the ds ϭc ⌳dt Ϫ⌳ dr Ϫr d⍀ , unstable orbital radius ͑i.e., the separatrix͒ summarizes the ⌳ϭ1Ϫr /r, d⍀2ϭd2ϩsin2 d2. ͑1͒ range of physically possible orbits, and demonstrates a S saddle-center bifurcation as a dimensionless parameter in- Equation ͑1͒ 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 light͒: ance between the coordinate and proper time reference r ϭ2MG/c2. ͑2͒ frames is given. Although the dynamical structure ͑i.e., the S effective potential͒ is invariant between the two reference The Lagrangian is a constant of the motion:
78 Am. J. Phys. 67 ͑1͒, January 1999 © 1999 American Association of Physics Teachers 78 1 2 1 2 Lϭ 2m0͑ds/d͒ ϭ 2m0c ; ϵproper time, ͑3͒ xЈϭ f ͑x,y͒ϭy, 3 2 ͑11͒ and if the orbit is confined to the equatorial plane, i.e., yЈϭg͑x,y͒ϭ 2x Ϫxϩ. ϭ/2, L takes the explicit form (˙tϭdt/d, etc.͒: To find the fixed points of ͑11͒͑i.e., equilibrium points of the 1 2 1 2 ˙2 1 Ϫ1 2 1 2 2 Lϭ 2m0c ϭ 2m0c ⌳t Ϫ 2m0⌳ r˙ Ϫ 2m0r ˙ . ͑4͒ solution͒ we solve simultaneously: xЈϭyЈϭ0, for x and y. From the Euler–Lagrange equations there are two additional Therefore, the fixed points of ͑11͒ are given by constants of motion ͓͑4͒ is trivial; ͑5͒ and ͑6͒ are first inte- 1ϩͱ1Ϫ6 1Ϫͱ1Ϫ6 grals͔: xជ*ϭ ,0 , xជ*ϭ ,0 . ͑12͒ 1 ͩ 3 ͪ 2 ͩ 3 ͪ tϭEϭm c2⌳˙t, ͑5͒˙ץ/Lץ tϭ0ץ/Lץ ⇒ 0 Alternatively, by expressing y in terms of x using ͑8͒: 2 ˙ ϭJϭm0r ˙ . ͑6͒ץ/Lץ ϭ0ץ/Lץ ⇒ xЈϭyϭϮ͓2Eˆ 2Ϫ͑2ϩx2͒͑1Ϫx͔͒1/2ϭ0, ͑13͒ 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: xЈϭyЈϭ0, 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 ͑4͒ and then combining with ͑5͒ in the Newtonian point are expressed solely in terms of : limit͔. J is the angular momentum of the system and since 2͓1Ϫ4Ϫ͑1Ϫ6͒1/2͔ this is constant, there will be no precession of the equatorial ˆ 2 E1Ϫ1ϭ 1/2 3 , plane. ͓͑1Ϫ6͒ Ϫ1͔ Continuing with the equations of motion, using ͑5͒ and ͑6͒ ͑14͒ 2͓Ϫ1ϩ4Ϫ͑1Ϫ6͒1/2͔ to eliminate ˙t and ˙ from 4 and then rearranging algebra- ˆ 2 ͑ ͒ E2Ϫ1ϭ 1/2 3 , ically gives the following result: ͓͑1Ϫ6͒ Ϫ1͔ 2 2 2 ˆ 2 2 2 2 2 respectively. Therefore, solving simultaneously for Eˆ 2 and x r˙ /c ϭ͑dr/ds͒ ϭE Ϫ͑1ϩJ /m0c r ͒⌳, ͑7͒ gives additional information on the dynamics. Furthermore, ˆ 2 where EϵE/m0c defines the total energy per unit rest en- the phase-plane equations analogous to ͑11͒ 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., rϭr() r˙ϭ(dr/d)˙ ͔ allows ͑7͒ to be further VI ͑and also in the Newtonian case͒ 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 r͒ is reduced by making the usual change the effective potential extrema͒. However, these additional ˆ 2 of variable to uϭrS /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 ͑12͒ a du/d 2ϭ2 Eˆ 2Ϫ 2 ϩu2 , 8 ͑ ͒ ͑ ͒⌳ ͑ ͒ linear stability analysis must be performed. Essentially, this where defines the dimensionless parameter: amounts to series expanding ͑11͒ about an arbitrary fixed 1 1 point in the small parameters: ␦xϭxϪx* and ␦yϭyϪy*. ϭ ͑m cr /J͒2ϭ2͑GMm /cJ͒2ϭ ͑r /J͒2. ͑9͒ 2 0 s 0 2 S Dropping second-order terms, the resulting first-order linear Equation ͑9͒ is expressed on the far right-hand side in the equations are expressed in matrix form: ‘‘geometrized’’ system of units ͑i.e., Gϭcϭ1, rSϭ2M; y f ␦xץ x fץ xЈ␦ 1 see, e.g., Shutz, p. 198͒ with m0 taken as unity for later Ϸ ͪ gͪͯ ͩ ␦y ץ g ץ ͩ yЈͪ␦ ͩ comparison with the standard results. x y xជϭxជ* Differentiating ͑8͒ with respect to then gives the familiar 0 1 ␦x second-order equation in dimensionless form: ϭ ϵA ␦xជ. ͑15͒ ͩ 3xϪ1 0ͪͯ ͩ ␦y ͪ ͉xជϭxជ* 2 2 3 2 xជϭxជ* d u/d ϩuϭϩ 2u . ͑10͒ The general solution of ͑15͒ is an exponential whose stability As previously discussed, there are two common procedures at each fixed point is analyzed by classifying the eigenvalues for calculating the value of perihelion precession. The first of the matrix A. Solving the eigenvalue problem, we find procedure ͓procedure ͑A͔͒ is to approximate an elliptic inte- roots to gral obtained by separating variables in ͑8͒͑see, for example, Ref. 5͒. In procedure ͑B͒, ͑10͒ is solved perturbatively or by ͉AϪI͉ϭ0, ͑16͒ using other approximation techniques. However, at this point we shall deviate from these approaches and consider ͑10͒ but since A is 2ϫ2, the characteristic polynomial may be from the viewpoint of phase-plane analysis. Additional dis- expressed: cussion and comparison with the Newtonian case is given in 2Ϫϩ⌬ϭ0, ͑17͒ the Appendix. where ϵtrace A, and ⌬ϵdeterminant A. The eigenvalues III. PHASE-PLANE ANALYSIS are roots of ͑17͒: ϭ 1 Ϯ 2Ϫ4⌬ , ͑18͒ To begin the phase-plane analysis of ͑10͒͑see, e.g., 2͑ ͱ ͒ Strogatz6 or Tabor7 for an introduction͒, let us convert this and accordingly, the exponential solutions of ͑15͒ are classi- second-order, nonlinear, inhomogeneous, differential equa- fied by various regions of Fig. 1 ͓dots mark the location of tion to two first-order equations by defining new variables: the fixed points given in ͑12͔͒. xϭu and yϭdu/d. In 2-d form, ͑10͒ is equivalent to Briefly, region I corresponds to a ‘‘saddle-node’’ fixed ͑primes will denote derivatives with respect to ͒ point, whose stable and unstable manifolds ͓corresponding to
