Free Hiding in the Mirror by Lawrence M. Krauss

ray’s trajectory bend downward. (Of course, the effect
would be very small, but since we are doing a thought experiment
here, we are free to imagine an arbitrarily accurate measuring
device.)
    But, special relativity tells us that light
rays move at constant speed in straight lines. How can we reconcile
this behavior with what you would measure in the elevator? Well,
one way to go in a straight line and also travel in a curve is to
travel on a straight line on a curved surface. This realization led
Einstein on a long mental journey in the course of which he was
drawn to the inescapable conclusion that space and time are not
only coupled together, but are also themselves dramatically
different than we perceive them to be. Space, and to some extent
time, can be curved in the presence of mass or energy. The result
was perhaps the most dramatic reformulation of our understanding of
the underlying nature of the physical universe in the history of
science.
    Einstein’s journey was replete with false
starts and dead ends, and the slowly dawning acceptance that
mathematical concepts that he had vaguely been exposed to while a
student might actually be useful for understanding the nature of
gravity. In 1912 Einstein finally realized that the mathematics of
Gauss, and then Riemann, which described the geometry of curved
surfaces and ultimately curved spaces, held the key to unlocking
the puzzle he had been wrestling with all those years. By November
1915, after almost having been scooped by the best mathematician of
that generation, David Hilbert, Einstein unveiled the final form of
his “gravitational field equations.” Einstein’s equations, as we
usually call them, provide a relation between the energy and
momentum of objects moving within space and the possible curvature
of that space. There are at least two fascinating and unexpected
facets of this relation. First, it turns out to be completely
independent of whatever system of coordinates one might use to
describe the position of objects within the curved space. Second,
and true to the spirit of special relativity—which by tying
together space and time also turned out to tie together mass and
energy—energy becomes the source of gravity. In general relativity,
however, such energy influences the very geometry of space itself—a
fact that makes general relativity almost infinitely more complex
and fascinating than Newton’s earlier law of gravitation. This is
because the energy associated with a gravitational field, and hence
with the curvature of space, in turn affects that curvature.
    In the jargon of mathematicians, general
relativity is a “nonlinear” theory. While technically speaking this
means that it is difficult to solve the relevant equations, in
physical terms it means that the distribution of mass and energy in
space determines the strength of the gravitational field at any
point, which in turn determines the curvature of space at any
point, which in turn determines subsequent distribution of masses
and energy, which in turn determines the curvature of space, and so
on. Nevertheless, in spite of the difficulty of dealing with these
equations, the single fact that affected Einstein during that
fateful November in 1915 more deeply than perhaps any other
discovery he had made in his lifetime was the realization that the
mathematical theory he had just proposed explained an obscure but
mysterious astronomical observation about the orbit of Mercury
around the sun. One of the most successful and stunning predictions
of Newton’s law of gravity is that the orbit of planets around a
central body such as the Sun should be described by mathematical
curves called ellipses. That the planetary motions were not perfect
circles had first been discovered, somewhat to his dismay, by
Johannes Kepler, and in short order Newton proved that his
universal law universally implied elliptical orbits. Nevertheless,
in 1859 the French astronomer Urbain Jean Joseph Le