Four-Vectors and Tensors

Lorentz covariance is cleanest in index notation: every physical law is written so both sides transform the same way under Lorentz transformations, making invariance manifest. This page sets up four-vectors, the metric, raising/lowering, tensors of higher rank, and the rules that keep equations covariant. Convention: ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$, $c=1$, Greek indices $0,1,2,3$.

Contravariant and covariant vectors

A contravariant four-vector transforms like the coordinates,

$$V^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{V}^{\nu }.$$

A covariant vector (one-form) transforms with the inverse,

$$W_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{W}_{\nu }.$$

The two are related by the metric, which lowers and raises indices:

$$V_{\mu }={\eta }_{\mu \nu }{V}^{\nu },V^{\mu }={\eta }^{\mu \nu }{V}_{\nu },{\eta }^{\mu \rho }{\eta }_{\rho \nu }={\delta }^{\mu }{}_{\nu }.$$

Invariant inner product

The contraction of a vector with itself is a Lorentz scalar:

$$V\cdot V={\eta }_{\mu \nu }{V}^{\mu }{V}^{\nu }=(V^0{)}^2-|\mathbf{V}{|}^2.$$

For the four-momentum this is the mass; for four-position it is the interval. Any fully contracted expression is frame-independent.

Higher-rank tensors

A rank-$(m,n)$ tensor carries $m$ upper and $n$ lower indices, each transforming with its own $\Lambda$:

$$T^{\prime \mu \nu }{}_{\rho }={\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }({\Lambda }^{-1}{)}^{\sigma }{}_{\rho }{T}^{\alpha \beta }{}_{\sigma }.$$

The metric ${\eta }_{\mu \nu }$ and the Kronecker ${\delta }^{\mu }{}_{\nu }$ are invariant tensors; the Levi-Civita symbol ${ϵ}^{\mu \nu \rho \sigma }$ is invariant under $S{O}^{+}(1,3)$ (pseudotensor under parity).

Covariance rules

• Free indices match on both sides; repeated indices (one up, one down) are summed.
• ${\partial }_{\mu }=\partial /\partial x^{\mu }$ transforms covariantly; ${\partial }^{\mu }$ contravariantly; $\square ={\partial }^{\mu }{\partial }_{\mu }$ is invariant.
• Any equation built from equal tensors holds in all frames — this is why physics is written tensorially.

Catalog

See invariants for building scalars and the dynamics in relativistic dynamics. Group structure: Lorentz & Poincaré.