Differential Geometry
General relativity describes gravity not as a force but as the curvature of spacetime. To make this precise, we need the language of differential geometry: the mathematics of curved spaces. This lesson develops the essential toolkit: manifolds, tensors, the metric, covariant derivatives, and curvature. These concepts are not mere formalism; they are the alphabet in which Einstein wrote his field equations.
The Core Insight
A manifold is a space that may be curved globally but looks flat locally. Physics on curved spacetime is built by demanding that the laws look the same in every local patch, this requirement forces us to invent connections, covariant derivatives, and curvature tensors.
Manifolds: Locally Flat Spaces
What is a manifold?
A manifold is a space that can have a complex overall shape (curved, twisted) but which, viewed from close enough, resembles ordinary flat space. The surface of the Earth is the best example: globally it is a sphere, but at the scale of a neighborhood it appears perfectly flat. You cannot represent it on a single flat map without distortion, but you can cover it with an atlas of overlapping maps.
A manifold is a topological space that, in a neighborhood of each point, looks like ordinary $\mathbb{R}^n$. The surface of the Earth is a familiar example: globally it is a sphere, but locally, on the scale of a city, it looks flat. We describe it using overlapping coordinate charts.
A coordinate chart is a smooth, invertible map from a patch of the manifold to $\mathbb{R}^n$. An atlas is a collection of charts that covers the entire manifold. Where two charts overlap, the transition function between them must be smooth. No single chart need cover the whole manifold, just as no single flat map can cover the whole Earth without distortion.
In general relativity, spacetime is a 4-dimensional manifold with coordinates $x^\mu = (x^0, x^1, x^2, x^3)$. The superscript $\mu$ is an index that runs from 0 to 3. We are free to choose any coordinates we like, the physics must not depend on this choice.
Tangent Vectors and Cotangent Vectors
At each point $p$ on a manifold, we can define a tangent space $T_p$: the vector space of all directions you can move from $p$. On a 2D surface embedded in 3D, the tangent space at each point is literally the tangent plane.
A tangent vector $V^\mu$ has components that transform under a change of coordinates $x^\mu \to x'^\mu$ as:
$$V'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} V^\nu$$Here and throughout, we use the Einstein summation convention: repeated upper and lower indices are summed over. A cotangent vector (or one-form) $\omega_\mu$ lives in the dual space and transforms with the inverse Jacobian:
$$\omega'_\mu = \frac{\partial x^\nu}{\partial x'^\mu} \omega_\nu$$The contraction of a vector with a one-form, $\omega_\mu V^\mu$, is a scalar, invariant under coordinate changes. This is by design: physics must be coordinate-independent.
Tensors: Multilinear Maps
A tensor of type $(p, q)$ has $p$ upper indices and $q$ lower indices, and transforms as:
$$T'^{\mu_1 \cdots \mu_p}{}_{\nu_1 \cdots \nu_q} = \frac{\partial x'^{\mu_1}}{\partial x^{\alpha_1}} \cdots \frac{\partial x'^{\mu_p}}{\partial x^{\alpha_p}} \frac{\partial x^{\beta_1}}{\partial x'^{\nu_1}} \cdots \frac{\partial x^{\beta_q}}{\partial x'^{\nu_q}} T^{\alpha_1 \cdots \alpha_p}{}_{\beta_1 \cdots \beta_q}$$Scalars are $(0,0)$ tensors, vectors are $(1,0)$, one-forms are $(0,1)$. The power of tensors is that any equation written in tensor form is automatically valid in all coordinate systems. This is exactly what we need for general covariance.
The Metric Tensor
What is the metric tensor?
The metric tensor is the "ruler" of spacetime. It tells you how to measure distances and angles at each point. In flat space, distances are computed with the Pythagorean theorem. In curved space, the metric changes from point to point, just as distances on a world map are distorted compared to actual distances on the globe. Knowing the metric everywhere means knowing the entire geometry of spacetime.
The most important tensor in general relativity is the metric tensor $g_{\mu\nu}$, a symmetric $(0,2)$ tensor that defines distances and angles on the manifold.
The Line Element
The infinitesimal squared distance between nearby points is:
$$ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu$$In flat Minkowski spacetime: $ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$, so $g_{\mu\nu} = \eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$ (in units where $c = 1$).
On a curved manifold, $g_{\mu\nu}$ varies from point to point. It encodes all the geometric information: distances, angles, volumes, and, as we shall see, curvature. The metric also raises and lowers indices: $V_\mu = g_{\mu\nu} V^\nu$ and $V^\mu = g^{\mu\nu} V_\nu$, where $g^{\mu\nu}$ is the matrix inverse of $g_{\mu\nu}$.
For example, on the surface of a sphere of radius $R$, the line element is:
$$ds^2 = R^2 (d\theta^2 + \sin^2\theta \, d\phi^2)$$The factor $\sin^2\theta$ encodes the fact that circles of constant latitude shrink as you approach the poles.
Christoffel Symbols and Covariant Derivatives
Why do we need the covariant derivative?
In flat space, computing how a quantity varies from one point to another is straightforward: you take the ordinary derivative. But in curved space, the reference axes themselves rotate from point to point. The covariant derivative corrects for this effect by accounting for the rotation of the axes, much like a navigator corrects a compass for the curvature of the Earth. The Christoffel symbols are the necessary corrections, computed from the metric.
