A temporal-spatial-entropic metric and its Kretschmann scalar

We present EinsteinKPy2Tex, an open-source Python package that 
computes fundamental geometric quantities for general relativistic metrics. 
Built upon the EinsteinPy library, the code calculates Christoffel symbols, 
Riemann curvature tensor, Ricci tensor and scalar, Einstein tensor, geodesic 
equations, and the Kretschmann scalar. Results are exported as professionally 
formatted \LaTeX\ documents with only non-zero components, facilitating 
direct inclusion in research papers and educational materials.

The software includes pre-configured implementations for two cornerstone 
metrics: the Friedmann-Lematre-Robertson-Walker (FLRW) metric for 
cosmology and the Schwarzschild metric for black hole physics. Built-in 
citations for Einstein (1915) and the EinsteinPy library (Ribeiro et al. 
2020) ensure proper academic attribution.

Building upon the Advanced Manifold-Metric Pairs (AMMP) framework recently 
developed by Ntelis (2025), we introduce and analyze a novel 5-dimensional 
metric that promotes entropy to a genuine spacetime coordinate—the txyzS 
metric. This represents the first concrete application of the AMMP 
framework, demonstrating that thermodynamic quantities can be treated as 
coordinates on equal footing with space and time. For this new metric, we 
compute all geometric quantities and derive the Kretschmann scalar, which 
acquires additional terms $4(\beta S)^4$ and $12H^2(\beta S)^2$ arising 
from the entropy dimension.

Numerical simulations of the Kretschmann scalar for matter-dominated, 
radiation-dominated, and $\Lambda$-dominated cosmologies reveal that the 
entropy dimension amplifies spacetime curvature by a factor ranging from 
$1.01$ to $4.15$ for entropy coordinates $S = 0.5$ to $2.0$, with the largest 
deviations occurring at late times ($t \sim 1$). The ratio 
$K_{\text{txyzS}}/K_{\text{FLRW}}$ exceeds unity for all $S > 0$, providing 
a clear observational target for constraining the AMMP framework using 
cosmic microwave background and large-scale structure data.

EinsteinKPy2Tex is designed for educators, students, and researchers needing rapid, accurate tensor calculations with publication-ready output. The entropy-dimension implementation establishes a new paradigm for incorporating thermodynamic degrees of freedom into general relativity, with potential applications to black hole thermodynamics, quantum gravity, late and early-universe cosmology. The software is open-source under an MIT license and permanently archived on Zenodo (DOI: 10.5281/zenodo.20563701), with the latest version available at https://github.com/lontelis/Einsteinpy_to_latex.