How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Tensors And Observers

1 · Prerequisites

2 · Summary

The page builds the tensor and observer language on the geometric arena fixed by gr-framework-and-geometry. Tensor fields of type $(r, s)$ are defined at a point and on a four-manifold by $C^{\infty}$ chart components over the maximal smooth atlas, with the tensor-bundle formulation recorded and the index conventions — Einstein summation and the musical movement of slots — fixed (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components). The component transformation law between overlapping charts is proved from the change-of-basis formulas for coordinate derivations and coordinate differentials rather than assumed, and the same computation shows that the smoothness condition is independent of the atlas used (The component transformation law of a smooth tensor field). Contraction is defined pointwise by the basis-independent mixed-tensor formula (The contraction of a mixed tensor, Contraction is independent of the basis formula), and the field it produces is proved smooth in every chart, with the explicit component formula $\sum_{i} T^{i}_{i}$ in the trace case (Contraction of a smooth tensor field is a smooth tensor field).

An observer at an event is a single future-directed unit timelike tangent vector; its rest space is the $g$ -orthogonal complement of the line it spans, and its spatial projection is defined by the formula $h_{n} (v) = v + g_{p} (n, v) n$ (Observers, rest spaces and the spatial projection at an event). The page proves that $h_{n}$ is the orthogonal projection onto the rest space with kernel the observer line, that the rest space has dimension three with positive-definite metric restriction, and that every tangent vector has a unique orthogonal decomposition $v = - g_{p} (v, n) n + h_{n} (v)$ (The observer rest-space decomposition and the spatial projection). The pairing of the observer with future causal vectors is analysed next: a nonzero future causal vector pairs negatively with every observer, and for unit timelike vectors the sign of the pairing decides the time direction (Pairing of an observer with future causal vectors). The relative velocity of a future unit timelike vector $u$ is then defined and classified: $u$ decomposes uniquely as $u = γ (n + w / c)$ with $γ \geq 1$ , $w$ in the rest space and $g_{p} (w, w) = c^{2} (1 - γ^{- 2})$ , and the assignment $u \mapsto (γ, w)$ is a bijection onto the pairs satisfying those conditions (Relative velocity decomposition of a future unit timelike vector).

Fields of observers and frames occupy the middle of the page. An observer congruence on an open set is a smooth field whose value at every point is an observer; its maximal integral curves are its observer curves, and charts adapted to a congruence are defined by constancy of the spatial coordinates along those curves (Observer congruences, their observer curves and adapted charts). Local frame fields, coordinate frames and tetrad fields — pointwise $g$ -orthonormal frames, with the future-directed and congruence-adapted variants — are defined and kept distinct, the pairwise commuting coordinate fields supplying the necessary condition that separates coordinate frames from general frames (Local frame fields, coordinate frames and tetrad fields). A smooth future-directed tetrad field exists in a neighborhood of every event: the proof normalizes a future-directed orthonormal basis of the tangent space at the point and orthogonalizes the remaining fields inside the moving rest spaces, with successive shrinking of neighborhoods and the injective-regular-map theorem supplying smoothness of the square roots (Existence of a smooth future-directed tetrad field near every event). The change between two future-directed tetrads is a Lorentz matrix $Λ$ with $Λ^{T} η Λ = η$ , $\det Λ = \pm 1$ and $Λ^{0}_{0} > 0$ , and the components of a tensor in one tetrad are those in the other transformed by $Λ$ and $Λ^{- 1}$ (The change between two future-directed tetrads is a Lorentz matrix).

Every congruence admits a local adapted chart, obtained by applying the flow-box theorem to the nowhere-vanishing congruence field (Every congruence admits a local adapted chart), while the family of rest spaces always forms a smooth rank-three distribution that is tangent to local hypersurfaces if and only if $α \land d α = 0$ for the flat form $α = n^{♭}$ of the congruence — an integrability criterion, not a universal property (Smoothness of the rest-space family and its Frobenius integrability criterion). Observer-relative quantities close the mathematical content: observer energy $E_{n} (P) = - c g_{p} (P, n)$ , spatial momentum $q_{n} (P) = h_{n} (P)$ and rest mass $m (P) = \sqrt{- g_{p} (P, P)} / c$ are defined by explicit formulas and the unique decomposition $P = (E_{n} (P) / c) n + q_{n} (P)$ is recorded (Observer-relative energy, spatial momentum and rest mass of a tangent vector); the page then proves the master identity $E^{2} = c^{2} (g_{p} (q, q) - g_{p} (P, P))$ , the positivity of $E$ for nonzero future causal $P$ , and the mass shell $E^{2} = m^{2} c^{4} + c^{2} g_{p} (q, q)$ for future timelike $P$ , including its null specialization (Positivity, relative-velocity identities and the mass shell of observer-relative energy and momentum). A closing audit keeps observer, congruence, frame, tetrad, adapted chart, rest-space integrability and observer-relative quantities distinct and points to the item that proves each property (Charts, frames, tetrads, congruences and observers are distinct objects).

Nothing on this page attaches a clock reading, a measurement, a dynamics or a claim about nature to these geometric objects; those readings belong to the physical postulates and unit conventions of the framework, recorded elsewhere on the track. The worked examples and counterexamples of the pair are collected on gr-tensors-and-observers-examples.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

The tensor bundle on a smooth four-manifold from coordinate evaluations

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $M$ be a smooth four-manifold without boundary and $r, s \geq 0$ integers. Write $E = ∐_{p \in M} Mult ((T_{p}^{*} M)^{r} \times (T_{p} M)^{s}, R)$ and $π (p, T) = p$ . For a smooth chart $(U, x)$ , let $C_{x} (T)$ be the array of the evaluations $T (d x^{a_{1}}, \dots, d x^{a_{r}}, \partial_{x^{b_{1}}}, \dots, \partial_{x^{b_{s}}})$ , with all indices in ${0, 1, 2, 3}$ . Then the bijections $Φ_{x} : π^{- 1} (U) \to U \times R^{4^{r + s}}$ , $Φ_{x} (p, T) = (p, C_{x} (T))$ , define a unique smooth vector-bundle structure on $E$ .

For $J_{y x} (p) = D (y \circ x^{- 1}) (x (p))$ , the transition $Φ_{y} Φ_{x}^{- 1}$ multiplies each upper component index by $J_{y x}$ and each lower component index by $J_{y x}^{- 1}$ , with the intermediate indices summed. These maps obey $g_{x x} = I$ and $g_{z y} g_{y x} = g_{z x}$ . A family $p \mapsto T_{p}$ is a smooth section exactly when its arrays $C_{x} (T_{p})$ are smooth in the charts of one smooth atlas, equivalently in every smooth chart. No metric or global frame is required.

Facts & Assumptions

Given: The smooth four-manifold $M$ , integers $r, s \geq 0$ and ${A C}_{ω}$ .

