Scalar Mach-Sciama Theory Advances Gravitation with FLRW Universe Temporal Kernel

The fundamental nature of gravity and its connection to the distribution of matter throughout the universe remain central questions in modern physics. A new approach to understanding these relationships comes from research led by A. M. Velásquez-Toribio of the Center for Astrophysics and Cosmology at the Federal University of Espirito Santo, alongside colleagues, who explore a scalar realization of Sciama’s Machian theory within a broader framework of scalar-tensor gravity. Their work reformulates field equations to isolate Machian requirements, independent of conformal frames, and implements Sciama’s causal postulate through a selection rule governing possible solutions. This research is significant because it offers a mechanism for determining the cosmological inertial scale based on the distribution of matter, while simultaneously aligning with established tests of gravity in weak field regimes, potentially bridging a gap between theoretical predictions and observational cosmology. By focusing on causal relationships and a specific temporal kernel, the team demonstrates how the background evolution of the universe links directly to the matter within our observable horizon.

Requirements can be stated independently of the conformal frame. Sciama’s causal postulate is implemented not by modifying the local dynamics, but as a selection rule on the solution space of the invariant scalar equation; the admissible configuration is the retarded response to the matter distribution in the causal past, with any source-free contribution removed. In a spatially flat Friedmann, Lemaître, Robertson, Walker (FLRW) universe, and in the light, slowly varying regime, the prescription reduces to an explicit temporal kernel. This kernel links the background scalar evolution to the matter content within the Hubble region, reproducing the expected Machian scaling in an expanding background. Universal coupling to a single physical metric is therefore maintained throughout the analysis.

Machian Principle via Invariant Scalar Field Equations

The research team engineered a novel approach to realising Sciama’s Machian principle within scalar-tensor gravity, formulating a scalar realisation within the Bergmann-Wagoner class. Scientists began by establishing a universally conformally coupled sector, then rewrote the field equations utilising an invariant set, I1, I2, I3, and the invariant metric ^gμν, to ensure Machian requirements remained independent of any chosen conformal frame. This innovative framing allowed for the implementation of Sciama’s causal postulate not through local dynamic modification, but as a selection rule applied directly to the solution space of the invariant scalar equation. The study pioneered a method for isolating the retarded response to matter distribution within the causal past, effectively removing any source-free contributions to the scalar field.

Working within a spatially flat Friedmann, Lemaître, Robertson, Walker (FLRW) universe and focusing on the light, slowly varying regime, the team derived an explicit temporal kernel linking background scalar evolution to content within the Hubble region. This kernel successfully reproduces the expected Machian scaling behaviour within an expanding cosmological background, demonstrating the method’s predictive power. The approach achieves a decoupling of the selection rule from specific conformal representations, ensuring its validity across different coordinate systems. Experiments employed a linear response analysis around a cosmological background, splitting the invariant scalar field into a homogeneous FLRW component and a perturbation, denoted as δI3.

The perturbation was then governed by a linear equation, ^DxδI3(x) = ^S(x), where ^D is a normally hyperbolic operator and ^S represents the linearized source term derived from the matter stress tensor. Crucially, the team harnessed the retarded Green function, ^Gret(x, x′), defined by ^Dx^Gret(x, x′) = δ(4)(x−x′) √ −^g(x), to construct the retarded particular solution, δIret 3 (x). This technique reveals that any solution to the linear equation can be decomposed into a sourced and a source-free component, with Sciama’s causal selection rule implemented by enforcing δIfree 3 (x) = 0, thereby ensuring the scalar perturbation is entirely generated by past matter sources. The team explicitly connected this to inertial calibration, demonstrating how the proper time is related by d τ= √ I1 d^τ, and deriving the invariant-units form of the point-particle action. This methodological innovation allows for a causal Machian determination of the cosmological inertial scale while maintaining consistency with standard weak-field tests, offering a robust framework for exploring the foundations of inertia and cosmology.

Machian Inertia from Scalar-Tensor Gravity

Scientists have formulated a scalar realization of Sciama’s Machian principle within the Bergmann-Wagoner class of scalar-tensor gravity, achieving a novel approach to understanding inertia. The research team began with a universally conformally coupled sector, rewriting field equations using an invariant set to ensure Machian requirements are independent of the chosen conformal frame. This work implements Sciama’s causal postulate not through alterations to local dynamics, but as a selection rule applied to the solution space of an invariant scalar equation, admitting only retarded responses to past distributions while excluding source-free contributions. Experiments revealed that in a spatially flat Friedmann, Lemaître, Robertson, Walker (FLRW) universe, and within a light, slowly varying regime, the prescription simplifies to a temporal kernel explicitly linking background scalar evolution to content within the Hubble radius.

Measurements confirm this kernel reproduces the expected Machian scaling in an expanding background, demonstrating a direct connection between the distribution of matter and the emergence of inertial properties. The team constructed the retarded Green-function solution on an FLRW background, establishing how it dictates the emergence of inertial mass through a field-dependent mass scale of the form m(φ) = m0A(φ). Tests prove that universal coupling to a single physical metric ensures structureless test bodies experience the same acceleration in a given external configuration, resulting in a vanishing Eötvs parameter at leading order. Data shows that non-universal effects only arise when gravitational binding energy significantly contributes to total mass, as observed in strongly self-gravitating objects.

The breakthrough delivers a causal Machian determination of the cosmological inertial scale while maintaining consistency with standard weak-field tests, a crucial validation of the theory’s applicability. Further analysis established a trace of the metric field equation, demonstrating that -A(Φ)R = T -3□A(Φ) – B(Φ)(∇Φ) -4U(Φ), providing a critical relationship for eliminating the Ricci scalar and simplifying calculations. Measurements of the scalar field equation, B(Φ)□Φ + B,Φ(Φ)(∇Φ) + A,Φ(Φ)R – U,Φ(Φ) = -α,Φ(Φ)T, further refine the understanding of scalar field interactions within the framework. The study successfully implements a conformal rescaling of the representative metric, combined with a scalar-field reparametrization, to maintain the action’s form while modifying the defining functions A, B, U, and α.

Causal Scalar Gravity and Inertial Scale Determination

This research presents a scalar realization of Sciama’s Machian principle within the Bergmann-Wagoner class of scalar-tensor gravity, successfully formulating the theory independently of conformal frame considerations. By focusing on an invariant set of quantities, the authors implemented Sciama’s causal postulate as a selection rule applied to the solution space, admitting only retarded responses to past source distributions and excluding source-free contributions. This approach establishes a mechanism for determining the cosmological inertial scale through causal principles, while simultaneously maintaining consistency with established weak-field tests of gravity. In a spatially flat Friedmann, Lemaître, Robertson, Walker universe, the resulting framework yields a temporal kernel that accurately reproduces the expected Machian scaling behaviour in an expanding background.

The work demonstrates that universal coupling to a single physical metric ensures test bodies experience the same acceleration, effectively eliminating leading-order violations of the Eötvös parameter. The authors acknowledge limitations stemming from the assumption of a slowly varying regime and the potential for non-universal effects in strongly self-gravitating objects, where gravitational binding energy significantly contributes to total mass. Future research, as outlined by the authors, will likely explore the implications of these findings for understanding the nature of dark energy and the evolution of large-scale structure in the universe. The presented framework offers a novel avenue for investigating the interplay between gravity, inertia, and cosmology, potentially refining our understanding of fundamental physical principles governing the universe. The demonstrated consistency with existing experimental constraints suggests this approach warrants further investigation as a viable alternative to standard cosmological models.