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One may ask whether the relations between energy and frequency and betweenmomentum and wave vector, introduced for matter waves by de Broglie, arerigorously valid in the presence of gravity. In this paper, we show this to betrue for Dirac equations in a background of gravitational and electromagneticfields. We first transform any Dirac equation into an equivalent canonicalform, sometimes used in particular cases to solve Dirac equations in a curvedspacetime. This canonical form is needed to apply the Whitham Lagrangianmethod. The latter method, unlike the WKB method, places no restriction on themagnitude of Planck's constant to obtain wave packets, and furthermorepreserves the symmetries of the Dirac Lagrangian. We show using canonical Diracfields in a curved spacetime, that the probability current has a Gordondecomposition into a convection current and a spin current, and that the spincurrent vanishes in the Whitham approximation, which explains the negligibleeffect of spin on wave packet solutions, independent of the size of Planck'sconstant. We further discuss the classical-quantum correspondence in a curvedspacetime based on both Lagrangian and Hamiltonian formulations of the Whithamequations. We show that the generalized de Broglie relations in a curvedspacetime are a direct consequence of Whitham's Lagrangian method, and not justa physical hypothesis as introduced by Einstein and de Broglie, and by manyquantum mechanics textbooks.

Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations