Mathematics

Differential Geometry

Differential geometry is the language of General Relativity. It provides the mathematical framework for describing curved spacetimes, geodesics, and the intrinsic geometry of manifolds without reference to an embedding space.

1. Manifolds and Coordinate Charts

A smooth manifold is a topological space that locally resembles Euclidean space. Formally, an n-dimensional manifold M is covered by coordinate charts.

Coordinate Charts

A chart is a homeomorphism from an open subset U ⊂ M to an open subset of ℝⁿ:

$$\phi: U \to \mathbb{R}^n, \quad p \mapsto (x^1(p), x^2(p), \ldots, x^n(p))$$

Smooth Atlas

A collection of charts $\{(U_\alpha, \phi_\alpha)\}$ is called an atlas if it covers M. The atlas is smooth if the transition functions are smooth:

$$\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta)$$

Example: The 2-sphere S² can be covered by stereographic projection charts from north and south poles. Spacetime in GR is a 4-dimensional Lorentzian manifold with coordinates $(t, x, y, z)$ or $(x^\mu)$.

2. Tangent Spaces and Tangent Vectors

At each point p ∈ M, we can define a tangent space T_p M, a vector space of all tangent vectors at p.

Tangent Vectors as Directional Derivatives

A tangent vector v at p is a linear map from smooth functions to ℝ that satisfies the Leibniz rule:

$$v: C^\infty(M) \to \mathbb{R}, \quad v(fg) = v(f)g(p) + f(p)v(g)$$

Coordinate Basis

In local coordinates $(x^1, \ldots, x^n)$, the coordinate basis for T_p M is:

$$\left\{\frac{\partial}{\partial x^1}\bigg|_p, \ldots, \frac{\partial}{\partial x^n}\bigg|_p\right\}$$

Any tangent vector can be written as:

$$v = v^\mu \frac{\partial}{\partial x^\mu}$$

Cotangent Space

The cotangent space T*_p M is the dual space to T_p M. Elements are called covectors or one-forms. The coordinate basis for T*_p M is $\{dx^1, \ldots, dx^n\}$ with:

$$dx^\mu\left(\frac{\partial}{\partial x^\nu}\right) = \delta^\mu_\nu$$

3. The Metric Tensor

The metric tensor g is a symmetric, non-degenerate (0,2)-tensor that defines distances and angles on the manifold.

Definition

At each point p, the metric is a bilinear map:

$$g_p: T_p M \times T_p M \to \mathbb{R}$$

In coordinates:

$$g = g_{\mu\nu} dx^\mu \otimes dx^\nu$$

Line Element

The infinitesimal distance (proper time in GR) is given by:

$$ds^2 = g_{\mu\nu} dx^\mu dx^\nu$$

Signature

- Riemannian manifold: signature (+,+,...,+) — all positive eigenvalues
- Lorentzian (pseudo-Riemannian) manifold: signature (-,+,+,+) or (+,-,-,-) — one timelike, rest spacelike
General Relativity uses Lorentzian manifolds.

Examples

Minkowski spacetime (flat):

$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$

Schwarzschild metric (black hole):

$$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2$$

4. Connections and Covariant Derivatives

On a curved manifold, we cannot simply take partial derivatives of vector fields—we need a connectionto define how vectors are transported and differentiated.

Covariant Derivative

The covariant derivative ∇ is a map that generalizes the directional derivative:

$$\nabla: \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M), \quad (X, Y) \mapsto \nabla_X Y$$

In components, for a vector field $V = V^\mu \partial_\mu$:

$$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda} V^\lambda$$

Christoffel Symbols

The connection coefficients $\Gamma^\lambda_{\mu\nu}$ are called Christoffel symbols. For the Levi-Civita connection (unique torsion-free, metric-compatible connection), they are:

$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} \left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\right)$$

Properties

- Torsion-free: $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$
- Metric-compatible: $\nabla_\lambda g_{\mu\nu} = 0$

Covariant derivative of a covector (one-form) $\omega = \omega_\mu dx^\mu$:

$$\nabla_\mu \omega_\nu = \partial_\mu \omega_\nu - \Gamma^\lambda_{\mu\nu} \omega_\lambda$$

5. Parallel Transport and Geodesics

Parallel Transport

A vector field V is parallel transported along a curve γ(λ) if:

$$\nabla_{\dot{\gamma}} V = 0$$

In components:

$$\frac{dV^\mu}{d\lambda} + \Gamma^\mu_{\nu\sigma} \frac{dx^\nu}{d\lambda} V^\sigma = 0$$

Geodesics

A geodesic is a curve whose tangent vector is parallel transported along itself—the "straightest possible" path.

$$\nabla_{\dot{\gamma}} \dot{\gamma} = 0$$

The geodesic equation:

$$\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\sigma} \frac{dx^\nu}{d\lambda} \frac{dx^\sigma}{d\lambda} = 0$$

Physical Interpretation

In General Relativity, free-falling particles (no external forces) follow geodesics of spacetime. Timelike geodesics correspond to massive particles, null geodesics to photons.

Example: In Minkowski spacetime (flat), geodesics are straight lines. In curved spacetime near massive objects, geodesics bend—this is gravity.

