Donato Bini

Black hole perturbations: a review of recent analytical results

@CNR, Rome, IT 

[Based on works done in collaboration with T. Damour and  A. Geralico]

bh perturbations in the literature:

Metric perturbations: Regge‐Wheeler‐Zerilli (especially in the case of static bh)

Curvature perturbations: Teukolsky formalism(for stationary bh and general type D background)

In both cases for bh…

In the end one has to solve a set of coupledOrdinary Differential Equations, rewritten in turn as a single Schrodinger‐like equation in the radialvariable.

Nonvacuum perturbationsWe are interested in perturbations induced by a test particle/testgyroscope moving on the equatorial plane of a Schwarzschild or a Kerr bhalong a circular or eccentric (bound) orbit.

This topic is very well known…and numerical studies/soultions to this problem exist sincemany years.

Analytical solutions (in PN form) exist since 2013  only! 

GSFBH perturbations

A small‐mass, small‐spin body orbiting a black hole produces ametric perturbation «whose regularized part» can be fullyanalytically computed (in PN sense).

EOBPN NR

GSF, GST, BH PERTURBATION THEORY

Synergies EOB formalism «easily» incorporates(and uses) information coming from other contexts.

Whatever new result is obtained in any of these contexts can be immediately converted in EOB!

PM

The analytically computed gauge‐invariant perturbation quantitiesof GSF theory are converted intoan analytical knowledge of some EOB potential.

A continuous flow of information betweenvarious contexts.

The full reconstruction of a gravitational perturbation consistsin a sum over an infinite number of multipolar contributions.  The difficulty is to «control» each n‐pole among the infinite ones. Therefore, one is mostlydepending on the machine doingthe explicit computation.

Created in 1999 by A. Buonanno and T. Damour

What has been analytically computedin GSF theory in the last 4 y

• Schwarzschild bh, of mass m2, perturbed by a small mass m1orbiting

1) along a circular geodesic; 

2) along an eccentric geodesic;

• Kerr bh, of mass m2 and rotation a, perturbed by a small mass m1(in the small rotation limit or to low PN expansion) orbiting

1) along an equatorial circular geodesic; 

2) along an equatorial eccentric geodesic.

Skeleton: From a particle perturbing theSchwarzschild spacetime to the fullreconstruction of the perturbed metric

R

More properly…

GSF theory has computed certain gauge‐invariant quantities, both in Schwazrschild andKerr bh spacetimes, in PN‐expanded form andwithin certain limits (e.g., small –mass, small‐spin, small‐eccentricity limit, etc.)

Background metric (Schwarzschild and Kerr)

Metric perturbation

(small mass ratio)

Cause of the perturbation

Parametric equations of the  (geodesic) orbit.

Metric perturbationsConsider the linearized Einstein equations for the perturbed metric

where is the background metric. Expandthe Einstein tensor as

Metric perturbations in Schwarzschild

Decompose the perturbed metric and energy‐momentum tensorin tensorial spherical harmonics

RW gauge

are 10 tensor harmonics (Zerilli notation) 

General source terms, separated in their even and odd parts

Specialization of general results to the case in which the perturbing mass moves along a circular geodesic

Metric perturbation equations in Schwarzschild are usually studied in the RW gauge. 

Even and odd perturbations form a coupled system of PDE

In the RW gauge for the ODD part, which has the odd parity , one chooses

There are then 3 coupled equations for the two functions and    

For the EVEN part, which has the parity , one chooses

There are then 7  coupled equations for the four functions

Perturbations of defined parity…

ODD

3 eqs

EVEN

7 eqs

EVEN

Only coupled ODE (radial) equations

Trivial Fourier spectrum for circular orbitsFor a particle moving along a circular orbit there is a single  frequency in the Fourier spectrum

For a particle moving along bound eccentric equatorial orbits if oneworks up to e2 terms there are 3  frequencies; up to e4 terms there are 5  frequencies, etc. 

A single (RW) equation for the oddpart

3 eqs collapse into 1: a lucky circumstance!

A single (Z) equation for the even part7 eqs collapse into 1: a lucky circumstance!

Putting the odd and even partstogether: a single RWZ equation

Chandrasekhar transformation

2 eqs (odd‐even)  collapse into 1: a lucky circumstance!

Source terms and fundamental equation

Similarly for the even sources, with one more derivative of Dirac delta

Fundamental equation to be solved with the Green functionmethod

Now we have reduced the problem to a «single» ODE…let us proceed with its solution…

R in and R up are two independent solutions of the homogeneous RW equation which arepurely ingoing at the bh horizon and purely outgoing at spatial infinity.

