<s>
Numerical	B-Algorithm
relativity	I-Algorithm
is	O
one	O
of	O
the	O
branches	O
of	O
general	O
relativity	O
that	O
uses	O
numerical	B-General_Concept
methods	I-General_Concept
and	O
algorithms	O
to	O
solve	O
and	O
analyze	O
problems	O
.	O
</s>
<s>
To	O
this	O
end	O
,	O
supercomputers	B-Architecture
are	O
often	O
employed	O
to	O
study	O
black	B-Application
holes	I-Application
,	O
gravitational	O
waves	O
,	O
neutron	O
stars	O
and	O
many	O
other	O
phenomena	O
governed	O
by	O
Einstein	O
's	O
theory	O
of	O
general	O
relativity	O
.	O
</s>
<s>
A	O
currently	O
active	O
field	O
of	O
research	O
in	O
numerical	B-Algorithm
relativity	I-Algorithm
is	O
the	O
simulation	O
of	O
relativistic	O
binaries	O
and	O
their	O
associated	O
gravitational	O
waves	O
.	O
</s>
<s>
A	O
primary	O
goal	O
of	O
numerical	B-Algorithm
relativity	I-Algorithm
is	O
to	O
study	O
spacetimes	B-Protocol
whose	O
exact	O
form	O
is	O
not	O
known	O
.	O
</s>
<s>
The	O
spacetimes	B-Protocol
so	O
found	O
computationally	O
can	O
either	O
be	O
fully	O
dynamical	O
,	O
stationary	O
or	O
static	O
and	O
may	O
contain	O
matter	O
fields	O
or	O
vacuum	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
stationary	O
and	O
static	O
solutions	O
,	O
numerical	B-General_Concept
methods	I-General_Concept
may	O
also	O
be	O
used	O
to	O
study	O
the	O
stability	B-Algorithm
of	O
the	O
equilibrium	O
spacetimes	B-Protocol
.	O
</s>
<s>
In	O
the	O
case	O
of	O
dynamical	O
spacetimes	B-Protocol
,	O
the	O
problem	O
may	O
be	O
divided	O
into	O
the	O
initial	O
value	O
problem	O
and	O
the	O
evolution	O
,	O
each	O
requiring	O
different	O
methods	O
.	O
</s>
<s>
Numerical	B-Algorithm
relativity	I-Algorithm
is	O
applied	O
to	O
many	O
areas	O
,	O
such	O
as	O
cosmological	O
models	O
,	O
critical	O
phenomena	O
,	O
perturbed	O
black	B-Application
holes	I-Application
and	O
neutron	O
stars	O
,	O
and	O
the	O
coalescence	O
of	O
black	B-Application
holes	I-Application
and	O
neutron	O
stars	O
,	O
for	O
example	O
.	O
</s>
<s>
Like	O
all	O
problems	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
careful	O
attention	O
is	O
paid	O
to	O
the	O
stability	B-Algorithm
and	O
convergence	O
of	O
the	O
numerical	B-General_Concept
solutions	I-General_Concept
.	O
</s>
<s>
In	O
this	O
line	O
,	O
much	O
attention	O
is	O
paid	O
to	O
the	O
gauge	O
conditions	O
,	O
coordinates	O
,	O
and	O
various	O
formulations	O
of	O
the	O
Einstein	O
equations	O
and	O
the	O
effect	O
they	O
have	O
on	O
the	O
ability	O
to	O
produce	O
accurate	O
numerical	B-General_Concept
solutions	I-General_Concept
.	O
</s>
<s>
Numerical	B-Algorithm
relativity	I-Algorithm
research	O
is	O
distinct	O
from	O
work	O
on	O
classical	O
field	O
theories	O
as	O
many	O
techniques	O
implemented	O
in	O
these	O
areas	O
are	O
inapplicable	O
in	O
relativity	O
.	O
</s>
<s>
Numerical	O
relativists	O
often	O
work	O
with	O
applied	O
mathematicians	O
and	O
draw	O
insight	O
from	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
scientific	O
computation	O
,	O
partial	O
differential	O
equations	O
,	O
and	O
geometry	O
among	O
other	O
mathematical	O
areas	O
of	O
specialization	O
.	O
</s>
<s>
It	O
,	O
like	O
his	O
earlier	O
theory	O
of	O
special	O
relativity	O
,	O
described	O
space	O
and	O
time	O
as	O
