Bending of electric field lines and light-ray trajectories in a static gravitational field

TL;DR

This paper theoretically investigates how electric field lines of a charge are bent in a gravitational field, demonstrating their similarity to light-ray trajectories and analyzing differences between stationary and freely falling charges.

Contribution

It establishes a theoretical link between the bending of electric field lines and light-ray trajectories in a gravitational field, extending the principle of equivalence to electrostatics.

Findings

Electric field lines of a stationary charge bend similarly to light rays in gravity.

Freely falling charges exhibit no bending of electric field lines, which remain radial.

Electric field configuration of a freely falling charge matches that of an inertial charge with equivalent velocity.

Abstract

It is well known that the light-ray trajectories follow a curved path in a gravitational field. This has been confirmed observationally where light rays coming from distant astronomical objects are seen to get bent in Sun's gravitational field. We explore here the bending of electric field lines due to gravity. We determine, from a theoretical perspective, not only the exact shapes of the bent trajectories of light rays, emitted isotropically by a source supported in a gravitational field, but also demonstrate that the electric field lines of a charge, supported in a gravitational field, follow exactly the trajectories of light rays emitted isotropically from a source at the charge location. From a detailed examination of the electrostatic field of a charge accelerated uniformly in the instantaneous rest frame, exploiting the strong principle of equivalence, we determine the bending of the electric field lines of a charge in a gravitational field. The fraction of electric field lines crossing a surface, stationary above or below the charge in the gravitational field, are shown to be exactly similar to the fraction of light-ray trajectories intersecting that surface, emanating from a source lying at the charge location. On the other hand, for a freely falling charge in the gravitational field there is no such bending of electric field lines. The field lines continue to extend in radial straight lines from the instantaneous 'present' position of the charge, as do the trajectories of light rays spreading away from the instantaneous position of a freely falling source in the gravitational field. The electric field configuration of a freely falling charge in the gravitational field is shown to be exactly the same as that of a charge moving uniformly in an inertial frame with velocity equal to the instantaneous ``present'' velocity of the freely falling charge.