📐Tensor Analysis Unit 5 – Metric Tensors and Inner Products

Key Concepts and Definitions

• Metric tensor $g_{ij}$ a symmetric, positive definite tensor that defines the inner product on a manifold
• Inner product $\langle u, v \rangle$ a generalization of the dot product that measures the "similarity" between vectors $u$ and $v$
  • Defined in terms of the metric tensor as $\langle u, v \rangle = g_{ij}u^iv^j$
• Manifold a topological space that locally resembles Euclidean space near each point (examples: sphere, torus)
• Tangent space $T_pM$ the vector space of all tangent vectors to a manifold $M$ at a point $p$
• Coordinate system a set of functions that assign unique labels (coordinates) to each point on a manifold
• Covariant components $v_i$ the components of a vector $v$ with respect to the basis of the cotangent space
• Contravariant components $v^i$ the components of a vector $v$ with respect to the basis of the tangent space

Metric Tensor Basics

• Metric tensor determines the geometry of a manifold by defining distances, angles, and volumes
• In local coordinates, the metric tensor is represented by a matrix $[g_{ij}]$
  • Matrix elements $g_{ij}$ are functions of the coordinates
• Metric tensor is used to raise and lower indices of tensors
  • Raising an index: $v^i = g^{ij}v_j$, where $g^{ij}$ is the inverse of $g_{ij}$
  • Lowering an index: $v_i = g_{ij}v^j$
• Metric tensor defines the line element $ds^2 = g_{ij}dx^idx^j$, which measures the infinitesimal distance between two nearby points
• Metric tensor is symmetric, i.e., $g_{ij} = g_{ji}$
• Metric tensor is positive definite, meaning $\langle v, v \rangle > 0$ for all non-zero vectors $v$

Inner Product Fundamentals

• Inner product is a bilinear operation that takes two vectors and returns a scalar
  • Bilinear: $\langle au + bv, w \rangle = a\langle u, w \rangle + b\langle v, w \rangle$ and $\langle u, av + bw \rangle = a\langle u, v \rangle + b\langle u, w \rangle$
• Inner product is symmetric: $\langle u, v \rangle = \langle v, u \rangle$
• Inner product is positive definite: $\langle v, v \rangle \geq 0$, with equality if and only if $v = 0$
• Norm (length) of a vector $v$ is defined as $\|v\| = \sqrt{\langle v, v \rangle}$
• Angle $\theta$ between two vectors $u$ and $v$ is given by $\cos\theta = \frac{\langle u, v \rangle}{\|u\|\|v\|}$
• Orthogonality: two vectors $u$ and $v$ are orthogonal if $\langle u, v \rangle = 0$
• Cauchy-Schwarz inequality: $|\langle u, v \rangle| \leq \|u\|\|v\|$

Geometric Interpretation

• Metric tensor defines the shape and curvature of a manifold
  • Flat manifold (Euclidean space): $g_{ij} = \delta_{ij}$ (Kronecker delta)
  • Curved manifold (non-Euclidean space): $g_{ij}$ varies with coordinates
• Inner product measures the projection of one vector onto another
  • Orthogonal vectors have zero projection (example: $\hat{x}$ and $\hat{y}$ in Cartesian coordinates)
• Norm of a vector represents its length or magnitude
  • Unit vectors have a norm of 1
• Angle between vectors characterizes their relative orientation
  • Parallel vectors have an angle of 0° or 180°, while orthogonal vectors have an angle of 90°
• Metric tensor determines the shape of the light cone in relativity
  • Timelike intervals: $ds^2 < 0$, spacelike intervals: $ds^2 > 0$, lightlike (null) intervals: $ds^2 = 0$

