Monotone Volume Formulas for Geometric Flows

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motivation for this work being List's extended Ricci flow system, the Ricci flow coupled with harmonic map heat flow and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.Comment: v2: final version (as published

Towards a general description of the interior structure of rotating black holes

The purpose of this paper is to present a number of proposals about the interior structure of a rotating black hole that is accreting slowly, but in an arbitrary time- and space-dependent fashion. The proposals could potentially be tested with numerical simulations. Outgoing and ingoing particles free-falling in the parent Kerr geometry become highly focused along the principal outgoing and ingoing null directions as they approach the inner horizon, triggering the mass inflation instability. The original arguments of Barrabes, Israel & Poisson (1990) regarding inflation in rotating black holes are reviewed, and shown to be based on Raychauduri's equation applied along the outgoing and ingoing null directions. It is argued that gravitational waves should behave in the geometric optics limit, and consequently that the spacetime should be almost shear-free. A full set of shear-free equations is derived. A specific line-element is proposed, which is argued should provide a satisfactory approximation during early inflation. Finally, it is argued that super-Planckian collisions between outgoing and ingoing particles will lead to entropy production, bringing inflation to an end, and precipitating collapse.Comment: 15 page

The interior structure of rotating black holes 3. Charged black holes

This paper extends to the case of charged rotating black holes the conformally stationary, axisymmetric, conformally separable solutions presented for uncharged rotating black holes in a companion paper. In the present paper, the collisionless fluid accreted by the black hole may be charged. The charge of the black hole is determined self-consistently by the charge accretion rate. As in the uncharged case, hyper-relativistc counter-streaming between ingoing and outgoing streams drives inflation at (just above) the inner horizon, followed by collapse. If both ingoing and outgoing streams are charged, then conformal separability holds during early inflation, but fails as inflation develops. If conformal separability is imposed throughout inflation and collapse, then only one of the ingoing and outgoing streams can be charged: the other must be neutral. Conformal separability prescribes a hierarchy of boundary conditions on the ingoing and outgoing streams incident on the inner horizon. The dominant radial boundary conditions require that the incident ingoing and outgoing number densities be uniform with latitude, but the charge per particle must vary with latitude such that the incident charge densities vary in proportion to the radial electric field. The sub-dominant angular boundary conditions require specific forms of the incident number- and charge-weighted angular motions. If the streams fall freely from outside the horizon, then the prescribed angular conditions can be achieved by the charged stream, but not by the neutral stream. Thus, as in the case of an uncharged black hole, the neutral stream must be considered to be delivered ad hoc to just above the inner horizon.Comment: Version 1: 12 pages, no figures. Version 2: Extensively revised, logic tightened, derivation more elegant. 18 pages, no figures. Version 3: Minor revisions to match published version. Mathematica notebook available at http://jila.colorado.edu/~ajsh/rotatinginflationary/rotatinginflationary.n

The interior structure of rotating black holes 2. Uncharged black holes

(Abridged) A solution is obtained for the interior structure of an uncharged rotating black hole that accretes a collisionless fluid. The solution is conformally stationary, axisymmetric, and conformally separable, possessing a conformal Killing tensor. Hyper-relativistic counter-streaming between collisionless ingoing and outgoing streams drives inflation at (just above) the inner horizon, followed by collapse. As ingoing and outgoing streams approach the inner horizon, they focus into twin narrow beams directed along the ingoing and outgoing principal null directions, regardless of the initial angular motions of the streams. The radial energy-momentum of the counter-streaming beams gravitationally accelerates the streams even faster along the principal directions, leading to exponential growth in the streaming density and pressure, and in the Weyl curvature and mass function. At exponentially large density and curvature, inflation stalls, and the spacetime collapses. As the spacetime collapses, the angular motions of the freely-falling streams grow. When the angular motion has become comparable to the radial motion, which happens when the conformal factor has shrunk to an exponentially tiny scale, conformal separability breaks down, and the solution fails. The condition of conformal separability prescribes the form of the ingoing and outgoing accretion flows incident on the inner horizon. The dominant radial part of the solution holds provided that the densities of ingoing and outgoing streams incident on the inner horizon are uniform, independent of latitude; that is, the accretion flow is "monopole." The sub-dominant angular part of the solution requires a special non-radial pattern of angular motion of streams incident on the inner horizon. The prescribed angular pattern cannot be achieved if the collisionless streams fall freely from outside the horizon.Comment: Version 1: 30 pages, 1 figure. Version 2: Extensively revised, logic tightened, derivation more elegant. 37 pages, 1 figure. Version 3: Minor revisions to match published version. Mathematica notebook available at http://jila.colorado.edu/~ajsh/rotatinginflationary/rotatinginflationary.n

The Black Hole Particle Accelerator as a Machine to make Baby Universes

General relativity predicts that the inner horizon of an astronomically realistic rotating black hole is subject to the mass inflation instability. The inflationary instability acts like a gravity-powered particle accelerator of extraordinary power, accelerating accreted streams of particles along the principal outgoing and ingoing null directions at the inner horizon to collision energies that would, if nothing intervened, typically exceed exponentially the Planck energy. The inflationary instability is fueled by ongoing accretion, and is occurring inevitably in essentially every black hole in our Universe. This extravagant machine, the Black Hole Particle Accelerator, has the hallmarks of a device to make baby universes. Since collisions are most numerous inside supermassive black holes, reproductive efficiency requires our Universe to make supermassive black holes efficiently, as is observed.Comment: 7 pages, 2 figures. NO honorable mention in the 2013 Essay Competition of the Gravity Research Foundatio