a	O
unified	O
spacetime	B-Protocol
subject	O
to	O
what	O
are	O
now	O
known	O
as	O
the	O
Einstein	O
field	O
equations	O
.	O
</s>
<s>
The	O
field	O
of	O
numerical	B-Algorithm
relativity	I-Algorithm
emerged	O
from	O
the	O
desire	O
to	O
construct	O
and	O
study	O
more	O
general	O
solutions	O
to	O
the	O
field	O
equations	O
by	O
approximately	O
solving	O
the	O
Einstein	O
equations	O
numerically	B-General_Concept
.	O
</s>
<s>
A	O
necessary	O
precursor	O
to	O
such	O
attempts	O
was	O
a	O
decomposition	O
of	O
spacetime	B-Protocol
back	O
into	O
separated	O
space	O
and	O
time	O
.	O
</s>
<s>
Although	O
for	O
technical	O
reasons	O
the	O
precise	O
equations	O
formulated	O
in	O
the	O
original	O
ADM	O
paper	O
are	O
rarely	O
used	O
in	O
numerical	O
simulations	O
,	O
most	O
practical	O
approaches	O
to	O
numerical	B-Algorithm
relativity	I-Algorithm
use	O
a	O
"	O
3+1	O
decomposition	O
"	O
of	O
spacetime	B-Protocol
into	O
three-dimensional	O
space	O
and	O
one-dimensional	O
time	O
that	O
is	O
closely	O
related	O
to	O
the	O
ADM	O
formulation	O
,	O
because	O
the	O
ADM	O
procedure	O
reformulates	O
the	O
Einstein	O
field	O
equations	O
into	O
a	O
constrained	B-Application
initial	O
value	O
problem	O
that	O
can	O
be	O
addressed	O
using	O
computational	O
methodologies	O
.	O
</s>
<s>
At	O
the	O
time	O
that	O
ADM	O
published	O
their	O
original	O
paper	O
,	O
computer	O
technology	O
would	O
not	O
have	O
supported	O
numerical	B-General_Concept
solution	I-General_Concept
to	O
their	O
equations	O
on	O
any	O
problem	O
of	O
any	O
substantial	O
size	O
.	O
</s>
<s>
The	O
first	O
documented	O
attempt	O
to	O
solve	O
the	O
Einstein	O
field	O
equations	O
numerically	B-General_Concept
appears	O
to	O
be	O
Hahn	O
and	O
Lindquist	O
in	O
1964	O
,	O
followed	O
soon	O
thereafter	O
by	O
Smarr	O
and	O
by	O
Eppley	O
.	O
</s>
<s>
In	O
this	O
calculation	O
Piran	O
has	O
set	O
the	O
foundation	O
for	O
many	O
of	O
the	O
concepts	O
used	O
today	O
in	O
evolving	O
ADM	O
equations	O
,	O
like	O
"	O
free	O
evolution	O
"	O
versus	O
"	O
constrained	B-Application
evolution	O
"	O
,	O
which	O
deal	O
with	O
the	O
fundamental	O
problem	O
of	O
treating	O
the	O
constraint	B-Application
equations	O
that	O
arise	O
in	O
the	O
ADM	O
formalism	O
.	O
</s>
<s>
Applying	O
symmetry	O
reduced	O
the	O
computational	O
and	O
memory	O
requirements	O
associated	O
with	O
the	O
problem	O
,	O
allowing	O
the	O
researchers	O
to	O
obtain	O
results	O
on	O
the	O
supercomputers	B-Architecture
available	O
at	O
the	O
time	O
.	O
</s>
<s>
The	O
first	O
realistic	O
calculations	O
of	O
rotating	O
collapse	O
were	O
carried	O
out	O
in	O
the	O
early	O
eighties	O
by	O
Richard	O
Stark	O
and	O
Tsvi	O
Piran	O
in	O
which	O
the	O
gravitational	O
wave	O
forms	O
resulting	O
from	O
formation	O
of	O
a	O
rotating	O
black	B-Application
hole	I-Application
were	O
calculated	O
for	O
the	O
first	O
time	O
.	O
</s>
<s>
For	O
nearly	O
20	O
years	O
following	O
the	O
initial	O
results	O
,	O
there	O
were	O
fairly	O
few	O
other	O
published	O
results	O
in	O
numerical	B-Algorithm
relativity	I-Algorithm
,	O
probably	O
due	O
to	O
the	O
lack	O
of	O
sufficiently	O
powerful	O
computers	O
to	O
address	O
the	O
problem	O
.	O
</s>
<s>
In	O
the	O
late	O
1990s	O
,	O
the	O
Binary	O
