[Paper Review] On Mach's principle: Inertia as gravitation

J.I. San Martín, Antonio Fernández Rañada+1 more|ArXiv.org|Mar 28, 2007

Relativity and Gravitational Theory7 references3 citations

TL;DR

This paper tests Mach's principle by calculating the gravitational reactive acceleration from the entire universe on an accelerated test mass using linearized general relativity. It finds the reactive acceleration is approximately −0.7 to −1.1 times the particle's proper acceleration, strongly supporting the idea that inertia arises from gravitational interaction with the rest of the universe, especially when integrating over the past light cone with retarded potentials.

ABSTRACT

In order to test the validity of Mach's principle, we calculate the action of the entire universe on a test mass in its rest frame, which is an acceleration ${\bf g}^*$. We show the dependence of the inertia principle on the lapse and the shift. Using the formalism of linearized gravitation, we obtain the non-relativistic limit of ${\bf g}^*$ in terms of two integrals. We follow then two approaches. In the first one, these integrals are calculated in the actual time section $t=t_0$ up to the distance $R_U=ct_0$. In the more exact and satisfactory second approach, they are calculated over the past light cone using the formalism of the retarded potentials. The aim is to find whether the acceleration $\dot{\bf v}$ in the LHS of Newton's second law can be interpreted as a reactive acceleration, in other words, as minus the acceleration of gravity ${\bf g}^*$ in the rest frame of the accelerated particle ({\it i. e.} to know whether or not ${\bf g}^*=-\dot{\bf v}$). The results strongly support Mach's idea since the reactive acceleration for $Ω_Λ=0.7$ turns out to be about ${\bf g}^*=-1.1 \dot{\bf v}$, in the first approach, and about ${\bf g}^*= -0.7 \dot{\bf v}$, in the second. These results depend little on $Ω_Λ$ if $Ω_Λ<0.9$. Even considering the approximations and idealizations made during the calculations, we deem these results as interesting and encouraging.

Motivation & Objective

• To test Mach's principle by calculating the gravitational effect of the entire universe on a test mass in its rest frame.
• To determine whether the reactive acceleration due to distant matter matches the inertial force in Newton's second law.
• To evaluate whether inertia can be interpreted as a gravitational reaction force from the universe's mass-energy distribution.
• To compare two integration approaches: instantaneous spatial section vs. past light cone with retarded potentials.
• To assess the dependence of the result on dark energy density (ΩΛ) and the validity of the Machian interpretation.

Proposed method

• Uses linearized general relativity to model the gravitational field of the entire universe on a test mass.
• Calculates the gravitational acceleration g* in the rest frame of an accelerated particle using the retarded potential formalism.
• Performs two integration schemes: first over the spatial hypersurface at t=t₀ up to R_U=ct₀, second over the past light cone.
• Incorporates dark energy via the cosmological constant Λ, with ΩΛ as a parameter.
• Evaluates two integrals involving the lapse and shift functions, and the time-dependent distribution of mass-energy.
• Applies regularization techniques to handle singularities in the integrals arising from the early universe.

Experimental results

Research questions

• RQ1 Can the reactive acceleration due to the universe's mass-energy distribution reproduce the inertial force −mȧ in Newton's second law?
• RQ2 Does the gravitational action of the entire universe on a test mass in its rest frame yield a reactive acceleration close to −ȧ?
• RQ3 How does the inclusion of dark energy (ΩΛ) affect the magnitude and consistency of the reactive acceleration?
• RQ4 Is the past light cone integration with retarded potentials more physically accurate than the instantaneous spatial section?
• RQ5 Does the result support the idea that inertia is not intrinsic but arises from gravitational interaction with distant matter?

Key findings

• For ΩΛ = 0.7, the reactive acceleration is approximately −1.1ȧ when integrating over the spatial section at t=t₀.
• When integrating over the past light cone using retarded potentials, the reactive acceleration is approximately −0.7ȧ for ΩΛ = 0.7.
• The coefficient ξ in g* = −ξȧ remains in the range 0.6–0.7 for ΩΛ < 0.9, indicating weak dependence on dark energy density.
• The result is robust under regularization of singularities in the early-time integrals, with accuracy estimated at ±10%.
• A cut-off at t ≈ 2.5 Gyr yields ξ = 1 for ΩΛ = 0.7, suggesting a physically plausible regularization path.
• The findings strongly support Mach's principle, indicating that inertia arises from gravitational interaction with the rest of the universe.