Abstract:
This paper proposes a theoretical framework wherein Newton’s gravitational constant, G, is not strictly constant but varies infinitesimally across spacetime due to a background scalar field ε(x). This field represents persistent, non-zero deviations from mathematical limits, such as 0.999… = 1. We explore the implications of these infinitesimal shifts on gravitational geometry, including potential effects on curvature, geodesic motion, and energy distribution. The model remains consistent with general relativity at macroscopic scales but introduces potential implications for phenomena such as dark energy, inflation, and gravitational anomalies at quantum or cosmological boundaries.
1. Introduction
The mathematical identity 0.999… = 1 holds exactly in the real number system via limits. However, if physical spacetime deviates from mathematical idealization by preserving infinitesimal differences, these micro-deviations might become physically meaningful. Motivated by non-standard analysis and recent developments in scalar-tensor gravity, if we examine the possibility that spacetime itself is structured by an ε-field: an infinitesimal scalar field that modulates gravitational behavior.
I do not seek to replace general relativity (GR) but to extend it by incorporating subtle corrections arising from sub-resolution geometric effects.
2. Defining the ε-Deformation
Let’s define a scalar field ε(x), where ε is a continuous function of spacetime coordinates such that:
0 < |ε(x)| ≪ 1
This field represents an infinitesimal deviation from mathematical exactness and modifies Newton’s gravitational constant as follows:
G(x) = G₀ × (1 − ε(x))
Where G₀ is the standard Newtonian gravitational constant.
Einstein’s field equations, with this substitution, become:
Gμν = (8πG(x)/c⁴) × Tμν = (8πG₀(1 − ε(x))/c⁴) × Tμν
This introduces a location-dependent shift in how spacetime responds to energy and momentum.
3. Physical Implications and Modeling
We’ll focus on one measurable consequence: how ε-deformation affects gravitational curvature near a massive object.
Modified Schwarzschild Metric (Weak-Field Approximation)
In general relativity, the Schwarzschild metric in weak-field approximation is:
ds² = −[1 − (2G(x)M / rc²)] · c²dt² + [1 − (2G(x)M / rc²)]⁻¹ · dr² + r²dΩ²
Substituting G(x) gives:
ds² = −[1 − (2G₀(1 − ε(x))M / rc²)] · c²dt² + …
This means:
- Gravitational time dilation now depends on ε(x)
- Light bending and redshift gain ε-dependent corrections
- Near black holes, even tiny ε(x) deviations may accumulate over distance or time
4. Cosmological Scale Speculation
On very large (cosmic) scales, if ε(x) varies slowly across spacetime, it may lead to:
- An average deviation, ⟨ε(x)⟩, acting as an effective cosmological constant
- Accelerated expansion (dark energy) emerging as global tension from ε(x)
- Inflation in the early universe triggered by rapid ε(x) fluctuations
While this resembles scalar-tensor gravity models, here it emerges from infinitesimal geometric structure, not massive fields or exotic particles.
5. Constraints and Justifications
We remain within known physical laws:
- ε(x) does not introduce superluminal propagation or violate causality
- Total energy-momentum conservation still holds, though local curvature varies infinitesimally
- Effects vanish smoothly as ε(x) → 0, recovering standard GR
The model is speculative but grounded in frameworks used in modern theoretical physics, such as scalar-tensor theories and vacuum fluctuation-driven gravity. Future work may explore links to string theory moduli fields or loop quantum gravity discretization.
6. Conclusion
The ε-deformed spacetime framework proposes that spacetime possesses an infinitesimal texture capable of subtly altering gravitational responses. While effects are negligible at everyday scales, they may accumulate in extreme environments or over cosmological distances, potentially contributing to dark energy, inflation, or gravitational anomalies.
We recommend further modeling of ε(x)-influenced gravitational waves and potential experimental bounds from precision orbital mechanics or cosmological surveys.
References:
- P.G.N. de Vegvar, “Commutatively deformed general relativity: foundations, cosmology, and experimental tests,” Eur. Phys. J. C 81, 786 (2021).
- M. Bojowald and G.M. Paily, “Deformed General Relativity and Effective Actions from Loop Quantum Gravity,” Phys. Rev. D 86, 104018 (2012).
- “Scalar–tensor theory,” Wikipedia.
