[Paper Review] Classical Solution of Field Equation of Gravitational Gauge Field and Classical Tests of Gauge Theory of Gravity

Ning Wu, Dahua Zhang|ArXiv.org|Aug 1, 2005

Relativity and Gravitational Theory8 references4 citations

TL;DR

This paper establishes a classical solution for the gravitational gauge field in spherical coordinates, deriving the Schwarzschild metric, and demonstrates that Newton’s second law of motion—when extended relativistically—yields identical predictions for classical tests of gravity (light deflection, perihelion precession, radar delay) as general relativity. The key contribution is showing that gravity can be consistently described as a physical interaction in flat Minkowski spacetime, with Newtonian mechanics extended relativistically, rather than as spacetime geometry.

ABSTRACT

A systematic method is developed to study classical motion of a mass point in gravitational gauge field. First, the formulation of gauge theory of gravity in arbitrary curvilinear coordinates is given. Then in spherical coordinates system, a spherical symmetric solution of the field equation of gravitational gauge field is obtained, which is just the Schwarzschild solution. In gauge theory of gravity, the equation of motion of a classical mass point in gravitational gauge field is given by Newton's second law of motion. A relativistic form of the gravitational force on a mass point is deduced in this paper. Based on the spherical symmetric solution of the field equation and Newton's second law of motion, we can discuss classical tests of gauge theory of gravity, including the deflection of light by the sun, the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun. It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity. From the study in this paper, an important qualitative conclusion on the nature of gravity is that gravity can be treated as a kind of physical interactions in flat Minkowski space-time, and the equation of motion of mass point in gravitational field can be given by Newton's second law of motion.

Motivation & Objective

• To develop a consistent formulation of gauge theory of gravity in arbitrary curvilinear coordinates, particularly spherical coordinates.
• To derive the classical solution of the gravitational gauge field equation, showing it yields the Schwarzschild metric.
• To establish Newton’s second law of motion as the equation of motion for a mass point in gravitational gauge field, in a relativistically covariant form.
• To test the gauge theory of gravity against classical general relativistic predictions, including light deflection, perihelion precession, and radar time delay.
• To demonstrate that despite differing foundational concepts—flat spacetime vs. curved spacetime—the gauge theory reproduces all classical results of general relativity.

Proposed method

• Formulate the gauge theory of gravity in arbitrary curvilinear coordinates using the gravitational gauge potential $ C_{ u}^{eta}(x) $ and the matrix $ G_{ u}^{eta} = \delta_{ u}^{eta} - gC_{ u}^{eta} $.
• Define the metric tensors $ g_{\alpha\beta} $ and $ g^{\alpha\beta} $ via $ G $ and its inverse, ensuring compatibility with flat Minkowski spacetime.
• Solve the field equation of the gravitational gauge field in spherical symmetry, obtaining a solution identical to the Schwarzschild metric.
• Derive a relativistic form of the gravitational force on a test particle using the gauge-covariant derivative and the field strength tensor.
• Apply Newton’s second law in the form $ \frac{d}{dt}(\gamma m \vec{v}) = \vec{F}_{\text{grav}} $, with the relativistic gravitational force derived from the gauge field.
• Calculate classical tests: light deflection, perihelion precession, and radar time delay, using the derived solution and equation of motion.

Experimental results

Research questions

• RQ1 Can the Schwarzschild solution be derived from the field equations of the gauge theory of gravity in spherical coordinates?
• RQ2 Does the gauge theory of gravity reproduce the classical predictions of general relativity for light deflection, perihelion precession, and radar time delay?
• RQ3 Is Newton’s second law of motion, when extended relativistically, a valid and consistent equation of motion for massive particles in the gravitational gauge field?
• RQ4 Can gravity be consistently described as a physical interaction in flat Minkowski spacetime without invoking spacetime curvature?
• RQ5 What is the role of the gravitoelectric and gravitomagnetic fields in the relativistic gravitational force within the gauge theory framework?

Key findings

• The classical solution of the gravitational gauge field equation in spherical coordinates yields the Schwarzschild metric, confirming consistency with general relativity in static, spherically symmetric spacetime.
• The relativistic form of the gravitational force derived from the gauge field matches the Newtonian limit and includes both gravitoelectric and gravitomagnetic contributions.
• The deflection of light by the Sun is predicted to be $ 4GM/c^2 r_0 $, identical to the general relativistic result.
• The perihelion precession of Mercury is calculated as $ 6\pi GM/c^2 a(1-e^2) $, matching the general relativistic prediction.
• The time delay of radar echoes is found to be $ (\Delta t)_{\text{max}} = 4GM \left(1 + \ln \frac{4r_\oplus r_M}{R_\odot^2} \right) $, exactly as in general relativity.
• Despite the absence of spacetime curvature and the use of Newton’s second law instead of geodesic motion, the gauge theory reproduces all classical tests of gravity identically to general relativity.