# Black Hole Geometry

The first science was geometry. Someone had discovered the right angle and the 3,4,5 rule and the Egyptians used it for surveying agricultural land following annual floods of the Nile. Indeed it became part of their culture; the pyramids are a celebration of it. As time went by people started to realize that there were many other interesting relationships between lengths and angles and these were explored further by the Greeks leading to Pythagoras’ Theorem and Euclid’s geometry. Geometry was a scientific exploration of the properties of space and numbers.

Unfortunately there was a downside: there is no rational number (i.e. vulgar fraction) which is the square root of 2 so that there is no number corresponding to the length of the hypotenuse of an equilateral, right angle triangle. This was devastating to the Greeks. Their wonderful, rational description of the physical world had holes in it! The problem was kept secret for many years and delayed the development of algebra by a millennium or so. The issue was resolved by the irrational numbers of Dedekind and Weierstrass in the modern era.

Just as Euclidean geometry is an exploration of the properties of 2 and 3 dimensional space, the tensor calculus of General Relativity is an exploration of the properties of 4 dimensional space-time. Space and time are described as a deformed, four dimensional, “rubber sheet” observed in various, deformed coordinate systems. The fundamental assumptions are (1) that the deformations (aka “the laws of physics”) must be identical in all coordinate systems and (2) that  they must become the classical, Newtonian laws of physics when the rubber sheet is “flat”. Like geometry it has theorems and proofs which purport to describe the real world from these axioms.

Two important theorems are those of Israel and of Birkhof

It appears to be a rule of thumb, reminiscent of Gödel’s Theorem, that any self-consistent and comprehensive description of the physical world  ultimately defines its own limitations, quantum theory leads to indeterminacy and Planck’s constant, differential equations lead to Chaos Theory and the tensor calculus of General Relativity leads to the event horizon.