[Paper Review] On Gravitational Motions

A. Loinger|ArXiv.org|Apr 24, 2008

Pulsars and Gravitational Waves Research10 references3 citations

TL;DR

This paper proves that all motions of bodies interacting solely through gravity are geodesic in general relativity, regardless of the specific form of the metric tensor, by deriving the geodesic equation from the first integral of the Lagrangian. It further demonstrates that gravitational waves do not exist in exact general relativity for purely gravitational systems, and critically analyzes the linearized approximation, showing its conceptual limitations and gauge dependence, while emphasizing that only coordinate-invariant statements have physical meaning in GR.

ABSTRACT

A new proof of the geodesic character of all motions of bodies that interact only gravitationally - and a detailed illustration of the real meaning of the linearized approximation of general relativity.

Motivation & Objective

• To establish that all gravitational motions in general relativity are geodesic, independent of the specific form of the metric tensor.
• To clarify the conceptual flaws and physical limitations of the linearized approximation of general relativity.
• To argue that gravitational waves cannot be emitted by purely gravitational systems, as such emissions would violate the geodesic nature of motion.
• To emphasize that only coordinate-invariant quantities have physical meaning in general relativity, following Hilbert’s principle of invariance.
• To correct widespread misconceptions about the physical reality of gravitational waves in the linearized approximation by exposing its gauge dependence and formal resemblance to Maxwell theory.

Proposed method

• Derives the geodesic equation from the first integral of the Lagrangian, where the Lagrangian is defined as $\mathcal{L} = g_{jk}(q(\tau)) \frac{dq^j}{d\tau} \frac{dq^k}{d\tau} = c^2 $, showing that this constraint leads to the geodesic equation.
• Uses Riemann-Fermi coordinates to show that geodesic solutions take the form $ y^j(\tau) = a^j \tau + b^j $, confirming the absence of damping terms and thus no gravitational radiation.
• Analyzes the linearized approximation of GR via $ g_{jk} \approx \eta_{jk} + h_{jk} $, with $ h_{jk} $ treated as small perturbations, and applies the harmonic gauge condition $ \partial_k \gamma^{jk} = 0 $ to derive the wave equation $ \eta^{mn} \partial_m \partial_n \gamma_{jk} = -2\kappa T_{jk} $.
• Applies Weyl’s 1944 derivation of the linearized Einstein equations as a self-consistent linear theory of gravity, showing its equivalence to the standard linearized GR.
• Introduces a gauge transformation $ \varphi_{jk} \rightarrow \varphi'_{jk} = \varphi_{jk} + \partial_j \xi_k + \partial_k \xi_j - \eta_{jk} \partial_m \xi^m $, with $ \xi_j $ satisfying the d’Alembert equation, to demonstrate the invariance of the field equations under such transformations.
• Applies Hilbert’s criterion: only statements invariant under arbitrary coordinate transformations have physical meaning, rejecting wave-like interpretations of $ g_{jk} $ as unphysical.

Experimental results

Research questions

• RQ1 Are all motions of bodies interacting only through gravity necessarily geodesic in general relativity, regardless of the specific form of the metric tensor?
• RQ2 Does the linearized approximation of general relativity correctly describe gravitational wave emission in purely gravitational systems?
• RQ3 What is the physical significance of the linearized approximation of GR, given its formal similarity to Maxwell’s equations and its gauge dependence?
• RQ4 Why do some authors incorrectly attribute physical reality to gravitational waves in the linearized approximation, despite the absence of such waves in the exact theory?
• RQ5 What criteria determine whether a physical statement in general relativity has real physical meaning, according to the principle of coordinate invariance?

Key findings

• All motions of bodies interacting only gravitationally are geodesic in general relativity, as a direct consequence of the first integral $ \mathcal{L} = c^2 $, which follows from the pseudo-Riemannian structure of spacetime.
• No gravitational waves are emitted by purely gravitational systems, as geodesic motion in Riemann-Fermi coordinates is free of damping terms and corresponds to uniform motion in straight lines.
• The linearized approximation of GR, while formally resembling Maxwell’s theory, is conceptually flawed in its interpretation of gravitational waves, as it depends on gauge choices and does not represent physical reality independently of coordinate systems.
• The equations of motion for matter in the linearized approximation are derived from the differential conservation law $ \partial_k T^k_j = 0 $, which holds due to the gauge condition $ \partial_k \gamma^{jk} = 0 $, and is consistent with the geodesic motion in the exact theory.
• The physical meaning of any statement in general relativity is only valid if it is invariant under arbitrary coordinate transformations, as emphasized by Hilbert; thus, properties like the wave character of $ g_{jk} $ or its propagation velocity are unphysical.
• Weyl’s 1944 derivation of the linearized Einstein equations shows that the linear theory is self-consistent and equivalent to the standard linearized approximation, but it does not imply the physical existence of gravitational waves in the exact theory.