Tensor Analysis - Tensors, Calculus, and Applications

Table of Contents

1. Introduction to Tensors
2. Tensor Algebra
3. Tensor Calculus
4. Covariant Derivative
5. Riemannian Geometry
6. Applications in Physics
7. Tensor Fields

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Introduction to Tensors

Motivation

Tensors generalize scalars, vectors, and matrices to higher dimensions

Coordinate-independent objects representing physical quantities

Transformation rules under coordinate changes

Tensor as Multilinear Map

Type $(p,q)$ tensor $T$: Maps $p$ covectors and $q$ vectors to $\mathbb{R}$

Contravariant order: $p$ Covariant order: $q$

Examples:

• Scalar (0,0): Real number
• Vector (1,0): Maps covector to number
• Linear form (0,1): Maps vector to number (covector)
• Linear map (1,1): Matrix (rank 2)

Metric tensor (0,2): Inner product

Component Form

In coordinates {x^\mu}:

$T = T^{\mu_1 \dots \mu_p}_{{\nu_1 \dots \nu_q}} \frac{\partial}{\partial x^{\mu_1}} \otimes \dots \otimes \frac{\partial}{\partial x^{\mu_p}} \otimes dx^{\nu_1} \otimes \dots \otimes dx^{\nu_q}$

Summation convention: Repeated indices summed

$T^{\mu …}_{{\nu …}}$: Components in coordinate system

Transformation Rules

Under coordinate change $x^\mu \to x'^\alpha$:

Contravariant (upper indices): $T'^{\alpha_1 \dots \alpha_p} = \frac{\partial x'^{\alpha_1}}{\partial x^{\mu_1}} \dots \frac{\partial x'^{\alpha_p}}{\partial x^{\mu_p}} T^{\mu_1 \dots \mu_p}$

Covariant (lower indices): $T'_{\beta_1 \dots \beta_q} = \frac{\partial x^{\nu_1}}{\partial x'^{\beta_1}} \dots \frac{\partial x^{\nu_q}}{\partial x'^{\beta_q}} T_{\nu_1 \dots \nu_q}$

Mixed tensor: Mixed product of transformations

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Tensor Algebra

Tensor Product

$u \otimes v$: Outer product

Components: $(u \otimes v)^{\mu\nu} = u^\mu v^\nu$

Properties:

• Not commutative: $u \otimes v \neq v \otimes u$ generally
• Associative: $(u \otimes v) \otimes w = u \otimes (v \otimes w)$
• Distributive: $u \otimes (v + w) = u \otimes v + u \otimes w$

Contraction

Sum over one upper and one lower index

Example: $T^\mu_\nu \to T^\mu_\mu$ (sum over $\mu$)

Decreases rank: $(p,q) \to (p-1,q-1)$

Raising/Lowering Indices

Metric tensor $g_{\mu\nu}$ (positive definite, symmetric)

Lower index: $T_\mu = g_{\mu\nu} T^\nu$

Raise index: $T^\mu = g^{\mu\nu} T_\nu$

where $g^{\mu\nu}$ inverse of $g_{\mu\nu}$

$g^{\mu\rho} g_{\rho\nu} = \delta^\mu_\nu$

Symmetrization and Antisymmetrization

Symmetric part: $T_{(\mu\nu)} = \frac{1}{2}(T_{\mu\nu} + T_{\nu\mu})$

Antisymmetric part: $T_{[μν]} = \frac{1}{2}(T_{μν} - T_{νμ})$

For p indices: $T_{(\mu_1 \dots \mu_p)} = \frac{1}{p!} \sum_{\sigma \in S_p} T_{\mu_{\sigma(1)} \dots \mu_{\sigma(p)}}$

$T_\epsilon$-notation: $\epsilon$-antisymmetric

Levi-Civita Symbol

$\epsilon^{\mu_1 \dots \mu_n}$: Totally antisymmetric

$\epsilon^{12 \dots n} = 1$ (even permutation) $\epsilon^{\mu …} = -1$ (odd permutation) $\epsilon = 0$ (repeated index)

$\epsilon^{\mu_1 \dots \mu_n} = (-1)^P$ where $P$ is permutation parity

Metric determinant connection: $\epsilon^{\mu_1 \dots \mu_n} = (\det g)^{-1/2} \tilde{\epsilon}^{\mu_1 \dots \mu_n}$

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Tensor Calculus

Partial Derivative

$\partial_\mu f = \partial f/\partial x^\mu$

For vectors: $\partial_\mu V^\nu = \partial V^\nu/\partial x^\mu$

Not a tensor! Transformation includes extra term

Reason: Vectors at different points live in different tangent spaces

Covariant Derivative

Correct derivative transforming as tensor

For vector field V: $\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\rho\mu} V^\rho$

Christoffel symbols $\Gamma^\nu_{\rho\mu}$: Connection coefficients

Determined by metric: $\Gamma^\mu_{\nu\rho} = \frac{1}{2}g^{\mu\sigma}(\partial_\nu g_{\rho\sigma} + \partial_\rho g_{\nu\sigma} - \partial_\sigma g_{\nu\rho})$

