Einstein’s equations of general relativity are like the
Himalayan mountains – beautiful and majestic when
viewed from a distance, but slippery and full of crevasses
when explored up close. Of those who venture into
them, not everyone comes back alive. A set of 10 inde-
pendent, nonlinear partial differential equations,
Einstein’s equations relate the energy and matter in a
region of space to its geometry. Astonishingly simple
when expressed in the geometric, coordinate-indepen-
dent language of tensors that Einstein ultimately hit
upon, the equations – when applied to real situations –
unfortunately become coupled beasts unlike anything
physicists had tamed since the days of Newton.
Einstein’s equations of general relativity can be
solved exactly only in a handful of cases – with one of
the first such solutions, and perhaps its most famous,
being that derived by the German astronomer Karl
Schwarzschild in 1916 for the simple case of a static,
spherical, uncharged mass in a vacuum. Schwarzs-
child’s assumptions, and his mathematical wizardry,
reduced the Einstein equations to a single, ordinary dif-
ferential equation that he was able to readily solve,
though even the master himself was surprised at the
possibility of an exact solution. The “Schwarzschild
solution” leads naturally to the concept of a black hole
(see box on p40), although Schwarzschild himself never
grasped the significance of the singularity in his solu-
tion, dying four months later on the Russian Front dur-
ing the First World War. Even Einstein thought the
Schwarzschild singularity – the radius where the solu-
tion is invalid because of division by zero – was physi-
cally meaningless, and it was only decades later that the
depths of the Schwarzschild solution were plumbed in
general relativity’s first golden age, which ran from
about 1960 to 1975, by Roger Penrose, Kip Thorne,
Stephen Hawking and many others besides.
As theories go, general relativity has been a great suc-
cess. Most famously, its early approximate solutions
accounted for a well-known discrepancy in the orbit of
the planet Mercury that could not be completely
accounted for using classical Newtonian physics, yield-
ing a value for the difference that agreed spot-on with
astronomical measurements. Einstein’s equations have
also predicted that light bends in a gravitational field and
even that radar signals are delayed when bounced off
one of our solar system’s inner planets. However, these
successes are all based on the “post-Newtonian” approx-
imation of the full Einstein equations, where speeds are
small compared with that of light and gravitational fields
are weak. Einstein’s general relativity has never been
tested in the vastly different “strong field” regime.
Thanks, however, to fast and powerful supercom-