Tensor Calculus and the Einstein Field Equation

July 2026

Original post: @cosmosarcive on X, 2026-07-19. If the embed above doesn’t load, everything it showed is written out below.

Why tensor calculus exists

Ordinary vector calculus works fine in flat space with a fixed set of coordinates, but it breaks down as soon as the underlying space itself is curved, or the coordinates change from point to point. Tensor calculus is the machinery built to fix that: a set of coordinate-independent rules for describing how quantities relate to each other regardless of which coordinate system happens to be in use. It’s the mathematical language General Relativity is written in — not a stylistic choice, but a genuine requirement once gravity is treated as spacetime curvature rather than a force.

The five pieces, in order

• Tensor — a multi-index object, $T^{i_1\ldots i_m}_{j_1\ldots j_n}$, that transforms consistently under a change of coordinates. Scalars (no indices) and vectors (one index) are just the simplest cases.
• Metric tensor — $g_{ij} = e_i \cdot e_j$, built from the dot products of the local basis vectors $e_i$ at a point. This is the object that actually defines distance and angle at that point — everything about a space’s geometry (flat, curved, how curved) is encoded in how $g_{ij}$ varies from point to point.
• Connection (Christoffel symbols) — $\Gamma^k_{ij} = \frac{1}{2}g^{k\ell}(\partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij})$, built directly from derivatives of the metric. This is what tells you how a vector’s components must change as you move it from one point to a neighboring one while keeping the vector itself genuinely unchanged — necessary precisely because “unchanged” isn’t a coordinate-independent notion once space is curved. (This is the same $\Gamma$ symbol PBT’s own papers reuse for an unrelated quantity, circulation/decay rate — see the Symbols & Units catalog for that collision.)
• Covariant derivative — $\nabla_k T^i_j = \partial_k T^i_j + \Gamma^i_{k\ell}T^\ell_j - \Gamma^\ell_{kj}T^i_\ell$, the actual generalization of “differentiate this” to curved space. The extra $\Gamma$ terms are exactly the correction needed to keep the result a genuine tensor — a plain partial derivative alone isn’t one, on curved space.
• The Einstein field equation — $G_{ij} = R_{ij} - \frac{1}{2}g_{ij}R = \frac{8\pi G}{c^4}T_{ij}$. Everything above exists to make this equation possible to write down: $G_{ij}$ (the Einstein tensor, built from curvature) on the left, $T_{ij}$ (the stress-energy tensor, describing the matter and energy actually present) on the right. In one line: the geometry of spacetime is determined by the matter and energy within it — the actual mathematical content of “mass tells spacetime how to curve, curvature tells mass how to move.”

Catalog status: Proven Systems

Tensor calculus itself is pure mathematics, not an empirical claim — it’s true independent of what physics uses it for. What it enables, the Einstein field equation, is a different matter entirely: see General Relativity for how extensively that specific equation’s predictions have been confirmed, from Mercury’s orbit to LIGO’s gravitational waves.

Where this touches PBT

This is worth being direct about. PBT’s own papers explicitly build their case against using this exact formalism — Paper 1’s introduction frames GR’s curvature as something that “leads to infinities at Planck scales,” and the whole series proposes mechanical pushes through a particle medium as an alternative to spacetime geometry doing the work. The Einstein field equation above is the actual, real mathematical object being offered an alternative to — not a strawman version of it. Paper 2’s light-bending calculation and this site’s Published Paper page both go out of their way to match this equation’s real, confirmed weak-field predictions rather than diverge from them, which only makes sense in light of how well-confirmed the equation itself is (see the catalog status above). Understanding what $G_{ij}=\frac{8\pi G}{c^4}T_{ij}$ actually says, and how well it’s held up, is a reasonable prerequisite for evaluating what a mechanical alternative to it would need to reproduce.