12.1 General relativity and spacetime geometry

General relativity revolutionized our understanding of gravity, describing it as the curvature of spacetime caused by mass and energy. This theory explains phenomena like black holes, gravitational waves, and the expansion of the universe.

Spacetime geometry is central to general relativity. It's described using a four-dimensional manifold with a Lorentzian metric, allowing us to mathematically represent the warping of space and time by massive objects.

Spacetime as a pseudo-Riemannian manifold

• Spacetime is a 4-dimensional manifold that combines the 3 spatial dimensions and 1 time dimension into a single geometric structure
• In general relativity, spacetime is described as a pseudo-Riemannian manifold, which is a generalization of a Riemannian manifold that allows for indefinite metrics
• The metric tensor of spacetime determines the geometry of the manifold and how distances and angles are measured between events

Metrics of spacetime

Minkowski metric for special relativity

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• The Minkowski metric is a flat spacetime metric used in special relativity to describe the geometry of spacetime in the absence of gravity
• It is given by the line element $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$, where $c$ is the speed of light and $dt$, $dx$, $dy$, and $dz$ are the infinitesimal changes in time and spatial coordinates
• The Minkowski metric has a signature of $(-,+,+,+)$, indicating one timelike dimension and three spacelike dimensions

Lorentzian metrics for general relativity

• Lorentzian metrics are used in general relativity to describe the geometry of spacetime in the presence of gravity
• They are pseudo-Riemannian metrics with a signature of $(-,+,+,+)$ or $(+,-,-,-)$, depending on the convention used
• The metric tensor in general relativity is determined by the distribution of matter and energy in spacetime through Einstein's field equations

Curvature of spacetime

Riemann curvature tensor

• The Riemann curvature tensor is a mathematical object that describes the intrinsic curvature of a manifold
• It measures how parallel transport of vectors around closed loops fails to return them to their original orientation
• In general relativity, the Riemann tensor is determined by the metric tensor and its derivatives

Ricci curvature tensor

• The Ricci curvature tensor is obtained by contracting the Riemann tensor over two of its indices
• It provides a measure of the average curvature in different directions at a given point in spacetime
• The Ricci tensor appears in Einstein's field equations and relates the curvature of spacetime to the distribution of matter and energy

Scalar curvature

• The scalar curvature, also known as the Ricci scalar, is a scalar quantity obtained by further contracting the Ricci tensor
• It gives a single number that characterizes the overall curvature of spacetime at a given point
• The scalar curvature appears in the Einstein-Hilbert action, which is used to derive Einstein's field equations from a variational principle

Geodesics in spacetime

Timelike vs spacelike geodesics

• Geodesics are the straightest possible paths between two points in a curved spacetime
• Timelike geodesics represent the worldlines of massive particles and always have a tangent vector with a negative norm ($ds^2 < 0$)
• Spacelike geodesics have a tangent vector with a positive norm ($ds^2 > 0$) and cannot be physically traversed by massive particles

Null geodesics for light paths

• Null geodesics have a tangent vector with a zero norm ($ds^2 = 0$) and represent the paths followed by light rays in spacetime
• They are the boundaries between timelike and spacelike regions in spacetime
• The paths of light are bent by the curvature of spacetime, leading to phenomena such as gravitational lensing

Geodesic deviation

• Geodesic deviation describes how initially parallel geodesics can diverge or converge due to the curvature of spacetime
• The relative acceleration between nearby geodesics is governed by the Riemann curvature tensor
• Geodesic deviation is responsible for the tidal forces experienced by extended objects in a gravitational field

Einstein's field equations

Energy-momentum tensor

• The energy-momentum tensor is a symmetric rank-2 tensor that describes the density and flux of energy and momentum in spacetime
• It includes contributions from matter, radiation, and other fields present in the universe
• The energy-momentum tensor acts as the source term in Einstein's field equations, determining how matter and energy curve spacetime

