Finally one arrives at the conceptual structure of a Lorentzian wormhole

Graphing one of these functions one is left with

to simulate the imaginary components of the space we add the functions

this will then be graphed with mesh with the following conditions

one then rotates this function 360 degrees, this can be done with the following functions, l will give the length of the wormhole, or the strength of the gravitational potential.

when graphed one has

The Geometry

Now what does a wormhole look like, sadly we can only give an approximation, as a wormhole is four-dimensional identity and graphs can only be done in 3-d. We begin with a Lorentzian-de Sitter metric

from here once build a wormhole geometry with a simple function p(x):

On Wormholes

5/2002

by: E. Halerewicz, Jr.

hal_warp@hotmail.com

produced and converted to HTML by Mathcad

abstract: This document describes the geometric formation of a wormhole, a theoretical object which allows for arbitrary shortcuts through space and time. As with all the files related to these pages this document is not meant to be a complete treatise, just a brief introduction into the subject at hand. It is recommend that the reader find other sources of reference on the material discussed. Also this file was produced in Mathcad's spread sheet, so expect grammatical and spelling errors, the emphasis is placed on the mathematics and graphics generation within this spreadsheet.

Introduction

A Wormhole is an extension of the Schwarzschild geometry, and is a direct result of General Relativity being a "dumb" theory. Dumb is a bit a harsh, but accurate it governs how space can bend with a set of initial conditions, however it is ignorant about if such a fields are possible. In the same sense that one car design a bridge made out of paper, but in reality the design would be completely impractical. The wormhole exists do to the symmetrical nature of General Relativity, the Schwarzschild geometry has a metric, or more precisely a line element of order

its signatures is derived from Minkowski Space, when one takes the square root of the function ds2 one finds that , which yields real and imaginary solutions . A Schwarzschild geometry with high density forms a black hole and a singularity, with the complex solutions however a second geometry is also created which prevents the formation of a singularity and forms a tunnel known as the Einstein-Rosen Bridge. Even though mathematically this bridge is allowed to exist, physically it is doomed to collapse, the gravitational forces are completely overwhelming. However if there is a large amount of negative energy, also referred to as exotic energy, then a wormhole may remain open. This would then allow for an arbitrary short travel time between two distant sources, or to even allow for the possibility of "time" travel. There are generally two classes of wormholes Lorentzian and Euclidean, Lorentzian are gravitational based wormholes, while as Euclidean are particle based. Interest in wormholes were rekindled by a work of Morris and Thorne. A general class wormhole solution is given by

from which more elaborate wormhole models can be constructed. As one can see this geometry is very nearly identical to the Schwarzschild geometry, with the exception of the first term. F (r) is known as a red shift function, and is sometimes called a flare out condition, as it causes an electromagnetic wave to be red shifted as it leaves the wormhole.