79 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 79 Fig. 3. Schematic of perihelion precession. Fig. 1. Eigenvalue classification. saddle node that appears is not predicted by Newtonian theory, but is due to an unstable orbital radius originating positive and negative ͑real͒ eigenvalues, respectively͔ are Ϫ3 given by the eigenvectors of ͑15͒. Region II represents an from the r term of the effective potential derived from ͑8͒ ‘‘unstable node,’’ i.e., two positive real eigenvalues with 2 ͓see also ͑33͒ and the Appendix͔: Ϫ4⌬Ͼ0; region III gives solutions having one ͑positive͒ ˆ 2 2 Veffϭ͑1ϩx /2͒͑1Ϫx͒. ͑20͒ real and one complex eigenvalue ͑‘‘unstable spirals’’͒, while regions IIЈ and IIIЈ are the complimentary stable solutions of As a result of this instability, there are orbital effects not present in the Newtonian theory which have been summa- regions II and III, respectively. The ‘‘boundary’’ cases are 1 given by 2ϭ4⌬͑degenerate nodes and lines of fixed points͒ rized in the literature ͑see, e.g., MTW, p. 637͒. But what is nice in the phase-plane approach is that this result comes out and ϭ0, ⌬Ͼ0 are ‘‘centers’’ giving periodic orbits in the very quickly in the analysis as a secondary fixed point ͑see phase plane. A complete discussion of these cases will not be Sec. V͒. given here, we simply use these results. Refer, for instance, to Strogatz6 for additional analysis and details. Evaluating the matrix A in ͑15͒ at each fixed point of ͑12͒, IV. PERIHELION PRECESSION the following classifications are obtained: Physically, perihelion precession means that the distance of closest approach ͑i.e., the perihelion; -helion refers to the sun͒ between two orbiting bodies begins to revolve about the orbit in the same sense as the orbiting body ͑see Fig. 3͒. Therefore, planetary orbits do not close but are separated by ͑19͒ a small correction, ⌬, after completing a single orbit. For the planet Mercury this is observed to be approximately 574 arcsec per century with 532Љ accounted for by a Newtonian analysis of external planetary perturbations. However, there are 43Љ not explained from a Newtonian analysis, but pre- dicted to within experimental error by Einstein’s theory. To perform the perihelion calculation simply solve ͑15͒ corresponding to a ‘‘saddle’’ and ‘‘center-node’’ fixed point, about xជ* . This first-order system is rewritten here as respectively ͑see Fig. 1 for the placement of these points͒.As 2 previously discussed, the linear stability analysis gives an ␦xЈϭ␦y, ␦yЈϭϪ2␦x, ϭ͑1Ϫ6͒1/4. ͑21͒ exponential solution about each fixed point with the phase- plane trajectories shown in Fig. 2 ͓the directions follow from The solutions are centers corresponding to precessing ellip- ͑11͒ and are indicated by arrows͔. tical orbits ͑see Fig. 2͒: Physically, trajectories about the center node correspond ␦x͑͒ϭA cos ϩB sin , to elliptical orbits and we will use this fact in a moment to ͑22͒ obtain the value for perihelion precession. However, the ␦y͑͒ϭϪA sin ϩB cos , with A and B arbitrary constants. Choosing initial conditions at the position of perihelion:
␦x͑0͒ϭu0 , ␦y͑0͒ϭdu͑0͒/dϭ0, ͑23͒ ͑22͒ becomes
␦x͑͒ϭu͑͒ϭu0 cos , ͑24͒ ␦y͑͒ϭuЈ͑͒ϭϪu0 sin ,
giving a typical ‘‘center’’ solution about the fixed point xជ 2* . As previously discussed, in ‘‘physical’’ space the orbit of m0 about M does not close. However, the phase-plane trajec- tory given by ͑24͒ must close after a single orbit since the Fig. 2. Linear stability phase plane. system is conservative ͑ignoring radiative effects͒. There-
80 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 80 fore, the period of a single orbit, ⌽, is defined from the period of the phase space trajectory given by ͑24͒: ⌽ϭ2. ͑25͒ Solving for ⌽, and then substituting for in the limit of small gives the following result: ⌽ϭ2Ϫ1Ϸ2ϩ3. ͑26͒ The Newtonian calculation gives only the first term, ⌽ ϭ2, as expected. However, as seen in ͑26͒, the Schwarzs- child solution gives the correction: 2 ⌬ϭ3ϭ6͑GMm0 /cJ͒ , ͑27͒ which is the usual value ͑expressed in MKS units͒ obtained from procedures ͑A͒ or ͑B͓͒compare also with ͑9͒ for the standard expression in the ‘‘geometrized’’ system of units͔.