Coupling of matter and geometry

• Einstein's field equations relate the curvature of spacetime, described by the Einstein tensor (constructed from the Ricci tensor and scalar curvature), to the energy-momentum tensor
• The equations take the form $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$, where $G_{\mu\nu}$ is the Einstein tensor, $T_{\mu\nu}$ is the energy-momentum tensor, $G$ is Newton's gravitational constant, and $c$ is the speed of light
• This coupling between matter and geometry is a fundamental principle of general relativity, showing that matter and energy determine the curvature of spacetime, which in turn affects the motion of matter and energy

Vacuum solutions

• Vacuum solutions to Einstein's field equations are spacetimes in which the energy-momentum tensor vanishes ($T_{\mu\nu} = 0$)
• These solutions describe regions of spacetime that are devoid of matter and energy, but can still have non-trivial curvature
• Examples of vacuum solutions include the Schwarzschild metric (describing the spacetime around a non-rotating, uncharged black hole) and the Kerr metric (describing the spacetime around a rotating black hole)

Schwarzschild solution

Spherical symmetry

• The Schwarzschild solution is a spherically symmetric, static solution to Einstein's field equations in vacuum
• It describes the spacetime geometry around a non-rotating, uncharged, spherically symmetric massive object, such as a black hole or a star
• The metric is given by $ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2 + r^2d\Omega^2$, where $M$ is the mass of the central object and $d\Omega^2$ is the line element on a unit 2-sphere

Event horizon and singularity

• The Schwarzschild solution has an event horizon at the Schwarzschild radius, $r_s = \frac{2GM}{c^2}$, which is the boundary of the black hole
• Inside the event horizon, the roles of time and radial coordinate are switched, making it impossible for anything, including light, to escape from the black hole
• At the center of the black hole ($r=0$), there is a gravitational singularity where the curvature of spacetime becomes infinite and general relativity breaks down

Orbits in Schwarzschild spacetime

• The orbits of particles and light in Schwarzschild spacetime are determined by the geodesic equations
• Stable circular orbits exist only for radii greater than $3r_s$, while unstable circular orbits exist between $1.5r_s$ and $3r_s$
• At the photon sphere, located at $1.5r_s$, light can orbit the black hole in unstable circular orbits
• Particles and light can also follow bound elliptical orbits or unbound hyperbolic trajectories, depending on their initial conditions

Kerr solution for rotating black holes

Stationary axisymmetric spacetimes

• The Kerr solution describes the spacetime geometry around a rotating, uncharged black hole
• It is a stationary and axisymmetric solution to Einstein's field equations in vacuum, meaning that the metric is independent of time and azimuthal angle
• The Kerr metric is given by $ds^2 = -\left(1-\frac{2GMr}{c^2\rho^2}\right)c^2dt^2 + \frac{\rho^2}{\Delta}dr^2 + \rho^2d\theta^2 + \left(r^2+a^2+\frac{2GMra^2\sin^2\theta}{c^2\rho^2}\right)\sin^2\theta d\phi^2 - \frac{4GMra\sin^2\theta}{c\rho^2}cdtd\phi$, where $M$ is the mass of the black hole, $a$ is the angular momentum per unit mass, $\rho^2 = r^2 + a^2\cos^2\theta$, and $\Delta = r^2 - \frac{2GMr}{c^2} + a^2$

Ergosphere and extracting energy

• The Kerr solution features an ergosphere, a region outside the event horizon where spacetime is dragged along with the rotation of the black hole
• Within the ergosphere, it is impossible for particles to remain stationary with respect to a distant observer, as they must rotate in the same direction as the black hole
• The existence of the ergosphere allows for the extraction of energy from a rotating black hole through the Penrose process, in which a particle enters the ergosphere, splits into two, and one of the resulting particles escapes with more energy than the original particle

