General Relativity Explained: Geodesics & Christoffel Symbols

Einstein's field equations describe how mass and energy curve space-time. These equations, which consist of 10 coupled partial differential equations, relate the geometry of space-time to the distribution of matter and energy. The equations show that matter tells space-time how to curve, and curved space-time tells matter how to move. These equations are extremely complex and require advanced mathematics to solve, often requiring computer simulations for practical applications.

The Einstein field equations are the fundamental equations of General Relativity, relating the geometry of spacetime to the distribution of matter and energy. These equations state that the curvature of spacetime (described by the Einstein tensor) is proportional to the energy-momentum tensor, which describes the density and flow of matter and energy. The equations are highly complex, involving 20 independent components that describe how spacetime curvature affects the motion of objects and how matter and energy affect spacetime curvature.

Einstein's field equations describe how matter and energy determine the curvature of spacetime. These equations consist of two parts: one describing the distribution of matter and energy, and another describing the resulting spacetime curvature. The equations are a family of differential equations that must be solved to understand gravitational phenomena. In flat spacetime, the spacetime interval ds² = -dt² + dx² measures distances between events, with light always traveling at 45-degree angles. Near massive objects, spacetime curves, requiring modified equations to account for this geometry.

Einstein's field equations relate the curvature of space-time to its energy content: R^μν - (1/2)Rg^μν = 8πGT^μν. The left side involves the Ricci tensor (contracted Riemann tensor) and Ricci scalar (further contraction), both derived from the metric tensor. The right side is the energy-momentum tensor multiplied by 8πG (Newton's gravitational constant). These equations replace Newton's F=ma and F=GMM/r² by stating that space-time curvature equals energy-momentum distribution. Solving them gives the metric tensor, which describes how objects move in gravitational fields.

General relativity describes gravity as spacetime curvature caused by mass. The Schwarzschild metric describes spacetime around a spherically symmetric, non-rotating mass: ds² = -c²(1 - 2GM/(c²r))dt² + (1 - 2GM/(c²r))⁻¹dr² + r²(dθ² + sin²θdφ²). The radial coordinate r measures distance from the center, and the metric coefficients depend on the Schwarzschild radius r_s = 2GM/c².

The Schwarzschild metric is a mathematical solution to Einstein's equations that describes the spacetime geometry around a spherical, non-rotating mass. When matter is compressed to sufficiently high density, the Schwarzschild metric predicts that spacetime becomes so curved that a black hole forms. The metric shows that at a certain critical density (the Schwarzschild radius), the curvature becomes infinite, creating a singularity.

Karl Schwarzschild found the first exact solution to Einstein's field equations in 1916, describing how spacetime curves around a spherical, non-rotating mass. This Schwarzschild metric shows that far from the mass, spacetime is nearly flat, but near the mass, time slows down and objects are pulled inward. At a critical distance called the Schwarzschild radius (R = 2GM/c²), spacetime curvature becomes so extreme that nothing, not even light, can escape. This defines the event horizon of a black hole, though most scientists initially doubted such objects could exist.

The Schwarzschild metric, solved by Karl Schwarzschild shortly after Einstein presented general relativity, describes the simplest possible black hole—a single unit of compressed mass within completely empty spacetime, unaffected by surroundings. This represents an eternal black hole that was never born and will never die, never growing or shrinking. From this earliest case study for general relativity, the concept of white holes was later extrapolated almost half a century later.

The Schwarzschild metric describes space-time geometry around a non-rotating black hole: c²dτ² = (1 - RS/R)c²dt² - R²/(1 - RS/R)dΩ²dr² - R²dΩ². This metric shows how space-time curvature affects measurements. At R = RS, the coefficient of dt² becomes zero, causing time dilation effects that explain why outside observers never see falling objects cross the event horizon.

To compare vectors at different locations on a manifold, we need parallel transport: imagine holding a vector like a harpoon and marching along a curve while keeping it as straight as possible. To compare two nearby vectors, we parallel transport one to the other so they share the same base point, then subtract them. If vectors are identical after parallel transport, the covariant derivative is zero (no change). If they differ, the covariant derivative is the difference vector, measuring how much the vector field deviates from parallel transport. This lets us track how vector fields change along curves.

Parallel transport along a curve γ: [0,ε] → X is defined by lifting γ to a curve γ̃ in E such that ∇_{γ'} γ̃ = 0. This condition defines a first-order ODE for γ̃ given γ. Local extensions exist via the inverse function theorem, and global extensions are constructed using partitions of unity. The covariant derivative ∇_V s along a vector field V combines directional derivatives with correction terms: ∇_V s = Σ[(∂f^β/∂x^a + γ^β_α^a f^α)V^a] s_β.

Parallel transport is the process of moving a vector along a curve while keeping it 'parallel' to itself. Given a curve parameterized by λ and a vector field, parallel transport produces a vector field that coincides with the original at λ = 0 but remains constant (covariantly) as we move along the curve. Mathematically, a vector u^a is transported parallelly if its covariant derivative with respect to λ is zero: D u^a / dλ = 0. The covariant derivative can be defined using parallel transport: Given a vector v at point x and nearby point x + δx, first transport v parallelly to x + δx to obtain v^∥. The covariant derivative is then lim(δx^μ→0) (v - v^∥) / δx^μ.

Parallel transport is the key concept for defining vector derivatives. In flat spacetime, vectors at different points can be compared directly. The covariant derivative is defined as: ∇_t V = lim(ε→0) [V(x+dx) - P(V(x))]/ε, where P(V(x)) is the vector V(x) parallel transported to x+dx. This definition ensures the derivative transforms as a tensor. In local inertial frames, this reduces to the ordinary partial derivative.

Parallel transport attempts to move vectors along curves while keeping them constant, but in curved space-time this is impossible without the vector changing direction. When transported around closed loops on curved surfaces like spheres, vectors return rotated, demonstrating curvature. The covariant derivative generalizes ordinary derivatives to curved coordinate systems, incorporating Christoffel symbols to account for changing basis vectors. Ordinary derivatives are not covariant—they don't transform properly under coordinate changes. Covariant derivatives replace ordinary derivatives in general relativity to obtain general covariant laws.