[F1]

A smooth manifold is Hausdorff and second countable, with smooth coordinate changes (Smooth manifolds and their smooth charts, Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces). Countable choice selects from an at most countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

Coordinate derivations form a basis and coordinate differentials its dual; on an overlap $\partial_{y^{b}} = \sum_{d} (J_{y x}^{- 1})^{d}_{b} \partial_{x^{d}}$ (Coordinate derivations form a basis of the tangent space, Coordinate differentials form the dual cotangent basis, Change-of-coordinate formula for tangent bases). Tensors are multilinear maps with $r$ covector and $s$ vector arguments (A type $(r, s)$ tensor on a finite-dimensional vector space).

[F3]

Derivatives of smooth coordinate changes and finite sums, products and composites of smooth functions are smooth, and the chain rule gives $J_{z y} J_{y x} = J_{z x}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[F4]

A smooth linear cocycle on a supplied countable open cover constructs a smooth vector bundle (Construction of a vector bundle from a smooth cocycle). A smooth section is a smooth map right-inverse to the projection (Smooth sections, local sections, and support).

Proof

1.1givenF1choose

If $M$ is empty, all conclusions hold for the empty bundle. Otherwise fix a countable basis $(B_{n})$ for its topology. For each nonempty $B_{n}$ contained in a smooth chart domain let $X_{n}$ be the nonempty set of such charts restricted to $B_{n}$ . By [F1], countable choice supplies one chart from each $X_{n}$ . These restricted charts cover $M$ : a point has a smooth chart neighbourhood and a basis neighbourhood containing it inside that neighbourhood. This is the only selection from infinitely many sets in the construction. Denote the supplied countable chart cover by $(U_{α}, x_{α})$ .

1.2F2algebraconstruct

For each chart, multilinearity and the bases in [F2] make $C_{x}$ a linear isomorphism on every fibre. Explicitly, a prescribed array $c^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}}$ recovers the tensor by $T (ξ_{1}, \dots, ξ_{r}, v_{1}, \dots, v_{s}) = \sum_{a_{1}, \dots, a_{r}, b_{1}, \dots, b_{s}} c^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}} \prod_{i} ξ_{i} (\partial_{x^{a_{i}}}) \prod_{j} d x^{b_{j}} (v_{j})$ . Evaluating this formula on the coordinate covectors and vectors recovers the array, and expanding arbitrary arguments in these bases recovers every original tensor. For $r = s = 0$ this is simply the one scalar value. Thus $Φ_{x}$ is the stated fibrewise linear bijection.

2.1F2F3step 1.2algebra

The covector change is $d y^{a} = \sum_{c} J_{y x}^{a}_{c} d x^{c}$ : evaluate both sides on each $\partial_{x^{c}}$ and use the coordinate derivative of $y^{a}$ . Together with the vector change of [F2], multilinearity gives $C_{y} (T)^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}} = \sum_{c_{1}, \dots, c_{r}, d_{1}, \dots, d_{s}} \prod_{i} J_{y x}^{a_{i}}_{c_{i}} \prod_{j} (J_{y x}^{- 1})^{d_{j}}_{b_{j}} C_{x} (T)^{c_{1} \dots c_{r}}_{d_{1} \dots d_{s}}$ . Call this linear map $g_{y x}$ . Its coefficients are smooth by [F3]; the inverse Jacobian entries are the derivatives of the smooth inverse coordinate change, evaluated at $y (p)$ . The maps are invertible, since exchanging $x, y$ gives their inverses.

3.1F3step 1.2step 2.1algebra

Fibrewise, $g_{y x} = C_{y} C_{x}^{- 1}$ , so $g_{x x} = I$ and $g_{z y} g_{y x} = C_{z} C_{y}^{- 1} C_{y} C_{x}^{- 1} = g_{z x}$ on every triple overlap. This proves the cocycle with its direction fixed, including the index contractions of step 2.1; it also agrees with the chain rule of [F3].

4.1F4step 1.1step 1.2step 2.1step 3.1construct

Apply [F4] to the countable cover of step 1.1, with rank $4^{r + s}$ and the cocycle of step 3.1. The resulting quotient bundle maps to the actual set $E$ by $[p, c]_{α} \mapsto (p, C_{x_{α}}^{- 1} (c))$ . The relation identifies precisely the representatives giving the same tensor, by $g_{β α} = C_{x_{β}} C_{x_{α}}^{- 1}$ ; hence this map is well defined and bijective. Transport the topology and smooth structure along it. The bundle charts on the chosen cover become exactly $Φ_{x_{α}}$ . Any further smooth chart has smooth linear transitions to these charts by step 2.1, so its $Φ_{x}$ is also a smooth bundle chart. Uniqueness follows because the charts on the cover determine the topology (a set is open exactly when its intersection with each chart domain has open chart image) and the generated smooth atlas. In particular another countable selection gives the same structure.

5.1F3F4step 2.1step 4.1∎

In the chart $x$ , a section has product-coordinate expression $p \mapsto (p, C_{x} (T_{p}))$ , so it is smooth exactly when these component functions are smooth, by [F4]. If the components are smooth on one smooth atlas, every other smooth chart is covered by overlaps with that atlas. Step 2.1 expresses its components on each overlap as finite sums of products of the original smooth components and smooth transition coefficients, composed with the base coordinate changes; [F3] makes them smooth. Smoothness is local, so they are smooth on the entire new chart. The converse is immediate. This proves the section criterion and all the claims.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Tensor fields of type (r,s) on a Lorentzian four-manifold and their components

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $r, s \geq 0$ be integers.

(a) Tensors at a point. For $p \in M$ , the tangent space $T_{p} M$ is the real vector space of derivations at $p$ (Derivations at a point and the tangent space) and the cotangent space is its algebraic dual, $T_{p}^{*} M = (T_{p} M)^{*}$ (Cotangent space and cotangent bundle as a disjoint union). A type $(r, s)$ tensor at $p$ is a multilinear map

$T_{p} : (T_{p}^{*} M)^{r} \times (T_{p} M)^{s} ⟶ R,$

in the sense of A type $(r, s)$ tensor on a finite-dimensional vector space: the first $r$ arguments are covectors and the last $s$ are vectors. In particular a type $(1, 0)$ tensor is a vector, a type $(0, 1)$ tensor is a covector and a type $(0, 2)$ tensor is a bilinear form, so the covariant $2$ -tensors of Smooth covariant 2-tensor fields on a smooth manifold are exactly the type $(0, 2)$ tensors of this definition.

