Download The Multipole Moments

Multipole moments as a tool to infer
from gravitational waves the geometry
around an axisymmetric body.
Thomas P. Sotiriou
SISSA, International School for Advanced
Studies, Via Beirut, 2-4 34014 Trieste,
Italy.
Motivation
tracer of the
geometry of
spacetime

small body orbiting
around a much more
massive one

geometry described by the mass and mass current multipole
moments

gravitational wave
(GW) observables

central object endowed
with an electromagnetic
field
two additional families of moments,
the electric and the magnetic field
moments

GWs carry all information
needed to infer these
moments as well
possible in principle to get valuable
information about the central object’s
electromagnetic field‡
geodesic
motion
infer the
moments
†F
information about the
central object†
D Ryan, PRD 52, 5707 (1995)
‡ T P Sotiriou and T A Apostolatos, PRD 71, 044005 (2005)
The Model
A test particle orbiting around a much more massive compact
object
 Realistic description for a binary system composed of a 10^4
M○ and a 10^6 M○ BH for example
 Detection:
LIGO, VIRGO etc.
up to ~ 300 M○
LISA
from ~ 3×105 M○ to ~3×107 M○
 Assumptions:

1. Spacetime is stationary, axisymmetric and reflection
symmetric with respect to the equatorial plane. Fμν respects
the reflection symmetry
2. The test particle does not affect the geometry and its motion
is equatorial
3. GW energy comes mainly from the quadrupole formula
(emission quadrupole related to the system as a whole); no
absorption by the central object
The quantities related to the test particle are:
 Energy per unit mass
 Orbital frequency
 Main frequency
We can relate these quantities to a number of observables
Observables

GW Spectrum
Periastron and Orbital
Precession Frequency

 Number of Cycles per Logarithmic
Interval of Frequency
where
The Multipole Moments
We have expressed all quantities of interest with respect to the
metric components. On the other hand the metric can be
expressed in terms of the multipole moments.
It has been shown by Ernst that
this metric can be fully
φδ~γ
determined by two complex functions.
One can use instead two complex functions that are more
directly related to the moments.
These functions can be written as power series expansions
at infinity
where
and can be evaluated from their value on the symmetry axis
The coefficients mi and qi are related to the mass moments, Mi,
the mass current moments, Si, the electric field moments, Ei,
and the magnetic field moments, Hi.
LOM stands for “Lower Order Multipoles”
Algorithm




Express the metric functions as power series in ρ and z,
with coefficients depending only on the multipole
moments
The observable quantities discussed depend only on the
metric functions. Thus they can also be expressed as
power series in ρ, with coefficients depending only on the
multipole moments
Use the assumed symmetry to simplify the computation.
Mi is zero when i is odd and Si is zero when i is even. For
e/m moments two distinct cases: Ei is zero for odd i and
Hi is zero for even i (es), or vice versa (ms)
Finally, Ω=Ω(ρ), so we can express ρ, and consequently
the observables, as a series of Ω, or
Power Expansion Formulas
Using this algorithm we get the observable quantities as power
series of u
where for example
The results are similar for Rn and Zn both in the electric
symmetric case (es) and the magnetic symmetric case (ms)
The most interesting observable is the number of cycles since
it is the most accurately measured
where
Conclusions

Each coefficient of the power expansions
includes a number of moments. At every
order, one extra moment is present
compared to the lower order coefficient

Accuracy of measuring the observed quantities affects the
accuracy of the moment evaluation. Ryan: LIGO is not
expected to provide sufficient accuracy but LISA is very
promising and we expect to be able to measure a few lower
moments
Lower order moments can give valuable information: Mass,
angular momentum, magnetic dipole, overall charge (e.g. KerrNewman BH).


possible to
evaluate the
moments
One can also check whether the moments are interrelated as
in a Kerr-Newman metric. A negative outcome can indicate
either that the central object is not a black hole or that the
spacetime is seriously affected by the presence of matter in
the vicinity of the black hole (e.g. an accretion disk)