We rewrite the Klein–Gordon equation in an arbitrary space-time transforming it into a generalized Schrödinger equation. Then, we take the weak field limit and show that this equation has certain differences with the traditional Schrödinger equation plus a gravitational field. Thus, this procedure shows that the Schrödinger equation derived in a covariant manner is different from the traditional one. We study the KG equation in a Newtonian space-time to describe the behavior of a scalar particle in an inertial system. This particle is immersed in a gravitational field with the new Schrödinger equation. We study particular physical systems given examples for which we find their energy levels, effective potential and the wave function of the systems. The results contain the gravitational effects due to the curvature of space-time. Finally, we discuss the possibility of the experimental verification of these effects in a laboratory using non-inertial reference frames.

我们在任意时空中重写Klein-Gordon方程，将其转换为广义Schrödinger方程。然后，我们采用弱场极限，表明该方程与传统的薛定ding方程加引力场有一定的差异。因此，此过程表明以协变方式导出的薛定ding方程与传统方程不同。我们研究牛顿时空中的KG方程，以描述惯性系统中标量粒子的行为。使用新的Schrödinger方程将该粒子浸没在重力场中。我们通过给出示例来研究特定的物理系统，我们可以找到它们的能级，有效电势和系统的波动函数。结果包含由于时空弯曲而产生的引力效应。最后，