TensoriaCalc: A User-Friendly Tensor Calculus Package for the Wolfram Language

TL;DR

TensoriaCalc is a Wolfram Language–integrated tensor-calculus package designed to perform concrete differential-geometry calculations by defining explicit metrics and computing objects such as the covariant derivative, Riemann/Ricci tensors, and Hodge duals. The paper demonstrates the framework across 3D Euclidean and 4D Minkowski settings, showing how isometries, Killing vectors, and Lie algebras arise in practice, and how standard operators like the exterior derivative, Lie derivative, and coordinate transformations are implemented. It also applies the toolkit to cosmology and black-hole physics, deriving FLRW curvature cases, linearized weak-field gravity, and Schwarzschild geometry, including geodesics, redshift, and light deflection. Collectively, TensoriaCalc provides a user-friendly, geometry-aware environment for GR and related theories, enabling efficient, symbolically exact tensor calculations directly within the Wolfram ecosystem, with support for multiple concurrent metrics and automatic extraction of curvature data.

Abstract

We describe TensoriaCalc, a tensor calculus package written to be smoothly consistent with the Wolfram Language, so as to ensure ease of usage. It allows multiple metrics to be defined in a given session; and, once a metric is computed, associated standard differential geometry operations to be carried out - covariant derivatives, Hodge duals, index raising and lowering, derivation of geodesic equations, etc. Other non-metric operations, such as the Lie and exterior derivatives, coordinate transformation on tensors, etc. are also part of its built-in functionality.

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