Spinning Charged Bodies and the Linearized Kerr Metric. Abstract

Transcription

1 Spinning Charged Bodies and the Linearized Kerr Metri J. Franklin Department of Physis, Reed College, Portland, OR 97202, USA. Abstrat The physis of the Kerr metri of general relativity (GR) an be understood qualitatively by analogy with the potentials of spinning harged spheres in eletrodynamis (E&M). We make this orrespondene expliit by omparing the Lagrangian for test partile motion in E&M with a spinning spherial soure to the Lagrangian for a test partile in GR under the influene of a linearized limit of the Kerr metri. The interpretation of Kerr as the metri appropriate to spinning massive bodies then emerges as a simple replaement of mass for harge in the E&M ase. 1

2 I. INTRODUCTION In E&M, as with all field theories, we have two sets of physial laws: 1. The field equations (Maxwell) that tell us how to generate E and B (or equivalently V and A) given a soure, and 2. An equation of motion (Lorentz) that tells us how test partiles respond to fields. GR is no different Einstein s equations gives us the onnetion between soures and fields, and the equation of motion for test partiles is just the geodesi equation desribing the straightest possible lines in a urved spae-time. Fields E&M GR 2 V 1 2 V = ρ 2 t 2 ɛ 0 G 2 A 1 2 µν = 8πG A = µ 2 t 2 0J 4 T µν Partiles L = 1 2 mv2 q(v v A) L = m ẋ µ g µν ẋ ν. (1) The field in GR is the metri g µν and the Einstein tensor G µν R µν 1 2 g µνr is a nonlinear ombination of g µν, and its first and seond derivatives (more ompliated than the left-hand side of Maxwell s equations, but in muh the same spirit). While E&M has ρ and J as soures for V and A, the metri in GR is generated by the full stress tensor of matter, just an expanded notion of what an generate fields. For the equations of motion, we have the Lagrangian assoiated with the Lorentz fore law for E&M parametrized by time t (so that v = d x ), and the geodesi Lagrangian for GR where x µ is the oordinate four-vetor with x 0 = t and the dots refer to derivatives with respet to the proper time τ. Indeed, if we take g µν = η µν, the Minkowski metri, we reprodue the (speial) relativisti Lagrangian for a free partile. Our goal is to take a speifi distribution of harge in E&M, find the Lagrangian governing test partile motion there, and ompare it to the Lagrangian for the Kerr metri of GR. In order to make the omparison, we will express the geodesi equation in terms of the oordinate time (as the parameter of the motion) and take a slow-motion limit, then input Kerr in linearized form as the metri. 2

3 ẑ ω = ωẑ R ŷ ˆx ρ FIG. 1: A uniformly harged spinning sphere with radius R. II. A CHARGED SPINNING SPHERE A sphere of radius R with onstant harge density ρ spinning about the ẑ axis with uniform angular veloity ω has well-known potentials 1. For r > R, V = ρr3 3ɛ 0 r A = µ 0ωρR 5 15 (2) sin θ ˆφ, (3) r 2 in SI units. With this set of potentials, in spherial oordinates, the Lagrangian reads L = 1 2 m ( ṙ 2 + r 2 θ2 + r 2 sin 2 θ φ 2) qρr3 3ɛ 0 r + µ 0ωρR 5 sin 2 θ φ, (4) 15 r and variation with respet to (r, θ, φ) will reprodue the Lorentz fore law for this distribution. The onfiguration is stati (we are not onsidering the spin-up of the harged body for us, it has been spinning forever with onstant ω), and the potential V is just that of a uniformly harged sphere. We know, from our early physis eduation, that the Newtonian potential for a sphere of uniform density would take the form (although with different onstants) of V with harge density ρ interpreted as mass density. But what about the other term in the Lagrangian? The magneti vetor potential has no analogue in Newtonian gravity whih is a salar theory, there is no notion that mass urrent hanges the motion of test bodies. Suppose that we take the eletrostati-newtonian orrespondene: qρr 3 3ɛ 0 r = qq 4πɛ 0 r GmM r (5) 3

