Question

In Schwarzschild geometry, the distance between two circles with different circumferences is larger than the Euclidean result, because the distance between the two circles is given by integrating dr/ (1 − rs/r) (where rs is a positive constant) along the radial path between them.

Invent an embedding of the radial part of the Schwarzschild spatial geometry in one extra Euclidean dimension, that we will call the z dimension. What is the relationship between dr and dz of the one-dimensional “surface” in this two-dimensional Euclidean space? Sketch the curve in an r > 0 quadrant of the r, z plane.

Answer 1

The Schwarzschild geometry describes the spacetime geometry of empty space surrounding any spherical mass. Karl Schwarzschild derived this geometry at the close of 1915, within a few weeks of Albert Einstein publishing his fundamental paper on the Theory of General Relativity. The history of this discovery and much more is wonderfully recounted in Kip Thorne’s book “Black Holes & Time Warps: Einstein’s Outrageous Legacy”.

One of the remarkable predictions of Schwarzschild’s geometry was that if a mass MM were compressed inside a critical radius rsrs, nowadays called the Schwarzschild radius, then its gravity would become so strong that not even light could escape. The Schwarzschild radius rsrs of a mass MM is given by rs=2GMc2 .rs=2GMc2 . where GG is Newton’s gravitational constant, and cc is the speed of light. For a 30 solar mass object, like the black hole in the fictional star system here, the Schwarzschild radius is about 100 kilometers. Curiously, the Schwarzschild radius had already been derived (with the correct result, but an incorrect theory) by John Michell in 1783 in the context of Newtonian gravity and the corpuscular theory of light. Michel derived the critical radius by setting the gravitational escape velocity vv equal to the speed of light cc in the Newtonian formula 12v2=GM/r12v2=GM/r for the escape velocity vv from the surface of a star of mass MM and radius rr.

Horizon The Schwarzschild surface, the sphere at 11Schwarzschild radius, is also called the horizon of a black hole, since an outside observer, even one just outside the Schwarzschild surface, can see nothing beyond the horizon.

Schwarzschild metric Schwarzschild’s geometry is described by the metric (in units where the speed of light is one, c=1c=1) ds2=−(1−r/rs)dt2+dr21−r/rs+r2do2 .ds2=−(1−r/rs)dt2+dr21−r/rs+r2do2 . The quantity dsds denotes the invariant spacetime interval, an absolute measure of the distance between two events in space and time, tt is a ‘universal’ time coordinate, rris the circumferential radius, defined so that the circumference of a sphere at radius rr is 2πr2πr, and dodo is an interval of spherical solid angle.