Radar Method in a Uniformly Accelerated Reference Frame

Abstract
The purpose of this work is to generalize the radar method known for the inertial frame of reference to the case of a uniformly accelerated frame of reference. The derivation of the corresponding formulas is based on the standard for the theory of relativity metric of a uniformly accelerated Möller frame of reference without applying any space-time transformation between some auxiliary inertial frame and the accelerated frame. To solve the problem of determining the trajectory of a light beam, depending on the initial direction of propagation, Fermat’s principle is used. To calculate the flight time of a photon to an object, knowing its coordinates, the condition of the light-likeness of the interval for the propagation of light is additionally introduced. The resulting trajectory of the light particle is an arc of a circle. For a small area near the source, the photon trajectory coincides with the parabolic trajectory of a classical corpuscle. An equation has been derived for the direction in which the radio signal is sent. The actual location of the object is not in the direction of the initial motion of the photon, but somewhat lower. The value of the angle of gravitational refraction for a closely spaced resting object is calculated. The further the object is in the “horizontal” direction, the greater the angle of refraction. The flight time of the light signal to the object is found. The signal emitted in the direction that forms an acute angle with the direction of acceleration leads the radio signal in the inertial frame of reference. Therefore, for a close object located above the radiation source, the calculated Shapiro delay time is negative. The coordinates of the remote object are also calculated. The totality of the obtained equalities completely determines the radar method. The resulting equalities, perhaps, allow for experimental verification.

Keywords
Fermat’s principle,Poincaré half-plane model, hyperbolic geometry,gravitational refraction, Shapiro delay
УДК 535.11, 535.22, 535.31

Radar Method in a Uniformly Accelerated Reference Frame