Orbits in Kerr spacetime

• The orbits of particles and light in Kerr spacetime are more complex than in Schwarzschild spacetime due to the presence of the black hole's rotation
• The innermost stable circular orbit (ISCO) depends on the black hole's angular momentum and the particle's orbital direction (prograde or retrograde)
• For a maximally rotating black hole, the ISCO is located at the event horizon for prograde orbits and at $9GM/c^2$ for retrograde orbits
• The Kerr metric also exhibits frame-dragging, or the Lense-Thirring effect, where a rotating black hole drags nearby spacetime and influences the orbits of particles and light

Cosmological solutions

Friedmann-Lemaître-Robertson-Walker metric

• The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is a solution to Einstein's field equations that describes a homogeneous and isotropic universe
• It is the foundation of modern cosmology and is given by $ds^2 = -c^2dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2d\Omega^2\right]$, where $a(t)$ is the scale factor that describes the expansion or contraction of the universe, and $k$ is the curvature parameter ($k=0$ for a flat universe, $k>0$ for a closed universe, and $k<0$ for an open universe)
• The FLRW metric is used to study the evolution of the universe on large scales and to derive the Friedmann equations that relate the expansion rate to the matter and energy content of the universe

Cosmological constant and dark energy

• The cosmological constant, denoted by $\Lambda$, is a term that can be added to Einstein's field equations to represent a constant energy density of the vacuum
• It was originally introduced by Einstein to achieve a static universe, but was later abandoned after the discovery of the expansion of the universe
• In the context of modern cosmology, the cosmological constant is often associated with dark energy, a hypothetical form of energy that permeates all of space and drives the accelerated expansion of the universe
• The presence of a positive cosmological constant leads to a repulsive gravitational effect that becomes dominant at large scales, causing the universe to expand at an increasing rate

Expansion history of the universe

• The expansion history of the universe describes how the scale factor $a(t)$ evolves with time
• It is determined by the Friedmann equations, which relate the expansion rate to the matter and energy content of the universe (including ordinary matter, radiation, and dark energy)
• The universe began with a Big Bang singularity at $t=0$, followed by a period of rapid inflation, during which the universe expanded exponentially
• After inflation, the universe underwent a period of radiation domination, followed by a period of matter domination
• Recent observations indicate that the universe is currently in a phase of accelerated expansion, driven by dark energy (which may be represented by a cosmological constant or a more general form of energy with negative pressure)

Gravitational waves

Linearized gravity

• Linearized gravity is an approximation to general relativity that is valid when the spacetime curvature is weak and the gravitational fields are small
• In this approximation, the metric tensor can be written as the sum of the Minkowski metric and a small perturbation: $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $\eta_{\mu\nu}$ is the Minkowski metric and $h_{\mu\nu}$ is the perturbation (with $|h_{\mu\nu}| \ll 1$)
• Linearized gravity is used to study the propagation of gravitational waves, which are small ripples in the fabric of spacetime that travel at the speed of light

Sources of gravitational waves

• Gravitational waves are generated by accelerating masses, similar to how electromagnetic waves are generated by accelerating charges
• The most significant sources of gravitational waves are compact binary systems, such as binary black holes, binary neutron stars, or a black hole-neutron star binary
• These systems emit gravitational waves as they orbit each other, losing energy and angular momentum in the process, which causes their orbits to decay and eventually leads to a merger
• Other sources of gravitational waves include supernovae, rapidly rotating neutron stars (pulsars), and the early universe (e.g., primordial gravitational waves from inflation)

Detection of gravitational waves

• Gravitational waves were first directly detected in 2015 by the Laser Interferometer Gravitational-Wave Observatory (LIGO), a century after their prediction by Einstein's general theory of relativity
• LIGO and other ground-based interferometers, such as Virgo and KAGRA, detect gravitational waves by measuring the tiny changes in the relative distances between suspended mirrors caused by the passing gravitational wave
• These interferometers are sensitive to gravitational waves in the frequency range of about 10 Hz to 10 kHz, which corresponds to the mergers of stellar-mass compact objects
• Future space-based interferometers, such as the Laser Interferometer Space Antenna (LISA), will be sensitive to gravitational waves in the millihertz range, which corresponds to the mergers of supermassive black holes and other extreme-mass-ratio inspirals