The in and up sols of the hom RWZ equation: I  PN solutions

Restore the gravitational constant and the speed of light in the units

Expand both the RWZ equation and its solution in powers of 

It is enough to compute the PN in solution: a lucky circumstance!

Necessary to build up the Green function

One does not need to compute the up solution since this is obtained from the in solutionby replacing l  ‐l‐1

First PN terms…as an example

In the up solution this term would have l‐2 in the denominator.Sum over l of cannot include this term: divergencies!

The magnetic numberm is inside 

Renormalization group equation

Sum over m«Available» in closed form in termsof hypergeometric functions, for any N

Explicitly computed for l=2,3,4,…

Etc. up to m20  

Apart from the l=0,1 (pure gauge modes) whichare computed separately, one cannot include  A8in the PN solution since this would diverge for l=2. If one need high‐accuracy PN solutions thenthe contributions due to l=2, i.e. order

should be computed differently!  MST technology

Sum over l

The in and up solutions of the homogeneous RWZ equation: II 

MST solutions

[second Kummer function,re‐expressable in terms of hypergeom]

The up l=2 MST solution as an example

Increasing transcendental structure

The final program/goalOnce one has obtained the solutions for the homogeneous (and then for the non‐homogeneous too)  RWZ equation, from the auxiliary function R (even, odd) one can Fourier‐antitransform and build back the perturbed metric h [Details still to be shown!]Subtleties Finite sum over m Infinite sum over l (analytically)

The aim of our calculation is to compute a GI function of h, at the source location (where by definition) the perturbed metric is singular.

Subtleties Identify the needed GI quantity. Regularize it conveniently.  Evaluate it at the  source location.

Not to forget: 

h with helicalsymmetry

PN solutions of the homogeneous RW 

equation

MST solutions of the homogeneous

RW equation

solutions to the recurrence system

Summation rules for(ordinary) sh

hkk Spin Orbit Tidal

low multipoles

Schwarzschild, RW gauge: Maple codes

JWKB solutions of the homogeneous

RW equation

Our PN and MSTsolutions of the RWand Teukolsky equationwill be put in the nextmonths in a publicrepository.

Low multipoles for h kk: l=0,1These are purely gauge modes which carry into the hole the energy and the angularmomentum of the particle. For Schwarzschild perturbations they can be explicitly computedfrom a linearized Kerr metric in the mass and rotational parameter of the hole, once thesegeneric parameters are replaced by the corresponding contributions due to the particle.

Mass and angular momentum carried by the particle to the hole

These are crucial and very delicate…because of their gauge‐dependence!

summed over m (depending on l only)

Even if one avoids divergencies in the solution of the RW equation the final form of this quantity contains constant terms (in l) and it cannot be summed for l=N up to infinity.

This difficulty is overtaken by using a regularization procedure, which actually consists in removing the singular part. The subtraction term can be pre‐determined analytically.For example

Regularization

Example: final GSF result

Thibault Damour @ IHES, Paris, has «skeletonized» allthis procedure and computed by hands in advance the coefficients of the first log terms appearing in thisexpression, based on his deepest knowledge of PN formalism!

Let us call a spade a spade

Example: its EOB transcriptionGSF

EOB

Available GSF results

• Along circular orbits: 

• Along eccentric orbits, averaged quantitiesover a period of radial motion: 

[roughly speaking]

Reminder of EOB

BDG

…In collaboration with T. Damour and A. Geralico… 

Whatever quantity we have computed withinGSF has been converted into the EOB, leading to new information for the EOB potentials

A(u), D(u), Q(u), GS(u), GS*(u)

A brief account is given below

Schwarzschild a5

Analytical determination of the two‐body gravitational interaction potential at the fourth post‐Newtonian approximationD. Bini, T.  Damour May 21, 2013. 6 pp.Phys.Rev. D87 (2013) no.12, 121501DOI: 10.1103/PhysRevD.87.121501e‐Print: arXiv:1305.4884 [gr‐qc]

Schwarzschild a6,6.5,7High‐order post‐Newtonian contributions to the two‐body gravitational interaction potential from analytical gravitational self‐force calculationsD. Bini, T. Damour. Dec 9, 2013. 21 pp.Phys.Rev. D89 (2014) no.6, 064063DOI: 10.1103/PhysRevD.89.064063e‐Print: arXiv:1312.2503 [gr‐qc]

Schwarzschild a7.5,8,8.5,9,9.5Analytic determination of the eight‐and‐a‐half post‐Newtonian self‐force contributions to the two‐body gravitational interaction potentialD. Bini, T. Damour Mar 10, 2014. 13 pp.Phys.Rev. D89 (2014) no.10, 104047DOI: 10.1103/PhysRevD.89.104047e‐Print: arXiv:1403.2366 [gr‐qc]

