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Oriented Lorentzian fields, observers and coordinate conventions

Definition

Let $(M, g, N)$ be a smooth, Hausdorff, second-countable four-manifold without boundary, with smooth Lorentzian metric $g$ of signature $(-, +, +, +)$ and smooth time-orientation field $N$ , as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Supply an independent orientation $o$ of $M$ in the sense of Oriented smooth manifolds and oriented charts. The metric and time orientation do not select $o$ . Write ${v o l}_{g}$ for the oriented metric volume form of The metric volume form, current flux through 3-faces and finite-chain charges; in any positively oriented chart it has components $\sqrt{∣ \det (g_{a b}) ∣} d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ .

A smooth real $k$ -form is a smooth section of $⋀^{k} T^{*} M$ , and its space is denoted $Ω^{k} (M)$ , as in A smooth differential $k$ -form. In this page, $F \in Ω^{2} (M)$ means an arbitrary smooth real two-form. No equation such as $d F = 0$ , no global potential, and no physical field law is included in this definition.

An observer at $p \in M$ is a vector $n \in T_{p} M$ with $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) < 0$ ; its rest space is $n^{⊥} = {v : g_{p} (v, n) = 0}$ , as in Observers, rest spaces and the spatial projection at an event. A smooth observer congruence on an open set $U \subseteq M$ is a smooth vector field whose value at every point is an observer. Its observer curves are the maximal integral curves, and a chart is adapted to it when its three spatial coordinate functions are constant along those curves, as in Observer congruences, their observer curves and adapted charts. Such a congruence need not be hypersurface-orthogonal.

A local frame field on $U$ is an ordered smooth basis field of $T M ∣_{U}$ . A coordinate frame is the basis of coordinate vector fields supplied by a chart. A tetrad field is a local frame $(E_{0}, E_{1}, E_{2}, E_{3})$ with $g (E_{A}, E_{B}) = diag (- 1, 1, 1, 1)_{A B}$ ; it is future-directed when $E_{0}$ is future-directed, as in Local frame fields, coordinate frames and tetrad fields. These are distinct objects: an observer congruence is a vector field, an adapted chart is a coordinate chart, and a tetrad is a basis field. A tetrad field is not presumed to be a coordinate frame. For every observer congruence $n$ , each point has a neighborhood carrying an $n$ -adapted tetrad; this is a local assertion only.

A worldline on this Lorentzian manifold means a $C^{1}$ or piecewise- $C^{1}$ regular curve with future-timelike tangent and specified interval domain, as defined precisely in Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. Its parameter is arbitrary; the proper-time and four-velocity constructions of that definition apply under their stated hypotheses. A worldline is a mathematical curve here and does not assert the existence or motion of a physical body.

For coordinate bookkeeping only, when a chart is chosen with all four coordinate values assigned the metre, write $x^{0} = c t$ , where $t$ is assigned seconds and $c > 0$ is the corresponding scale in $m s^{- 1}$ . Then $g_{a b}$ is recorded as dimensionless. This is a unit convention on a chart, not a claim that the chart is inertial, orthonormal, global, or physically realized; $c$ in this mathematical convention has no asserted measured value.

Use the unique torsion-free metric-compatible connection $\nabla^{g}$ and the curvature convention of Metric compatibility, torsion freedom and the metric-selected connection class and Tensoriality of the curvature, its coordinate formula and its point-normal metric formula. In a length-coordinate chart, the displayed Christoffel formula differentiates dimensionless metric components once, so $Γ^{a}_{b c}$ is recorded in $m^{- 1}$ . The curvature coordinate formula contains one derivative of $Γ$ or a product of two $Γ$ components, so $R^{a}_{b c d}$ and its Ricci and scalar contractions are recorded in $m^{- 2}$ . These component assignments apply only in length-coordinate charts; other coordinate units transform by their Jacobians.

Facts & Assumptions

Given: A smooth time-oriented Lorentzian four-manifold $(M, g, N)$ , an independent orientation $o$ , and, where specified, a smooth observer congruence $n$ on an open set.

[F1]

The geometric data consist of a smooth boundaryless four-manifold, a smooth nondegenerate metric of signature $(-, +, +, +)$ , and a smooth timelike field $N$ selecting future direction. The time orientation does not supply a manifold orientation. (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts)

[F2]

A pointwise observer is future unit timelike; its rest space is $n^{⊥}$ . The orthogonal projection $h_{n} (v) = v + g (n, v) n$ has image $n^{⊥}$ , and this three-dimensional subspace has positive-definite restricted metric. (Observers, rest spaces and the spatial projection at an event, The observer rest-space decomposition and the spatial projection)

[F3]

A smooth observer congruence is a smooth future unit timelike field; its maximal integral curves and adapted charts have the domains and spatial-coordinate condition stated in Observer congruences, their observer curves and adapted charts. A tetrad field and coordinate frame have the distinct meanings stated in Local frame fields, coordinate frames and tetrad fields.