(b) The tensor bundle and smooth fields. The type $(r, s)$ tensor bundle $T_{s}^{r} M$ is the disjoint union of the fibres $T_{s}^{r} M ∣_{p} = Mult ((T_{p}^{*} M)^{r} \times (T_{p} M)^{s}, R)$ over $p \in M$ (The type $(r, s)$ tensor bundle); it carries the smooth vector-bundle structure defined by the tensor-coordinate change rules (The tensor bundle on a smooth four-manifold from coordinate evaluations). A smooth tensor field of type $(r, s)$ on $M$ is a family $T = (T_{p})_{p \in M}$ assigning to each $p$ a type $(r, s)$ tensor at $p$ , such that for every smooth chart $(U, x)$ of the maximal smooth atlas the component functions

$T^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}} (q) : = T_{q} (d x^{a_{1}} ∣_{q}, \dots, d x^{a_{r}} ∣_{q}, \partial_{b_{1}} ∣_{q}, \dots, \partial_{b_{s}} ∣_{q}), q \in U,$

are $C^{\infty}$ on $U$ ; here the coordinate derivations $\partial_{b} ∣_{q} = \partial / \partial x^{b} ∣_{q}$ form a basis of $T_{q} M$ (Coordinate derivations form a basis of the tangent space) and the coordinate differentials $d x^{a} ∣_{q}$ form the dual basis of $T_{q}^{*} M$ (Coordinate differentials form the dual cotangent basis). This is equivalent to $T$ being a smooth section of $T_{s}^{r} M$ in the sense of A smooth tensor field: the component functions are the frame coefficients of the section in the local frame induced by the chart. The smoothness condition is independent of the chosen smooth atlas and the components in two overlapping charts are related by the transformation law proved in the lemma on this page; the definition is therefore chart independent rather than merely chart-wise.

(c) Index conventions. Indices are written $a, b, c, \dots$ for components in a chart, with the Einstein summation convention: a repeated index, once upper and once lower, is summed over $0, 1, 2, 3$ . An upper index is a contravariant slot in which a covector is evaluated, a lower index a covariant slot in which a vector is evaluated. Slots are moved between these positions with the musical isomorphisms $♭_{p}, ♯_{p}$ of The inverse metric, musical isomorphisms and index raising to change slot variance. For example, if $T$ is a type $(1, 1)$ field, lowering its contravariant slot gives the type $(0, 2)$ field with components $T_{a b} : = g_{a c} T^{c}_{b}$ ; if $S$ is a type $(0, 2)$ field, raising its first covariant slot gives the type $(1, 1)$ field with components $S^{a}_{b} : = g^{a c} S_{c b}$ . Each operation changes the tensor type and is invertible because $g_{p}$ is nondegenerate.

(d) Scope. The definition is purely local and chart-based; no global triviality of $T_{s}^{r} M$ , no global frame and no choice of metric-compatible fields is assumed or implied. All fields are smooth ( $C^{\infty}$ ); no lower-regularity or distributional tensor field is defined here. The smooth structure of the tangent bundle, and hence of $T_{s}^{r} M$ (The tensor bundle on a smooth four-manifold from coordinate evaluations), inherits the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) recorded with Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; the component formulation of smoothness itself makes no choice.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Observers, rest spaces and the spatial projection at an event

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold and let $p \in M$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(a) Observers. An observer at $p$ is a tangent vector $n \in T_{p} M$ such that $g_{p} (n, n) = - 1$ and $n$ is future-directed in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, that is, $g_{p} (n, N_{p}) < 0$ . Because $g_{p} (n, n) = - 1 \neq 0$ , an observer is timelike and in particular nonzero (Timelike, null, spacelike and causal vectors of a Lorentz vector space). Observers at $p$ exist: the vector $N_{p}$ is timelike, and $n : = N_{p} / \sqrt{- g_{p} (N_{p}, N_{p})}$ satisfies $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) = - \sqrt{- g_{p} (N_{p}, N_{p})} < 0$ , where the square root is the nonnegative root of Rational powers $a^{r}$ of a positive base; then $- n$ is a unit timelike vector with $g_{p} (- n, N_{p}) > 0$ , so it is not an observer, and the two vectors $n, - n$ point in opposite time directions.

(b) Rest space. For an observer $n$ at $p$ , the rest space of $n$ is the $g_{p}$ -orthogonal complement of the line $R n$ ,

$n^{⊥} = {v \in T_{p} M : g_{p} (v, n) = 0},$

in the sense of eta-orthogonality and orthogonal complements in a Lorentz vector space. It is a linear subspace of $T_{p} M$ . The further facts that $\dim n^{⊥} = 3$ and that the restriction of $g_{p}$ to $n^{⊥}$ is positive definite are proved in The observer rest-space decomposition and the spatial projection ↗; they are consequences, not part of the definition.

(c) Spatial projection. The spatial projection of the observer $n$ is the map

$h_{n} : T_{p} M ⟶ T_{p} M, h_{n} (v) : = v + g_{p} (n, v) n .$

The map is linear because $g_{p}$ is bilinear, its image is contained in $n^{⊥}$ , and it fixes $n^{⊥}$ pointwise; that it is the $g_{p}$ -orthogonal projection onto $n^{⊥}$ with kernel $R n$ is proved in The observer rest-space decomposition and the spatial projection ↗.

(d) Scope. An observer at $p$ is a single tangent vector. It is not an ordered basis or frame of $T_{p} M$ , not a chart, not a field of observers such as the congruences defined later on this page, and not a curve or a worldline: no parametrized curve, clock reading, proper time or measurement is attached to $n$ by this definition. The physical reading of an observer as the direction of an ideal clock's worldline, and the measurement interpretation of the observer-relative quantities introduced later on this page, belong to the physical postulates and unit conventions of the framework and are not asserted here.

Remarks

Both conditions are used. The condition $g_{p} (n, n) = - 1$ makes $n$ a unit timelike vector of a fixed normalization and forces $n \neq 0$ ; the condition $g_{p} (n, N_{p}) < 0$ selects one of the two time directions and is not implied by the first, since replacing $n$ by $- n$ preserves $g_{p} (n, n) = - 1$ and reverses the sign of $g_{p} (n, N_{p})$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). No positivity of any observer-relative energy is asserted or used in this definition.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Observer congruences, their observer curves and adapted charts

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $U \subseteq M$ be a nonempty open subset, carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure.

(a) Congruences. An observer congruence on $U$ is a smooth vector field $n$ on $U$ (A smooth vector field is a smooth section of the tangent bundle) such that $n_{q}$ is an observer at $q$ for every $q \in U$ (Observers, rest spaces and the spatial projection at an event); explicitly,

$g_{q} (n_{q}, n_{q}) = - 1 and g_{q} (n_{q}, N_{q}) < 0 for every q \in U .$

Since $g_{q} (n_{q}, n_{q}) = - 1$ , the field $n$ is nowhere vanishing. A single observer at a point is therefore not a congruence, and a congruence is not an ordered basis or frame of $T_{q} M$ .