Schwarzschild a10,10.5Detweiler’s gauge‐invariant redshift variable: Analytic determinationof the nine and nine‐and‐a‐half post‐Newtonian self‐force contributionsD. Bini, T. Damour. Feb 9, 2015. 4 pp.Phys.Rev. D91 (2015) 064050DOI: 10.1103/PhysRevD.91.064050e‐Print: arXiv:1502.02450 [gr‐qc]

[C. Kavanagh and coll.have raised the presentknowledge of a(u) up to 22.5PN]

The EOB‐SO sectorTwo‐body gravitationalspin‐orbit interaction atlinear order in the mass ratioD. Bini, T. Damour Apr 10, 2014. 22 pp.Phys.Rev. D90 (2014) no.2, 024039DOI: 10.1103/PhysRevD.90.024039e‐Print: arXiv:1404.2747 [gr‐qc]

The EOB‐SO sector

Spin‐dependent two‐body interactions from gravitational self‐force computationsD. Bini, T. Damour, A. Geralico. Oct 21, 2015. 18 pp.Phys.Rev. D92 (2015) no.12, 124058, Erratum: Phys.Rev. D93 (2016) no.10, 109902DOI: 10.1103/PhysRevD.93.109902, 10.1103/PhysRevD.92.124058e‐Print: arXiv:1510.06230 [gr‐qc]

Gravitational self‐force corrections to two‐body tidalinteractions and the effective one‐body formalismD. Bini, T. Damour. Sep 24, 2014. 39 pp.Phys.Rev. D90 (2014) no.12, 124037DOI: 10.1103/PhysRevD.90.124037e‐Print: arXiv:1409.6933 [gr‐qc]

Confirming and improving post‐Newtonian and effective‐one‐body results from self‐force computationsalong eccentric orbits around a Schwarzschild blackholeD. Bini, T. Damour, A. Geralico. Nov 14, 2015. 15 pp.Phys.Rev. D93 (2016) no.6, 064023DOI: 10.1103/PhysRevD.93.064023e‐Print: arXiv:1511.04533 [gr‐qc]

New gravitational self‐force analytical results for eccentric orbits around a Schwarzschild black holeD. Bini, T. Damour, A. Geralico. Jan 12, 2016. 27 pp.Phys.Rev. D93 (2016) no.10, 104017DOI: 10.1103/PhysRevD.93.104017e‐Print: arXiv:1601.02988 [gr‐qc]

High post‐Newtonian order gravitational self‐force analytical results for eccentric equatorial orbitsaround a Kerr black holeD. Bini, T. Damour, A. Geralico.Feb 26, 2016. 13 pp.Phys.Rev. D93 (2016) no.12, 124058DOI: 10.1103/PhysRevD.93.124058e‐Print: arXiv:1602.08282 [gr‐qc]

Alternative/competitor  numerical‐analytic methods

The idea of Abhay Shah

From Shah slides at Capra 2014

From Shah slides at Capra 2014

From Shah slides at Capra 2014

From Shah slides at Capra 2014

From Shah slides at Capra 2014

But one should know this in advance!

PSLQ algorithm

An algorithm which can be used to find integer relations between real numbers

such that

with not all . 

Although the algorithm operates by manipulating a lattice, it does not reduce it to a short vector basis, and is therefore not a lattice reduction algorithm. 

PSLQ is based on a partial sum of squares. 

It was developed by Ferguson and Bailey (1992). 

The present analytical knowledge of the main EOB potential

Even if these potentials have a known analyticalform in PN sense up to very high PN orders, theyare often used in a resummed form in the EOBnumerical/companion codes, for an obviousconvenience.

a(x)

d(x)_

q(x)

(x)

GS and f A

Working with MapleTM… while mostof people uses MathematicaTM

Mathematica seems more efficient than Maple in GSF computations, especially whenworking at very high orders. 

Maple is still more convenient when working with an explicitly given metric, since anytensorial calculus on a given manifold can be performed easily in Maple.

The ideal situation is when one is comparing results obtained by using two codes developedin the two different languages, Mathematica and Maple, especially when one ismanipulating very large expressions, using pre‐defined simplifying routines, consummingalmost all the available memory resources of the machines, etc.

Future GSF works should involve group of researchers.

The main open problems in GSFTODAY

The analytical, explicit expression of the gauge‐dependent low multipoles l=0,1 along genericorbits in Schwarzschild and Kerr.

Slow progress in 1 y.

Second order perturbation equations?

Thanks for your kind attention!