Supplemental Lecture 6 Rotating Sources, Gravito-Magnetism and The Kerr Black Hole Abstract This lecture consists of several topics in general relativity dealing with rotating sources, gravito- magnetism and a rotating black hole described by the Kerr metric. We begin by studying slowly rotating sources, such as planets and stars where the gravitational fields are weak and linearized gravity applies. We find the metric for these sources which depends explicitly on their angular momentum J. We consider the motion of gyroscopes and point particles in these spaces and discover “frame dragging” and Lense-Thirring precession. Gravito-magnetism is also discovered in the context of gravity’s version of the Lorentz force law. These weak field results can also be obtained directly from special relativity and are consequences of the transformation laws of forces under boosts. However, general relativity allows us to go beyond linear, weak field physics, to strong gravity in which space time is highly curved. We turn to rotating black holes and we review the phenomenology of the Kerr metric. The physics of the “ergosphere”, the space time region between a surface of infinite redshift and an event horizon, is discussed. Two appendices consider rocket motion in the vicinity of a black hole and the exact redshift in strong but time independent fields. Appendix B illustrates the close connection between symmetries and conservation laws in general relativity. Keywords: Rotating Sources, Gravito-Magnetism, frame-dragging, Kerr Black Hole, linearized gravity, Einstein-Maxwell equations, Lense-Thirring precession, Gravity Probe B (GP-B). Introduction. Weak Field General Relativity. In addition to strong gravity, the textbook studied space times which are only slightly curved. There were several reasons for this emphasis, including, 1. compatibility of general relativity and Newton’s theory of gravitation in the limit of weak field, static, non-relativistic applications, and 2. gravitational waves. “Slightly curved” means that throughout a region of space time there exist coordinate systems where, = + , 1 (1.) where is the Minkowski metric, that readsℎ in a Cartesian�ℎ � ≪ coordinate system, = 1, = = = 1 with all other components vanishing. 00 11 22 33 Some care− must be used in this formalism because the decomposition Eq. 1 is not generally covariant [1,2,3]. There are two families of coordinate transformations which play an important role here: 1. Lorentz transformations (boosts) of special relativity, and 2. Infinitesimal general coordinate transformations. Under boosts in Minkowski space time,, = ′ � where the coefficients of the Lorentz transformation depend on the velocity between frames S and S’. The explicit formulas for were derived , discussed and applied extensively in the textbook. Under the action of a Lorentz boost the Minkowski metric is left invariant = ′ = while the dynamical gravitational field transforms as a second rank ∑tensor = in a static Minkowski space time.ℎ This fact was used in the ′ discussionsℎ of ∑gravitational ℎ waves in Chapter 12 of the textbook to view problems in linearized, weak field general relativity as problems of special relativity. In this approach is considered ( ) to be a field, analogous to of electrodynamics, which propagates in Minkowskiℎ space time and satisfies the principles of special relativity. Chapter 12 also considered the invariance of linearized general relativity under infinitesimal coordinate transformations, = + ( ) (2) ′ where ( ) is comparable in size to . It is easy to calculate the transformation law for the metric and deduce the transformation ℎlaws for (unchanged) and , ℎ = = + ′ � ′ �� − � � − �� ℎ� ′ = + + ℎ − − ⋯ + (3) ′ ≡ ℎ where raising and lowering of indices is done with the Minkowski metric, = . We learn from Eq. 3 that, ∑ = (4) ′ Again, viewing these formulas from theℎ perspectiveℎ − of static− Minkowski space time, Eq. 4 is analogous to a gauge transformation in electrodynamics: it leaves the physics unchanged but describes it with a transformed tensor . Furthermore, Chapter 12 derived the linear form of ′ the Einstein field equations, = ℎ = where is the Ricci tensor, 1 8 4 is the Ricci scalar, = , and − is 2the energy −-momentum tensor of the matter fields in space time. We found in∑ Chapter 12 that the field equations simplified if we defined = , where = . Furthermore, we found that if we choose the “Lorenzℎ� gauge” 1 ℎfor − 2, ℎ ℎ ∑ ℎ ℎ = 0 (5) � then the Einstein field equations reduce,∑ to linearℎ order in , ℎ� = (6) 16 4 □ℎ� − = = where □ is the wave operator, 2 . We used this equation to study 1 2 2 2 gravitational waves in Chapter□ 12 of∑ the textbook. The− choice∇ of the Lorenz gauge, Eq. 5, was essential to obtain a tractable, appealing wave equation, Eq. 6. As with most low order approximations to differential equations in physics, linearized gravity has several limitations [1,2,3]. We see from the formulas above (and will see in our applications below) that the linear formulation satisfies the linear superposition principle, just like electrodynamics. Therefore, the non-linearities and the foundations of general relativity, that all forms of energy-momentum experience universal gravitational attraction, are compromised. However, in linearized gravity we can compute the motion of test particles in the weak gravitational fields generated by masses whose motion is prescribed externally. In other words one typically does not calculate how the masses that generate the gravitational fields themselves move under their own gravitational fields. One treats sources of electric and magnetic fields in electrodynamics by this rule and we apply it here. This is how we illustrated gravitational wave generation in the textbook: we let describe two orbiting masses (“Neutron Stars”) and saw that their time varying quadrupole moment generates gravitational waves described by . In a quantitative calculation the full Einstein equations, = = , ℎmust� be 1 8 4 used everywhere and a fully consistent calculation results where − the2 dynamical − metric describes both the propagation of gravitational waves far from the source as well as the internal dynamics of the source itself. Far from the source the Einstein field equations reduce to linearized gravity. These limitations of the linearized gravity approach show up in Eq. 6 [1,2,3]. The Lorenz gauge condition reads = 0 which implies = 0 which represents energy- momentum conservation∑ in flat,ℎ� static Minkowski space∑ time. But the exact conservation law should read = 0, where is the covariant derivative, = + ∑+ , where the Christoffel symbols bring the curvature of space time into ∑the Γproblem. The∑ curvatureΓ may be strong in the vicinity of a radiating source and the hypothesis = + , 1, may not apply there. The full formalism of general relativity is requ ired. Or oneℎ can�ℎ, and� ≪frequently does, proceed phenomenologically and treat as an external source and only use Eq. 6 far from the source in analogy to radiation problems in electrodynamics. Slowly Rotating Stars, Frame-Dragging and Lense-Thirring Precession Now let’s apply this formalism to some simple but instructive situations [1,2,3]. What is the metric, to first order in , outside a rotating mass (a star or planet)? Although the source is rotating let’s suppose thatℎ there is no explicit time dependence. Examples might be a rigid sphere rotating at a constant angular velocity . Or perhaps the star only has cylindrical symmetry about a fixed axis of rotation. In either case, = 0 and = 0. The wave equation Eq. 6 reduces to Poisson’s equation, 0 0ℎ� ( ) = ( ) (7) 2 −16 4 ∇ ℎ� ⃗ ⃗ which was discussed and solved in the textbook in the context of electrodynamics, ( ) ( ) = (8) | | 4 �⃗ 3 4 ℎ� ⃗ ∫ ⃗−�⃗ Furthermore, recall from Chapter 12 that the simplest ( ) reads, ( ) = ( ) ( )⃗ ( ) (9) where ( ) is the proper mass density of the⃗ source ⃗ and ⃗ ( ) is⃗ its four velocity. If we apply this source ⃗ to non-relativistic stars and planets where ( )⃗ 1, then to first order = ( , ). And � ⃗ ⁄� ≪ ⁄ ≈ �⃗ = = = 0 (10) 00 2 0 where we neglect because it is a second order correction and the formalism ≈ here is accurate only to first order. The expressions for the metric follow, ( ) ( ) ( ) = ( ) = (11) | | | | 00 4 �⃗ 3 0 4 �⃗ 3 2 3 ℎ� ⃗ − ∫ ⃗−�⃗ ℎ� ⃗ − ∫ ⃗−�⃗ and ( ) 0. To bring out the similarities to problems of rotating charge distributions in electroℎ� dynamics,⃗ ≈ it is convenient to define, ( ) ( ) ( ) = ( ) = (12) | | | | �⃗ 3 4 �⃗ 3 2 Φ ⃗ − ∫ ⃗−�⃗ ⃗ − ∫ ⃗−�⃗ So, = ( ) = 0 (13) 00 4Φ 0 2 ℎ� ℎ� ⃗ ℎ� ≈ In order to retrieve the metric = + , we recall, from Chapter 12 of the textbook, that = , where = . Noteℎ that one can invert this expression by noting 1 ℎ� ℎ − 2 ℎ ℎ ∑ ℎ that = = , so = . From Eq. 13 we read off = = 1 00 , ℎ�so ∑ ℎ� −ℎ ℎ ℎ� − 2 ℎ� ℎ� ℎ� − ∑ ℎ� 00 ℎ� 1 1 = = 2 2 00 �00 00 � � ℎ 1ℎ − 1 ℎ ℎ = = = = 2 2 ℎ11 ℎ�11 − 11ℎ� ℎ� ℎ22 ℎ33 = (14) 1 ℎ01 2 This produces the metric to first order in | | 1, �⃗⁄ ≪ = 1 + + 1 ( + + ) (15) 2 2Φ 2 2 2 2Φ 2 2 2 2 2 � � ∑ − � − � This generalizes the weak field isotropic metric studied in the textbook to stationary rotating sources which produces the new term. 0 These expressions can be simplified further for a rotating uniform rigid sphere of radius R. Suppose that one samples the gravitational field far from the source, . Then the leading term