## Einstein's Equations

Einstein
Equation's describe the dynamic of
spacetime. They couple
geometry to a matter source, i.e., the distribution of matter
determines how the spacetime is curved. In turn, the motion of matter
is affected by the geometry of spacetime. The situation is similar to
electrodynamics where the
electric and magnetic fields affect the motion of charged particles
which in turn generate electric and magnetic fields. Although one
can (formally) write the
EE in a very compact form they are in fact
very complex
and constitute a system of coupled non-linear ellipto-hyperbolic PDE.
Even without presence of any matter the solutions of EE can be very
interesting and non-trivial (e.g., gravitational
waves, black
holes).
Another peculiar property of EEs is the possibility
of formation of singularities
from an originally
regular matter distribution.

Due to the inherent complexities of the EE, they are exactly solvable
only for limited class of problems - typically exhibiting high degree
of symmetry and describing simple matter (or no matter at all).