Glossary

8.2 Geodesic equations and their derivation

3 min read•august 9, 2024

Geodesics are the curved space equivalent of straight lines, defining the shortest path between two points on a surface or manifold. They're crucial in general relativity, describing the motion of free-falling particles and light rays in curved spacetime.

The , derived from Euler-Lagrange equations, mathematically describes these curves. It uses to account for the curvature of space, showing how geodesics deviate from straight lines due to the underlying geometry.

Geodesic Equations

Fundamental Concepts of Geodesics

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  • Geodesic defines shortest path between two points on a curved surface or manifold
  • Represents straightest possible line in curved space
  • Generalizes concept of straight lines from Euclidean geometry to curved spaces
  • Geodesic equation describes mathematical formulation of geodesic curves
  • Affine parameter serves as natural parametrization along geodesic curves
    • Measures distance along curve in units proportional to proper time or arc length
    • Allows for simplified form of geodesic equation

Derivation and Formulation

  • Euler-Lagrange equations form basis for deriving geodesic equation
    • Originate from calculus of variations
    • Provide method for finding stationary points of functionals
  • Principle of least action applied to derive geodesic equation
    • Action defined as integral of Lagrangian along curve
    • Minimizing action leads to equations of motion for geodesics
  • Geodesic equation in terms of affine parameter λ: d2xμdλ2+Γαβμdxαdλdxβdλ=0\frac{d^2x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda} = 0
    • xμx^\mu represents coordinates of geodesic
    • Γαβμ\Gamma^\mu_{\alpha\beta} denotes Christoffel symbols

Applications and Significance

  • Geodesics play crucial role in general relativity
    • Describe motion of free-falling particles in curved spacetime
    • Determine path of light rays in gravitational fields
  • Used in differential geometry to study properties of curved surfaces
  • Applied in various fields (computer graphics, robotics, navigation)
  • Solving geodesic equations reveals intrinsic geometry of manifold
  • Understanding geodesics essential for analyzing gravitational lensing effects

Metric and Connections

Metric Tensor and Its Properties

  • gμνg_{\mu\nu} defines notion of distance and angle in curved space
  • Symmetric tensor field on manifold
  • Components of metric tensor vary with position in curved space
  • Inverse metric tensor gμνg^{\mu\nu} satisfies gμνgνρ=δμρg_{\mu\nu}g^{\nu\rho} = \delta^\rho_\mu
  • Metric tensor used to raise and lower indices of tensors
  • Determinant of metric tensor g=det⁡(gμν)g = \det(g_{\mu\nu}) important for volume elements

Christoffel Symbols and Their Role

  • Christoffel symbols Γαβμ\Gamma^\mu_{\alpha\beta} represent connection coefficients
  • Define how vectors parallel transport on curved manifold
  • Express how coordinate bases change from point to point
  • Symmetric in lower two indices: Γαβμ=Γβαμ\Gamma^\mu_{\alpha\beta} = \Gamma^\mu_{\beta\alpha}
  • Calculated from metric tensor and its derivatives: Γαβμ=12gμν(∂αgβν+∂βgαν−∂νgαβ)\Gamma^\mu_{\alpha\beta} = \frac{1}{2}g^{\mu\nu}(\partial_\alpha g_{\beta\nu} + \partial_\beta g_{\alpha\nu} - \partial_\nu g_{\alpha\beta})
  • Not tensors, but used to construct tensorial quantities (curvature tensor)

Covariant Derivative and Parallel Transport

  • Covariant derivative ∇μ\nabla_\mu generalizes ordinary derivative to curved spaces
  • Accounts for changes in coordinate bases when differentiating tensor fields
  • For a vector VμV^\mu, covariant derivative given by: ∇αVμ=∂αVμ+ΓαβμVβ\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha\beta}V^\beta
  • Parallel transport defined using covariant derivative
    • Vector VμV^\mu parallel transported along curve xμ(λ)x^\mu(\lambda) if: DVμDλ=dVμdλ+ΓαβμdxαdλVβ=0\frac{DV^\mu}{D\lambda} = \frac{dV^\mu}{d\lambda} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\lambda}V^\beta = 0
  • Covariant derivative preserves tensor character of objects
  • Used to formulate equations of motion and field equations in general relativity

Geodesic Properties

Autoparallel Curves and Their Characteristics

  • Autoparallel curves maintain constant direction in curved space
  • Tangent vector to autoparallel curve remains parallel to itself under parallel transport
  • Mathematically expressed as: DDλ(dxμdλ)=0\frac{D}{D\lambda}\left(\frac{dx^\mu}{d\lambda}\right) = 0
  • Coincide with geodesics in spaces with metric-compatible connection
  • Generalize concept of straight lines to curved spaces
  • Autoparallel property independent of parametrization

Extremal Curves and Variational Principles

  • Extremal curves minimize or maximize length between two points
  • Geodesics as extremal curves of the action integral: S=∫abgμνdxμdλdxνdλdλS = \int_a^b \sqrt{g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}}d\lambda
  • leads to geodesic equation
  • Not all extremal curves are necessarily shortest paths (analogous to great circles)
  • Extremal property connects geodesics to principle of least action in physics
  • Local nature of extremal property (may not be globally shortest path)
  • Importance in studying minimal surfaces and soap film problems

Key Terms to Review (18)

