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Newtonian limit of Extended Theories of Gravity

Salvatore Capozziello

TL;DR

The paper investigates how Extended Theories of Gravity alter the Newtonian limit, focusing on higher order curvature terms and scalar-tensor components and their compatibility with current observations. It uses the PPN framework and GR as a benchmark to classify deviations, deriving Newtonian and weak-field corrections such as Yukawa terms and quadratic potentials in four major ETG channels: fourth-order gravity, scalar-tensor gravity, and string-dilaton gravity. The results show that ETG naturally reproduce GR at short scales but introduce scale dependent corrections that can address cosmological and galactic phenomena, while remaining testable via solar system experiments, pulsar timing, lensing, and gravitational wave observations. The analysis highlights parameter sensitivities (masses $m_2$, $m_0$, scalar couplings) and suggests a geophysical window where deviations might be detectable.

Abstract

Newtonian limit of Extended Theories of Gravity (in particular, higher--order and scalar--tensor theories) is theoretically discussed taking into account recent observational and experimental results.

Newtonian limit of Extended Theories of Gravity

TL;DR

The paper investigates how Extended Theories of Gravity alter the Newtonian limit, focusing on higher order curvature terms and scalar-tensor components and their compatibility with current observations. It uses the PPN framework and GR as a benchmark to classify deviations, deriving Newtonian and weak-field corrections such as Yukawa terms and quadratic potentials in four major ETG channels: fourth-order gravity, scalar-tensor gravity, and string-dilaton gravity. The results show that ETG naturally reproduce GR at short scales but introduce scale dependent corrections that can address cosmological and galactic phenomena, while remaining testable via solar system experiments, pulsar timing, lensing, and gravitational wave observations. The analysis highlights parameter sensitivities (masses , , scalar couplings) and suggests a geophysical window where deviations might be detectable.

Abstract

Newtonian limit of Extended Theories of Gravity (in particular, higher--order and scalar--tensor theories) is theoretically discussed taking into account recent observational and experimental results.

Paper Structure

This paper contains 17 sections, 140 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The foundation of metric theories: The Equivalence Principle
  3. The Parametrized Post Newtonian (PPN) limit
  4. Basics of General Relativity
  5. The weak field approximation and the linearized Einstein equations
  6. Gravitational Waves
  7. Discussion of Classical Tests of General Relativity
  8. Extended theories of gravity
  9. General Relativity with a cosmological constant
  10. Higher-order-scalar-tensor theories: the general scheme
  11. Scalar--tensor gravity
  12. Higher-order gravity
  13. Newtonian limit in extended theories of gravity
  14. Recovering the Newtonian limit in fourth-order theories of gravity
  15. Newtonian limit in scalar-tensor gravity
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Measurements of the coefficient $(1+\gamma)/2$ from light deflection and time delay measurements -- the GR value is unity. Shapiro time-delay measurements using Viking spacecraft and Very Large Baseline Interferometry (VLBI) light deflection measurements are in agreement with GR to $0.1 \%$. Results from Cassini mission are in agreement at the level of $5 \times 10^{-5}$.