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 small body orbiting

around a much more massive one

† F D Ryan, PRD 52, 5707 (1995)‡ T P Sotiriou and T A Apostolatos, PRD 71, 044005 (2005)

geometry described by the mass and mass current multipole moments

gravitational wave (GW) observables

central object endowed with an electromagnetic field

GWs carry all information needed to infer these moments as well

geodesic motion

tracer of the geometry of spacetime

infer the moments

information about the central object†

two additional families of moments, the electric and the magnetic field moments

possible in principle to get valuable information about the central object’s electromagnetic field‡

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:

Main frequency

Energy per unit mass

Orbital 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 MomentsWe 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 determined by two complex functions.

One can use instead two complex functions that are moredirectly related to the moments.

~~ γφδ

These functions can be written as power series expansionsat 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 FormulasUsing this algorithm we get the observable quantities as powerseries 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 sinceit 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

possible to evaluate the moments

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. Kerr-Newman BH).

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)