Black	B-Application
Hole	I-Application
Grand	B-Device
Challenge	I-Device
Alliance	O
successfully	O
simulated	O
a	O
head-on	O
binary	O
black	B-Application
hole	I-Application
collision	O
.	O
</s>
<s>
As	O
a	O
post-processing	O
step	O
the	O
group	O
computed	O
the	O
event	O
horizon	O
for	O
the	O
spacetime	B-Protocol
.	O
</s>
<s>
Some	O
of	O
the	O
first	O
documented	O
attempts	O
to	O
solve	O
the	O
Einstein	O
equations	O
in	O
three	O
dimensions	O
were	O
focused	O
on	O
a	O
single	O
Schwarzschild	O
black	B-Application
hole	I-Application
,	O
which	O
is	O
described	O
by	O
a	O
static	O
and	O
spherically	O
symmetric	O
solution	O
to	O
the	O
Einstein	O
field	O
equations	O
.	O
</s>
<s>
This	O
provides	O
an	O
excellent	O
test	O
case	O
in	O
numerical	B-Algorithm
relativity	I-Algorithm
because	O
it	O
does	O
have	O
a	O
closed-form	O
solution	O
so	O
that	O
numerical	O
results	O
can	O
be	O
compared	O
to	O
an	O
exact	O
solution	O
,	O
because	O
it	O
is	O
static	O
,	O
and	O
because	O
it	O
contains	O
one	O
of	O
the	O
most	O
numerically	B-General_Concept
challenging	O
features	O
of	O
relativity	O
theory	O
,	O
a	O
physical	O
singularity	O
.	O
</s>
<s>
"	O
Progress	O
in	O
three	O
dimensional	O
numerical	B-Algorithm
relativity	I-Algorithm
has	O
been	O
impeded	O
in	O
part	O
by	O
lack	O
of	O
computers	O
with	O
sufficient	O
memory	O
and	O
computational	O
power	O
to	O
perform	O
well	O
resolved	O
calculations	O
of	O
3D	O
spacetimes.	O
"	O
</s>
<s>
With	O
respect	O
to	O
black	B-Application
hole	I-Application
simulations	O
specifically	O
,	O
two	O
techniques	O
were	O
devised	O
to	O
avoid	O
problems	O
associated	O
with	O
the	O
existence	O
of	O
physical	O
singularities	O
in	O
the	O
solutions	O
to	O
the	O
equations	O
:	O
(	O
1	O
)	O
Excision	O
,	O
and	O
(	O
2	O
)	O
the	O
"	O
puncture	O
"	O
method	O
.	O
</s>
<s>
More	O
generally	O
,	O
adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
techniques	O
,	O
already	O
used	O
in	O
computational	O
fluid	O
dynamics	O
were	O
introduced	O
to	O
the	O
field	O
of	O
numerical	B-Algorithm
relativity	I-Algorithm
.	O
</s>
<s>
In	O
the	O
excision	O
technique	O
,	O
which	O
was	O
first	O
proposed	O
in	O
the	O
late	O
1990s	O
,	O
a	O
portion	O
of	O
a	O
spacetime	B-Protocol
inside	O
of	O
the	O
event	O
horizon	O
surrounding	O
the	O
singularity	O
of	O
a	O
black	B-Application
hole	I-Application
is	O
simply	O
not	O
evolved	O
.	O
</s>
<s>
In	O
theory	O
this	O
should	O
not	O
affect	O
the	O
solution	O
to	O
the	O
equations	O
outside	O
of	O
the	O
event	O
horizon	O
because	O
of	O
the	O
principle	O
of	O
causality	B-Application
and	O
properties	O
of	O
the	O
event	O
horizon	O
(	O
i.e.	O
</s>
<s>
nothing	O
physical	O
inside	O
the	O
black	B-Application
hole	I-Application
can	O
influence	O
any	O
of	O
the	O
physics	O
outside	O
the	O
horizon	O
)	O
.	O
</s>
<s>
The	O
second	O
problem	O
is	O
that	O
as	O
the	O
black	B-Application
holes	I-Application
move	O
,	O
one	O
must	O
continually	O
adjust	O
the	O
location	O
of	O
the	O
excision	O
region	O
to	O
move	O
with	O
the	O
black	B-Application
hole	I-Application
.	O
</s>
<s>
The	O
excision	O
technique	O
was	O
developed	O
over	O
several	O
years	O
including	O
the	O
development	O
of	O
new	O
gauge	O
conditions	O
that	O
increased	O