Covariant Derivative of Covector

$\nabla_\mu V_\nu = \partial_\mu V_\nu - \Gamma^\rho_{\nu\mu} V_\rho$

More minus sign for lower indices

Covariant Derivative of General Tensor

$\nabla_\mu T^{\mu_1 \dots \mu_p}_{{\nu_1 \dots \nu_q}} = \partial_\mu T^{\mu_1 \dots \mu_p}_{{\nu_1 \dots \nu_q}} + \sum_{i=1}^{p} \Gamma^{\mu_i}_{{\rho\mu}} T^{\dots \rho \dots}_{{\dots}} - \sum_{j=1}^{q} \Gamma^{\rho}_{{\nu_j \mu}} T^{\dots}_{{\dots \rho \dots}}$

One +$\Gamma$ for each upper index, one -$\Gamma$ for each lower index

Commutation

Second derivatives don’t commute:

$[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_{\sigma\mu\nu} V^\sigma$

Riemann tensor measures curvature

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Covariant Derivative

Parallel Transport

Vector parallel transported if covariant derivative vanishes

Dependence on path in curved spaces

Geodesics: Shortest paths where tangent vector parallel transported along itself

Geodesic Equation

Minimizing action or parallel transport:

$\frac{d^2x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\rho} \frac{dx^\nu}{d\lambda} \frac{dx^\rho}{d\lambda} = 0$

where $\lambda$ is affine parameter

In flat space: Straight lines ($\Gamma = 0$)

Parallel Vector Fields

Vector field V: $\nabla_\mu V^\nu = 0$ everywhere

Impossible in general curved spaces (except on geodesics)

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Riemannian Geometry

Metric Tensor

$g_{\mu\nu}(x)$: Defines geometry locally

Riemannian metric: Positive definite

Lorentzian metric (relativity): Signature $(+---)$ or $(-+++)$

Line element: $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$

Riemann Tensor

Measures curvature of manifold

$R^\rho_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}$

Symmetries:

• $R^\rho_{\sigma\mu\nu} = -R^\rho_{\sigma\nu\mu}$
• $R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$
• $R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$
• $R_{\rho[\sigma\mu\nu]} = 0$ (Bianchi identity)

Number of independent components: $n^2(n^2-1)/12$ in $n$ dimensions

For $n=4$: 20 independent components

Ricci Tensor

$R_{\mu\nu} = R^\rho_{\mu\rho\nu}$

Symmetric: $R_{\mu\nu} = R_{\nu\mu}$

Trace of Riemann tensor

Scalar Curvature

$R = g^{\mu\nu} R_{\mu\nu}$

Single number characterizing curvature

Positive: Spherical Negative: Hyperbolic Zero: Flat

Einstein Equations

Relating geometry to matter:

$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8\pi G T_{\mu\nu}$

$T_{\mu\nu}$: Stress-energy tensor

In vacuum: $R_{\mu\nu} = 0$ (Ricci-flat)

Vacuum Einstein equations

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Applications in Physics

Electromagnetism

Field strength tensor:

$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$

Antisymmetric: $F_{\mu\nu} = -F_{\nu\mu}$

Maxwell equations in curved spacetime:

$\nabla_\mu F^{\mu\nu} = J^\nu$ $\nabla_{[\rho} F_{\mu\nu]} = 0$

General Relativity

Stress-energy tensor: $T_{\mu\nu}$

Energy density: $T_{00}$ Momentum density: $T_{0i}$ Stress tensor: $T_{ij}$

Conservation: $\nabla_\mu T^{\mu\nu} = 0$

Continuum Mechanics

Strain tensor: $\epsilon_{ij}$

Stress tensor: $\sigma_{ij}$

Hooke’s law: $\sigma_{ij} = C_{ijkl} \epsilon_{kl}$

Elasticity tensor: $C_{ijkl}$ (symmetric under various index permutations)

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Tensor Fields

Definitions

Tensor field: Tensor assignment to each point in space

Scalar field: Function

Vector field: $V^\mu(x)$

Metric field: $g_{\mu\nu}(x)$

Tensor density: Tensors weighted by powers of metric determinant

Lie Derivative

Infinitesimal transformation along vector field

$L_X T = \lim_{\epsilon\to0} \frac{T(\phi_\epsilon(x)) - T(x)}{\epsilon}$

where $\phi_\epsilon$ is flow of $X$

Measures change under infinitesimal diffeomorphism

Differential Forms

Antisymmetric covariant tensors

$k$-form: Antisymmetric tensor of type $(0,k)$

Exterior derivative $d$: Maps $k$-form to $(k+1)$-form

$d^2 = 0$

Stokes’ Theorem: $\int_M d\omega = \int_{\partial M} \omega$

Hodge Star

Map between $k$-forms and $(n-k)$-forms

Depends on orientation and metric

$* \omega = \frac{1}{k!(n-k)!} \omega_{\mu_1 \dots \mu_k} \epsilon^{\mu_1 \dots \mu_n} dx^{\mu_{k+1}} \otimes \dots \otimes dx^{\mu_n}$

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