# Weak gravitational quantum effects in boson particles

## TL;DR

This paper derives a covariant Schr"odinger equation from the Klein-Gordon equation in curved space-time, revealing subtle gravitational effects on scalar particles that could be experimentally observed in non-inertial laboratory settings.

## Contribution

It introduces a new covariant Schr"odinger equation from the Klein-Gordon equation and analyzes gravitational effects on scalar particles in weak gravitational fields.

## Key findings

- Differences between traditional and covariant Schr"odinger equations in gravitational fields
- Potential to observe gravitational effects in laboratory non-inertial frames
- Energy levels and wave functions affected by space-time curvature

## Abstract

We rewrite the Klein-Gordon (KG) equation in an arbitrary space-time transforming it into a generalized Schr\"odinger equation. Then we take the weak field limit and show that this equation has some differences with the traditional Schr\"odinger equation plus a gravitational field. Thus, this procedure shows that the Schr\"odinger equation derived in a covariant manner is different from the traditional one. With this new Schr\"odinger equation, we study the KG equation in a Newtonian space-time to describe the behavior of a scalar particle in an inertial system. We give some examples where it is possible to study the energy levels, effective potential and the wave function of the systems, these results contain the gravitational effects due to the curvature of space-time. We show that it is possible to verify experimentally these effects in a laboratory using non-inertial reference frames.