The eth formalism in numerical relativity R GomezL Lehner P Papadopoulos z and J Winicour Department of Physics and Astronomy University of Pittsburgh Pittsburgh PA USA Abstract We present a nite dierence version of the eth formalism which allows use of tensor elds in spherical co ordinates in a manner which avoids p olar singularities The metho d employs two overlapping stereographic co ordinate patches with interpolations b etween the patches in the regions of overlap It provides a new and eective computational to ol for dealing with a wide variety of systems in which spherical co ordinates are natural such as the generation of radiation from an isolated source We test the formalism with the evolution of waves in three spatial dimensions and the calculation of the curvature scalar of arbitrarily curved geometries on top ologically spherical manifolds The formalism is applied to the solution of the RobinsonTrautman equation and reveals some new features of gravitational waveforms in the nonlinear regime PACS numbers x Cb gr-qc/9702002 31 Jan 1997 Short title The eth formalism in numerical relativity August z Present address Department of Astronomy and Astrophysics Pennsylvania State University University Park PA USA Introduction The inspiral and merger of a binary black hole system is anticipated to b e the prime source of gravitational waves for future wave detectors Calculation of the emitted waveform by means of emerging sup ercomputer technology is the goal of the Binary Black Hole Grand Challenge In this pap er we present a formalism which plays a strategic part in extracting the waveform pro duced in this three dimensional problem At large distances from a compact source the wavefronts of any radiation eld b ecome spherical this leads naturally to the use of a spherical co ordinate system Indeed spherical co ordinates and spherical harmonics are standard analytic to ols in the description of radiation But the use of spherical co ordinates in numerical work leads invariably to the vexing problem of co ordinate singularities at the origin and along the p olar axis Finite dierence approximations suer particularly b ecause they have no natural way of enforcing the correct b oundary conditions on their solutions As a result spherical co ordinates have mainly b een used in axisymmetric systems where the p olar singularities may b e regularized by standard tricks In the absence of symmetry these techniques do not easily generalize and they would b e esp ecially prohibitive to develop for tensor elds This pap er presents an approach that should b e of interest to researchers working in numerical relativity in problems where spherical co ordinates are a natural to ol The eth formalism and the asso ciated spinweighted spherical harmonics allow a simple and unied description of vector and tensor elds without the undue complexity of traditional vector and tensor harmonics Unfortunately these techniques have remained obscure and unfamiliar to many researchers who would b enet from them We present here a nite dierence version of the eth formalism with applications to numerical relativity The framework for the computational application of the eth formalism is developed in Sec Although our presentation is on the algorithmic level we present enough detail to direct a well dened transition from algorithm to co de development In Sec we present tests and applications of the formalism which demonstrate b oth its accuracy and usefulness The rst application the evolution of scalar waves is illustrative of the techniques necessary to carry out a dimensional evolution using spherical co ordinates The second application the numerical calculation of the curvature scalar of a top ologically spherical manifold illustrates the global and tensorial features of the metho d This calculation is at the heart of several imp ortant problems such as the computation of the Hawking mass The third application solution of the RobinsonTrautman equation reveals new and unexp ected nonlinear prop erties of the gravitational waveform This work illustrates how computational relativity can greatly b enet from approaches based up on the b est analytic to ols The Eth Formalism We present here a nite dierence treatment of the sphere based up on i the standard metho d of describing the global dierentiability of functions by means of lo cal co ordinate patches and iithe eth formalism for describing dierentiation of elds on the sphere Patching Let and lab el the p oints on the sphere Then the real and imaginary parts of i the stereographic co ordinate tan e provide a smo oth co ordinatization of the N sphere excluding the p oint at the south p ole Similarly provides a S N smo oth co ordinatization except at the north p ole These two stereographic co ordinate patches are sucient to cover the sphere It would b e p ossible in principle to cover the sphere with a number of nonsingular patches by selecting several directions on the sphere and asso ciating with them a lo cal co ordinate patch and then i i excluding from those patches the lo cal p olar region Stereographic co ordinates provide the most economical choice where only two nonsingular patches are used A smo oth scalar eld on the sphere may b e represented in terms of smo oth functions of on the lower hemisphere j j and as smo oth functions of S S S N on the upp er hemisphere j j The continuity of is ensured by requiring N N at the equator but in order to ensure smo othness this equality must S S N N extend to a common overlap region ab out the equator Our rst task is to implement this in a nite dierence treatment 1 2 1 2 In order to construct the grid let S S and N N b e the real and imaginary i parts of and resp ectively Let the grid consist of the p oints S s M and S N i i N n M where s and n are each pairs of integers with M s M and i i i i M n M Thus the grid p oints at the intersection of the real and imaginary i axes with the equator corresp ond to s M and s M and n M i i i and n M The grid p oints at the p oles corresp ond to s and n Note i i i that our choice of rectangular grid domains is not unique Other strategies can b e followed to economize the number of p oints in the overlap region b etween the S and N grids but the rectangular choice lends itself to a simple implementation at the grid b oundaries Given a smo oth function on the sphere we now give instructions for representing it on the grid Denote the values of at p oints on the S grid by and the S S S s s 1 2 values of at p oints on the N grid by In the overlap region the grid N N N n n 1 2 representation must b e consistent with the condition However S S N S only in exceptional cases will a p oint that lies on the S grid also lie on the N grid To deal with this we introduce n in M 1 2 S N n n 1 2 2 2 n n 1 2 which gives the value of at a p oint which lies on the N grid Similarly we dene S s is M 1 2 N S s s 1 2 2 2 s s 1 2 In the same way in the overlap region and determine the values and S N S s s 1 2 at p oints which lie on the S grid and and at p oints on the N N S s s N n n S N n n 1 2 1 2 1 2 grid However the grid values are weaker information than the smo oth function S s s 1 2 and do not by themselves determine the value of at p oints on the N grid Similarly S do not by themselves determine at a p oint on the S grid In these the values N n n 1 2 cases interpolation is necessary to evaluate functions on one grid at p oints lying on the other grid In order to see how this applies to dierentiation consider the gradient of with comp onents i in the S patch and i in the N patch Then at p oints S S N N S N which do not lie on the grid b oundary the partial derivatives can b e approximated by i centered nite dierences in the standard way For example at the p oint S s the i appropriate nite dierence approximation is S s +1s S s 1s 1 2 1 2 2 1 O S S 1 For a p oint on the grid b oundary eg the p oint S s we use the same 2 approximation except that the eld at the virtual grid p oint s M s is i 2 approximated by the value obtained from interpolating In order to N NS (M +1)s 2 achieve an error of second order in the derivative the interpolation error must b e of third order This provides second order accurate nite dierence representations of the gradient of at all grid p oints and in fact two representations corresp onding to i S S S i at p oints which lie in the overlap b etween the two grids The analytic and N N N expressions are related by the transformation N S S N N S N S As a check on the nite dierence approximation the nite dierence equivalent of must hold to second order accuracy Equivalently the gradient may b e converted into a scalar eld by taking basis comp onents and the scalar eld compared b etween patches This is the strategy of the eth formalism that we shall consider later Second derivatives of a scalar eld with resp ect to the co ordinates may also b e approximated to second order by central dierencing The interior p oints in each grid patch are handled by the standard techniques and the p oints on the grid b oundary are handled by introducing a virtual grid p oint as ab ove The following fourth order interpolation scheme ensures that b oth the rst and second derivatives are approximated to second order accuracy at b oundary p oints First we extend the grid by one p oint on each co ordinate direction The p oints at the S grid with co ordinates