Mathematical Properties and Operations

• Metric tensor and its inverse satisfy $g_{ik}g^{kj} = \delta_i^j$
• Determinant of the metric tensor $\det(g_{ij}) = g$ is used in volume integration $dV = \sqrt{|g|}d^nx$
• Christoffel symbols $\Gamma^k_{ij}$ are derived from the metric tensor and its derivatives
  • Used to define covariant derivatives and parallel transport
• Riemann curvature tensor $R_{ijkl}$ measures the curvature of a manifold and is constructed from the Christoffel symbols and their derivatives
  • Vanishes identically for flat manifolds
• Covariant derivative $\nabla_i$ generalizes the partial derivative to curved manifolds
  • Defined using Christoffel symbols: $\nabla_iV^j = \partial_iV^j + \Gamma^j_{ik}V^k$
• Parallel transport moves vectors along curves while preserving their inner product
  • Governed by the equation $\frac{dV^i}{dt} + \Gamma^i_{jk}V^j\frac{dx^k}{dt} = 0$

Applications in Physics and Engineering

• General relativity: metric tensor describes the geometry of spacetime
  • Einstein field equations relate the metric tensor to the energy-momentum tensor
• Geodesics: shortest paths between points on a manifold, determined by the metric tensor
  • Geodesic equation: $\frac{d^2x^i}{dt^2} + \Gamma^i_{jk}\frac{dx^j}{dt}\frac{dx^k}{dt} = 0$
• Fluid dynamics: metric tensor appears in the equations of motion for fluids on curved surfaces
• Elasticity theory: metric tensor describes the deformation of elastic materials
  • Strain tensor: $\varepsilon_{ij} = \frac{1}{2}(g_{ij} - \delta_{ij})$
• Electromagnetism: metric tensor is used in the formulation of Maxwell's equations on curved spacetime
• Quantum mechanics: inner product defines the probability amplitudes and expectation values of observables
  • Hilbert space: a complete inner product space used in quantum theory

Common Challenges and Misconceptions

• Distinguishing between contravariant and covariant components
  • Contravariant components transform oppositely to coordinate differentials, while covariant components transform in the same way
• Understanding the role of the metric tensor in raising and lowering indices
  • The metric tensor and its inverse are used to convert between contravariant and covariant components
• Interpreting the physical meaning of the metric tensor components
  • The components $g_{ij}$ encode information about the geometry of the manifold, such as distances, angles, and curvature
• Recognizing the difference between coordinate-dependent and coordinate-independent quantities
  • Tensors are coordinate-independent objects, while their components are coordinate-dependent
• Applying the correct index notation and summation convention
  • Repeated indices are summed over, with one upper and one lower index (Einstein summation convention)
• Distinguishing between flat and curved manifolds
  • Flat manifolds have a constant metric tensor, while curved manifolds have a metric tensor that varies with coordinates

Practice Problems and Examples

1. Given the metric tensor $g_{ij} = \begin{pmatrix} 1 & 0 \\ 0 & \sin^2\theta \end{pmatrix}$ on a 2-sphere, find the components of the inverse metric tensor $g^{ij}$.
2. Compute the inner product $\langle u, v \rangle$ of the vectors $u = (1, 2, -1)$ and $v = (3, 0, 4)$ in Euclidean 3-space with the metric tensor $g_{ij} = \delta_{ij}$.
3. Determine the norm of the vector $v = (2, -1, 3)$ in Minkowski spacetime with the metric tensor $g_{ij} = \text{diag}(-1, 1, 1, 1)$.
4. Find the angle between the vectors $u = (1, 1)$ and $v = (-1, 1)$ in the Euclidean plane using the inner product.
5. Given the metric tensor $g_{ij} = \begin{pmatrix} 1 & 0 \\ 0 & r^2 \end{pmatrix}$ in polar coordinates $(r, \theta)$, compute the Christoffel symbols $\Gamma^r_{rr}$, $\Gamma^r_{\theta\theta}$, and $\Gamma^\theta_{r\theta}$.
6. Consider the metric tensor $g_{ij} = \text{diag}(1, r^2, r^2\sin^2\theta)$ in spherical coordinates $(r, \theta, \phi)$. Calculate the volume element $dV$.
7. A particle moves along a geodesic in a 2D space with the metric tensor $g_{ij} = \begin{pmatrix} 1 & 0 \\ 0 & e^{2x} \end{pmatrix}$. Write down the geodesic equations for the coordinates $x(t)$ and $y(t)$.