Dr. Jonathan Kenigson, MNatSci, FRSA
Acting Don of Research
Athanasian Hall, Cambridge LTD
1 & 3 Kings Meadow, Osney Mead, Oxford OX2 0DP
Einstein’s Field Equations are highly nonlinear and extremely difficult to solve exactly. They permit many different solutions given physically reasonable initial conditions. One way to partially solve the equations in the presence of – say – an electromagnetic field is to impose Maxwell conditions on the free space around a black hole. The Maxwell conditions introduce another layer of complexity because the Theory of Fields as contrived during the early 20th century was only equipped to handle Euclidean geometry, where space is regular and flat. Minkowski’s Geometry and the ensuing development of Tensor Calculus heralded a new epoch in Differential Geometry and Relativity. From roughly 1905 to 1920, these fields developed rapidly in tandem, co-informing each other and making reciprocal, intuitive advances. The first use of comoving coordinates in Relativity had been employed in 1916 to demonstrate the geometric tenability of the Schwarzschild black hole. This solution is static and has no electric charge. The RN solution came with advances in the coordinate geometry necessary to model stationary magnetic fields in highly curves spacetimes.
It would be half a century before mathematicians could unravel the details of rotating black holes. The first hints of such advance came in the early 1960s’ during the era of the Mercury programme. The general theory of Differential Geometry on Pseudo-Riemannian Manifolds (places where calculus can be done, but which are not “flat” except in small neighbourhoods) permits the development of the notion of a “Weyl Tensor.” The Weyl Tensor permits the quantification of tidal forces felt relative to an observer moving on a geodesic toward a gravitating body. In other words, Weyl Tensors measure the “stress and strain” that the gravitational force produces on bodies moving toward other bodies in the possible presence of other forces. It can be shown that the Weyl tensor vanishes on any Pseudo-Riemannian manifold in dimension 3 or less. The 4th dimension defined by Spacetime is the first dimension in which Weyl Curvature can be given any formal meaning in terms of the Ricci Tensor. The Ricci component of gravitation defines the deformation in volume experienced by a hypersphere in the local Spacetime. Euclidean spheres are not “spherical” near massive gravitating bodies, but rather more oblong, stretched in the direction of a dominant gravitating body. Aleksei Petrov introduced the Type Theory of Weyl Tensors on a general Pseudo-Riemannian Manifold in 1954. The Tensor Type gives an algebraic classification of a geometric phenomenon and permits the categorization of Weyl Curvatures almost everywhere.
Petrov Degeneration is a key aspect of the Penrose programme and defines the process by which a given Weyl Tensor changes type. Informally, Petrov degeneration occurs when different models of Spacetime are considered as solutions of Einstein’s equations. Any Big-Bang universe governed by Relativity that is the result of FRLW initial conditions (like our universe is often taken to be) must – except for possibly infinitely many points that are defined by Black holes or other singularities – have Petrov Type O, wherein all local curvature is defined by fields acting on the surrounding Spacetime. There exists a finite Euclidean distance past which all information must be lost to local observers, and they cannot “see” a signal broadcast from the more distant regions. Relativity asserts that this distance is proportional to the speed of light in a vacuum. The Weyl Tensor surrounding a rotating black hole is not O, however. It is type D – a type in which the deformation of Spacetime produced by its own rotation causes the vacuum to produce tidal effects and so-called “Gravitometric Effects”. Tops spin differently because space is spinning. If there is a magnetic field nearby, perhaps induced by a plasma or some other electromagnetic entity, this field spins, too. In fact, any field that interacts with matter to produce a change in energy must undergo some such rotation by the Relativistic equivalence of mass and energy.
