Cohomology in algebra, geometry, physicsand statistics

Hertz potentials and the decay of higher-spin fields

The purpose of the talk is to illustrate how differential complexes can be used in relativity. Electromagnetism and linearized gravity (more generally higher-spin fields) are governed by hyperbolic systems of partial differential equations. Solutions to these systems can be generated by the mean of potentials (here, Hertz potentials) satisfying a wave equation. It is possible to recast the problem of representing a solution to these higher-spin fields by Hertz potentials in the context of the initial value problem. Initial data for higher-spin fields satisfy constraint equations, and cannot be chosen freely. The integrability conditions for these constraints are described by elliptic complexes. These elliptic complexes also happen to be those describing the relation between initial data for higher-spin fields and those for their Hertz potentials. The problem of describing the asymptotic behavior of generic solutions to higher-spin fields can then be completely deduced from the asymptotic behavior of solutions to the scalar wave equations on flat spacetime.

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