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Physics is simple only when analyzed locally

Matteo Luca Ruggiero

TL;DR

The paper addresses how General Relativity's 4D spacetime geometry can be translated into locally measurable quantities by introducing a spacetime splitting based on a timelike congruence. It develops a projection framework with time and space projectors $T(u)$ and $P(u)$ to obtain a local Minkowski structure, yielding decompositions such as $P^{\alpha}=E(P,u)u^{\alpha}+p^{\alpha}$ and $U^{\alpha}=\gamma(u,U)[u^{\alpha}+v^{\alpha}]$, and shows how relative kinematics emerge from these projections. Key results reveal that Newtonian gravity arises as a gravito-inertial effect in non-geodesic frames, and the dynamics can be expressed in Newtonian-like form with gravito-electric and gravito-magnetic contributions, illustrating a gravity–inertia equivalence within GR. The framework is demonstrated through Schwarzschild-like scenarios, including gravitational redshift between static observers, and is argued to have pedagogical and foundational benefits by clarifying how GR measurements connect to familiar SR concepts in a locally Lorentzian setting.

Abstract

The definition of a reference frame in General Relativity is achieved through the construction of a congruence of time-like world-lines. In this framework, splitting techniques enable us to express physical phenomena in analogy with Special Relativity, thereby realizing the local description in terms of Minkowski spacetime in accordance with the equivalence principle. This approach holds promise for elucidating the foundational principles of relativistic gravitational physics, as it illustrates how its 4-dimensional mathematical model manifests in practical measurement processes conducted in both space and time. In addition, we show how, within this framework, the Newtonian gravitational force naturally emerges as an effect of the non-geodesic path of the reference frame.

Physics is simple only when analyzed locally

TL;DR

The paper addresses how General Relativity's 4D spacetime geometry can be translated into locally measurable quantities by introducing a spacetime splitting based on a timelike congruence. It develops a projection framework with time and space projectors and to obtain a local Minkowski structure, yielding decompositions such as and , and shows how relative kinematics emerge from these projections. Key results reveal that Newtonian gravity arises as a gravito-inertial effect in non-geodesic frames, and the dynamics can be expressed in Newtonian-like form with gravito-electric and gravito-magnetic contributions, illustrating a gravity–inertia equivalence within GR. The framework is demonstrated through Schwarzschild-like scenarios, including gravitational redshift between static observers, and is argued to have pedagogical and foundational benefits by clarifying how GR measurements connect to familiar SR concepts in a locally Lorentzian setting.

Abstract

The definition of a reference frame in General Relativity is achieved through the construction of a congruence of time-like world-lines. In this framework, splitting techniques enable us to express physical phenomena in analogy with Special Relativity, thereby realizing the local description in terms of Minkowski spacetime in accordance with the equivalence principle. This approach holds promise for elucidating the foundational principles of relativistic gravitational physics, as it illustrates how its 4-dimensional mathematical model manifests in practical measurement processes conducted in both space and time. In addition, we show how, within this framework, the Newtonian gravitational force naturally emerges as an effect of the non-geodesic path of the reference frame.

Paper Structure

This paper contains 9 sections, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Reference Frames
  3. Reference Frames in Flat Spacetime
  4. Reference Frames in Curved Spacetime
  5. The Projection Technique
  6. Projection of Kinematical Quantities
  7. Properties of the Congruence
  8. Relative Description of Dynamics
  9. Discussion and Final Remarks

Figures (3)

  • Figure 1: On the right, a reference frame in SR; on the left, along a world-line $\ell$ at each event we can define a (local) reference frame.
  • Figure 2: The reference frame $\mathcal{C}_{u}$ is defined by the integral curves of the vector field $u$; the reference frame $\mathcal{C}_{u'}$ is defined by the integral curves of $u'$.
  • Figure 3: At each point along $\ell$, $\Sigma$ is the 3-dimensional space orthogonal to $u$.