For a pseudo-Riemannian manifold and its Laplace operator, the wave equation on is the linear differential equation
where denotes the wave operator /Laplace operator, a hyperbolic differential operator.
For the inhomogenous equation
is called the Klein-Gordon equation.
On a globally hyperbolic spacetime the wave equation/Klein-Gordon equation has unique advanced and retarded Green functions.
Their difference is the Peierls bracket which gives the Poisson bracket on the covariant phase space of the free scalar field. This in turn defines the Wick algebra of the free scalar field, which yields the quantization of the free scalar field to a quantum field theory.
The bicharacteristic strips of the Klein-Gordon operator are cotangent vectors along lightlike geodesics (this example).
F. Friedlander, The Wave Equation on a Curved Space-Time, Cambridge: Cambridge University Press, 1975
Howard Georgi, The Physics of Waves, Prentice Hall (1993) web, pdf
Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)
Nicolas Ginoux, Linear wave equations, Ch. 3 in Christian Bär, Klaus Fredenhagen, Quantum Field Theory on Curved Spacetimes: Concepts and Methods, Lecture Notes in Physics, Vol. 786, Springer, 2009
Sergiu Klainerman, chapter 4, section 3 of Lecture notes in analysis, 2011 (pdf)
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