(b) Observer curves. Because $n$ is a smooth vector field on $U$ , the theorem on maximal integral curves (Through each point there is a unique maximal integral curve, Integral curves of a vector field) assigns to every $q \in U$ a unique maximal integral curve $γ_{q} : I_{q} \to U$ of $n$ with $γ_{q} (0) = q$ , defined on an open interval $I_{q} \subseteq R$ . We call these curves the observers (or observer curves) of the congruence. Each curve is regular: for every $s \in I_{q}$ , $γ_{q}^{'} (s) = n_{γ_{q} (s)}$ and

$g_{γ_{q} (s)} (n_{γ_{q} (s)}, n_{γ_{q} (s)}) = - 1,$

so its tangent is nonzero, unit timelike and future-directed. Completeness of the maximal intervals $I_{q}$ , and global product or foliation structure of $U$ , are not asserted. The parameter $s$ is the flow parameter of $n$ ; it is not identified with any clock reading here.

(c) Adapted charts. An $n$ -adapted chart is a smooth chart $(u^{0}, u^{1}, u^{2}, u^{3}) : U^{'} \to R^{4}$ of $M$ whose domain $U^{'}$ is a nonempty open subset of $U$ and whose spatial coordinates $u^{1}, u^{2}, u^{3}$ are constant along every integral curve $γ : J \to U^{'}$ of the restricted field $n ∣_{U^{'}}$ , where $J$ is an interval. Equivalently, for each maximal observer curve $γ_{q} : I_{q} \to U$ , each connected component $K$ of $γ_{q}^{- 1} (U^{'})$ (a maximal nonempty interval contained in this open subset of $R$ ), and $i = 1, 2, 3$ , the function $u^{i} \circ γ_{q}$ is constant on $K$ . If $q \in U^{'}$ and $0 \in K$ , this constant is $u^{i} (q)$ . This condition does not identify labels on different components when a maximal curve leaves and re-enters $U^{'}$ . No global adapted chart, and no adapted chart on all of $U$ , is asserted by this definition; one exists near every point of $U$ by the existence corollary proved on this page.

(d) Scope. A congruence is a field of observer directions on an open set; it is not a chart, not a tetrad field, and not a family of parametrized worldlines with specified domains. The physical reading of its observer curves as ideal-clock worldlines, and the identification of the flow parameter with a measurable time, belong to the physical postulates of the framework and are not asserted here.

Remarks

Choice convention. The smooth structure used for the vector field $n$ is the tangent-bundle structure of A smooth vector field is a smooth section of the tangent bundle, which assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ); this definition inherits that assumption and uses no further choice principle. The restricted structure on $U$ is the canonical one of the cited proposition.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The component transformation law of a smooth tensor field

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $r, s \geq 0$ , and let $T$ be a family assigning to each $p \in M$ a type $(r, s)$ tensor at $p$ (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components). Let $(U, x)$ and $(V, y)$ be overlapping smooth charts, and write $T^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}}$ and $T^{' a_{1} \dots a_{r}}_{b_{1} \dots b_{s}}$ for the components of $T$ in the charts $x$ and $y$ . Then for every $q \in U \cap V$ ,

$T^{' a_{1} \dots a_{r}}_{b_{1} \dots b_{s}} (y (q)) = \sum_{c_{1}, \dots, c_{r}} \sum_{d_{1}, \dots, d_{s}} (\prod_{i = 1}^{r} \frac{\partial y^{a_{i}}}{\partial x^{c_{i}}} (x (q))) (\prod_{j = 1}^{s} \frac{\partial x^{d_{j}}}{\partial y^{b_{j}}} (y (q))) T^{c_{1} \dots c_{r}}_{d_{1} \dots d_{s}} (x (q)),$

where $\partial y^{a} / \partial x^{c}$ and $\partial x^{d} / \partial y^{b}$ are the entries of the total derivatives of $y \circ x^{- 1}$ and $x \circ y^{- 1}$ . Moreover $T$ is a smooth tensor field of type $(r, s)$ if and only if its components are $C^{\infty}$ in the charts of some one smooth atlas of $M$ ; in that case the components in every smooth chart are $C^{\infty}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , integers $r, s \geq 0$ , a family $T$ of type $(r, s)$ tensors, and overlapping smooth charts $(U, x)$ , $(V, y)$ .

[L1]

The components of $T$ in a chart are the evaluations $T^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}} (q) = T_{q} (d x^{a_{1}} ∣_{q}, \dots, \partial_{b_{1}} ∣_{q}, \dots)$ , and $T$ is a smooth tensor field of type $(r, s)$ when these are $C^{\infty}$ in the charts of the maximal smooth atlas (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L2]

The coordinate derivations of one chart expand in those of another by $\partial_{x^{i}} ∣_{q} = \sum_{j} \frac{\partial (y^{j} \circ x^{- 1})}{\partial x^{i}} (x (q)) \partial_{y^{j}} ∣_{q}$ (Change-of-coordinate formula for tangent bases).

[L3]

The coordinate differentials of a chart form the dual basis of its coordinate derivations, and every covector expands in a dual basis (Coordinate differentials form the dual cotangent basis, Coordinate derivations form a basis of the tangent space).

[L4]

For a chart $(U, x)$ and $q \in U$ , the coordinate derivation acts on a smooth function by $\partial_{x^{i}} ∣_{q} f = D_{i} (f \circ x^{- 1}) (x (q))$ ; in particular $\partial_{x^{i}} ∣_{q} (y^{j}) = \frac{\partial y^{j}}{\partial x^{i}} (x (q))$ (Coordinate derivations are well-defined derivations).

[L5]

Smoothly compatible charts have smooth transition maps in both directions, and the entries of their total derivatives are smooth functions (Smoothly compatible charts and the smoothness of Euclidean transition maps, $C^{r}$ and smooth maps between smooth manifolds).

[L6]

Matrix inversion preserves $C^{k}$ regularity on the invertible matrices, so the entries $\frac{\partial x^{d}}{\partial y^{b}}$ of the inverse Jacobian are $C^{\infty}$ (Matrix inversion preserves $C^{k}$ regularity where the determinant is nonzero).

[L7]

Finite sums and products of $C^{\infty}$ Euclidean maps are $C^{\infty}$ , and composites of $C^{\infty}$ maps are $C^{\infty}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

Proof

technique · direct

1.1L2L3L4given

Fix $q \in U \cap V$ and write $\partial_{x^{i}} : = \partial_{x^{i}} ∣_{q}$ , $\partial_{y^{j}} : = \partial_{y^{j}} ∣_{q}$ , $d x^{i} : = d x^{i} ∣_{q}$ , $d y^{j} : = d y^{j} ∣_{q}$ . By [L2] applied to the two charts, $\partial_{y^{b}} = \sum_{d} \frac{\partial x^{d}}{\partial y^{b}} (y (q)) \partial_{x^{d}}$ . By [L3] and [L4], $d y^{a}$ is the dual basis element of $\partial_{y^{a}}$ and its coefficients in the dual basis $(d x^{c})$ are $d y^{a} (\partial_{x^{c}}) = \partial_{x^{c}} (y^{a}) = \frac{\partial y^{a}}{\partial x^{c}} (x (q))$ ; hence $d y^{a} = \sum_{c} \frac{\partial y^{a}}{\partial x^{c}} (x (q)) d x^{c}$ .