4 at fae value and apply it to the magneti term in (4), this would give us the Lagrangian for a theory of gravity that inluded the spin of the entral body as a soure. Again, this is absent in Newtonian theory, so for now, the proedure is speulative. The above suggests that the orret replaement is qq 4πɛ 0 qµ 0 ωρr 5 15 sin 2 θ r φ = GmM, so for the magneti portion, we have qωr2 Q 5r(4πɛ 0 2 ) sin2 θ φ mωr2 MG sin 2 θ 5r φ, (6) 2 representing the predited gravitational oupling of a spinning massive entral body to a test body. III. LINEARIZED KERR The Einstein equations in vauum (away from soures) redue to R µν = 0 for the Rii tensor. There are only a few known solutions to this equation, but in terms of (astro)physial relevane, the Kerr solution 2 is the most useful. The Kerr metri in Boyer-Lindquist 3,4 oordinates reads: ( ) 1 2MGr 2 ρ 0 0 2aMGr sin 2 θ 2 3 ρ 2 ρ 0 2 g µν = ρ 2 0 ( 2aMGr sin2 θ 0 0 (r 2 + ( ) a 2) ) 3 ρ 2 + 2a 2 MGr sin 2 θ sin 2 θ 4 ρ 2 ( ) a 2 ρ 2 r 2 + os 2 θ ( ) a 2 MGr 2 + r 2. 2 for x µ =(t, r, θ, φ). This metri desribes a spae-time whih is not flat, and therefore, the oordinates, while we have written them with familiar names, do not represent the usual spherial oordinates. There are two parameters in the metri: a and M, whose physial signifiane will beome lear as we go. For now, it is interesting to note that a = M = 0 reprodues the spherial Minkowksi metri of speial relativity, and a = 0 gives Shwarzshild (what about M = 0?). It is also relatively lear that for large r, the above redues to flat spae. This metri does indeed have vanishing Rii tensor, and therefore is an exat solution of Einstein s equations in vauum. Beause we want to ompare with our eletrostati Lagrangian, written in terms of a Eulidean three-dimensional spae in spherial oordinates with time as a parameter, we 4 (7)

5 expand the above metri in powers of the fundamental lengths: a taking a and MG 2 that is, we are MG r and r. Under this assumption: MG 0 0 2aMG sin2 θ 2 r 3 r 2MG g µν 0 0 r r. (8) r 2 sin 2 θ 2aMG sin2 θ }{{} 3 r η µν This metri is only an approximate solution to the vauum Einstein equations, there are orretions of order ( ) 2 ( ) a and MG 2. We have given up exatness in favor of interpretabil- 2 ity the form of (8) allows us to view the oordinates (r, θ, φ) as the usual flat spherial ones, that s the role of the Minkowski metri as the dominant term. Now we want to use the linearized metri (8) in the geodesi Lagrangian to understand how test bodies move. Our approah will be to keep terms appropriate to the linearization proedure, assume that the test body motion is slow (ompared to ) and parametrize the motion by the oordinate time rather than proper time. The end result will be a Lagrangian in whih the seond term in (8) (representing deviation from flat spae) appears as an effetive potential whih we an ompare with (4). The ation for geodesi deviation reads: dx S = m ẋµ g µν ẋ ν dτ = m µ g dx ν µν ( ) 2 dτ dτ = m dx µ g µν dx ν (9) whih gives us the geodesi Lagrangian using t as the parameter for the test body motion. In terms of general metri omponents, we have L = m g g 0j dxj dxj g dx k jk where we let the j and k indies run from 1 3 (the spatial omponents), and note that dx 0 =. Using (8) as the metri, the Lagrangian reads ( L = m 1 2MG 2 r ) amg sin2 θ r φ ( r MG 2 r ) (ṙ ) 2 (10) ( ) 2 r θ 2 2 sin 2 θ ( ) 2 r φ. (11) 5