V. AN ‘‘EXACT’’ PHASE PLANE
In Sec. IV a linear stability analysis has been considered about each fixed point. However, this procedure gives only ‘‘local’’ information on the general relativistic orbits, and is Fig. 4. An ‘‘exact’’ phase plane for ϭ1/9. in fact one shortcoming of the linear stability analysis. Therefore, no correspondence can be made with parabolic, hyperbolic, or orbits near the black hole event horizon using or for the binary pulsar system discovered by Hulse and Fig. 2 alone. However, since the equations of motion are 9 integrable due to so many constants of motion ͓i.e., ͑4͒–͑6͔͒, Taylor: an exact phase plane can be constructed and then several 3Ϸ4° Ϸ7.4ϫ10Ϫ3. ͑32͒ ⇒ ‘‘global’’ features of these orbits may be deduced as a result. 1 In addition, other qualitative features of the Schwarzschild To check that ϭ 9 is a reasonable value in Fig. 4, an upper orbital dynamics may be derived from this diagram ͑Fig. 4͒ bound may be placed on for the existence of stable or as discussed below. Note: Since the equations are integrable unstable orbits from either phase-plane fixed point. By in- ͑ 1 no chaos exists here. However, if additional degrees of free- spection of ͑12͒,ifϾ6 then no ͑real͒ fixed points exist for dom are allowed, the possibility for chaos exists; see, e.g., a given value of energy and angular momentum. To trace the Ref. 8 for a discussion of chaos in relativistic orbital dynam- physical origin of this value and to understand the topologi- ics͒. cal structure of Fig. 4 from a more general viewpoint, note To obtain an exact phase-plane diagram, consider the that when yϭ0in͑30͒ the effective potential is obtained: ‘‘level curves’’ found by taking the ratio of xЈ and yЈ from 2 2 3 Vˆ Ϫ1ϭ͑x Ϫx ͒/2Ϫx. ͑33͒ ͑11͒, and then integrating to get a conserved quantity: eff The locations of the stable and unstable orbits are found as 3 2 2 3 2 dy/dxϭ͑ 2x Ϫxϩ͒/y y ϭϩx Ϫx ϩ2x. ͑28͒ ˆ .xVeffϭ0 for x, which gives identically ͑12͒ץ usual by solving ⇒ 1 The value of the constant  is easily found by comparison From ͑12͒, no extrema exist for Ͼ 6, establishing an upper 1 with ͑8͒: bound on for stable or unstable orbits. For ϭ 6, stable ˆ 2 orbits ͑smallest value of͒ and unstable orbits ͑largest value ϭ2͑E Ϫ1͒, ͑29͒ of͒ coincide at so that ͑28͒ may be alternatively expressed: r1,2ϭ3rS , ͑34͒ 2 2 2 3 Eˆ Ϫ1ϭ͑y ϩx Ϫx ͒/2Ϫx. ͑30͒ ˆ 2 providing an inflection point in the plot of VeffϪ1vsxas In Fig. 4, the level curves corresponding to different values shown in Fig. 5 for several values of ͑Ref. 10͒͑Note: The of Eˆ in ͑30͒ are shown with the effective potential ͑20͒͑with standard presentations on this diagram are commonly dis- 1 ˆ 2 ϭ 9͒. These curves are exact solutions of ͑11͒ for various played as VeffϪ1vs1/x; see, e.g., Wald, Ohanian, and 1 energies and initial conditions, and should be compared with Ruffini, or MTW for an alternative parametrization using rS 2 the approximate solutions given by the linear stability analy- and J/m0c ͒. But another critical value of occurs when sis of Fig. 2. The vertical dotted line at r /rϭ1, labels the ˆ 2 ˆ 2 1 S E ϭ1. To see this, solve E1Ϫ1ϭ0 using ͑14͒ to get ϭ 8, black hole event horizon. as displayed in Fig. 5. Therefore, qualitatively distinct orbits The value of used in ͑30͒ has been greatly exaggerated exist based upon the following values of : to better illustrate the qualitative features of the exact phase 1 1 1 1 1 1 plane. For a more realistic value of consider Mercury’s 0ϽϽ 8, ϭ 8, 8ϽϽ 6, ϭ 6, Ͼ 6. ͑35͒ orbit—taking the value of ⌬ over a single orbit and then 1 For Ͼ 6 there is insufficient angular momentum for m0 to using ͑27͒: sustain an orbit, therefore the mass simply falls into M and 3Ϸ0.104Љ Ϸ5.4ϫ10Ϫ8, ͑31͒ correspondingly, Vˆ has no extrema. The physical signifi- ⇒ eff
81 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 81 Fig. 7. Schwarzschild bifurcation diagram. Fig. 5. Schwarzschild effective potential.
