Null Lagrangian, Gauge Field Theory and Novel Klein-Gordon Equation in Curved Spacetime

A gauge theory of massive complex scalar fields on curved spacetime with a given metric is developed. The theory is based on null Lagrangians and naturally accounts for effects of gravitationally-caused curvature of spacetime on the field by imposing its gauge invariance. The derived conditions for gauge invariance result in a novel Klein-Gordon equation, whose form is unique for a given metric. A specific form of this metric-dependent Klein-Gordon equation in the Schwarzschild spacetime is presented, and its physical implications for the fields outside and inside black holes are discussed.