9.1 Geometry of curved spacetime

Curved spacetime is a key concept in general relativity. It describes how gravity warps the fabric of space and time, creating a geometry that deviates from flat Euclidean space.

This section explores the mathematical tools used to describe curved spacetime. We'll look at manifolds, metrics, curvature tensors, and parallel transport, which are essential for understanding gravity's effects on spacetime.

Spacetime Geometry

Manifold and Metric Tensor

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• Spacetime can be described mathematically as a 4-dimensional manifold, a continuous space that locally resembles Euclidean space
• The geometry of spacetime is determined by the metric tensor, a mathematical object that defines the distance between points in spacetime
• The metric tensor $g_{\mu\nu}$ is a symmetric 4x4 matrix that encodes information about the geometry of spacetime
• In flat spacetime, the metric tensor reduces to the Minkowski metric $\eta_{\mu\nu}$, which describes the geometry of special relativity

Minkowski and Schwarzschild Spacetimes

• Minkowski spacetime is a flat spacetime described by the Minkowski metric, where the line element is given by $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$
• The Schwarzschild metric describes the geometry of spacetime around a spherically symmetric, non-rotating mass (such as a black hole or a star)
• In Schwarzschild coordinates, the line element is given by $ds^2 = -(1-\frac{2GM}{c^2r})c^2dt^2 + (1-\frac{2GM}{c^2r})^{-1}dr^2 + r^2d\Omega^2$, where $M$ is the mass of the central object and $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the line element on a 2-sphere

Curvature

Riemann and Ricci Tensors

• Curvature is a measure of how much a manifold deviates from being flat (Euclidean)
• The Riemann curvature tensor $R_{\mu\nu\rho\sigma}$ is a mathematical object that encodes information about the curvature of spacetime
• The Riemann tensor has 20 independent components in 4 dimensions and satisfies certain symmetry properties
• The Ricci tensor $R_{\mu\nu}$ is obtained by contracting the Riemann tensor over two of its indices, $R_{\mu\nu} = R^{\rho}_{\mu\rho\nu}$, and has 10 independent components in 4 dimensions

Scalar Curvature and Einstein Tensor

• The scalar curvature $R$ is obtained by contracting the Ricci tensor with the metric tensor, $R = g^{\mu\nu}R_{\mu\nu}$, and is a single scalar value that characterizes the overall curvature of spacetime
• The Einstein tensor $G_{\mu\nu}$ is a symmetric 4x4 tensor constructed from the Ricci tensor and scalar curvature, given by $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$
• The Einstein tensor appears in the Einstein field equations, which relate the curvature of spacetime to the presence of matter and energy

Parallel Transport

Parallel Transport and Geodesics

• Parallel transport is a way of moving a vector along a curve in a manifold while preserving its direction relative to the curve
• In flat spacetime, parallel transport is trivial, as vectors maintain their direction when moved along straight lines
• In curved spacetime, parallel transport is non-trivial and depends on the path taken
• Geodesics are curves that parallel transport their own tangent vector, meaning that the tangent vector remains parallel to itself along the curve
• In the absence of external forces, particles in spacetime follow geodesics, which are the straightest possible paths in curved spacetime

Christoffel Symbols and Covariant Derivative

• The Christoffel symbols $\Gamma^{\mu}_{\nu\rho}$ are connection coefficients that describe how vectors change under parallel transport
• The Christoffel symbols can be computed from the metric tensor and its derivatives, $\Gamma^{\mu}_{\nu\rho} = \frac{1}{2}g^{\mu\sigma}(\partial_{\nu}g_{\rho\sigma} + \partial_{\rho}g_{\nu\sigma} - \partial_{\sigma}g_{\nu\rho})$
• The covariant derivative $\nabla_{\mu}$ is a generalization of the partial derivative that takes into account the curvature of spacetime and the change in basis vectors
• The covariant derivative of a vector $V^{\mu}$ is given by $\nabla_{\nu}V^{\mu} = \partial_{\nu}V^{\mu} + \Gamma^{\mu}_{\nu\rho}V^{\rho}$, which includes a term involving the Christoffel symbols to account for the change in the vector under parallel transport