1 1 for Ͻ ͒ corresponds to a critically unstable hyperbolic or- cance of ϭ is discussed above ͑34͒. The physical meaning 8 6 bit that separates trajectories spiraling into M ͑above the of the other values in ͑35͒ are understood by analyzing the separatrix͒ or escaping to infinity ͑below the separatrix͒.13 separatrix11 structure of ͑30͒. Essentially, this corresponds to Similarly, as illustrated in Fig. 6 these separatrices are a limitation placed upon the types of orbits that may exist summarized according to the following values of as dis- before an unstable orbit is reached and the kinematic classi- tinct unstable orbits: fication of the separatrices as distinct unstable orbits. 1 In essence, the separatrix gives a graphic representation of 0ϽϽ 8 unstable hyperbolic, the critical relationship between energy and angular momen- ⇒ ϭ 1 unstable parabolic, ͑36͒ tum at the unstable orbital radius ͑see Fig. 6͒. For a given 8⇒ angular momentum ͑͒, the critical energy of the unstable 1ϽϽ 1 unstable elliptic. ˆ 8 6⇒ orbit is calculated from E1 of ͑14͒. For the values of plot- 1 1 ted in Fig. 5, these energies are computed and marked with It is obvious from Fig. 6 that for in the range: 8рϽ 6, ˆ 2 only elliptical orbits are possible ͑about xជ 2*͒ before the un- horizontal lines. Substituting these values of E Ϫ1 into ͑30͒, 1 the separatrices corresponding to ͑35͒ are plotted in Fig. 6. stable orbit is reached, while the case 0ϽϽ 8 allows all These distinct separatrices divide the phase plane into four three: hyperbolic, parabolic, and elliptic as discussed above. 1 1 But these results are consistent with the orbital motion ob- regions of motion for 0ϽϽ 6 ͑ϭ 9 is just one special case in Fig. 4͒. To begin, consider Fig. 4 in the region surround- tained from inspection of the effective potential for different ing the stable fixed point xជ* . The oval trajectory in this values of in Fig. 5. These qualitative differences over the 2 range of unstable orbits have not been pointed out in the region corresponds to an elliptical orbit and was used earlier literature, but this is not to imply that the Schwarzschild to find the value for perihelion precession. A unique para- orbital dynamics are poorly understood; see, e.g., bolic orbit occurs as the phase-plane trajectory just touches Chandrasekhar14 for an alternative but lengthy analysis. the y axis and separates the hyperbolic and elliptic orbits. It should be noted that a physical orbit corresponding to The hyperbolic orbit12 is characterized by a trajectory ap- the separatrix can never be achieved in finite proper time. To proaching M from infinity, but then returning to infinity with do so would imply that the phase-plane trajectories change constant dr/d. Therefore, the separatrix of Fig. 4 ͑typical direction at xជ 1* , which is not possible in a deterministic sys- tem. To see this, consider the proper time equivalent of ͑30͒ ͓this is ͑7͒ after rewriting the equation using the definition of in ͑9͒ and again using xϭrS /r͔: ͑dr/ds͒2ϭEˆ 2Ϫ1ϩ͑x3Ϫx2͒/2ϩx. ͑37͒ Separating variables gives an elliptic integral: cϭϮ͵ dr/ͱ͑Eˆ 2Ϫ1͒ϩxϪx2/2ϩx3/2, ͑38͒ which diverges to Ϯϱ as r approaches the unstable orbital ˆ 2 radius r1 of ͑12͓͒and ͑14͒ is substituted for E Ϫ1͔, i.e., for a particle approaching the saddle point along the separatrix. From the separatrix analysis it is apparent that a bifurca- 1 tion occurs at the critical value ϭ 6, i.e., the topological structure of the phase plane changes as the two fixed points move together, coalesce into a single fixed point, and then disappear from the phase plane as is further increased above the critical value 1/6. Therefore, the Schwarzschild orbital dynamics may be interpreted and analyzed as a con- servative 2-d bifurcation phenomena. Specifically, this bifur- cation is a saddle-center bifurcation15 ͑see Fig. 7͒, and sum- Fig. 6. Separatrices for selected values of . marizes the range of physically possible orbits that may
82 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 82 occur as the energy and angular momentum are varied for Ͼ0. But from a more general viewpoint one should also consider negative values of ͓although it is clear that Ͻ0 has no physical interpretation since must be positive definite according to ͑9͒; note also that ϭ0in͑11͒ gives the phase-plane equations for light rays—see ͑44͒ below͔. For Ͻ0 the two fixed points ͓Eq. ͑12͔͒ change stability at ϭ0 as shown in Fig. 7. Therefore, another ͑transcritical͒ bifurcation occurs at ϭ0 ͑see, e.g., Strogatz,6 pp. 50–52͒, 1 followed by the saddle-center bifurcation at ϭ 6. Finally, an interpretation of the phase-plane trajectories to the right of the separatrix should be given, namely those trajectories leaving and then returning through the event ho- rizon. These trajectories are clearly nonphysical since it is impossible for any classical particle or light ray to escape from within the black hole horizon. The origin of these tra- jectories may be understood as a consequence of the symme- try of ͑11͒ under the interchange: Ϫ; y Ϫy, where Ϫ is due to the time-reversal→ symmetry→ of the Schwarzschild→ dynamics. As a consequence, this system is classified as reversible and gives the symmetry of Fig. 4 ͑and Fig. 6͒ about the x axis, but with the vector field below the x axis reversing direction.16
