[Paper Review] Reference Frame in General Relativity

Aleks Kleyn|ArXiv.org|May 6, 2004

Relativity and Gravitational Theory10 references3 citations

TL;DR

This paper introduces a reference frame formalism in general relativity using smooth fields of orthonormal bases and anholonomic coordinates to define observer-relative measurements. It derives Lorentz transformations in curved spacetime, applies them to calculate relative speeds and Doppler shifts, and shows that gravitational time dilation near Earth causes a 0.297 Myr age difference for a star orbiting Sgr A* over 10 Myr of observer time.

ABSTRACT

A reference frame in event space is a smooth field of orthonormal bases. Every reference frame is equipped by anholonomic coordinates. Using anholonomic coordinates allows to find out relative speed of two observers and appropriate Lorentz transformation. Synchronization of a reference frame is an anholonomic time coordinate. Simple calculations show how synchronization influences time measurement in the vicinity of the Earth. Measurement of Doppler shift from the star orbiting the black hole helps to determine mass of the black hole. According observations of Sgr A, if non orbiting observer estimates age of S2 about 10 Myr, this star is 0.297 Myr younger.

Motivation & Objective

• To extend the invariance principle of special relativity to general relativity by defining reference frames via orthonormal bases on curved manifolds.
• To provide a geometric framework for measuring relative speeds and Lorentz transformations between observers in arbitrary spacetime geometries.
• To analyze how synchronization of reference frames affects time measurements in gravitational fields, particularly near Earth.
• To apply the formalism to estimate black hole mass using Doppler shift from stars orbiting Sgr A*.
• To clarify the role of anholonomic coordinates and metric structure in defining observer-dependent physical quantities.

Proposed method

• Defining a G-reference frame as a smooth field of orthonormal vector fields on a manifold, with the symmetry group G of the tangent space.
• Using anholonomic coordinates—particularly anholonomic time coordinates—for synchronization, which directly influences time measurements in gravitational fields.
• Deriving Lorentz transformations between observers by expressing orthonormal bases in different frames using the metric and velocity parameters.
• Applying the formalism to Schwarzschild spacetime to compute relative velocity and transformation matrices between static and moving observers.
• Analyzing the Friedman metric to compute redshift due to cosmological expansion, showing that $ a au = \text{const} $ leads to observed redshift.
• Using the Doppler shift of light from a star orbiting Sgr A* to infer black hole mass, based on time dilation effects in strong gravity.

Experimental results

Research questions

• RQ1 How can the invariance principle of special relativity be generalized to curved spacetime using orthonormal reference frames?
• RQ2 What is the role of anholonomic coordinates in defining observer-dependent time and synchronization in gravitational fields?
• RQ3 How do Lorentz transformations between observers differ in curved spacetime compared to Minkowski space?
• RQ4 What is the quantitative effect of gravitational time dilation on the age difference between a distant observer and a star orbiting Sgr A*?
• RQ5 How does cosmological expansion lead to redshift, and can this be derived from the geometry of the Friedman metric?

Key findings

• In the vicinity of Earth, gravitational time dilation causes a non-inertial observer to age 0.297 Myr less than a distant observer over a 10 Myr interval.
• The Doppler shift from a star orbiting Sgr A* can be used to infer the black hole’s mass, assuming the star’s orbital period and redshift are measurable.
• The Lorentz transformation between a static observer and a radially moving observer in Schwarzschild spacetime takes the standard relativistic form, with corrections from the metric factor $ \frac{r}{r - r_g} $.
• In Friedman spacetime, the observed redshift arises from the scale factor $ a(t) $, with $ a\omega = \text{const} $, leading to $ K(t_1) = \frac{a(t_1)}{a(t_2)} $, where $ K $ is the redshift factor.
• The time derivative of the redshift factor $ K $ in a closed Friedman model is $ \dot{K} = \frac{\sinh(t_1 - t_2)}{\cosh^2 t_2} $, showing that $ K $ decreases as $ t_1 $ increases.
• The formalism confirms that the speed of light is invariant in the chosen orthonormal frame, and that coordinate-dependent speed variations are artifacts of coordinate choice, not physical.