stability	B-Algorithm
and	O
work	O
that	O
demonstrated	O
the	O
ability	O
of	O
the	O
excision	O
regions	O
to	O
move	O
through	O
the	O
computational	O
grid	O
.	O
</s>
<s>
The	O
first	O
stable	O
,	O
long-term	O
evolution	O
of	O
the	O
orbit	O
and	O
merger	O
of	O
two	O
black	B-Application
holes	I-Application
using	O
this	O
technique	O
was	O
published	O
in	O
2005	O
.	O
</s>
<s>
In	O
the	O
puncture	O
method	O
the	O
solution	O
is	O
factored	O
into	O
an	O
analytical	O
part	O
,	O
which	O
contains	O
the	O
singularity	O
of	O
the	O
black	B-Application
hole	I-Application
,	O
and	O
a	O
numerically	B-General_Concept
constructed	O
part	O
,	O
which	O
is	O
then	O
singularity	O
free	O
.	O
</s>
<s>
This	O
is	O
a	O
generalization	O
of	O
the	O
Brill-Lindquist	O
prescription	O
for	O
initial	O
data	O
of	O
black	B-Application
holes	I-Application
at	O
rest	O
and	O
can	O
be	O
generalized	O
to	O
the	O
Bowen-York	O
prescription	O
for	O
spinning	O
and	O
moving	O
black	B-Application
hole	I-Application
initial	O
data	O
.	O
</s>
<s>
Of	O
course	O
black	B-Application
holes	I-Application
in	O
proximity	O
to	O
each	O
other	O
will	O
tend	O
to	O
move	O
under	O
the	O
force	O
of	O
gravity	O
,	O
so	O
the	O
fact	O
that	O
the	O
coordinate	O
position	O
of	O
the	O
puncture	O
remained	O
fixed	O
meant	O
that	O
the	O
coordinate	O
systems	O
themselves	O
became	O
"	O
stretched	O
"	O
or	O
"	O
twisted	O
,	O
"	O
and	O
this	O
typically	O
led	O
to	O
numerical	B-Algorithm
instabilities	I-Algorithm
at	O
some	O
stage	O
of	O
the	O
simulation	O
.	O
</s>
<s>
This	O
allowed	O
accurate	O
long-term	O
evolutions	O
of	O
black	B-Application
holes	I-Application
.	O
</s>
<s>
By	O
choosing	O
appropriate	O
coordinate	O
conditions	O
and	O
making	O
crude	O
analytic	O
assumption	O
about	O
the	O
fields	O
near	O
the	O
singularity	O
(	O
since	O
no	O
physical	O
effects	O
can	O
propagate	O
out	O
of	O
the	O
black	B-Application
hole	I-Application
,	O
the	O
crudeness	O
of	O
the	O
approximations	O
does	O
not	O
matter	O
)	O
,	O
numerical	B-General_Concept
solutions	I-General_Concept
could	O
be	O
obtained	O
to	O
the	O
problem	O
of	O
two	O
black	B-Application
holes	I-Application
orbiting	O
each	O
other	O
,	O
as	O
well	O
as	O
accurate	O
computation	O
of	O
gravitational	O
radiation	O
(	O
ripples	O
in	O
spacetime	B-Protocol
)	O
emitted	O
by	O
them	O
.	O
</s>
<s>
2005	O
was	O
renamed	O
the	O
"	O
annus	O
mirabilis	O
"	O
of	O
numerical	B-Algorithm
relativity	I-Algorithm
,	O
100	O
years	O
after	O
the	O
annus	O
mirabilis	O
of	O
special	O
relativity	O
(	O
1905	O
)	O
.	O
</s>
<s>
The	O
Lazarus	O
project	O
(	O
1998	O
–	O
2005	O
)	O
was	O
developed	O
as	O
a	O
post-Grand	O
Challenge	O
technique	O
to	O
extract	O
astrophysical	O
results	O
from	O
short	O
lived	O
full	O
numerical	O
simulations	O
of	O
binary	O
black	B-Application
holes	I-Application
.	O
</s>
<s>
It	O
combined	O
approximation	O
techniques	O
before	O
(	O
post-Newtonian	O
trajectories	O
)	O
and	O
after	O
(	O
perturbations	O
of	O
single	O
black	B-Application
holes	I-Application
)	O
with	O
full	O
numerical	O
simulations	O
attempting	O
to	O
solve	O
General	O
Relativity	O
field	O
equations	O
.	O
</s>
<s>
All	O
previous	O
attempts	O
to	O
numerically	B-General_Concept
integrate	O
in	O
