Characteristic Surface Data for the Eikonal Equation
Ezra T. Newman, Alejandro Perez
TL;DR
This paper extends methods for solving the eikonal equation to characteristic surfaces, revealing a Legendre transformation relationship between Cauchy and characteristic data, and analyzing wave-front singularities.
Contribution
It introduces a novel approach to solve the eikonal equation with characteristic data and uncovers a Legendre transformation linking different data types.
Findings
Relationship between Cauchy and characteristic data via Legendre transformation
Solutions describe wave-front singularities
Extension of solution methods to curved space-times
Abstract
A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe their associated wave-front singularities.
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