In order to compactly define a distance between two events in a Petrov Type D paradigm like that imposed by a rotating black hole, it is expedient to change the notion of distance using a change of coordinates. Instead of real distances (like 5 km or 50 m), one permits complex distances (those involving imaginary numbers) and asserts the equivalence of these distances modulo equality of complex norms. There is really nothing counterintuitive about complex distances. After all, the solution of the simple Quadratic equation defined by stating that the square of a number is equal to negative one produces the imaginary unit. Once derided for its non-physicality, this unit, and its generalizations in Quaternionic and n-Onionic Algebraic Field Theory, is responsible for the entire edifice that defines symmetry and supersymmetry in mathematical terms. One sees a physical need for a given entity and breathes it into existence, marveling at the same instant how one can define a quantity abstractly in terms of pure logic and create a theory that makes novel physical predictions. The resulting coordinate system is defined by so-called Boyer-Linquist Coordinates, which we briefly met in the previous article. In this metric, the Hamiltonian Formalism that abstractly quantifies the energies of test particles entering the black hole event horizon becomes much simpler. A “line element” is a representation of an infinitesimal distance travelled along a surface. Every reasonable coordinate system has one. In the case of Euclidean geometry, it is defined by the Theorem of Pythagoras and is learned in undergraduate Classical Mechanics in terms of 1-variable parametric integration.
In the most complex case of a Kerr black hole, One maps a triad (M, J, r) consisting of the black hole mass (M), the rotational momentum (J), and r (the distance modulus in Oblate Spherical Coordinates) to the line element defined by relations that can concisely and uniquely recover the Euclidean coordinates of an infalling test particle. It is expedient to introduce a spin parameter (typically called “a”) that represents the angular momentum per unit-mass of the black hole. This parameter simplifies the Boyer-Lindquist coordinates without disrupting the Hamiltonian Dynamics of the local test particles, showing that it is a “natural” transformation that is physically well-defined. Coordinate choices should not influence the energy of a system, but only its representation. The resulting line element is a perturbation of the original case of Schwarzschild from 1916 which reassuringly simplifies to the same when the angular momentum is set to 0. One would suspect this behaviour to obey such a limiting process wherein successively slower rotation produces successfully closer qualitative approximations of the original 1916 metric. The resulting metric derived by Kerr possesses many interesting properties that the original 1916 metric does not possess. These properties are non-exhaustively enumerated below.
The Relativistic Rotational Energy of the Kerr Black Hole is directly proportional to the difference (M-M*), where M* is a mass that is “irreducible” and cannot be extracted through any Penrose process. The constant of proportionality is precisely the square of the speed of light in a vacuum. In other words, no matter how fast the black hole spins, it reaches an irreducible mass that cannot be dissipated by non-Quantum mechanisms. Additionally, the total mass M of the black hole is affected by its rotation in a predictable manner. In fact, the irreducible mass M* is related to the total mass M by a polynomial equation of degree 4. This equation must have at least one real solution. Because complex solutions must occur as conjugates, there must be exactly 4 real solutions or 2 real and two complex conjugate solutions. The two real solutions can be discriminated as physical or non-physical by the physical constraint that M>M* in all circumstances.
Kerr Theory predicts that there are inner and outer event-horizons whose radii are Quadratic functions of the Schwarzschild radius and are physical if and only if GM > aK, where K is the square of the speed of light c in a vacuum and a is the parameter J/(Mc). Black holes that would have “naked” singularities may violate the Cosmic Censorship Hypothesis as defined by Penrose in 1969. Regions of infinite information density violate many established precepts of computing and would have novel implications for the theory of Quantum computation, in which infinitely dense data compression could possibly be recovered by a reversible process. Furthermore, there should exist an outer surface known as an “Ergosphere” in which speed-of-light frame-dragging occurs. A Penrose Process could theoretically extract energy from this region, although tidal forces would be immense, and any possible practical application would be negated by intense radiation and gravitational forces. A further set of surfaces known as Photospheres are mathematically predicted to exist in which the curvature of Spacetime permits photons (radiation) to inhabit infinitely many possible “orbits” between inner and outer spheres. The Photospheres interact with the Ergosphere in nonlinear and unpredictable ways. One can prove with little effort that a maximal Photosphere must exist whose radius is strictly a function of (M, J), where M is the total Mass-Energy and J is the Rotational Momentum.
In our next BBC submission, we shall consider the most general case of Relativistic black hole. This sort of black hole possesses electric charge as well as spin. Due to the economy of Boyer-Lindquist Coordinates, the arithmetic is not much more complicated. As in any well-defined theory, every case should reduce to every other case at points of degeneracy. A rotating black hole with zero charge should reduce to the Kerr metric. A Kerr black hole with zero spin should reduce to the classical 1916 metric of Schwarzschild.
Editorial Note: I would like to thank undergraduate Ricky Capps of Tennessee, USA for his work in reviewing this article and for his enduring interest in Cosmology and Mathematics.