Symmetry operators

Thomas Bäckdahl

Thomas Bäckdahl is a Post-Doctoral Research Assistant in the School of Mathematics at the University of Edinburgh.

Conserved quantities, for example energy and momentum, play a fundamental role in the analysis of dynamics of particles and fields. For field equations, one manifestation of conserved quantities in a broad sense is the existence of symmetry operators, i.e. linear differential operators which take solutions to solutions. A well-known example of a symmetry operator for the scalar wave equation is provided by the Lie derivative along a Killing vector field.

It is important to note that other kinds of objects can generate symmetry operators. For waves in the Kerr spacetime there is a symmetry operator associated with Carter’s constant. This symmetry, which is “hidden” in the sense that it arises from a Killing spinor, satisfying a generalization of the Killing vector equation, rather than a Killing vector, was an essential ingredient in a proof of decay of scalar waves on the Kerr background by my co-authors.

In this paper we consider what conditions on a spacetime are necessary for the existence of symmetry operators and determine the general form of such operators for the conformal wave equation, the Dirac-Weyl equation, and the Maxwell equation, i.e. for massless fields of spins 0, 1/2 and 1.

Other work in this direction has been done before. However, for the Maxwell equation, the results were so complicated that a geometric interpretation has been difficult to find. Now, we have found a systematic way to reduce the problem to a nice, compact form. Our results make clear that in general, second-order symmetry operators are generated by Killing spinors, satisfying some fairly simple auxiliary conditions involving the curvature. We also develop a unified treatment of the symmetry operators for all three field equations.

To achieve this reduction, we have developed computer algebra tools which allow us to handle the very complex commutator identities and index manipulations needed in the proofs. The computer algebra tools which were used are now available in the SymManipulator package, part of xAct suite for Mathematica.  These tools will also allow us to handle more complicated systems like linearized gravity, and we expect them to be valuable in proving the stability of black holes.

Read the full article in Classical and Quantum Gravity:
Second order symmetry operators
Lars Andersson, Thomas Bäckdahl, and Pieter Blue
Class. Quantum Grav. 31 135015

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