VI. PROPER AND COORDINATE TIME PHASE DIAGRAMS Fig. 8. Proper and coordinate time phase diagrams.
In the standard analysis on relativistic orbital dynamics, the proper time parameter is replaced by the equatorial angle as the independent variable. One advantage of this replace- fixed point that appears in the coordinate reference frame at ment is to simplify the algebra of a perturbative analysis, and the event horizon ͑see Fig. 8͒. is a carryover from the standard techniques applied in the To summarize these results, the corresponding phase- Newtonian case ͑see the Appendix͒. However, as far as the plane equations analogous to ͑11͒ in both the proper and phase-plane analysis is concerned, there are no essential dif- coordinate time reference frames are derived by differentiat- ficulties analyzing the dynamics using the proper time ͑or ing ͑39͒ and ͑40͒, respectively. In each case the results are coordinate time͒ as independent variables. In fact, there is given by additional information available which also gives a nontrivial dx/dϭyϭϮx2͓Eˆ 2Ϫ͑1ϩx2/2͒⌳͔1/2, introduction to dynamical invariance. ͑42͒ To demonstrate the invariance of the effective potential dy/dϭx3͓7x3Ϫ6x2ϩ10xϩ8͑Eˆ 2Ϫ1͔͒/4, between the proper and coordinate time reference frames, and start with ͑7͒ to obtain the proper time result ͑Note: Using rather than eliminates the x4 leading term appearing be- dx/dtϭyϭϮx2⌳1/2͓Eˆ 2Ϫ͑1ϩx2/2͒⌳͔1/2/Eˆ , low͒: 43 dy/dtϭx3⌳͓9x4Ϫ15x3ϩ2x2͑3ϩ7͒ ͑ ͒ 2 2 4 ˆ 2 2 ͑rS /c͒ x˙ ϭx ͓E Ϫ͑1ϩx /2͒⌳͔. ͑39͒ ϩ2x͑6Eˆ 2Ϫ11͒Ϫ8͑Eˆ 2Ϫ1͔͒/4. The corresponding coordinate time expression is obtained using: x˙ ϭ(dx/dt)˙t, in combination with ͑5͒ which gives Although ͑42͒ and ͑43͒ are more complicated algebraically than ͑11͒, the simultaneous solution of x˙ ϭy˙ ϭ0 for Eˆ 2 and x 2 2 4 ˆ 2 ˆ 2 2 ͑rS /c͒ ͑dx/dt͒ ϭx ͑⌳/E͒ ͓E Ϫ͑1ϩx /2͒⌳͔. ͑40͒ in each case reduces to ͑12͒ and ͑14͒ identically, but with Solving ͑40͒ for Eˆ 2 gives the coordinate frame expression another fixed point, xϭ0, at infinity and at xϭ1 in the case for the total energy: of ͑43͒. However, the fixed point at infinity exists for the Newtonian case as well, and is discussed in the Appendix. x4͑1ϩx2/2͒⌳3 ˆ 2 The fixed point at the event horizon is obviously coordinate E ϭ 4 2 2 2 . ͑41͒ x ⌳ Ϫ͑rS /c͒ ͑dx/dt͒ dependent and does not correspond to any extrema of the effective potential. Nevertheless, this fixed point has physical By inspection of ͑41͒,asdx/dt 0, the effective potential consequences for observers in the coordinate reference ˆ 2 ˆ 2→ˆ 2 ͑20͒ is recovered, i.e., Veff VeffЈ ϭVeff is invariant between frame—explaining the slowing down of objects and redshift the proper and coordinate time→ reference frames. Therefore, of signals approaching the event horizon. the dynamical structure is invariant, or alternatively stated, As discussed below ͑14͒, there are additional nonphysical ˆ 2 ˙ ϭ ˙ ϭ the extrema of Veff are identical in either reference frame. roots obtained when solving x y 0, only for x and y. The However, the phase diagrams in each case are not identical nonphysical nature of these fixed points is due to the fact that due to the existence of an additional ‘‘frame-dependent’’ there must be a constraint placed upon Eˆ when or t is used
83 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 83 as the independent variable. Solving simultaneously the ex- pressions for y given in ͑42͒ or ͑43͒ gives the proper con- straint on Eˆ and as a result forces these fixed points to coin- cide with the extrema of the effective potential. This is also a feature of the Newtonian dynamics when using t as the in- dependent variable.
