NYU Black Renaissance Noire NYU Black Renaissance Noire V. 16.1 | Page 17

When massive clouds of interstellar
dust coalesce, condense, and begin
to radiate, a star is born. Within a
few billion years from the time of
their birth, all stars age and eventually
die. But they have a very interesting
afterlife. After a life of burning, stars
use up their fuel and cool, and with the
lack of outward radiation pressure,
they eventually collapse under their
own inward gravitational pull. In 1931,
Nobel Prize-winning Indian physicist
Subrahmanyan Chandrasekhar
showed that when all the mass of a
dying star collapsed to within a s mall
enough volume, it formed an enchanting
object called a white dwarf — a quiet
remnant of the former star with the
pressure of its own constituent electrons
holding it up against gravity. One day,
our sun will become a white dwarf,
shrinking roughly to the size of Earth.
In 1939, Robert Oppenheimer and
George Volkoff, with the work of
Richard Tolman, showed that for
stars more massive than the sun, even
just one and a half times bigger, their
gravity would be too great for their
constituent electrons to hold them up.
These stellar remnants collapse further
until finally their neutrons take up
the slack, pushing back against gravity.
The result? Neutron stars. For stars
greater yet, three times or more massive
than the sun, even neutrons can’t fight
gravity. The nuclei collapse — and
then our theories teeter on the edge of
our understanding. In step black holes.

BLACK RENAISSANCE NOIRE

The great power of mathematical
symmetry is that it can reduce the
complexity of the equations. Imagine
there are two separate equations that
describe the oscillation of two particles,
particle X and particle Y. One example
of a “symmetric” situation would
be if the behavior of X was exactly the
same as Y. The two differential
equations could thus be reduced to
one, and once a solution for either
X or Y was found, the solution of the
other would follow.

Sometimes, nature actually provides
these serendipitous situations of high
symmetry, and physicists can delight
in discovering the solutions. In the
case of Einstein’s equations, spherical
symmetry was a good place to start.
Spheres could model the structure of
stars, like our sun. The geometry of
spheres allowed gravity to be reduced
to a radially uniform field around a
compact central source. It was such a
natural and simple idea that, within a
few months of Einstein developing his
theory, Karl Schwarzschild, German
physicist and astronomer, found a
spherically symmetric solution to the
equations. But there was a glitch.
As smaller and smaller radii were
considered, a radius was reached, now
known as the Schwarzschild radius,
where the equations revealed something
called a singularity — mathematically
the sort of thing you get if you divide by
zero. Physicists don’t like singularities.
They usually imply regions of infinite
energy or force. Really, most singularities
tell us that something is wrong with
our theory in the regions where they
show their face. But this singularity
was pointing to something new and
downright awe inspiring about our
spherical friends, stars.

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To really understand the magic behind
Einstein’s ten coupled differential
equations, it is useful to begin
by considering a solution to them.
But given their complexity, it is no easy
task to dream up a physical space-time
configuration that satisfies them.
It’s no longer a case of studying a graph
and guessing the form of the function,
as we did with Newton’s equations.
Even today, with the help of powerful
computers, we still cannot find exact
solutions of the gravitational field
for interesting astrophysical systems.
Nevertheless, just after Einstein
developed his theory, physicists were
buzzing with curiosity about his new
space-time concept and eager to find
solutions. For starters, they armed
themselves with Dirac’s trusty method:
using the power of symmetry.