#
This Is Why
Scientists Will Never Exactly Solve General Relativity

[[It
is worth looking at the original for the excellent graphics.]]

In theory, Einstein's equations
are deterministic as well, so you can imagine something similar would occur: if
you could only know the mass, position, and momentum of each particle in the
Universe, you could compute anything as far into the future as you were willing
to look. But whereas you can write down the equations that would govern how
these particles would behave in a Newtonian Universe, we can't practically
achieve even that step in a Universe governed by General Relativity. Here's
why.

But in General Relativity, the
challenge is much greater. Even if you knew those same pieces of
information — positions, masses, and momenta of each particle — plus
the particular relativistic reference frame in which they were valid, that
wouldn't be enough to determine how things evolve. The structure of Einstein's greatest
theory is too complex even for that.

In General Relativity, it isn't
the net force acting on an object that determines how it moves and accelerates,
but rather the curvature of space (and spacetime) itself. This immediately
poses a problem, because the entity that determines the curvature of space is
all of the matter and energy present within the Universe, which includes a lot
more than merely the positions and momenta of the massive particles we have.

In General Relativity, unlike
Newtonian gravity, the interaction of any mass you consider also plays a role:
the fact that it also has energy means that it also deforms the fabric of
spacetime. When you have any two massive objects moving and/or accelerating
relative to one another in space, it causes the emission of gravitational
radiation, too. That radiation isn't instantaneous, but only propagates
outwards at the speed of light. This is an enormously difficult factor to
account for.

Perhaps the most demonstrative
example is to imagine the simplest Universe possible: one that was empty, with
no matter or energy, and that never changed with time. That's completely
plausible, and is the special case that gives us plain old special relativity
and flat, euclidean space. It's the simplest, most uninteresting case possible.

Instead of flat, euclidean space,
we find that space is curved, no matter how far away you get from the mass. We
find that the closer you get, the faster the space beneath you "flows"
towards the location of that point mass. We find that there's a specific
distance at which you'll cross the event horizon: the point-of-no-return, where
you cannot escape even if you were to move arbitrarily close to the speed of
light.

This spacetime is much more
complicated than empty space, and all we did was add one mass. This was the
first exact, non-trivial solution ever discovered in General Relativity: the
Schwarzschild solution, which corresponds to a non-rotating black hole.

·
perfect fluid
solutions, where the energy, momentum, pressure, and shear stress of
the fluid determine your spacetime,

·
electrovacuum
solutions, where gravitational, electric and magnetic fields can
exist (but not masses, electric charges or currents),

·
scalar field
solutions, including a cosmological constant, dark energy,
inflationary spacetimes, and quintessence models,

·
solutions with
one point mass that rotates (Kerr), has charge (Reissner-Nordstrom),
or rotates and has charge (Kerr-Newman),

·
or
a fluid solution
with a point mass (e.g., Schwarzschild-de Sitter space).

You might notice that these solutions
are also extraordinarily simple, and don't include the most basic
gravitational system we consider all the time: a Universe where two masses are
gravitationally bound together.

Instead, all we can do is make
assumptions and either tease out some higher-order approximate terms (the post-Newtonian
expansion) or to examine the specific form of a problem and attempt to solve
it numerically. Advances in the science of numerical relativity,
particularly in the 1990s and later, are what enabled astrophysicists to
calculate and determine templates for a variety of gravitational wave
signatures in the Universe, including approximate solutions for two merging
black holes. Whenever LIGO or Virgo make a detection, this is the theoretical
work that makes it possible.

We can extract how the behavior
of a solvable system differs from Newtonian gravity and then apply those
corrections to a more complicated system that perhaps we cannot solve.

Or we can develop novel numerical
methods for solving problems that are entirely intractable from a theoretical
point of view; so long as the gravitational fields are relatively weak (i.e.,
we aren't too close to too large a mass), this is a plausible approach.

·
the
curvature of space is continuously changing,

·
every
mass has its own self-energy that also changes spacetime's curvature,

·
objects
moving through curved space interact with it and emit gravitational radiation,

·
all
the gravitational signals generated only move at the speed of light,

·
and
the object's velocity relative to any other object results in a relativistic
(length contraction and time dilation) transformation that must be accounted
for.

When you take all of these into
account, it all adds up to most spacetimes that you can imagine, even
relatively simple ones, leading to equations that are so complex that we cannot
find a solution to Einstein's equations.

We cannot even write down the
Einstein field equations that describe most spacetimes or most Universes we can
imagine. Most of the ones we can write down cannot be solved. And most of the
ones that can be solved cannot be solved by me, you, or anyone. But still, we
can make approximations that allow us to extract some meaningful predictions
and descriptions. In the grand scheme of the cosmos, that's as close as
anyone's ever gotten to figuring it all out, but there's still much farther to
go. May we never give up until we get there.