VII. LIGHT RAYS The analysis of photon orbits in the Schwarzschild space- time is a straightforward application of the techniques dis- cussed for timelike orbits. For light rays, dϭ0, which in turn implies that both E and J are divergent from ͑5͒ and ͑6͒, although their ratio remains finite. As a result, 0, and the phase-plane equations for light rays follow as a→ special case of ͑11͒: xЈϭyϭϮ͓1/b2Ϫx2͑1Ϫx͔͒1/2,
3 2 ͑44͒ yЈϭ 2 x Ϫx, where 1/b2ϵ2Eˆ 2 is a constant expressing the dimension- ˆ Fig. 9. Null geodesics phase plane. less impact parameter, b, as the finite ratio of E, J, and rS . The simultaneous solution of xЈϭyЈϭ0 for 1/b2 and x results in two fixed points and the corresponding values of 1/b2ϭ0, these orbits just graze the event horizon from the the impact parameter: inside and simultaneously ͑in an unrelated trajectory͒ reach 2 2 4 2 the center node fixed point of the effective potential ͑see Fig. ͕x1ϭ 3;1/b ϭ 27͖, ͕x2ϭ0;1/b ϭ0͖, ͑45͒ 9͒. For 1/b2Ͻ0, b loses its interpretation as an impact pa- giving the standard results for the unstable orbital radius, x1 , rameter since the trajectories in this case originate from the and the impact parameter at which this instability occurs. singularity at rϭ0 and lie within the horizon. For 1/b2 The fixed point, x , is a center node ͑at infinity͒ about which 4 2 Ͻ 27, the trajectories are confined to within the separatrix and the hyperbolic orbits ‘‘precess’’ ͑see Ref. 12 for a comment correspond to the light rays arriving from infinity, reaching a on the timelike case͒ and gives the standard result for light turning point ͓given by the appropriate root of the first equa- bending. Therefore, the perihelion precession of timelike or- tion in ͑44͔͒, and then return to infinity as discussed below bits and light bending are actually special cases of one an- 2 4 ͑45͒. For 1/b Ͼ 27, a photon arrives from infinity ͑above the other: In the timelike case this center node fixed point is at separatrix͒ and then falls through the event horizon. The cor- finite r and allows ‘‘real’’ circular orbits; but for light rays responding time-reversed trajectories are given below the this fixed point moves to infinity and gives the precessing separatrix. hyperbolic orbits noted above. However, a phase-plane cal- The trajectories to the right of the unstable orbital radius culation of light bending analogous to that discussed in Sec. of Fig. 9 are also interpreted as time-reversed paths that III does not work here. This is due to the fact that a linear reach a maximum distance from the event horizon and then stability analysis ͑15͒ ‘‘kills’’ the necessary terms; namely, return to the singularity. But another interpretation is pos- the impact parameter disappears from the matrix A ͑a similar sible using simple energy considerations: A photon and its result occurs when calculating the period of a simple pendu- time-reversed counterpart originate from a point of maxi- lum for large angles using this technique͒. mum distance from the horizon, and then both proceed si- The phase-plane level curves for light rays in Fig. 9 cor- 2 multaneously from this point into the horizon. The analogous respond to different values of 1/b . These are shown to- interpretation is also possible in the timelike case for massive gether with the locations of the fixed points and photon ef- particles ͑see Fig. 4 and the discussion in the final paragraph 2 fective potential: x (1Ϫx). The most striking difference of Sec. V͒. However, these are classical interpretations and between the photon and timelike phase-plane dynamics should not be identified with quantum phenomena. ͑comparing Figs. 9 and 4͒ is that the center node fixed point moves to the origin as 0 ͑as discussed above͒.Asa VIII. DISCUSSION result, circular photon orbits→ do not exist in any ‘‘dynami- cal’’ sense, but become circular in geometry as the orbits We have considered an alternative procedure for calculat- approach the separatrix. To see this use the definition of y in ing the value of perihelion precession and have summarized the first equation of ͑44͒, and then separating variables the Schwarzschild orbital dynamics in the modern setting of 2 4 shows that ϱ as x x1 and 1/b 27 ͓this result is phase-plane analysis. Contrasting these calculations with the analogous to the→ proper time→ divergence→ pointed out in ͑38͔͒. standard textbook procedures, the main results are obtained Therefore, the separatrix corresponds to the unstable ‘‘pho- very quickly while minimizing the algebra, but placing more ton sphere’’ that is commonly discussed in the literature ͑see, emphasis on the physics. For example, by calculating the e.g., Ohanian and Ruffini,1 p. 410͒. value for perihelion precession using a perturbative solution, The physical interpretation of the various phase-plane re- a departure is made from an analysis based on physical con- gions of Fig. 9 is similar to that of Fig. 4, but there are cepts to an exercise in algebra. However, in the phase-plane important differences. For light rays with impact parameter approach, the physical concepts are given greater emphasis
