[Paper Review] Interpretation of intuitionistic solution of the vacuum Einstein equations in smooth topos

А.К. Гуц, Artem A. Zvyagintsev|ArXiv.org|Jan 24, 2000

Relativity and Gravitational Theory9 references3 citations

TL;DR

This paper develops an intuitionistic approach to gravity using synthetic differential geometry within a smooth topos framework, showing that vacuum solutions of the Einstein equations can exhibit non-classical behavior—such as strong infinitesimal gravitational fields and variable cosmological constants—depending on the topos stage. The key contribution is a non-classical Schwarzschild-like solution where the cosmological constant and matter density are infinitesimal, leading to metric signatures that depend on hidden dimensions and non-Archimedean structures.

ABSTRACT

The topos theory is a theory which is used for deciding a number of problems of theory of relativity, gravitation and quantum physics. In the article spherically symmetric solution of the vacuum Einstein equations in the Intuitionistic theory of Gravitation at different stages of smooth topos ${\bf Set}^{\bf L_{op}}$ is considered. Infinitesimal "weak" gravitational field can be strong at some stagies, for which we have the additional dimensions. For example, the cosmological constant is not constant with respect to additional dimensions. Signature of space-time metric can depend of density of vacuum and cosmological constant.

Motivation & Objective

• To extend the classical vacuum Einstein equations into an intuitionistic framework using synthetic differential geometry (SDG) and topos theory.
• To investigate how gravitational fields behave in different topos stages, particularly those involving infinitesimal quantities and additional dimensions.
• To explore the physical implications of non-classical solutions where the cosmological constant and matter density are not constant but depend on the topos stage.
• To analyze how the signature of the spacetime metric can vary based on the local structure of the topos and the behavior of infinitesimal fields.
• To demonstrate that what appears as a weak gravitational field in 4D spacetime may be strong in higher-dimensional or non-Archimedean geometric contexts.

Proposed method

• Formalizing gravity in the smooth topos $\mathbf{Set}^{\mathbf{L}^{\mathrm{op}}}$, where $\mathbf{L}$ is the opposite category of finitely generated $C^\infty$-rings, enabling infinitesimal reasoning.
• Applying the Kock-Lawvere axiom to define differentiable functions on a ring $\mathbf{R}$ that includes infinitesimals $D = \{x \in \mathbf{R} \mid x^2 = 0\}$, replacing classical logic with intuitionistic logic.
• Deriving vacuum Einstein equations in SDG with a non-zero energy-momentum tensor $\frac{8\pi G}{c^2} d u_i u_k$, where $d \in D$ is an infinitesimal density.
• Analyzing spherically symmetric solutions in different topos stages, including $\ell C^\infty(\mathbb{R})/\{a^2\}$, $\ell C^\infty(\mathbb{R}^n)/I$, and $\ell C^\infty(U)$, to study metric behavior under infinitesimal conditions.
• Using generalized elements and morphisms between stages (e.g., $\ell B \to \ell A$) to model transitions and derive stage-dependent metrics via pullbacks.
• Applying the condition $2\Lambda\rho = \kappa c^2 \rho^2$ in non-classical settings to maintain consistency with vacuum equations despite non-zero infinitesimal densities.

Experimental results

Research questions

• RQ1 How do vacuum solutions of the Einstein equations behave in an intuitionistic framework with infinitesimal matter densities?
• RQ2 What physical consequences arise when the cosmological constant $\Lambda$ is not constant but varies across topos stages?
• RQ3 Can a gravitational field that is weak in 4D spacetime appear strong in higher-dimensional or non-Archimedean geometric contexts?
• RQ4 How does the signature of the spacetime metric depend on the topos stage and the form of $\Lambda$, $\rho$, and $C$?
• RQ5 What is the role of hidden parameters $a \in \mathbb{R}^n$ in the topos model, and how do they relate to additional dimensions and non-classical field behavior?

Key findings

• A non-classical Schwarzschild-like solution is derived where $\Lambda$ and $\rho$ are infinitesimals, leading to a metric with the form $ds^2 = \left(1 + \frac{(\kappa c^2\rho - 2\Lambda)r^2}{6} + \frac{C}{r}\right)dt^2 - \cdots$, valid in stages where $C$ is also infinitesimal.
• At the stage $\ell C^\infty(\mathbb{R})/\{a^2\}$, the metric components depend on linear terms in $a$, resulting in a non-weak gravitational field in five dimensions $(t,r,\theta,\phi,a)$, even though the 4D projection appears weak.
• At the stage $\ell C^\infty(\mathbb{R}^2)/\{a_1 - a_2\}$, the condition $\rho^2 \equiv 0 \mod I$ forces $\rho = 0$, $\Lambda = 0$, and the metric reduces to the Minkowski form.
• In the stage $\ell C^\infty(U)$ for bounded $U$, the functions $\rho$, $\Lambda$, and $C$ can be small, preserving the Minkowski signature and ensuring a weak field in the 4D limit.
• The transition between stages via morphisms $\psi: \ell B \to \ell A$ leads to a transformed metric where $\rho$ and $\Lambda$ are pulled back, and infinitesimality is preserved modulo the ideal $J$.
• The condition $2\Lambda\rho = \kappa c^2 \rho^2$ holds modulo ideals in all stages, ensuring consistency with vacuum equations despite non-zero infinitesimal densities.