6. Curvature Tensors

Curvature measures how much a manifold deviates from being flat. It is encoded in several related tensors.

Riemann Curvature Tensor

The Riemann tensor measures the failure of second covariant derivatives to commute:

$$(\nabla_\mu \nabla_\nu - \nabla_\nu \nabla_\mu) V^\lambda = R^\lambda_{\sigma\mu\nu} V^\sigma$$

In terms of Christoffel symbols:

$$R^\rho_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}$$

Ricci Tensor

The Ricci tensor is the contraction of the Riemann tensor:

$$R_{\mu\nu} = R^\lambda_{\mu\lambda\nu}$$

Ricci Scalar

The Ricci scalar (scalar curvature) is the trace of the Ricci tensor:

$$R = g^{\mu\nu} R_{\mu\nu}$$

Einstein Tensor

The Einstein tensor appears in Einstein's field equations:

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$$

Einstein's field equations relate spacetime curvature to matter-energy:

$$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

Physical Meaning

- Riemann tensor: Full curvature information (20 independent components in 4D)
- Ricci tensor: Traces "volume distortion" (10 independent components)
- Ricci scalar: Single number characterizing overall curvature

7. Fiber Bundles and Gauge Theory

Fiber bundles generalize the concept of manifolds by attaching a "fiber" space to each point of a base manifold. They are essential for modern gauge theories and quantum gravity.

Fiber Bundle Structure

A fiber bundle consists of:

  • Total space E
  • Base manifold M
  • Fiber F (space attached to each point)
  • Projection π: E → M

Locally: $E \cong U \times F$ (looks like product space), but globally may have non-trivial topology.

Tangent Bundle

The tangent bundle TM is a fiber bundle where the fiber at each point is the tangent space:

$$TM = \bigcup_{p \in M} T_p M$$

Principal Bundle and Gauge Fields

A principal G-bundle has a Lie group G as its fiber. Gauge theories are formulated on principal bundles.

The connection on a principal bundle is a gauge field (e.g., electromagnetic potential A_μ for U(1), gluon fields for SU(3)).

The curvature of the connection is the field strength tensor F_μν (electromagnetic field, gluon field strength).

Connection to General Relativity

In GR, the connection (Christoffel symbols) can be viewed as a gauge field for the diffeomorphism group. Loop quantum gravity reformulates GR as an SU(2) gauge theory on a principal bundle.

8. Differential Forms

Differential forms provide a coordinate-free way to express integration and calculus on manifolds.

k-Forms

A k-form ω is a totally antisymmetric (0,k)-tensor:

$$\omega = \frac{1}{k!} \omega_{\mu_1 \ldots \mu_k} dx^{\mu_1} \wedge \cdots \wedge dx^{\mu_k}$$

The wedge product ∧ is antisymmetric:

$$dx^\mu \wedge dx^\nu = - dx^\nu \wedge dx^\mu$$

Exterior Derivative

The exterior derivative d maps k-forms to (k+1)-forms:

$$d: \Omega^k(M) \to \Omega^{k+1}(M)$$

Key property: $d^2 = 0$ (applying d twice gives zero).

For a 0-form (function) f:

$$df = \partial_\mu f \, dx^\mu$$

For a 1-form $\omega = \omega_\mu dx^\mu$:

$$d\omega = \partial_\mu \omega_\nu \, dx^\mu \wedge dx^\nu = \frac{1}{2}(\partial_\mu \omega_\nu - \partial_\nu \omega_\mu) dx^\mu \wedge dx^\nu$$

Stokes' Theorem

The fundamental theorem of calculus on manifolds:

$$\int_M d\omega = \int_{\partial M} \omega$$

This unifies the gradient theorem, Green's theorem, Stokes' theorem, and divergence theorem.

Application to Physics

- Maxwell's equations can be elegantly written using 2-forms
- The electromagnetic field strength F is a 2-form: $F = dA$ where A is the potential 1-form
- $dF = 0$ (Bianchi identity) and $d\star F = \star J$ (Maxwell's equations with sources)

9. Applications to General Relativity

Spacetime as a Lorentzian Manifold

General Relativity models spacetime as a 4-dimensional Lorentzian manifold (M, g) with metric signature (-,+,+,+).

Key Differential Geometry Concepts in GR

  • Metric g_μν: Determines distances, angles, and causal structure
  • Levi-Civita connection: Unique connection compatible with metric, torsion-free
  • Geodesics: Free-fall trajectories of particles and photons
  • Riemann tensor: Encodes tidal forces and spacetime curvature
  • Einstein equations: Relate geometry (G_μν) to matter (T_μν)

Examples

Schwarzschild solution: Ricci tensor R_μν = 0 (vacuum), spherically symmetric

FLRW cosmology: Homogeneous, isotropic spacetime with R_μν proportional to g_μν

Gravitational waves: Ripples in curvature h_μν propagating at speed c, solutions to linearized Einstein equations

Why Differential Geometry?

GR cannot be formulated without differential geometry. The equivalence principle demands gravity be geometric, not a force—curvature of spacetime itself. The mathematical language needed to describe this is precisely the differential geometry of Lorentzian manifolds.

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