84 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 84 and made more accessible to beginning students. This is due than using . Furthermore, there are results shared by the to the fact that the phase-plane technique itself is based es- relativistic case ͑discussed in Sec. VI͒ that are clarified in sentially on the ‘‘energy-method’’ diagrams taught in intro- this analysis. ductory mechanics courses. The analysis presented in Sec. To begin, consider the Newtonian limit of the equations VI demonstrates that important topics such as dynamical in- derived in Sec. II. The effective potential/͑unit rest energy͒ is variance are easily handled using the phase-plane techniques. ˆ 2 given in ͑20͒, and is defined as Veff which gives the proper These provide nontrivial and physically interesting examples Newtonian limit for Vˆ ͑to within an additive constant͒ in which normally are difficult conceptually for beginning stu- eff the limit of large r. As a result, the Newtonian limit of ͑20͒ dents. is given by In addition, the traditional analysis of the effective poten- tial could be augmented with discussion on the ‘‘exact’’ Vˆ ϭ͓1Ϫxϩ͑x2Ϫx3͒/2͔1/2Ϸ1Ϫx/2ϩx2/4, ͑46͒ Schwarzschild phase plane, or more specifically its separatrix eff structure. Essentially, the separatrix gives a geometric repre- which differs from the standard Newtonian form by an addi- sentation of the critical relationship occurring between en- tive constant ͑corresponding to the rest mass energy of m0͒. ergy and angular momentum, and as such, divides the phase The standard Newtonian effective potential energy is chosen plane into physically distinct regions of motion. By varying a to be zero at infinity, giving the usual expression: dimensionless parameter involving the angular momentum, a ˆ 2 saddle-center bifurcation occurs as the two fixed points coa- Veffϭx /4Ϫx/2, ͑47͒ lesce and disappear—altering the phase-plane topology. For compared to the relativistic limit where the energy at infinity the case of light rays the separatrix corresponds physically to corresponds to the rest mass energy. But this additive con- an unstable ‘‘photon sphere’’ as discussed earlier. As a spe- stant is of little consequence insofar as the dynamics are cial case of ͑11͒, the photon orbits also provide a transcritical concerned, and so we adopt 47 for the remaining discus- bifurcation point of the dynamics—exchanging stability at ͑ ͒ sion ͓Note: From a certain viewpoint ͑see Kompaneyets,4 p. ϭ0. 44͒ one may regard this difference as a choice of ‘‘gauge:’’ For additional applications it would be interesting to ana- ˆ lyze solutions other than the Schwarzschild case, e.g., the i.e., the ‘‘Newtonian gauge’’ takes Veffϭ0 at infinity, while ˆ Reissner–Nordstrom ͑a charged, spherically symmetric the special ‘‘relativistic gauge’’ is Veffϭ1͔. black hole͒, the Kerr solution ͑a rotating black hole͒,orthe The corresponding Newtonian expression for ͑39͒ is de- Kerr–Newman solution ͑a charged, rotating black hole͒. Fur- rived using the standard Lagrangian and Hamiltonian results: ther applications would include an analysis of cosmological 2 ˙ 2 4 ˆ ˆ solutions and nonconservative orbital dynamics ͑i.e., systems ͑rS /c͒ x ϭ2x ͓EϪVeff͔, ͑48͒ emitting gravitational radiation͒, and also solutions stem- where Vˆ is given by 47 and xϭr /r. Although the ming from alternative theories of general relativity. Analysis eff ͑ ͒ S of these topics will appear elsewhere. choice of units seems odd at first, this form gives the most In summary, constructing an exact phase plane for an ar- straightforward comparison with the relativistic case. As a bitrary solution will only be possible if the fixed point alge- check, ͑48͒ reduces to ͓after substituting ͑2͒, ͑47͒, and then ͑9͔͒ braic equation, xЈϭyЈϭ0, is of fourth order or less ͑and in addition if a sufficient number of first integrals exists͒. Oth- 2 4 2 2 2 ͑du/dt͒ ϭϪu ͑J/m͒ ͓u Ϫ2u0uϪbu͔, ͑49͒ erwise, finding roots will be difficult if not impossible. How- 2 2 ever, a numerical approach could always be taken, and where uϭ1/r and u0ϭGMm /J gives the standard radius 2 2 2 would be motivated by the interesting pictures that result of a circular orbit. The constant: buϵ1/b ϭ2mE/J ex- from combining the fixed point structure of general relativity presses the impact parameter ͑for a particle approaching state space into a diagram that includes the event horizon. from infinity͒ in terms of E and J. The zeroes of ͑49͒ give the standard turning points of the effective potential ͑aside from ACKNOWLEDGMENTS uϭ0͒. Furthermore, substituting uϭu() and then ͑6͒ into ͑49͒͑the Newtonian expression for J is identical in form to I thank Jim Crawford and Richard Treat for many helpful the relativistic case͒ leads to the standard second-order dif- comments and discussion on the analysis. I also thank ferential equation that is commonly evaluated for the analy- Charles Jaffe and Steve Strogatz for pointing out references, sis of these orbits. and Boyd Edwards for discussion in Physics 401A, Spring Continuing with the analysis, differentiating ͑48͒ gives the 1997. In addition, I thank Abhay Ashtekar, Beverly Berger, dimensionless phase-plane equations expressed using t as the Bryce DeWitt, Bill Hiscock, Ted Newman, Jorge Pullin, independent variable: Carlo Rovelli, Lee Smolin, and Clifford Will for helpful 2 ˆ 2 1/2 comments on the existing literature. x˙ ϭyϭϮx c͓4Eϩ2xϪx ͒] /rSͱ2, ͑50͒ y˙ ϭϪx3c2͑3x2Ϫ5xϪ8Eˆ ͒/2r2. APPENDIX: NEWTONIAN PHASE-PLANE S ANALYSIS Solving simultaneously, x˙ ϭy˙ ϭ0 for Eˆ 2 and x then gives the two fixed points: As discussed earlier in Sec. I, the standard analysis of the Newtonian orbital dynamics is based on the change of inde- ˆ ͕x1ϭ;EϭϪ/4͖, x2ϭ0. ͑51͒ pendent variable, t , for the purpose of finding a closed form solution describing→ the orbital geometry. But a phase- The first gives the standard results: a center node correspond- plane analysis of the differential equations using time as the ing to a Newtonian circular orbit with radius r1 and energy independent variable is no more complicated in principle given by
