On the discrete version of the black hole solution

A Schwarzschild-type solution in Regge calculus is considered. Earlier, we considered a mechanism of loose fixing of edge lengths due to the functional integral measure arising from integration over connection in the functional integral for the connection representation of the Regge action. The length scale depends on a free dimensionless parameter that determines the final functional measure. For this parameter and the length scale large in Planck units, the resulting effective action is close to the Regge action. Earlier, we considered the Regge action in terms of affine connection matrices as functions of the metric inside the 4-simplices and found that it is a finite-difference form of the Hilbert–Einstein action in the leading order over metric variations between the 4-simplices. Now we take the (continuum) Schwarzschild problem in the form where spherical symmetry is not set a priori and arises just in the solution, take the finite-difference form of the corresponding equations and get the metric (in fact, in the Lemaitre or Painlevé–Gullstrand like frame), which is nonsingular at the origin, just as the Newtonian gravitational potential, obeying the difference Poisson equation with a point source, is cutoff at the elementary length and is finite at the source.

BOUNDARY LAYER CALCULATIONS ON CYLINDERS OF RANKINE-OVAL SECTIONS

An approximate calculation of the boundary layer parameters around a long cylinder of Rankine-oval section is carried out. The calculations are undertaken as a step-by-step analysis using the finite-difference form of the integral formulations of the boundary layer equations. The separation line of the boundary layer for an oval of thickness = 0.2 is located in the rear portion of the oval in a plane making an angle of 137° with the direction of the flow.

An Electrical Potential Analyser

The electrical potential analyser provides, in effect, an electrical analogy to the solution of Poisson's and Laplace's equations when expressed in finite difference form. The principal object in designing the instrument was to provide an experimental alternative to the “relaxation” technique which, although invaluable in the solution of many problems, is often very lengthy and tedious. The potential analyser consists essentially of a square mesh, the nodal points of which are interconnected by low value wire resistances, and a base. Corresponding nodes on the mesh and base are connected through high value resistances. A current is applied in such a manner that an electrical potential is provided, as required, to points on the mesh and the mesh boundary; the base being either used or discarded according to the nature of the problem. Readings of potential at each mesh node are recorded by a suitable galvanometer circuit. An analyser having a square mesh, such that the mesh separation is 1/12 the side length, has been constructed and encouraging results have been obtained with its use. The instrument is easily operated and results are rapidly obtained. The use of the analyser is not limited to the solution of Poisson's and Laplace's equations. By a suitable interchange of network resistances, other equations which can be expressed in finite difference form may be solved and solutions to problems involving irregular boundaries present no undue difficulty.