Border regions of gravity

So far in the pages of Elementary Einstein, there were two examples for general relativity reaching its limits. Both cases involved space-time-singularities. The first example lurked in the interior of a black hole. As briefly described in the chapter on black holes, inside every black hole, there is a space-time boundary where the journey of an infalling object ends abruptly, a singularity, where space-time curvature becomes infinitely strong. Also, we encountered a kind of mirror image to this singularity in connection with the big bang models, in the section where the mysterious beginning of the universe was mentioned: according to the classical cosmological models, the universe began with the big bang, another space-time boundary, a singularity where all matter of the universe is compressed to infinitely high density.

Ill-behaved space-time boundaries with bizarre infinintely high densities are a sure indication that we have reached the boundaries of Einstein´s general theory of relativity. In a way, that does not come as a surprise: With energy concentrations so high and matter compressed to microscopically small scales, one should expect the laws of the micro-world to come into play, the laws of quantum theory. But in Einstein´s theory, these laws are not included. That is why, when describing the earliest stages of our universe or the interior of a black hole, Einstein´s general relativity is not enough. The proper model would incorporate both Einstein´s geometrical gravity and the laws of the quantum world. Put in a different way: a proper model would have to rely on a theory of quantum gravity.

From an analysis of the fundamental constants of gravity and quantum theory, it follows that the natural length scale at which quantum gravity effects can be expected to become important is what is called the Planck length - roughly 10^-35 meters. The following illustration will hopefully give you an inkling of how small this is:

From everyday lengths of about one meter, you would have to zoom in by a factor of 10 billion to come to the typical scale of atoms. Only if you zoom again by a factor of 1 trillion trillion will you reach the Planck length. Two of the strange objects that might await us there - spin networks and strings - are sketched in the illustration.

So far, so good. Unfortunately, formulating a theory of quantum gravity has turned out to be one of the hardest problems of modern physics. Attempts to incorporate gravity into the quantum framework as if it were just another force like electromagnetism or the nuclear forces have failed utterly: the result is a model in which unphysical results (physical quantities predicted as having infinite values) abound, and which have no predictive power whatsoever. Until now, the question of what a proper quantum theory of gravity should look like has not found a complete answer. There are, however, promising candidates for a theory of quantum theory. The most important candidate theories will be introduced briefly on the following two pages.

Next page: Loop quantum gravity