[F4]

The metric-selected connection is torsion-free and metric-compatible, with $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ . The curvature convention is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ , $R (\partial_{c}, \partial_{d}) \partial_{b} = R^{a}_{b c d} \partial_{a}$ , and (Metric compatibility, torsion freedom and the metric-selected connection class, Tensoriality of the curvature, its coordinate formula and its point-normal metric formula)

$R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b}) .$

The Ricci and scalar contractions are ${Ric}_{b d} = \sum_{a} R^{a}_{b a d}$ and $S = \sum_{b, d} g^{b d} {Ric}_{b d}$ .

[F5]

For each $m \geq 0$ , repeated differentiation of $x^{1 / 2}$ gives $D^{m} (x^{1 / 2}) = (\prod_{j = 0}^{m - 1} (\frac{1}{2} - j)) x^{1 / 2 - m}$ on $(0, \infty)$ ; each derivative is continuous there, so $x \mapsto \sqrt{x}$ is smooth. For $m = 0$ the product is empty and equals $1$ ; differentiating the formula at order $m$ by Continuity and derivatives of positive-base real powers multiplies its coefficient by $\frac{1}{2} - m$ , proving the next order by finite induction. The same induction with $\frac{1}{2}$ replaced by any fixed real exponent $α$ makes $x^{α}$ smooth on $(0, \infty)$ ; in particular $α = - \frac{1}{2}$ covers reciprocal square-root normalizers. For each finite $k$ , $C^{k}$ Euclidean maps are closed under componentwise algebra and composition gives closure of $C^{k}$ maps under finite componentwise sums, products and composition on open Euclidean domains. Applying this for every finite $k$ shows that composing the square root with a smooth positive function preserves smoothness, as used to normalize a smooth timelike field.

[A1]

The smooth tangent-bundle structure used by the existing Lorentzian and observer definitions is recorded under the Axiom of Countable Choice ${A C}_{ω}$ . Its exact statement is that each countable family of nonempty sets has a choice function. No full Axiom of Choice is used by the finite local construction below. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1F2F5algebra

The normalization $n : = N / \sqrt{- g (N, N)}$ is smooth because $- g (N, N)$ is a smooth positive function and [F5] makes its positive square root smooth. Bilinearity gives $g (n, n) = - 1$ and $g (n, N) = - \sqrt{- g (N, N)} < 0$ , so [F2] makes $n$ an observer at every point and hence a smooth observer congruence. This supplies an observer field without identifying it with a chart or tetrad.

1.2F1F2F3F5A1constructalgebra

Fix $p \in U$ and a connected chart around $p$ . The four coordinate vectors form a basis at $p$ . By [F2], the linear map $h_{n_{p}}$ has image the three-dimensional space $n_{p}^{⊥}$ , so three projected coordinate vectors form a basis of $n_{p}^{⊥}$ . Extend the chosen coordinate fields over the chart and project them by $h_{n}$ ; after shrinking the chart, their projections remain independent by continuity. They lie in the positive-definite rest spaces by [F2]. Apply the three finite Gram-Schmidt operations in these rest spaces; every denominator is the positive square root of a positive smooth squared norm, so [F5] makes the resulting fields smooth. Together with $E_{0} = n$ , the three fields form a local tetrad. If needed, reverse one spatial member so the tetrad has the supplied orientation from [F1]; the sign remains fixed on the connected chart. This is a local tetrad field as in [F3], not a coordinate frame or an assertion of an adapted coordinate chart. Thus an $n$ -adapted tetrad exists locally, with no global tetrad or hypersurface-orthogonality conclusion.

2.1F4algebra∎

In a length-coordinate chart, each first coordinate derivative of $g_{a b}$ has dimension $m^{- 1}$ , so the Christoffel formula in [F4] has dimension $m^{- 1}$ . Each term in the curvature formula in [F4] has dimension $m^{- 2}$ ; contraction with dimensionless $g^{a b}$ preserves that dimension for Ricci and scalar curvature. This is coordinate-unit bookkeeping, not a physical field equation.

Oriented Lorentzian fields, observers and coordinate conventions

Definition

Facts & Assumptions

Proof

Depends on

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