<p>We investigate the Kretschmann scalar within the framework of parametrised absolute parallelism (PAP) geometry, extending the classical understanding of space–time curvature in general relativity by incorporating torsion. This extension introduces a dimensionless parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ b $</EquationSource> <EquationSource Format="MATHML"><math> <mi>b</mi> </math></EquationSource> </InlineEquation>, allowing a seamless interpolation between Riemannian geometry (RG) (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ b = 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and Weitzenböck geometry (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ b = 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>). Using the pseudo-Riemannian metric associated with Friedmann–Lemaître–Robertson–Walker cosmology, we derive an explicit expression for the modified Kretschmann scalar, which captures contributions from standard curvature, curvature–torsion interactions and pure torsion. Our analysis reveals that the modified scalar reduces to the classical value under specific conditions (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ b = 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ b = \pm \sqrt{2} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mo>±</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation>), while deviations occur for other <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ b $</EquationSource> <EquationSource Format="MATHML"><math> <mi>b</mi> </math></EquationSource> </InlineEquation> values. This work highlights the potential role of torsion in the early Universe dynamics and provides a framework for exploring deviations from standard general relativity in cosmological models, such as the Big Bang–Big-Rip model (BRM).</p>

Parametrised evolution of the Kretschmann scalar in Friedmann–Lemaître–Robertson–Walker cosmology with torsion contributions and Big-Rip model

期刊论文
发表日期：

2026年2月16日

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Tahia F Dabash,
M A Bakry

摘要

We investigate the Kretschmann scalar within the framework of parametrised absolute parallelism (PAP) geometry, extending the classical understanding of space–time curvature in general relativity by incorporating torsion. This extension introduces a dimensionless parameter $ b $ $b$ , allowing a seamless interpolation between Riemannian geometry (RG) ( $ b = 0 $ $b = 0$ ) and Weitzenböck geometry ( $ b = 1 $ $b = 1$ ). Using the pseudo-Riemannian metric associated with Friedmann–Lemaître–Robertson–Walker cosmology, we derive an explicit expression for the modified Kretschmann scalar, which captures contributions from standard curvature, curvature–torsion interactions and pure torsion. Our analysis reveals that the modified scalar reduces to the classical value under specific conditions ( $ b = 0 $ $b = 0$ or $ b = \pm \sqrt{2} $ $b = \pm \sqrt{2}$ ), while deviations occur for other $ b $ $b$ values. This work highlights the potential role of torsion in the early Universe dynamics and provides a framework for exploring deviations from standard general relativity in cosmological models, such as the Big Bang–Big-Rip model (BRM).