6 To expand this, we use our linearization assumption and keep only terms of order a r, MG 2 r and a r MG, where the metri is still a valid vauum solution. In addition, we keep only linear 2 r and quadrati veloities, this limits us to test bodies that move muh less than the speed of light. Under these assumptions, the Lagrangian is approximately: L m 2 + mmg 2 ammg sin2 θ φ + 1 }{{} r r 2 ignorable }{{} 2 m ( ṙ 2 + r 2 θ2 + r 2 sin 2 θ φ 2), (12) }{{} potential kineti where the first term is a onstant that will not show up in the equations of motion, the seond term is an effetive potential and the third term is the usual kineti energy. Compare this Lagrangian with (4) the shared kineti term tells us we are talking about motion in a three-dimensional flat spae. The potential above has a Newtonian omponent that we expet, and a term proportional to φ whih is new. From the Newtonian term, we learn that M appearing in (7) is the entral body s mass. The new term is preisely of the form predited by our eletrostati example. Referring to (6), if we assoiate (modulo some purely numerial fators) a ωr 2, the linearized Kerr metri inludes a ontribution that we an interpret as the spin of a massive body. The a parameter has units of angular momentum per unit mass, and its presene in the Lagrangian tells us that GR predits, at this linearized level (and in the slow-motion approximation) the existene of a gravito-magneti potential that ouples to test partiles in the same way as the magneti ontribution from spinning harged spheres. IV. SUMMARY In introdutory physis, Newtonian gravity is usually a student s first exposure to 1 r 2 fores. In addition, it is a natural plae to disuss the spherial symmetry of soures and the onnetion between fields and fores. When the Coulomb field is introdued later on, during E&M, it is often ompared to the Newtonian spherially symmetri fore law. During a more advaned ourse in E&M, students learn about the magneti vetor potential, and so it is natural to ask if the simplest stationary onfiguration in E&M that supports both eletri and magneti potentials, namely the harged spinning sphere, has any natural ousin on the gravitational side. The prime motivation for general relativity is the idea that Newtonian gravity requires modifiation, and looking for orretions from E&M provides a nie symmetry to the original, 6

7 freshman physis notion that gravity and eletrostatis have something in ommon. That they do at the linearized GR level is well known 5, of ourse, but the very expliit parallel treatment provided here is meant to be aessible to students of E&M, before the details of the linearized Einstein equations have been presented. In addition, the parameters of the Kerr metri an be given real physial meaning by this diret analogy, again, before the metri has been disussed expliitly as a vauum solution. V. CLASS ROOM DISCUSSION Following are some questions that an be posed to the students, and some ideas for additional disussion that an be used to introdue related topis in GR that follow naturally from the analogy presented here: We assumed, based on the form of (8), that we ould interpret the oordinates as flat. That approah will not reprodue the orret numerial expression for perihelion preession in a linearized Shwarzshild limit, one must keep the non-flat radial oordinate. What are the defiienies in the linearized Kerr setting? How an one visualize a oordinate like the Shwarzshild r? Why are diagrams of orbital motion relevant? These lead naturally to a disussion of asymptoti flatness and observers. For E&M, we know that vauum solutions to the field equations are onstrained by boundaries at infinite and some region enlosing the soure. Vauum Einstein equations have the same feature, and the E&M approah here makes this point. A general treatment of linearization and oordinate hoies at the level of Einstein s equation an stem from this example. Given the similarities of E&M and GR, why is a vetor theory of gravity insuffiient? The geodesis of Kerr an also be treated from here, partiularly the equatorial ase 6,7. Spinning test partiles are often enountered in E&M, and with the stati example given here, it is relatively simple to ouple test-body spin, whih suggests the oupling in GR (GPB experiment and Lense-Thirring preession follow). 7

8 The linearized limit here laks the additional Carter s onstant of full Kerr, where did it go? We have gone through the parallels forward here, prediting a orretive term and then verifying it. But one ould also start with the spinning harged sphere, and demand that there be a gravitational analogue to the magneti term. Then the Kerr metri, one derived, appears more familiar, and a and M whih show up as onstants of integration in the derivation an be given immediate interpretation. Eletroni address: jfrankli@reed.edu 1 David J. Griffiths. Introdution to Eletrodynamis. Prentie-Hall, Roy P. Kerr. Gravitational Field of a Spinning Mass as an Example of Algebraially Speial Metris. Physial Review Letters, 11(5): , Robert H. Boyer and Rihard W. Lindquist. Maximal Analyti Extension of the Kerr Metri. Journal of Mathematial Physis, 8(2): , Kip S. Thorne Misner, Charles W. and John Arhibald Wheeler. Gravitation. W. H. Freeman and Company, Robert M Wald. General Relativity. The University of Chiago Press, James B. Hartle. Gravity: An Introdution to Einstein s General Relativity. Addison Wesley, S. Chandrasekhar. The Mathematial Theory of Blak Holes. Oxford University Press,