Curvature-induced dynamical effective spacetime dimension in an extension of general relativity

Abstract We propose a covariant extension of general relativity in which the local effective dimension of spacetime is promoted to a dynamical, curvature-induced field on an underlying four-dimensional manifold. Deviations from four dimensions are encoded in a scalar degree of freedom $$\varepsilon (x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , defining $$D_{\textrm{eff}}(x)=4-\varepsilon (x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which enters the gravitational action through a dimension weight $$v(\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> multiplying the Einstein–Hilbert term and a scalar potential $$U(\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . At the level of the field equations, these ingredients combine into a curvature-sensitive effective potential $$V(\varepsilon ,R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for the dynamical dimension field, where R is the Ricci scalar. In the limit $$\varepsilon \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$v(\varepsilon )\rightarrow 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the weak-field regime of general relativity is continuously recovered. We derive the modified Einstein equations and the equation of motion for $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> , showing that the resulting framework is scalar–tensor-like in structure, with the additional field controlling the local effective dimensionality of spacetime rather than describing an independent matter component. As benchmark applications, we study static, spherically symmetric configurations and a spatially flat Friedmann–Lemaître–Robertson–Walker background, where curvature-induced dimensional effects lead to controlled deformations of the mass–radius relation in compact objects and small corrections to the background expansion history.