2.1step 1.1L1algebra

Since $T_{q}$ is multilinear in its $r$ covector and $s$ vector arguments, substituting the expansions of step 1.1 into the definition of the $y$ -components gives the displayed transformation law: each contravariant slot contributes one factor $\frac{\partial y^{a_{i}}}{\partial x^{c_{i}}} (x (q))$ and each covariant slot one factor $\frac{\partial x^{d_{j}}}{\partial y^{b_{j}}} (y (q))$ , and the sums over the intermediate indices are finite.

3.1step 2.1L1algebra

In the special case $r = s = 0$ the double sum is empty and the law reduces to $T^{'} (y (q)) = T (x (q))$ ; in the case $r = 1, s = 0$ it reduces to the vector transformation law and in the case $r = 0, s = 1$ to the covector law of step 1.1.

3.2step 2.1L1L5L6L7given

Suppose $T$ has $C^{\infty}$ components in the charts of one smooth atlas $A$ . Let $(V, y)$ be any smooth chart and $q \in V$ ; choose a chart $(U, x)$ of $A$ with $q \in U$ (the charts of $A$ cover $M$ ). By step 2.1 the components of $T$ in $y$ equal the finite sum of step 2.1, whose factors are: the numbers computed from the components of $T$ in $x$ , which are $C^{\infty}$ functions of $x$ by hypothesis, evaluated along the smooth map $x$ ; and the Jacobian entries $\frac{\partial y^{a}}{\partial x^{c}} \circ x$ and $\frac{\partial x^{d}}{\partial y^{b}} \circ y$ , which are $C^{\infty}$ on the overlap by [L5] and [L6]. By [L7] the sum is $C^{\infty}$ on $y (U \cap V)$ . Since $q$ and $(V, y)$ were arbitrary, the components of $T$ are $C^{\infty}$ in every smooth chart, so $T$ is a smooth tensor field by [L1]. The converse direction is immediate from [L1], since the maximal atlas contains every smooth atlas.

4.1step 1.1step 2.1step 3.1step 3.2∎

Steps 1.1, 2.1 and 3.1 prove the transformation law and step 3.2 proves the smoothness criterion.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Contraction of a smooth tensor field is a smooth tensor field

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $r, s \geq 1$ , and let $T$ be a smooth tensor field of type $(r, s)$ on $M$ (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components). For each $p \in M$ let $(ctr T)_{p}$ be the contraction of $T_{p}$ over its first contravariant slot and its first covariant slot (The contraction of a mixed tensor). Then:

$ctr T$ is a smooth tensor field of type $(r - 1, s - 1)$ on $M$ , and its definition is independent of every basis.
In every smooth chart $(U, x)$ the components of $ctr T$ are

$(ctr T)^{a_{2} \dots a_{r}}_{b_{2} \dots b_{s}} = \sum_{i = 0}^{3} T^{i a_{2} \dots a_{r}}_{i b_{2} \dots b_{s}},$

and the field obtained by contracting any other pair consisting of one contravariant and one covariant slot has the analogous component formula after permuting the slots.
If $r = s = 1$ , then $tr T : = ctr T$ is a smooth function on $M$ and $tr T = \sum_{i = 0}^{3} T^{i}_{i}$ in every smooth chart.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , integers $r, s \geq 1$ , and a smooth type $(r, s)$ tensor field $T$ on $M$ .

[L1]

Contraction of a type $(r, s)$ tensor $T_{p}$ over its first contravariant and first covariant slot is the type $(r - 1, s - 1)$ tensor given in a basis $(e_{i})$ with dual basis $(e^{i})$ by $(ctr T_{p}) (α_{2}, \dots, α_{r}, v_{2}, \dots, v_{s}) = \sum_{i} T_{p} (e^{i}, α_{2}, \dots, α_{r}, e_{i}, v_{2}, \dots, v_{s})$ (The contraction of a mixed tensor).

[L2]

The contraction formula has the same value for every basis and its dual basis (Contraction is independent of the basis formula).

[L3]

For a tensor field $T$ of type $(r, s)$ , the component functions in a chart are $T^{a_{1} \dots a_{r}}_{b_{1} \dots b_{s}} (q) = T_{q} (d x^{a_{1}} ∣_{q}, \dots, \partial_{b_{1}} ∣_{q}, \dots)$ , and $T$ is smooth exactly when these are $C^{\infty}$ in every smooth chart (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L4]

Finite sums and products of $C^{\infty}$ maps on chart domains are $C^{\infty}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

Proof

technique · direct

1.1L1L2given

For each $p \in M$ the assignment $(ctr T)_{p} : = ctr (T_{p})$ is a type $(r - 1, s - 1)$ tensor at $p$ by [L1], and it does not depend on any basis by [L2]; hence $ctr T$ is a well-defined family of type $(r - 1, s - 1)$ tensors on $M$ .

2.1step 1.1L1L3algebra

Let $(U, x)$ be a smooth chart and $q \in U$ . Applying [L1] with the coordinate basis $e_{i} = \partial_{i} ∣_{q}$ and its dual basis $e^{i} = d x^{i} ∣_{q}$ (which is the dual coordinate basis) and evaluating on the arguments $d x^{a_{2}} ∣_{q}, \dots, d x^{a_{r}} ∣_{q}, \partial_{b_{2}} ∣_{q}, \dots, \partial_{b_{s}} ∣_{q}$ gives $(ctr T)^{a_{2} \dots a_{r}}_{b_{2} \dots b_{s}} (q) = \sum_{i = 0}^{3} T_{q} (d x^{i} ∣_{q}, d x^{a_{2}} ∣_{q}, \dots, \partial_{i} ∣_{q}, \partial_{b_{2}} ∣_{q}, \dots) = \sum_{i = 0}^{3} T^{i a_{2} \dots a_{r}}_{i b_{2} \dots b_{s}} (q)$ , which is the displayed formula.

3.1step 2.1L3L4given

The right-hand side of step 2.1 is a finite sum of the component functions $T^{i a_{2} \dots a_{r}}_{i b_{2} \dots b_{s}}$ of the smooth field $T$ , which are $C^{\infty}$ on $U$ by [L3]; hence by [L4] the components of $ctr T$ are $C^{\infty}$ on $U$ . Since the chart was arbitrary, [L3] now shows that $ctr T$ is a smooth type $(r - 1, s - 1)$ field, proving claims 1 and 2 for the first slot pair.

