Transformation Between the Schwarzschild Coordinates and Local Inertial Coordinates

TL;DR

This paper derives a transformation between Schwarzschild and local inertial coordinates and demonstrates how gravitational bending affects light near a massive body, showing relativistic effects are significantly larger than Newtonian predictions.

Contribution

It provides a specific coordinate transformation and quantifies the gravitational bending of light near Earth, highlighting relativistic effects in a local region.

Findings

Photon trajectory shows about three times greater vertical deflection than Newtonian prediction.

Transformation enables better understanding of light behavior in curved spacetime.

Relativistic effects are significant in local gravitational fields.

Abstract

We present a transformation between the Schwarzschild coordinates and local inertial coordinates, and demonstrate the effect of gravitational bending of light near a massive body in a small region. When a photon is emitted from a point near the Earth's surface with an initial horizontal direction, its parabolic trajectory would have a vertical deflection that is about three times that of a non-relativistic Newtonian particle would trace.