Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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Date Created: 10/21/15
Lecture 11 Orbiting a Black Hole 0 Using the principle of extremal proper time we can nd another constant of the motion when the motion is not purely radial ie Agb is not zero This turns out to be the relativistic generalization of angular momentum L Lmrj 1 0 To show that this the correct generalization of angular momen tum for motion in a spherically symmetric spacetime we can employ the principle of maximal ageing again 0 Consider a motion comprising two sections A and B delineated by three events with coordinates in the Schwarzschild frame A lt13 39r A710 gt 71 the A section with mean r coordinate m and elasped proper time 04 71 gt 39r Arltlgt ggt the B section with mean r coordinate TB and elapsed proper time 7393 The nal angle lt13 is considered xed and only the intermediate angle q will be varied We will set 2 O and employ the Schwarzschild metric cl2 r2dggt2 stuff independent of q 2 29 2 1 o This yields T143 TB 0 Thus we see that W constant and can be identi ed with a conserved angular momentum per unit mass for motion in a spherically symmetric spacetime 0 General comment Notice that we now have two conserved quantities energy E m02AT and angular momentum L m rgg The fundamental reason why these two quantities do not change with time is related to the fact that the compo nents of the Schwarzschild metric do not depend explicitly on either time t or angle q This independence of the metric on certain coordinates is referred to as a symmetry isometry and is intimately connected to the existence of conservation laws So now there are 2 constants of the motion energy and angular momentum Using these and the Schwarzschild metric we can compute the orbit of any body near the black hole ie motions which are not just radial Starting from some initial position rggt and the constants L and E we can imagine computing changes in t and q using the equations Emcg At 2 M L51 A7 3 and using the form of the Schwarzschild metric leads to an ex pression for the change in r coordinate Ar 1 Em022 1 23231 1 A computer can be programmed to compute the trajectory 7 739 7 5t739 for any E and L The possible orbits may be best Visualized by considering the e ective potential i 2 AT 4 Effective Potential Just examine the form for 2 it Em nomeg ed where 1 V39r 2GrlI Lm2 i 1 1 me2 027quot 027quot Choose some angular momentum and plot this If L is su icently small the potential has a monotonic behavior with 39r and an inwardly moving particle experiences an increasing radial velocity as it is inevitably pulled down to the event horizon However if we increase L sufficiently the curve develops more structure If we want to gure out some motion we should consider the in tersection of the horizontal line y E me2 with this curve The radial velocity squared 57 is proportional to the difference in the squares of E inc2 and Vmcg Thus the possible motions are con ned to the region where E 2 V When E V the ra dial velocity is zero Such points in general correspond to points at which the radial velocity is about to change sign 7 they corre spond to points at which the radial distance is either a maximum or a minimum On the graph they correspond to intersections of V with the line E A particle set off so that it lies between two such points will move so as its r coordinate oscillates back and forth betweenn these two limits In general these motions will correspond to some sort of elliptical precessing orbit The one exception to this occurs when the two intersection points merge into one i this occurs when the energy matches a turning point on the Vr curve The case of a minimum corresponds to a stable circular orbit For a maximum it would be an unstable circular orbit The question of stability is decided by imagining perturbing the orbit by a small amount and seeing what hap pens in the resulting motion It should be clear that for orbits around a maximum of the potential the object will eventually be captured by the black hole 0 Notice that small enough r there is a pit if the particle has enough energy it will reach this region and be swallowed by the black hole 0 This situation is helped if the initial angular momentum is small 7 which decreases the potential barrier 0 Circular orbits If we set 2 O we can nd the radii of any turning points In class we found that these are given by 2 l 12 l 7 1 a 1 TE 2 l2 0 Thus the minimum angular momentum to avoid capture is given by mem xgcrg 2L where l men Initial conditions 0 Determining L and E Imagine a shell observer launching a satellite with a certain speed perpendicular to the radial direc tion Can we use this info to determine L and E and hence predict the motion 2 dt dt 2 1 o Emc Am AdtshellJMdT This is then E mc AA Mme Thus Emc2 1473h6ll dt d 0 Similarly L me Tofffa Howie Thus Lm To ysheu vsheu o This can be trivially generalized to arbitrary initial directions of motion Effective Potential for Photons 0 Now the motion is described not in terms of E and L but a single impact parameter b which measures the perpendicular distance from the initial photon trajectory to the black hole I for vanishing mass m 0 To see this consider the radial and angular equations for motion in a Schwarzschild spacetime for arbitrary mass First replace derivatives with respect to proper time by derivatives with re spect to faraway time since if O for massless particles Then rearrange expressions to be functions of b and mo2 E Take limit m gt 0 We nd LH ff r and dqs b 39r Tmz l fl To derive an effective potential picture of the photon motion consider changing variables from act to r8wglt8wu so as to get a pure constant as the rst term on the RHS of the radial equation 1 drshell 2 1 2 72 Cdtshell b2 where 1 2 7 E v 7quot T2 1 T Vertical aXis is a measure of radial velocity as before The issue of whether a given photon is captured or merely deflected and its possible orbital motion is revealed by plotting this effective potential and comparing with 1152 For large values of this pa rameter a photon travelling inward is captured For small values it cannot penetrate over the peak in the effective potential and is merely deflected Notice that for 39r ZiGr lIc2 unstable circular orbits are possible this is the photon sphere Photons impact ing at this critical b will circle the hole many times before either falling inward or escaping Notice that all local observers shell or FF observe photons to be moving at the speed of light but the Schwarzschild or faraway observer can measure quite different speeds can see this easily by setting the spacetime distance to zero in the Schwarzschild metric formula corresponding to photons Imagine sitting at the photon sphere and looking around First imagine the trajectories of photons wrt faraway coordinates Notice that every star produces some light with the critical im pact parameter This will lead to multiple images of the star corresponding to photons that traverse several times around the photon sphere before entering the observers eyes Thus such an observer sitting on the photon sphere will see additional images of these stars scattered on a bright ring which extends all around him transverse to the radial outward direction Its like a halo around the black hole image In addition the observer will pick up further images correspond ing to photons that pass different ways around the black hole 7 the phenomenon of gravitational lensing