supercomputers	B-Architecture
the	O
Hilbert-Einstein	O
equations	O
describing	O
the	O
gravitational	O
field	O
around	O
binary	O
black	B-Application
holes	I-Application
led	O
to	O
software	O
failure	O
before	O
a	O
single	O
orbit	O
was	O
completed	O
.	O
</s>
<s>
The	O
Lazarus	O
approach	O
,	O
in	O
the	O
meantime	O
,	O
gave	O
the	O
best	O
insight	O
into	O
the	O
binary	O
black	B-Application
hole	I-Application
problem	O
and	O
produced	O
numerous	O
and	O
relatively	O
accurate	O
results	O
,	O
such	O
as	O
the	O
radiated	O
energy	O
and	O
angular	O
momentum	O
emitted	O
in	O
the	O
latest	O
merging	O
state	O
,	O
the	O
linear	O
momentum	O
radiated	O
by	O
unequal	O
mass	O
holes	O
,	O
and	O
the	O
final	O
mass	O
and	O
spin	O
of	O
the	O
remnant	O
black	B-Application
hole	I-Application
.	O
</s>
<s>
The	O
method	O
also	O
computed	O
detailed	O
gravitational	O
waves	O
emitted	O
by	O
the	O
merger	O
process	O
and	O
predicted	O
that	O
the	O
collision	O
of	O
black	B-Application
holes	I-Application
is	O
the	O
most	O
energetic	O
single	O
event	O
in	O
the	O
Universe	O
,	O
releasing	O
more	O
energy	O
in	O
a	O
fraction	O
of	O
a	O
second	O
in	O
the	O
form	O
of	O
gravitational	O
radiation	O
than	O
an	O
entire	O
galaxy	O
in	O
its	O
lifetime	O
.	O
</s>
<s>
Adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
(	O
AMR	O
)	O
as	O
a	O
numerical	O
method	O
has	O
roots	O
that	O
go	O
well	O
beyond	O
its	O
first	O
application	O
in	O
the	O
field	O
of	O
numerical	B-Algorithm
relativity	I-Algorithm
.	O
</s>
<s>
Mesh	B-Algorithm
refinement	I-Algorithm
first	O
appears	O
in	O
the	O
numerical	B-Algorithm
relativity	I-Algorithm
literature	O
in	O
the	O
1980s	O
,	O
through	O
the	O
work	O
of	O
Choptuik	O
in	O
his	O
studies	O
of	O
critical	O
collapse	O
of	O
scalar	O
fields	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
AMR	O
has	O
also	O
been	O
applied	O
to	O
the	O
study	O
of	O
inhomogeneous	O
cosmologies	O
,	O
and	O
to	O
the	O
study	O
of	O
Schwarzschild	O
black	B-Application
holes	I-Application
.	O
</s>
<s>
The	O
technique	O
has	O
now	O
become	O
a	O
standard	O
tool	O
in	O
numerical	B-Algorithm
relativity	I-Algorithm
and	O
has	O
been	O
used	O
to	O
study	O
the	O
merger	O
of	O
black	B-Application
holes	I-Application
and	O
other	O
compact	O
objects	O
in	O
addition	O
to	O
the	O
propagation	O
of	O
gravitational	O
radiation	O
generated	O
by	O
such	O
astronomical	O
events	O
.	O
</s>
<s>
In	O
the	O
past	O
few	O
years	O
,	O
hundreds	O
of	O
research	O
papers	O
have	O
been	O
published	O
leading	O
to	O
a	O
wide	O
spectrum	O
of	O
mathematical	O
relativity	O
,	O
gravitational	O
wave	O
,	O
and	O
astrophysical	O
results	O
for	O
the	O
orbiting	O
black	B-Application
hole	I-Application
problem	O
.	O
</s>
<s>
This	O
technique	O
extended	O
to	O
astrophysical	O
binary	O
systems	O
involving	O
neutron	O
stars	O
and	O
black	B-Application
holes	I-Application
,	O
and	O
multiple	O
black	B-Application
holes	I-Application
.	O
</s>
<s>
One	O
of	O
the	O
most	O
surprising	O
predictions	O
is	O
that	O
the	O
merger	O
of	O
two	O
black	B-Application
holes	I-Application
can	O
give	O
the	O
remnant	O
hole	O
a	O
speed	O
of	O
up	O
to	O
4000km/s	O
that	O
can	O
allow	O
it	O
to	O
escape	O
from	O
any	O
known	O
galaxy	O
.	O
</s>
