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ReadMe (Zenodo Dataset): Einstein Tensor Cycle (ETC) Transformations for FLRW and de Sitter Solutions Hirokazu Maruyama October 27, 2025 1. Overview This dataset accompanies the paper “Einstein Tensor Cycle (ETC) Transformation and Invariance in FLRW Cosmology.” It contains both Mathematica notebook files (.nb) and generated output files (.pdf) that demonstrate the symbolic computation of the Einstein Tensor Cycle (ETC) transformation: g(n+1) µν =Gµν[g(n)]. Through these computations, the fixed–point relation Gµν[g] = Λgµν is verified for three de Sitter metrics with different curvature parameters (k=±1,0) and for the matter–dominated Friedmann universe. 2. Repository Contents This dataset includes both source notebooks (.nb) and output PDFs: (a) Notebooks •ETC deSitter k1.nb ―Mathematica source code for the closed de Sitter case (k= +1); •ETC deSitter k0.nb ―Mathematica source code for the flat de Sitter case (k= 0); •ETC deSitter k-1.nb ―Mathematica source code for the open de Sitter case (k=−1); •ETC Friedmann.nb ―Mathematica source code for the matter-dominated Friedmann universe; (b) Outputs •ETC deSitter k1.pdf ―Output log for the closed de Sitter universe (k= +1); •ETC deSitter k0.pdf ―Output log for the flat de Sitter universe (k= 0); •ETC deSitter k-1.pdf ―Output log for the open de Sitter universe (k=−1); •ETC Friedmann.pdf ―Output log for the matter-dominated Friedmann universe (k= 0); •README.pdf ―The present document. 1
Each notebook calculates the metric tensor, Christoffel symbols, Ricci tensor, Ricci scalar, and Einstein tensor for the corresponding FLRW/de Sitter metric. It then performs three successive ETC iterations and checks the invariance condition Gµν = Λgµν. 3. Computational Environment •Software: Wolfram Mathematica 14.0 (64-bit) •Operating System: Windows 11 •Encoding: UTF-8 •Additional Packages: None required (core functions only) Each notebook uses the built-in functions: FullSimplify,PowerExpand, and Chop for algebraic simplification and numerical stabilization at each iteration. 4. How to Reproduce the Results 1. Open the corresponding .nb notebook in Mathematica. 2. Assign the curvature parameter kas needed (1,0,−1). 3. Define the scale factor: a(t) = H−1cosh(Ht) (k= 1), eHt (k= 0), H−1sinh(Ht) (k=−1), H=√Λ/3. 4. Run the code (3 iterations) and verify that G(n+1) µν =G(n) µν , confirming the ETC invariance. 5. For the Friedmann case (matter-dominated universe), set a(t) = exp(√8πGρ 3t), and confirm that G00 = 8πGρ/c2, recovering the standard Friedmann relation. 5. Citation If you use this dataset or computational method, please cite: H. Maruyama, Einstein Tensor Cycle Transformations for FLRW and de Sitter Solutions, Zenodo (2025). DOI: [assigned upon upload]. 6. License This dataset is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Users may freely copy, redistribute, and modify the files, provided that proper credit is given to the author. 7. Contact Questions or feedback: [email protected] 2
8. Acknowledgments The author thanks Wolfram Research for the Mathematica platform, and acknowledges Zenodo for providing an open-access DOI repository to ensure the reproducibility and preservation of scientific data. 3