3.2step 2.1L1L3algebra

For another pair of slots, permute the arguments of $T_{p}$ by the corresponding finite permutation of the $r + s$ slots; the permutation is a linear isomorphism of the space of multilinear maps, so the permuted family is again a smooth tensor field (its components are components of $T$ with permuted indices), and contraction of the chosen pair is the first-pair contraction of the permuted field; the component formula is the one of step 2.1 with the contraction index read in the chosen slots.

4.1step 1.1step 2.1step 3.1algebra∎

If $r = s = 1$ , then $ctr T$ is a type $(0, 0)$ tensor at each $p$ , that is, a real number depending on $p$ ; by steps 1.1-2.1 it is a smooth function and its value in any chart is $\sum_{i} T^{i}_{i}$ by step 2.1. This proves claim 3.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The observer rest-space decomposition and the spatial projection

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , and let $n$ be an observer at $p$ (Observers, rest spaces and the spatial projection at an event). Write $h : = h_{n}$ , $n^{⊥} = {v \in T_{p} M : g_{p} (v, n) = 0}$ (eta-orthogonality and orthogonal complements in a Lorentz vector space) and $R n = {λ n : λ \in R}$ . Then:

$h : T_{p} M \to T_{p} M$ is linear, $h (n) = 0$ , $h (v) = v$ for every $v \in n^{⊥}$ and $h \circ h = h$ ; its kernel is $\ker h = R n$ , its image is $im h = n^{⊥}$ , and $T_{p} M = R n \oplus n^{⊥}$ is an orthogonal direct sum: every $v \in T_{p} M$ has the unique decomposition

$v = - g_{p} (v, n) n + h (v),$

in which $- g_{p} (v, n) n \in R n$ and $h (v) \in n^{⊥}$ .
For all $v, w \in T_{p} M$ the projection identity

$g_{p} (h (v), h (w)) = g_{p} (v, w) + g_{p} (v, n) g_{p} (n, w)$

holds; in particular $g_{p} (h (v), h (v)) = g_{p} (v, v) + g_{p} (v, n)^{2}$ .
$\dim n^{⊥} = 3$ and the restriction of $g_{p}$ to $n^{⊥}$ is positive definite.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a point $p \in M$ and an observer $n$ at $p$ .

[L1]

An observer at $p$ is a vector $n \in T_{p} M$ with $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) < 0$ , and the spatial projection of an observer is $h_{n} (v) = v + g_{p} (n, v) n$ (Observers, rest spaces and the spatial projection at an event).

[L2]

There are ordered $g_{p}$ -orthonormal bases of $T_{p} M$ , i.e. ordered bases $(f_{0}, f_{1}, f_{2}, f_{3})$ with $g_{p} (f_{a}, f_{b}) = diag (- 1, 1, 1, 1)_{a b}$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)).

[L3]

In an $η$ -orthonormal ordered basis of a Lorentz vector space, the coordinate expression is $η (v, v) = - (v^{0})^{2} + ∣ v_{v e c} ∣^{2}$ with $v = v^{0} e_{0} + v^{1} e_{1} + v^{2} e_{2} + v^{3} e_{3}$ , the timelike set is $T = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ the two connected components, and these descriptions coincide with the two components in some order in every orthonormal basis (The timelike set has exactly two convex components, described in orthonormal coordinates).

[L4]

In the time-oriented manifold, $C_{p}^{+} = {v \in T_{p} M : g_{p} (v, v) < 0 and g_{p} (v, N_{p}) < 0}$ and $C_{p}^{-} = {v \in T_{p} M : g_{p} (v, v) < 0 and g_{p} (v, N_{p}) > 0}$ ; a nonzero causal vector is future-directed exactly when it lies in $cl (C_{p}^{+})$ , and $g_{p} (v, N_{p}) \neq 0$ for every nonzero causal $v$ (Classification of future and past causal vectors by a timelike field).

[L5]

A vector is timelike when $g_{p} (v, v) < 0$ , null when $v \neq 0$ and $g_{p} (v, v) = 0$ , spacelike when $g_{p} (v, v) > 0$ , and causal when it is timelike or null (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L6]

Every vector of a finite-dimensional space has a unique coordinate list in an ordered basis (A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis).

[L7]

The Euclidean inner product of $x, y \in R^{3}$ is $⟨ x, y ⟩ = \sum_{i = 1}^{3} x_{i} y_{i}$ with norm $∥ x ∥_{2}$ , and $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

technique · direct

1.1L1algebra

The map $h$ is linear because $v \mapsto g_{p} (n, v)$ is linear and $n$ is fixed. Moreover $h (n) = n + g_{p} (n, n) n = n - n = 0$ , and if $v \in n^{⊥}$ then $g_{p} (n, v) = 0$ and $h (v) = v$ ; hence for every $v$ one has $g_{p} (h (v), n) = g_{p} (v, n) + g_{p} (v, n) g_{p} (n, n) = g_{p} (v, n) - g_{p} (v, n) = 0$ , so $h (v) \in n^{⊥}$ , and then $h (h (v)) = h (v) + g_{p} (h (v), n) n = h (v)$ , that is, $h \circ h = h$ .

1.2L2L3L4L5L6given

By [L2] choose an ordered $g_{p}$ -orthonormal basis $(f_{0}, f_{1}, f_{2}, f_{3})$ . The vector $f_{0}$ is timelike, and [L4] gives $g_{p} (f_{0}, N_{p}) \neq 0$ ; replacing $f_{0}$ by $- f_{0}$ if necessary, which preserves orthonormality, we may assume $g_{p} (f_{0}, N_{p}) < 0$ , so that $f_{0} \in C_{p}^{+}$ and $f_{0}$ is future-directed. By [L3] the two components of the timelike set are $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ in this basis, in some order; since $f_{0}$ has coordinates $(1, 0, 0, 0) \in T_{+}$ and lies in $C_{p}^{+}$ , the identification is $C_{p}^{+} = T_{+}$ . Writing $n = n^{0} f_{0} + n^{1} f_{1} + n^{2} f_{2} + n^{3} f_{3}$ by [L6] and using that $n$ is future unit timelike, we get $n^{0} > ∣ n_{v e c} ∣ \geq 0$ and $(n^{0})^{2} - ∣ n_{v e c} ∣^{2} = 1$ .

2.1step 1.1L1algebra

From $h (v) = 0$ we get $v = - g_{p} (v, n) n \in R n$ , and conversely $h (n) = 0$ , so $\ker h = R n$ ; since $h$ fixes every vector of $n^{⊥}$ pointwise and its image lies in $n^{⊥}$ , its image is exactly $im h = n^{⊥}$ .

