Outline Motivation Self Force Singular Field Calculation Mode Sum - PowerPoint PPT Presentation

H IGH - ORDER E XPANSIONS OF THE S INGULAR F IELD Anna Heffernan University College Dublin Collaborators: Adrian Ottewill, University College Dublin Barry Wardell, University College Dublin and AEI, Berlin arXiv1204.xxxx BritGrav 12, Southampton

Outline • Motivation • Self Force • Singular Field Calculation • Mode Sum Decomposition • Results • Future Work

Motivation • Verification of Einstein’s Theory of Relativity through gravitational wave detection • EMRI (Extreme Mass Ratio Inspiral) Sources are essentially a “test mass” falling in the geometry of a Kerr Black Hole (BH) Image Credit: NASA JPL • Waveform required for matched filtering • Self-force required for accurate waveform • Mode-sum regularisation parameters radically speed up these self-force calculations

Self Force • Smaller mass does not follow a geodesic of the background • Its mass curves the space-time itself - it follows a geodesic of an effective space-time • 3 main approaches all require subtraction of an appropriate singular component from the retarded field to leave a finite regular field which is solely responsible for self-force.

Self Force Retarded field satisfies The retarded solution to the above equation gives rise to a field which gives the self-force, Given the Detweiler-Whiting Green Function, We may define the Detweiler-Whiting singular field,

Singular Field The scalar singular field and self-force are x A x x The EM singular field and self-force are x' x R � The gravitational singular field and self-force are

Singular Field For coordinate expansion, we introduce the notation We expand the Synge world function We use to solve for coefficients We can differentiate this easily to obtain Once we know , we can calculate the Van Vleck Determinant

Singular Field We expand around the point . Representing the worldline in terms of porper time gives us We want to determining the points on the world line that are connected by a null geodesic. That is we want to solve Use to calculate bivector of parallel transport. Writing , this gives us

Singular Field 0 � 2 2. � 10 � 6 0.004 � 0.00001 � 4 0.002 1. � 10 � 6 � S � � 1 � � 0.00002 � S � 0 � � S � 1 � � S � 2 � � 6 0.000 0 � 0.00003 � 0.002 � 1. � 10 � 6 � 8 � 0.004 � 2. � 10 � 6 � 0.00004 � 10 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 � r � r � r � r 0 0 1. � 10 � 8 3. � 10 � 11 � 1. � 10 � 8 � 2. � 10 � 10 2. � 10 � 11 5. � 10 � 9 � 2. � 10 � 8 � 4. � 10 � 10 1. � 10 � 11 � 3. � 10 � 8 � S � 4 � � S � 5 � � S � 6 � � S � 3 � 0 0 � 6. � 10 � 10 � 4. � 10 � 8 � 1. � 10 � 11 � 8. � 10 � 10 � 5. � 10 � 9 � 2. � 10 � 11 � 5. � 10 � 8 � 1. � 10 � 9 � 3. � 10 � 11 � 1. � 10 � 8 � 6. � 10 � 8 � 1.2 � 10 � 9 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 � r � r � r � r

Mode Sum To obtain expressions which are readily expressed as mode sums, it is useful to work in a rotated coordinate frame. We introduce Riemann normal coordinates on the 2-sphere at , in the form where The Schwarzschild metric is now given by the line element

Mode Sum Barack and Ori (2000, 2002) first looked at the multipole decomposition of the self-force, The mode contribution at is given by With particle on the pole, for all , so We find the self-force has the form where and

Mode Sum Explicitly in our coordinates, takes the form We use our definition of to rewrite our ‘s in an alternate form where As , it can now be shown that

Mode Sum For the higher order terms we can write where a and We note that will simply integrate to 1 We note that integrating over is the same as averaging over the angles , so Tidy up resulting equations with a

Results - Scalar Regularised l-modes for Radial Self-Force for Circular Schwarzschild 0.001 10 � 7 a � f r 10 � 11 10 � 15 10 � 19 1 2 5 10 20 50 �

Results - EM Regularised l-modes for Radial Self-Force for Elliptic Schwarzschild 0.1 0.001 a 10 � 5 1.5 2.0 3.0 5.0 7.0 10.0

Results - Gravity Regularised l-modes for Radial Self-Force for Circular Schwarzschild 0.1 0.001 a 10 � 5 10 � 7 10 � 9 1.5 2.0 3.0 5.0 7.0 10.0 15.0

Results - Kerr Regularised l-modes for Radial Self-Force for Circular Kerr 0.001 without D r l -2 reference 0.0001 with D r l -4 reference 1e-05 1e-06 l F r a 1e-07 1e-08 1e-09 1e-10 1 10 l Plot provided by Niels Warburton

Future Work • Higher orders in Schwarzschild • Eccentric orbits in scalar Kerr • Spheroidal harmonic decomposition • Kerr gravity????