📐Metric Differential Geometry Unit 6 – Isometries and Killing Vector Fields

Key Concepts and Definitions

• Metric tensor $g_{ij}$ defines the inner product on a Riemannian manifold $M$
• Isometry is a diffeomorphism $f: M \to M$ that preserves the metric tensor $g$
  • Satisfies $f^*g = g$, where $f^*$ denotes the pullback of $f$
• Killing vector field $X$ is a vector field that generates a one-parameter family of isometries
  • Satisfies the Killing equation $\mathcal{L}_X g = 0$, where $\mathcal{L}_X$ is the Lie derivative along $X$
• Lie derivative $\mathcal{L}_X$ measures the change of a tensor field along the flow of a vector field $X$
• Lie bracket $[X, Y]$ of two vector fields $X$ and $Y$ is another vector field that measures their non-commutativity
• Exponential map $\exp_p: T_pM \to M$ maps tangent vectors to points on the manifold along geodesics

Isometries: Fundamentals and Types

• Isometries preserve distances, angles, and geodesics on a Riemannian manifold
• Types of isometries include translations, rotations, and reflections
  • Translations shift points by a constant vector (Euclidean space)
  • Rotations preserve the origin and rotate points around it (spheres, SO(3))
  • Reflections flip points across a hyperplane or mirror (hyperbolic space)
• Isometry group $\text{Isom}(M)$ is the set of all isometries of $M$ with composition as the group operation
• Isometries can be classified as direct (orientation-preserving) or indirect (orientation-reversing)
• Infinitesimal isometries are vector fields that generate isometries and satisfy the Killing equation
• Isometries preserve curvature invariants such as the Ricci scalar and Kretschmann scalar

Killing Vector Fields: Introduction

• Killing vector fields (KVFs) are infinitesimal generators of isometries on a Riemannian manifold
• The flow of a KVF preserves the metric tensor $g$, i.e., $\mathcal{L}_X g = 0$
• In local coordinates, the Killing equation is $\nabla_i X_j + \nabla_j X_i = 0$, where $\nabla$ is the Levi-Civita connection
• KVFs form a Lie algebra under the Lie bracket operation $[X, Y]$
  • The Lie algebra of KVFs is isomorphic to the Lie algebra of the isometry group $\text{Isom}(M)$
• The maximum number of linearly independent KVFs on an $n$-dimensional manifold is $\frac{n(n+1)}{2}$
  • Achieved by maximally symmetric spaces (Euclidean, spherical, hyperbolic)

Properties of Killing Vector Fields

• KVFs are divergence-free, i.e., $\nabla_i X^i = 0$, which follows from the Killing equation
• The Lie bracket of two KVFs is another KVF, forming a closed Lie algebra structure
• KVFs preserve the Riemann curvature tensor $R_{ijkl}$, Ricci tensor $R_{ij}$, and scalar curvature $R$
  • $\mathcal{L}_X R_{ijkl} = 0$, $\mathcal{L}_X R_{ij} = 0$, and $\mathcal{L}_X R = 0$
• The integral curves of KVFs are geodesics, as they satisfy the geodesic equation $\nabla_X X = 0$
• KVFs are related to conserved quantities along geodesics via Noether's theorem
  • If $X$ is a KVF and $\gamma$ is a geodesic, then $g(X, \dot{\gamma})$ is constant along $\gamma$

Connections to Symmetry and Conservation Laws

• Isometries and KVFs are intimately related to the symmetries of a Riemannian manifold
• Each continuous symmetry of the metric tensor corresponds to a conserved quantity (Noether's theorem)
  • Translational symmetry leads to conservation of linear momentum
  • Rotational symmetry leads to conservation of angular momentum
  • Time translation symmetry leads to conservation of energy
• The number of linearly independent KVFs determines the degree of symmetry of the manifold
  • Highly symmetric spaces (Euclidean, spherical, hyperbolic) have the maximum number of KVFs
• Homogeneous spaces are manifolds where the isometry group acts transitively, i.e., any two points can be connected by an isometry
• Isotropy group at a point $p$ is the subgroup of isometries that fix $p$, related to the stabilizer in group theory

Applications in Physics and Geometry

• KVFs play a crucial role in general relativity and gravitational physics
  • Spacetime symmetries and conservation laws are encoded by KVFs of the spacetime metric
  • Static and stationary spacetimes possess timelike or spacelike KVFs, respectively
• Black hole spacetimes (Schwarzschild, Kerr) have KVFs corresponding to time translation and axial rotation
• Cosmological spacetimes (FLRW) have KVFs related to the homogeneity and isotropy of the universe
• KVFs are used in the classification of exact solutions to Einstein's field equations
• In Riemannian geometry, KVFs are employed in the study of symmetric spaces and homogeneous manifolds
  • Symmetric spaces have a KVF corresponding to the involutive isometry at each point
• KVFs are relevant in the context of geometric flows, such as the Ricci flow, where they are preserved under the flow

Problem-Solving Techniques

• To find KVFs, solve the Killing equation $\nabla_i X_j + \nabla_j X_i = 0$ in local coordinates
  • This yields a system of coupled partial differential equations for the components of $X$
• Exploit the symmetries of the metric tensor to simplify the Killing equation
  • Look for adapted coordinates that reflect the symmetries (spherical, cylindrical)
• Use the properties of KVFs (divergence-free, Lie algebra, curvature preservation) to constrain the solutions
• Employ tensor calculus and differential geometry techniques to manipulate the equations
  • Christoffel symbols, covariant derivatives, Lie derivatives, etc.
• Utilize computational tools (symbolic math software, numerical methods) for complex metrics
• Verify the obtained KVFs by checking the Killing equation and their Lie bracket relations

Advanced Topics and Extensions

• Conformal Killing vector fields (CKVFs) generalize KVFs by allowing conformal transformations of the metric
  • Satisfy the conformal Killing equation $\mathcal{L}_X g = \lambda g$, where $\lambda$ is a scalar function
• Affine Killing vector fields (AKVFs) preserve the connection instead of the metric
  • Satisfy the equation $\mathcal{L}_X \Gamma^i_{jk} = 0$, where $\Gamma^i_{jk}$ are the Christoffel symbols
• Homothetic Killing vector fields (HKVFs) are a special case of CKVFs with constant $\lambda$
  • Generate self-similar transformations of the metric, relevant in cosmology
• Killing tensors are generalizations of KVFs to higher-rank tensor fields
  • Satisfy the equation $\nabla_{(i_1} K_{i_2 \ldots i_r)} = 0$, where $K$ is a rank-$r$ symmetric tensor
• Killing-Yano tensors are antisymmetric generalizations of Killing tensors
  • Related to hidden symmetries and conserved quantities in black hole spacetimes
• Studying KVFs in the presence of matter fields and energy-momentum tensors
  • Coupling between symmetries and matter distribution
• KVFs in alternative gravity theories (f(R), scalar-tensor, Lovelock) and their physical implications