2.2step 1.1L1algebra

Using bilinearity and symmetry of $g_{p}$ , $g_{p} (h (v), h (w)) = g_{p} (v + g_{p} (n, v) n, w + g_{p} (n, w) n) = g_{p} (v, w) + g_{p} (n, w) g_{p} (v, n) + g_{p} (v, n) g_{p} (n, w) + g_{p} (v, n) g_{p} (n, w) g_{p} (n, n)$ , and the last term is $- g_{p} (v, n) g_{p} (n, w)$ because $g_{p} (n, n) = - 1$ ; hence $g_{p} (h (v), h (w)) = g_{p} (v, w) + g_{p} (v, n) g_{p} (n, w)$ . Taking $w = v$ gives the stated special case.

2.3step 1.2L7algebra

Let $v \in n^{⊥}$ , with coordinates $v = v^{0} f_{0} + v_{v e c}$ as in step 1.2. Since $0 = g_{p} (v, n) = - v^{0} n^{0} + v_{v e c} \cdot n_{v e c}$ and $n^{0} \geq 1 > 0$ by step 1.2, we have $v^{0} = (v_{v e c} \cdot n_{v e c}) / n^{0}$ , and by [L7] $∣ v_{v e c} \cdot n_{v e c} ∣ \leq ∣ v_{v e c} ∣ ∣ n_{v e c} ∣$ ; therefore $g_{p} (v, v) = - (v^{0})^{2} + ∣ v_{v e c} ∣^{2} \geq ∣ v_{v e c} ∣^{2} (1 - ∣ n_{v e c} ∣^{2} / (n^{0})^{2}) = ∣ v_{v e c} ∣^{2} ((n^{0})^{2} - ∣ n_{v e c} ∣^{2}) / (n^{0})^{2} = ∣ v_{v e c} ∣^{2} / (n^{0})^{2} \geq 0$ . Equality forces $∣ v_{v e c} ∣ = 0$ and then $v^{0} = 0$ , that is, $v = 0$ . Hence $g_{p}$ is positive definite on $n^{⊥}$ .

3.1step 2.1L1algebra

The displayed formula $v = - g_{p} (v, n) n + h (v)$ is a rearrangement of the definition of $h$ , and the two summands lie in $R n$ and $n^{⊥}$ . If $v = λ n + w$ with $λ \in R$ and $w \in n^{⊥}$ , then applying $g_{p} (\cdot, n)$ gives $g_{p} (v, n) = - λ$ , so $λ = - g_{p} (v, n)$ and $w = v + g_{p} (v, n) n = h (v)$ ; hence the decomposition is unique and $T_{p} M = R n \oplus n^{⊥}$ . It is orthogonal because $g_{p} (λ n, w) = λ g_{p} (n, w) = 0$ for $w \in n^{⊥}$ , and $R n \cap n^{⊥} = {0}$ since $g_{p} (λ n, n) = - λ$ vanishes only for $λ = 0$ .

3.2step 1.1step 2.1step 1.2L6algebra

The vectors $h (f_{1}), h (f_{2}), h (f_{3})$ lie in $n^{⊥}$ by step 1.1 and are linearly independent: if $\sum_{i} c^{i} h (f_{i}) = 0$ , then $h (\sum_{i} c^{i} f_{i}) = 0$ , so $\sum_{i} c^{i} f_{i} \in R n$ by step 2.1. This vector has zero first coordinate in the basis of step 1.2, while a vector $λ n$ has first coordinate $λ n^{0}$ with $n^{0} > 0$ . Hence $λ = 0$ and all $c^{i} = 0$ by [L6]. The four vectors $n, h (f_{1}), h (f_{2}), h (f_{3})$ are linearly independent: pairing a linear relation with $n$ first makes the coefficient of $n$ zero, and the just-proved independence makes the others zero. Since $T_{p} M$ has dimension $4$ , they form a basis. For $v \in n^{⊥}$ , expand in this basis and pair with $n$ again; the coefficient of $n$ is zero. Thus the three projected spatial vectors span $n^{⊥}$ and form its basis, giving dimension $3$ . Together with step 2.3 this proves claim 3 without assuming the generic basis vector $f_{0}$ equals $n$ or $h (f_{0}) = 0$ .

4.1step 1.1step 2.1step 3.1step 2.2step 2.3step 3.2∎

Claims 1 and 2 are steps 1.1, 2.1, 3.1 and 2.2, and claim 3 is steps 2.3 and 3.2.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Pairing of an observer with future causal vectors

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , and let $n$ be an observer at $p$ (Observers, rest spaces and the spatial projection at an event). For $w \in T_{p} M$ write $h = h_{n}$ and $γ (w) : = - g_{p} (n, w)$ .

If $w$ is nonzero future causal, then $γ (w) > 0$ .
If $w$ is future unit timelike, then $γ (w) \geq 1$ , with $γ (w) = 1$ if and only if $w = n$ , and $g_{p} (h (w), h (w)) = γ (w)^{2} - 1$ .
If $w$ is unit timelike, then $w$ is future-directed if and only if $g_{p} (n, w) < 0$ , and past-directed if and only if $g_{p} (n, w) > 0$ ; in particular $g_{p} (n, w) \neq 0$ for such $w$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a point $p \in M$ , an observer $n$ at $p$ , and a vector $w \in T_{p} M$ .

[L1]

An observer at $p$ is a vector $n$ with $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) < 0$ ; units mean $g_{p} (n, n) = - 1$ , so a future unit timelike vector $w$ is exactly a future-directed vector with $g_{p} (w, w) = - 1$ (Observers, rest spaces and the spatial projection at an event, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L2]

The projection identity $g_{p} (h (v), h (v^{'})) = g_{p} (v, v^{'}) + g_{p} (v, n) g_{p} (n, v^{'})$ holds for all $v, v^{'}$ , $\ker h = R n$ , and $g_{p}$ is positive definite on $n^{⊥} = im h$ (The observer rest-space decomposition and the spatial projection).

[L3]

There are ordered $g_{p}$ -orthonormal bases of $T_{p} M$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)).

[L4]

In an orthonormal ordered basis the timelike set is $T = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ the two components, in some order in every orthonormal basis (The timelike set has exactly two convex components, described in orthonormal coordinates).

[L5]

In the time-oriented manifold, $C_{p}^{+} = {v : g_{p} (v, v) < 0, g_{p} (v, N_{p}) < 0}$ and $C_{p}^{-} = {v : g_{p} (v, v) < 0, g_{p} (v, N_{p}) > 0}$ are the two components of the timelike set; a nonzero causal vector is future-directed exactly when it lies in $cl (C_{p}^{+})$ , and $g_{p} (v, N_{p}) \neq 0$ for every nonzero causal $v$ (Classification of future and past causal vectors by a timelike field, Time orientation, future and past timelike/null/causal vectors, and the closed cones).