Affine connection: An affine connection is a mathematical tool that defines how to differentiate vectors along curves on a manifold. It allows for the comparison of vectors at different points and enables the concept of parallel transport, which is crucial for understanding geometric properties in physics and geometry.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work laid the foundations for modern differential geometry and contributed significantly to the understanding of curved spaces. His ideas on manifolds and metrics have profoundly influenced various areas of mathematics and physics, particularly in understanding gravitational fields and the geometric properties of space-time.
Christoffel Symbols: Christoffel symbols are mathematical objects used in differential geometry to describe how vectors change as they are parallel transported along curves in a manifold. They play a crucial role in formulating the geodesic equations, which represent the shortest paths in curved space, and are fundamental to understanding how curvature affects the behavior of physical systems described by tensor fields.
Curvilinear coordinates: Curvilinear coordinates are a coordinate system where the coordinate lines can be curved, allowing for a more flexible representation of shapes and spaces. They provide an alternative to Cartesian coordinates, making it easier to describe complex geometries and surfaces. This system is particularly useful in fields like physics and engineering, where it is essential to analyze objects that do not conform to straight lines or flat planes.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various areas of mathematics, including geometry, algebra, and mathematical logic. His work laid the groundwork for modern theoretical physics, particularly through the formulation of the Hilbert space concept, which has implications in quantum mechanics and the mathematical framework of general relativity.
Euler-Lagrange Equation: The Euler-Lagrange equation is a fundamental equation in calculus of variations, used to find the path that minimizes or maximizes a functional. This equation is crucial in deriving geodesic equations, which represent the shortest paths between points in curved spaces, such as those described by Riemannian geometry. The connection between the Euler-Lagrange equation and geodesics helps in understanding how physical systems evolve in the context of general relativity and geometric mechanics.
Free-fall path: A free-fall path is the trajectory followed by an object moving solely under the influence of gravity, without any other forces acting on it. This concept is crucial in understanding how objects behave in curved spacetime, where the path taken can be described by geodesics, which represent the shortest distance between points in this geometric framework.
Gauss's Lemma: Gauss's Lemma is a fundamental result in differential geometry that relates the geodesics of a surface to the curvature and the connection defined on that surface. It establishes how the Christoffel symbols, which represent the connection in a curved space, are crucial for determining the behavior of curves and their derivatives in a way that respects the geometry of the space. This lemma is essential for deriving the equations of geodesics, showing how they are influenced by the underlying metric and how they minimize distances.
Geodesic curvature: Geodesic curvature refers to the measure of how a curve deviates from being a geodesic on a surface. It quantifies the bending of a curve relative to the surface itself, helping to describe how paths behave in curved spaces. This concept is important when analyzing geodesics, which are the shortest paths between points on a surface, as it connects geometric intuition with physical applications, such as understanding forces in physics and properties in differential geometry.
Geodesic Curve: A geodesic curve is the shortest path between two points on a curved surface or manifold, representing the generalization of the concept of a straight line in Euclidean space. These curves arise naturally in differential geometry and are crucial for understanding how objects move in a curved space, often characterized by their intrinsic properties and metrics.
Geodesic Deviation Equation: The geodesic deviation equation describes how nearby geodesics in a curved spacetime diverge or converge due to the presence of gravitational fields. This equation reveals how the curvature of spacetime influences the relative motion of freely falling particles and is pivotal in understanding the effects of gravity on trajectories. It directly connects to the foundational aspects of geodesic equations and their derivation, illustrating how geometry shapes the paths of objects moving under gravity.
Geodesic Equation: The geodesic equation describes the path that a particle follows when moving through a curved space without any external forces acting on it. This equation is central to both physics and geometry as it provides a way to understand the motion of objects in a gravitational field and defines how distances are measured on curved surfaces.
Metric Tensor: The metric tensor is a mathematical construct that describes the geometric properties of a space, including distances and angles between points. It serves as a fundamental tool in general relativity, allowing for the understanding of how curvature affects the geometry of spacetime, and relates to other essential concepts like curvature, gravity, and tensor analysis.
Polar Coordinates: Polar coordinates are a two-dimensional coordinate system that represents points in terms of their distance from a reference point and the angle from a reference direction. This system is especially useful in contexts where circular or rotational symmetry is present, making it easier to describe the position of points and the relationships between them in fields like mechanics and differential geometry.
Riemannian manifold: A Riemannian manifold is a real, smooth manifold equipped with an inner product on the tangent space at each point, allowing for the measurement of lengths and angles. This structure provides a way to generalize the notions of geometry, including curvature, distance, and geodesics, enabling deeper insights into both mathematical and physical applications.
Sectional Curvature: Sectional curvature is a geometric concept that measures how a particular two-dimensional plane section of a Riemannian manifold curves in relation to the manifold's overall shape. This curvature can provide insight into the manifold's geometric properties, influencing how geodesics behave, and is closely related to concepts like the Ricci tensor and scalar curvature, as well as the Riemann curvature tensor.
Trajectory of a particle: The trajectory of a particle refers to the path that the particle follows as it moves through space over time. This concept is crucial for understanding how particles interact with forces and how they travel along geodesics, which are the shortest paths between points in a curved space. In the context of geodesic equations, the trajectory is determined by the geometry of the space and the forces acting on the particle, leading to insights into how particles behave in various spacetime structures.
Variational Principle: The variational principle is a fundamental concept in physics and mathematics that states that the path taken by a system between two states is the one for which a certain quantity, typically an action or energy, is minimized or maximized. This principle is crucial in deriving equations of motion, such as the geodesic equations, which describe the shortest path between points in curved space. It connects physical phenomena with geometric interpretations, making it a key tool in understanding the behavior of systems in both physics and geometry.
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