10.2 Covariant and Contravariant Tensors

Tensors are powerful mathematical tools that generalize scalars, vectors, and matrices. They're essential in physics for describing complex relationships between quantities, especially in relativity and electromagnetism.

Covariant and contravariant tensors transform differently under coordinate changes. The metric tensor allows us to switch between these types, crucial for expressing physical laws consistently across different reference frames.

Tensor Types and Transformations

Covariant vs contravariant tensors

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• Tensors generalize scalars, vectors, and matrices geometric objects characterized by rank (type) and transformation properties under coordinate changes
• Contravariant tensors have upper indices components transform oppositely to basis vectors
  • Contravariant vector $A^i$ transforms as $A'^i = \frac{\partial x'^i}{\partial x^j} A^j$
• Covariant tensors have lower indices components transform same way as basis vectors
  • Covariant vector $B_i$ transforms as $B'_i = \frac{\partial x^j}{\partial x'^i} B_j$
• Mixed tensors have both upper and lower indices each index transforms according to its respective rule (contravariant or covariant)

Index manipulation with metric tensor

• Metric tensor $g_{ij}$ symmetric, non-degenerate, rank-2 covariant tensor defines inner product and geometry of space
• Raise index by multiplying covariant tensor with inverse metric tensor $g^{ij}$
  • $A^i = g^{ij} A_j$ raises index of covariant vector $A_j$
• Lower index by multiplying contravariant tensor with metric tensor $g_{ij}$
  • $B_i = g_{ij} B^j$ lowers index of contravariant vector $B^j$
• Euclidean space metric tensor Kronecker delta $\delta_{ij}$
• Minkowski spacetime (special relativity) metric tensor Minkowski metric $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$

Applications in Physics

Tensor expression of physical laws

• Maxwell's equations in covariant form
  • $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ (inhomogeneous equation)
  • $\partial_{[\alpha} F_{\beta\gamma]} = 0$ (homogeneous equation)
  • $F_{\mu\nu}$ electromagnetic field tensor, $J^\nu$ four-current density
• Einstein's field equations in general relativity
  • $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$ relates geometry ($G_{\mu\nu}$ Einstein tensor) to matter-energy content ($T_{\mu\nu}$ stress-energy tensor)
• Stress-energy tensor $T_{\mu\nu}$ describes density and flux of energy and momentum in spacetime
  • Components include energy density, pressure, and stress (shear and normal)

Covariance in relativity applications

• Special relativity
  1. Lorentz transformations and four-vectors
     • Position four-vector $x^\mu = (ct, \vec{x})$
     • Momentum four-vector $p^\mu = (E/c, \vec{p})$
  2. Invariant spacetime interval $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$
• General relativity
  1. Geodesic equation $\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$ describes motion of particles in curved spacetime
     • $\Gamma^\mu_{\alpha\beta}$ Christoffel symbols (connection coefficients) related to metric tensor
  2. Parallel transport and covariant derivative generalize partial derivatives to curved spaces
     • $\nabla_\mu A^\nu = \partial_\mu A^\nu + \Gamma^\nu_{\mu\lambda} A^\lambda$ defines covariant derivative of contravariant vector $A^\nu$