Four-Vectors and Lorentz-Invariant Decomposition

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Invariant content is separated from measured components by keeping the tensor itself fixed and specifying an observer before projecting it. A Lorentz scalar is independent of inertial coordinates; a measured temporal or spatial part is generally observer-dependent but is defined invariantly from the tensor and the observer’s four-velocity.

Required background. Calculus and Linear Algebra provides dual spaces, contractions, linear maps, and basis changes; Variational Mechanics and Symmetries supplies the distinction between a geometric object and its coordinate description.

Vectors, covectors, and contractions

A four-vector $A^\mu$ has components that transform as

A'^\mu=\Lambda^\mu{}_\nu A^\nu.

A covector $\alpha_\mu$ is a linear map from vectors to scalars and transforms with the inverse matrix,

\alpha'_\mu=(\Lambda^{-1})^\nu{}_\mu\alpha_\nu,

so $\alpha_\mu A^\mu$ is unchanged. The metric identifies vectors and covectors:

A_\mu=\eta_{\mu\nu}A^\nu.

In inertial Cartesian components with $(-+++)$ , lowering an index gives $A_0=-A^0$ and $A_i=A^i$ . Index position is therefore not decoration. General tensors transform with one appropriate matrix factor per index, while fully contracted expressions are scalars. Hughes 2024, Lecture 9, §§9.1–9.3, printed pp. 71–78 (PDF) develops the metric, dual vectors, and general tensors in this order.

Choose an observer, then project

At an event, an observer is represented by a future-directed unit timelike vector

u^\mu u_\mu=-1.

The observer’s local rest-space projector is

h^\mu{}_\nu=\delta^\mu{}_\nu+u^\mu u_\nu, \qquad h^\mu{}_\nu u^\nu=0, \qquad h^\mu{}_\alpha h^\alpha{}_\nu=h^\mu{}_\nu.

Thus $h$ annihilates the observer’s time direction and acts as the identity on the three-dimensional subspace orthogonal to $u^\mu$ . The metric restricted to that rest space is positive definite.

For any vector $A^\mu$ , define its observer-temporal scalar and spatial part by

A_{(u)}=-u_\mu A^\mu, \qquad A^{\langle\mu\rangle}=h^\mu{}_\nu A^\nu.

Then

A^\mu=A_{(u)}u^\mu+A^{\langle\mu\rangle}, \qquad u_\mu A^{\langle\mu\rangle}=0,

and its invariant norm becomes

A^\mu A_\mu=-A_{(u)}^2 +A^{\langle\mu\rangle}A_{\langle\mu\rangle}.

Two operations must not be conflated. A passive coordinate change transforms the components of both $A^\mu$ and $u^\mu$ and leaves $A_{(u)}$ unchanged. Replacing $u^\mu$ by a different physical observer changes the decomposition and can change $A_{(u)}$ , even if one continues to calculate in the same coordinates.

A covector decomposes analogously: with $\alpha_{(u)}=-u^\mu\alpha_\mu$ and $\alpha_{\langle\mu\rangle}=h^\nu{}_\mu\alpha_\nu$ , one has $\alpha_\mu=\alpha_{(u)}u_\mu+\alpha_{\langle\mu\rangle}$ . These definitions use no preferred chart. Gourgoulhon 2007, §2.4.1, printed pp. 29–31, Eqs. (2.60)–(2.71) (arXiv v1 PDF) gives the general timelike-normal projector algebra; Lim 2019, Definitions 1.4–1.5, printed p. 4 (PDF) gives the underlying vector/covector transformation rules with the same $(-+++)$ signature.

Rank-two observer decomposition

For a symmetric tensor $T^{\mu\nu}$ , the same projector defines

\rho_{(u)}=T^{\mu\nu}u_\mu u_\nu, \qquad q^\mu_{(u)}=-h^\mu{}_\alpha T^{\alpha\nu}u_\nu, \qquad S^{\mu\nu}_{(u)} =h^\mu{}_\alpha h^\nu{}_\beta T^{\alpha\beta}.

Here $\rho_{(u)}$ is the observer’s temporal-temporal component, $q^\mu_{(u)}$ is spatial, and $S^{\mu\nu}_{(u)}$ is spatial in both indices. The reconstruction is

T^{\mu\nu} =\rho_{(u)}u^\mu u^\nu +u^\mu q^\nu_{(u)} +q^\mu_{(u)}u^\nu +S^{\mu\nu}_{(u)}.

For a stress-energy tensor these pieces are energy density, momentum density/energy flux, and spatial stress relative to $u^\mu$ . A different observer obtains different pieces from the same $T^{\mu\nu}$ . If the tensor is not symmetric, its two mixed projections must be kept separately. Gourgoulhon 2007, §4.1.2, printed pp. 52–53, Eqs. (4.3)–(4.10) (arXiv v1 PDF) derives the corresponding rank-two decomposition; its $n$ , $\gamma$ , and spatial momentum covector become the page’s $u$ , $h$ , and raised flux vector.

Transfer check: two observers split one vector

In $(t,x)$ components, let $A^\mu=(5,3)$ , so $A^2=-16$ . Decompose it first relative to $u_0^\mu=(1,0)$ and then relative to $u_1^\mu=(5/4,3/4)$ .

Solution

Both candidate observers must be checked. Clearly $u_0^2=-1$ , while

u_1^2=-\frac{25}{16}+\frac9{16}=-1.

For $u_0$ ,

A_{(u_0)}=-u_0\cdot A=5, \qquad A^{\langle\mu\rangle}_{(u_0)}=(0,3).

The norm reconstruction gives $-5^2+3^2=-16$ .

For $u_1$ ,

A_{(u_1)} =-u_1\cdot A =-\left(-\frac54\,5+\frac34\,3\right)=4.

Moreover, $A^\mu=4u_1^\mu$ , so

A^{\langle\mu\rangle}_{(u_1)} =A^\mu-A_{(u_1)}u_1^\mu=0.

The second observer is aligned with the timelike vector and measures no spatial part; the invariant check is now $A^2=-4^2=-16$ . The temporal numbers $5$ and $4$ differ, but each is a well-defined scalar after its observer is supplied.

Common failures are to transform components without transforming the basis convention, to use $p^0$ as every observer’s energy, to lower a time index without the minus sign, or to use $\delta^\mu{}_\nu-u^\mu u_\nu$ as the rest projector in the $(-+++)$ convention. A null direction cannot itself be normalized into an observer four-velocity. Indeed, for any nonzero future null vector $k^\mu$ ,

k_{\langle\mu\rangle}k^{\langle\mu\rangle}=k_{(u)}^2,

so every timelike observer sees a spatial part whose magnitude equals the measured temporal component; no null rest frame exists.

Relativistic Energy, Momentum, and Mass Shells applies this decomposition to four-momentum and separates observer energy from invariant mass.

References

Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. §2.4.1, printed pp. 29–31, and §4.1.2, printed pp. 52–53 (arXiv:gr-qc/0703035v1 PDF), 2007.
Hughes, Scott A. Introduction to Relativity and Spacetime Physics. MIT 8.033 Lecture 9, §§9.1–9.3, printed pp. 71–78 (PDF), Fall 2024.
Lim, Ian. General Relativity. Notes for Malcolm Perry’s Cambridge Mathematical Tripos Part III course, Michaelmas 2018, last updated January 25, 2019. Definitions 1.4–1.5, printed p. 4 (PDF).