On a curved manifold, the ordinary partial derivative of a tensor is not a tensor, it does not transform correctly under coordinate changes. To fix this, we introduce the covariant derivative $\nabla_\mu$, which requires a connection.
For the unique torsion-free, metric-compatible connection (the Levi-Civita connection), the connection coefficients are the Christoffel symbols:
Christoffel Symbols
$$\Gamma^\lambda{}_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right)$$They are built entirely from the metric and its first derivatives. They are symmetric in their lower indices: $\Gamma^\lambda{}_{\mu\nu} = \Gamma^\lambda{}_{\nu\mu}$.
The covariant derivative of a vector is:
$$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu{}_{\mu\lambda} V^\lambda$$and for a one-form:
$$\nabla_\mu \omega_\nu = \partial_\mu \omega_\nu - \Gamma^\lambda{}_{\mu\nu} \omega_\lambda$$The key property is that $\nabla_\mu g_{\alpha\beta} = 0$: the covariant derivative of the metric vanishes. This means the connection is compatible with the notion of distance defined by $g_{\mu\nu}$.
Parallel Transport
How do you compare vectors at different points on a curved manifold? You cannot just subtract them, they live in different tangent spaces. The answer is parallel transport: slide a vector along a curve while keeping it "as straight as possible."
A vector $V^\mu$ is parallel transported along a curve $x^\mu(\lambda)$ if:
$$\frac{DV^\mu}{d\lambda} = \frac{dx^\nu}{d\lambda} \nabla_\nu V^\mu = 0$$On a flat space, parallel transport is trivial, the vector components don't change. On a curved space, something remarkable happens: if you parallel transport a vector around a closed loop, it comes back rotated. The amount of rotation directly measures the curvature enclosed by the loop.
The Riemann Curvature Tensor
The rotation of a vector under parallel transport around an infinitesimal loop is measured by the Riemann curvature tensor:
Riemann Tensor
$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho{}_{\nu\sigma} - \partial_\nu \Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$$It measures the failure of covariant derivatives to commute: $[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho{}_{\sigma\mu\nu} V^\sigma$.
In $n$ dimensions, the Riemann tensor has $\frac{n^2(n^2-1)}{12}$ independent components. In 4D spacetime, that is 20 independent components. In 2D, it has only 1 independent component, all curvature information is captured by a single number, the Gaussian curvature.
A space is flat if and only if $R^\rho{}_{\sigma\mu\nu} = 0$ everywhere.
Ricci Tensor and Scalar
By contracting the Riemann tensor, we obtain objects that appear directly in Einstein's equations.
The Ricci tensor is a contraction on the first and third indices:
$$R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu}$$It is symmetric: $R_{\mu\nu} = R_{\nu\mu}$. The Ricci scalar (or scalar curvature) is a further contraction:
$$R = g^{\mu\nu} R_{\mu\nu}$$The Ricci tensor captures the part of curvature that causes volumes to change. A positive Ricci scalar means space is "converging" (like a sphere), while a negative one means "diverging" (like a saddle).
Geodesics
In flat space, the shortest path between two points is a straight line. On a curved manifold, the analogous curves are geodesics: curves that extremize the proper distance (or proper time in spacetime).
The Geodesic Equation
$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu{}_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$A freely falling particle follows a geodesic. Gravity is not a force, it is the curvature of spacetime guiding objects along geodesics.
This equation can be derived from the action principle. The proper time along a path is:
$$\tau = \int \sqrt{-g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} \, d\lambda$$Extremizing this action gives the geodesic equation. We have come full circle: the principle of least action, which we began with in classical mechanics, now determines the motion of particles in curved spacetime.
On the surface of the Earth, geodesics are great circles (the shortest paths between cities on a globe). In spacetime, geodesics are the worldlines of freely falling objects, particles moving under gravity alone, with no other forces.
Key Insights
- A manifold is a space that is locally flat but may be globally curved, described by overlapping coordinate charts
- Tensors are the natural objects on manifolds: they transform covariantly under coordinate changes, ensuring physics is coordinate-independent
- The metric tensor $g_{\mu\nu}$ defines distances, angles, and volumes on the manifold
- Christoffel symbols encode how the coordinate basis vectors change from point to point, enabling the covariant derivative
- Parallel transport around a closed loop reveals curvature, a vector returns rotated
- The Riemann tensor $R^\rho{}_{\sigma\mu\nu}$ is the complete measure of curvature, built from the metric and its first two derivatives
- Geodesics are the straightest possible paths on a curved manifold, determined by the same variational principle that governs all of physics
Looking Ahead
We now have the full mathematical toolkit of differential geometry. In the next lesson, we use it to write down the central equation of general relativity: Einstein's field equations. These equations relate the curvature of spacetime (via the Ricci tensor and scalar) to the distribution of matter and energy (via the stress-energy tensor). The result is a self-consistent theory where matter tells spacetime how to curve, and spacetime tells matter how to move.
- A manifold is locally flat but globally curved, and the metric tensor $g_{\mu\nu}$ encodes all geometric information including distances, angles, and curvature.
- The covariant derivative and Christoffel symbols correct for the changing coordinate basis on curved spaces, enabling meaningful differentiation of tensors.
- Parallel transporting a vector around a closed loop on a curved manifold rotates it, and the Riemann curvature tensor quantifies this effect.
- Geodesics are the straightest possible paths in curved spacetime, and freely falling particles follow them, making gravity a consequence of geometry rather than a force.