Description of Research Projects - Melanie Chanona
Boson
Stars
and
Black
Holes
in
2+1
Dimensions
In
the
summer
of
2012,
I
investigated
a
special
class
of
black
hole
and
boson
star
solutions
that
only
exist
in
a
spacetime
that
has
two
spatial
dimensions
and
one
time
dimension.
Boson
stars
are
theoretical
objects
that,
in
contrast
to
black
holes,
have
no
event
horizon.
Solutions
in
this
type
of
spacetime
are
of
considerable
interest
as
it
has
developed
into
a
fascinating
test
bed
for
understanding
quantum
gravity
and
has
the
potential
to
increase
our
understanding
of
the
famous
“AdS/CFT”
correspondence
conjecture.
My
work
focused
particularly
on
finding
solutions
with
helical
symmetry
that
were
hitherto
unknown.
I
was
successful
in
fully
constructing
numerical
boson
stars
solutions
and
additionally
was
able
to
find
strong
evidence
to
support
the
existence
of
black
hole
solutions,
which
had
been
conjectured
in
earlier
work
by
my
supervisor
Robert
Mann
and
PhD
student
Sean
Stotyn.
Angular
Momentum
Conservation
in
the
Earth-Moon
System:
A
Quasilocal
Approach
In
the
summer
of
2013,
I
switched
my
focus
within
the
field
of
general
relativity
from
numerical
to
theoretical
work
and
helped
develop
a
novel
theory
that
expresses
the
fundamental
properties
of
the
universe
such
as
conservation
of
energy
and
momentum
in
a
more
elegant
way
than
has
been
done
before.
The
key
concept
in
this
approach
is
to
enclose
the
system
of
interest
in
a
two
dimensional
quasilocal
rigid
surface
(called
an
RQF)
in
order
to
establish
a
mechanism
for
measuring
the
changing
energy
fluxes
as
the
system
evolves.
My
main
goal
was
to
demonstrate
the
applicability
and
efficacy
of
this
approach
by
accurately
predicting
the
rate
of
recession
of
the
Moon,
which
is
spiraling
away
from
the
Earth
due
to
tidal
effects,
thus
causing
a
transfer
of
angular
momentum.
The
RQF
approach
precisely
reproduced
the
observed
result
of
3.8cm/year
and,
most
importantly,
I
was
able
to
show
this
without
requiring
the
restrictive
conditions
and
assumptions
of
traditional
methods.
Thus
my
work
was
successful
in
motivating
the
use
of
RQFs
and
supported
the
claim
that
it
is
a
superior
method
for
understanding
and
describing
gravitational
effects
in
a
broad
variety
of
astrophysical
systems.
Linear
Stability
of
Oceanic
Fronts
in
a
Shallow
Water
Model
In
the
summer
of
2014,
I
explored
the
field
of
geophysical
fluid
dynamics.
My
research
involved
the
growth
of
instabilities
in
currents
under
a
variety
of
physical
constraints.
I
began
by
looking
at
a
three-layer
shallow
water
model
that
simulated
the
Antarctic
Shelf
Front,
which
has
striking
topographical
features
that
affect
the
evolution
of
the
current
that
travels
across
it.
The
complexity
of
this
current
motivated
a
closer
examination
of
the
dynamics
of
each
layer
independently
and
I
was
able
to
determine
that
the
most
interesting
features
occurred
when
the
surface
layer
was
similar
to
a
hyperbolic
tangent
profile.
Using
spectral
collocation
methods
to
numerically
solve
the
equations
that
govern
the
dynamics
of
the
current,
I
was
able
to
solve
for
both
large
scale
and
small
scale
instabilities
and
see
how
the
growth
rates
depended
on
the
depth
of
the
layer.
Under
specific
conditions,
we
can
allow
the
surface
profile
to
‘outcrop’
on
one
side,
in
other
words,
we
can
consider
a
vanishing
layer
depth.
While
previous
work
has
speculated
that
a
two-layer
model
was
needed
to
find
certain
unstable
modes,
I
was
able
to
demonstrate
that
they
in
fact
can
develop
in
a
one-layer
model
only.
Finally,
I
examined
a
parabolic
profile
and
was
able
to
categorize
the
instabilities
into
pairs
that
have
peculiar
asymmetries
on
either
side
of
the
current
jet.
This
also
had
not
been
addressed
in
earlier
explorations
of
this
type
of
setup.