85 Am. J. Phys., Vol. 67, No. 1, January 1999 Bruce Dean 85 Cambridge, 1990͒, pp. 105–108; R. Wald, General Relativity ͑University of Chicago Press, Chicago, 1984͒, pp. 139–143; S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity ͑Wiley, New York, 1972͒, pp. 194–197, 230–233. 2N. Anderson and G. R. Walsh, ‘‘Phase-Plane Methods for Central Or- bits,’’ Am J. Phys. 58 ͑6͒, 548–551 ͑1990͒. 3B. Davies, ‘‘Derivation of Perihelion Precession,’’ Am. J. Phys. 51 ͑10͒, 909–911 ͑1983͒; D. Ebner, ‘‘Comment on B. Davies’s ‘Derivation of Perihelion Precession, ’ ’’ ibid. 53 ͑4͒, 374 ͑1985͒; Daniel R. Stump, ‘‘Precession of the Perihelion of Mercury,’’ ibid. 56 ͑12͒, 1097–1098 ͑1988͒. 4A. P. Arya, Introduction to Classical Mechanics ͑Prentice–Hall, Engle- wood Cliffs, NJ, 1990͒, pp. 222–252; H. C. Corben and P. Stehle, Clas- sical Mechanics ͑Dover, Mineola, NY, 1994͒, 2nd ed., pp. 90–100; G. Fowles and G. Cassiday, Classical Dynamics of Particles and Systems ͑Harcourt-Brace, Orlando, FL, 1993͒, 5th ed., pp. 191–216; H. Goldstein, Classical Mechanics ͑Addison–Wesley, Reading, MA, 1980͒, 2nd ed., pp. 94–102; A. S. Kompaneyets, Theoretical Mechanics ͑Dover, New York, 1962͒, 2nd ed., pp. 41–48; J. Marion and S. Thornton, Analytical Mechan- ics ͑Harcourt–Brace, Orlando, FL, 1995͒, 4th ed., pp. 291–321; A. Roy, Orbital Dynamics ͑Hilger, Redcliffe Way, Bristol, 1982͒, 2nd ed., pp. 69–100; K. Symon, Mechanics ͑Addison–Wesley, Reading, MA, 1971͒, pp. 128–134. 5J. L. Martin, General Relativity—A First Course for Physicists ͑Prentice– Hall International, Hertfordshire, UK, 1996͒, Revised ed., pp. 58–60. 6 Fig. 10. Newtonian phase-plane diagram with t as the independent variable S. Strogatz, Nonlinear Dynamics and Chaos—with Applications to Phys- (ϭ1). ics, Biology, Chemistry, and Engineering ͑Addison–Wesley, New York, 1994͒. Also see problem #6.5.7, p. 186. 7M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduc- tion ͑Wiley, New York, NY, 1989͒, pp. 13–30. 2 2 8 x1ϭ r1ϭrS /ϭJ /GMm , L. Bombelli and E. Calzetta, ‘‘Chaos around a Black Hole,’’ Class. Quan- ⇒ tum Grav. 9 ͑12͒, 2573–2599 ͑1992͒; S. Suzuki and K. Maeda, ‘‘Chaos in 2 ͑52͒ Eˆ ϭϪ/4 EϭϪm͑GMm/J͒ /2. Schwarzschild Space-Time: The Motion of a Spinning Particle,’’ Phys. ⇒ Rev. D 55, 4848–4859 ͑1997͒. The second fixed point at infinity simply expresses the fact 9R. A. Hulse and J. H. Taylor, ‘‘Discovery of a Pulsar in a Binary Star that it takes an infinite amount of time for the orbiting par- System,’’ Astrophys. J. 195, L51–L53 ͑1975͒. ticle, m, to reach the turning point at infinity ͑in the case of 10Using x rather than 1/x for the horizontal axis pushes the singularity at r parabolic and hyperbolic orbits͒—a fixed point that is shared ϭ0 to infinity. As a result, the relative locations of fixed points are more in the relativistic orbital dynamics. For comparison with the easily scaled and plotted in the phase plane. relativistic phase-plane results the Newtonian phase diagram 11Also termed homoclinic orbit in the literature on nonlinear analysis. for ͑50͒ is shown in Fig. 10. 12These are actually precessing hyperbolic orbits if one allows negative r values ͑although nonphysical—shown to the left of the y axis in Fig. 4͒. 13 a͒Present address: NASA Goddard Space Flight Center, Mailstop: 551.0, But for an ordinary massive object M, the trajectories never reach the Greenbelt, MD 20771. event horizon as it lies within the object’s surface. 14 1M. Carmeli, Classical Fields: General Relativity and Gauge Theory S. Chandrasekhar, The Mathematical Theory of Black Holes ͑Oxford U.P., ͑Wiley, New York, 1982͒, pp. 218–220; A. Eddington, The Mathematical Clarendon, New York, 1983͒, pp. 96–122. 15 Theory of Relativity ͑Cambridge U.P., Cambridge, 1924͒, 2nd ed., pp. J. Hale and H. Kocak, Dynamics and Bifurcations ͑Springer-Verlag, New 88–90; C. Misner, K. Thorne, and J. Wheeler, Gravitation ͑Freeman, San York, 1991͒, p. 425; J. M. Mao and J. B. Delos, ‘‘Hamiltonian Bifurcation Francisco, 1973͒, pp. 659–670, 1110–1116; H. Ohanian and R. Ruffini, Theory of Closed Orbits in the Diamagnetic Kepler Problem,’’ Phys. Rev. Gravitation and Spacetime ͑Norton, New York, 1994͒, pp. 401–408; B. A 45, 1746–1761 ͑1992͒. 16 Schutz, A First Course in General Relativity ͑Cambridge U.P., Cambridge, Incidentally, xជ 2* of Fig. 4 ͑and Fig. 3͒ is classified as a nonlinear center; 1990͒, pp. 275–284; H. Stephani, General Relativity ͑Cambridge U.P., see Strogatz ͑Ref. 6, p. 114͒.
THE AWE FACTOR The other aspect of science, the one that I am more concerned with, is the wonder, the ‘‘awe factor.’’ I ask myself, what is the appeal of religion, what is the appeal of UFOs, what is the appeal of von Da¨niken or Velikovsky, all that nonsense? I suspect that a part of it is the kind of awesome romance that science ought to be the master of. Don’t let us allow religion to walk away with the awe factor. Science has orders of magnitude more to offer in this field. Black holes are incomparably more wondrous, more romantic, than anything you read in the pseudoscientific literature, in New Age drivel, in the ‘‘occult,’’ in the Bible. Let’s not sell science short.
Richard Dawkins, ‘‘The ‘Awe’ Factor,’’ Skeptical Inquirer 17͑3͒, 242–243 ͑1993͒.
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