Chapter 14: Tensors, Manifolds & General Relativity
1907–1915
Einstein's Struggle: 1907–1915
In 1905 Einstein had overturned mechanics with special relativity. By 1907 he realized his theory had a fatal flaw: it could not accommodate gravity. Newton's gravitational force acted instantaneously across all space — impossible in a world where nothing travels faster than light.
That same year, while working at the patent office in Bern, Einstein had what he called “the happiest thought of my life”: the equivalence principle. A person in free fall feels no gravity. A person in an accelerating rocket feels an artificial gravity indistinguishable from the real thing. Therefore, gravity is not a force — it is the curvature of spacetime.
This insight was clear. The mathematics to express it was not. Einstein spent eight years struggling to find the right equations. The missing ingredient was a branch of mathematics he had never studied: Riemannian tensor calculus.
Grossmann, Ricci & the Tensor Calculus
In 1912 Einstein returned to Zürich and turned to his old friend Marcel Grossmann, now a professor of mathematics. Einstein told him: “Grossmann, you must help me or I will go crazy.” Grossmann went to the library and returned with the answer: the absolute differential calculus of Gregorio Ricci-Curbastro and Tullio Levi-Civita, developed between 1884 and 1901.
Ricci and Levi-Civita had built a calculus of objects — tensors — that transform in a specific, covariant way under arbitrary changes of coordinates. A tensor equation valid in one coordinate system is automatically valid in all. This was exactly what Einstein needed: laws of physics that hold in any reference frame, accelerating or not.
The key object is the metric tensor \(g_{\mu\nu}\), which encodes distances and angles in curved spacetime:
\( ds^2 = g_{\mu\nu}\, dx^\mu\, dx^\nu \)
The Metric Tensor as Gravitational Potential
In Newtonian gravity, a single scalar \(\Phi\) is the gravitational potential. In general relativity, the ten independent components of the symmetric tensor\(g_{\mu\nu}\) replace it. The Christoffel symbols
\( \Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma}\!\left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\right) \)
encode how basis vectors change from point to point — the connection. Free-falling particles follow geodesics, the straightest possible paths in curved spacetime:
\( \frac{d^2 x^\rho}{d\tau^2} + \Gamma^\rho_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0 \)
What Newton called “the force of gravity” is, in this language, simply the straightest possible motion through a curved geometry.
Einstein's Field Equations
By November 1915, after years of false starts, Einstein had the complete theory. The field equations relate the geometry of spacetime (left side) to the distribution of matter and energy (right side):
\( G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}R\,g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu} \)
Here \(R_{\mu\nu}\) is the Ricci curvature tensor (a contraction of Riemann's curvature tensor), \(R\) is the Ricci scalar (its trace), and\(T_{\mu\nu}\) is the stress-energy tensor encoding the density and flux of energy and momentum. Ten equations in ten unknowns — but diffeomorphism invariance reduces this to six independent equations for the six independent components of\(g_{\mu\nu}\).
John Wheeler later summarized it: “Spacetime tells matter how to move; matter tells spacetime how to curve.”
Levi-Civita's Parallel Transport
Levi-Civita introduced a beautifully geometric concept: parallel transport. To carry a vector along a curve on a curved surface while keeping it “as parallel as possible,” you must account for the curvature of the surface itself. After parallel transporting a vector around a closed loop, it returns rotated by an angle proportional to the enclosed curvature.
The covariant derivative \(\nabla_\mu\) captures this:
\( \nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\rho}\, V^\rho \)
A geodesic is the curve along which the tangent vector is parallel-transported to itself:\(\nabla_{\dot\gamma}\dot\gamma = 0\). The geometry of parallel transport is the geometric meaning of the connection, and the holonomy it generates is precisely the Riemann curvature tensor.
Cartan's Differential Forms & Hilbert's Action Principle
Élie Cartan reformulated the same theory in a coordinate-free language ofdifferential forms and moving frames. In this approach the connection becomes a Lie-algebra-valued 1-form and curvature a 2-form; the Einstein equations emerge from Cartan's structure equations. This reformulation would later prove essential for Yang–Mills gauge theory and modern geometry.
Remarkably, on November 20, 1915 — five days before Einstein's own submission — David Hilbert derived Einstein's field equations from a variational principle. The Einstein–Hilbert action is
\( S = \frac{c^4}{16\pi G}\int R\,\sqrt{-g}\;d^4x \)
Varying this action with respect to \(g_{\mu\nu}\) yields Einstein's equations. The simplest diffeomorphism-invariant action for a metric gives the richest theory of gravitation.
The Mathematical Beauty: Geometry Is Gravity
Riemann imagined in 1854 that the geometry of physical space might be determined by physical forces. Einstein proved it in 1915: the gravitational field is the metric, the curvature of spacetime. Pure Riemannian geometry, developed with no physical application in mind, turned out to be the exact mathematical structure of gravity.