[L6]

The Euclidean inner product of $x, y \in R^{3}$ satisfies $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

technique · direct

1.1L3L4L5givenalgebra

By [L3] choose an ordered orthonormal basis $(f_{0}, f_{1}, f_{2}, f_{3})$ ; replacing $f_{0}$ by $- f_{0}$ if necessary we may assume $g_{p} (f_{0}, N_{p}) < 0$ , so $f_{0}$ is future-directed and, by [L4] and [L5] as in the rest-space lemma, the future timelike component is $C_{p}^{+} = T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and the future causal vectors are the nonzero points of $cl (T_{+}) = {v : v^{0} \geq ∣ v_{v e c} ∣}$ . Write $n = n^{0} f_{0} + n_{v e c}$ and $w = w^{0} f_{0} + w_{v e c}$ ; then $n^{0} > ∣ n_{v e c} ∣ \geq 0$ , $(n^{0})^{2} - ∣ n_{v e c} ∣^{2} = 1$ , and for nonzero future causal $w$ we have $w^{0} \geq ∣ w_{v e c} ∣$ with $w^{0} > 0$ (because $w \neq 0$ forces either $∣ w_{v e c} ∣ > 0$ or $w^{0} > 0 \geq ∣ w_{v e c} ∣$ ).

2.1step 1.1L6algebra

With the coordinates of step 1.1, $g_{p} (n, w) = - n^{0} w^{0} + n_{v e c} \cdot w_{v e c}$ , so by [L6] $γ (w) = n^{0} w^{0} - n_{v e c} \cdot w_{v e c} \geq n^{0} w^{0} - ∣ n_{v e c} ∣ ∣ w_{v e c} ∣ \geq w^{0} (n^{0} - ∣ n_{v e c} ∣)$ . Since $n^{0} - ∣ n_{v e c} ∣ = (n^{0} + ∣ n_{v e c} ∣)^{- 1} > 0$ and $w^{0} > 0$ , we get $γ (w) > 0$ ; this proves claim 1.

3.1step 2.1L2givenalgebra

Let $w$ be future unit timelike. By claim 1, $γ (w) > 0$ ; by [L2], $g_{p} (h (w), h (w)) = g_{p} (w, w) + γ (w)^{2} = - 1 + γ (w)^{2}$ , and the left side is $\geq 0$ because $h (w) \in n^{⊥}$ and $g_{p}$ is positive definite there, so $γ (w)^{2} \geq 1$ and hence $γ (w) \geq 1$ . Equality holds exactly when $h (w) = 0$ , i.e. $w \in \ker h = R n$ by [L2]; a future unit timelike vector proportional to $n$ must equal $n$ . This proves claim 2.

3.2step 2.1L1L5algebra

Let $w$ be unit timelike. It is timelike and nonzero, so by [L5] it is future-directed or past-directed according to membership in $C_{p}^{+}$ or $C_{p}^{-}$ , and $g_{p} (w, N_{p}) \neq 0$ . If $w$ is future-directed, step 2.1 applied to $w$ gives $- g_{p} (n, w) > 0$ , i.e. $g_{p} (n, w) < 0$ . If $w$ is past-directed, then $- w$ is future unit timelike, so step 2.1 gives $- g_{p} (n, - w) = g_{p} (n, w) > 0$ . Conversely, if $g_{p} (n, w) < 0$ and $w$ were past-directed, the previous case would give $g_{p} (n, w) > 0$ , a contradiction; and if $g_{p} (n, w) > 0$ and $w$ were future-directed, the first case would give $g_{p} (n, w) < 0$ , a contradiction. Hence $w$ is future-directed exactly when $g_{p} (n, w) < 0$ , and past-directed exactly when $g_{p} (n, w) > 0$ ; in particular $g_{p} (n, w) \neq 0$ . This proves claim 3.

4.1step 2.1step 3.1step 3.2∎

Claims 1, 2 and 3 are steps 2.1, 3.1 and 3.2 respectively.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Local frame fields, coordinate frames and tetrad fields

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold and let $U \subseteq M$ be a nonempty open subset (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Write $η : = diag (- 1, 1, 1, 1)$ for the four-by-four diagonal matrix with diagonal entries $- 1, 1, 1, 1$ , and use capital indices $A, B = 0, 1, 2, 3$ for frame members.

(a) Local frame fields. A local frame field on $U$ is a tuple $E = (E_{0}, E_{1}, E_{2}, E_{3})$ of smooth vector fields on $U$ (A smooth vector field is a smooth section of the tangent bundle) such that for every $q \in U$ the vectors $E_{0} (q), E_{1} (q), E_{2} (q), E_{3} (q)$ form an ordered basis of $T_{q} M$ , equivalently a pointwise independent tuple of four vector fields.

(b) Coordinate frames. If $(U, x)$ is a smooth chart, the tuple $(\partial / \partial x^{0}, \partial / \partial x^{1}, \partial / \partial x^{2}, \partial / \partial x^{3})$ of coordinate fields is a local frame field on $U$ , its coordinate frame. Two coordinate fields of one chart commute: by the coordinate formula for the Lie bracket (Coordinate formula for the Lie bracket, The Lie bracket of smooth vector fields) applied to constant coordinate components, $[\partial / \partial x^{a}, \partial / \partial x^{b}] = 0$ for all $a, b$ . Consequently a local frame field $(E_{0}, \dots, E_{3})$ with $[E_{A}, E_{B}] \neq 0$ for some $A, B$ is not the coordinate frame of any chart; this necessary condition is the only one asserted here, and no converse is claimed by this definition.

(c) Tetrad fields. A tetrad field (or $g$ -orthonormal frame field) on $U$ is a local frame field $(E_{0}, E_{1}, E_{2}, E_{3})$ on $U$ such that

$g_{q} (E_{A} (q), E_{B} (q)) = η_{A B} for all A, B and all q \in U .$

A tetrad field is future-directed when in addition $E_{0} (q)$ is future-directed for every $q \in U$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model); then each $E_{0} (q)$ is an observer at $q$ (Observers, rest spaces and the spatial projection at an event), so $E_{0}$ is an observer congruence on $U$ (Observer congruences, their observer curves and adapted charts), while $E_{1}, E_{2}, E_{3}$ are pointwise an orthonormal basis of the rest space of that observer. Neither existence of a tetrad field on a prescribed open set, nor global existence on $M$ , is asserted by this definition; local existence near every point is proved in the existence theorem on this page.

(d) Observer-adapted tetrads. Given an observer $n$ at a point $p$ , a tetrad at $p$ is $n$ -adapted when its first member is $n$ . Given an observer congruence $n$ on $U$ , a tetrad field on $U$ is $n$ -adapted when $E_{0} = n$ ; its remaining members are then $g$ -orthonormal fields in the moving rest spaces of the congruence.

(e) Scope. A tetrad field is not assumed to be a coordinate frame, and a frame field is not assumed to be orthonormal. Components of a tensor field in a tetrad and components in a chart are different objects; the transformation rules for the two are distinguished later on this page. No relation to the physical interpretation of a reference frame is asserted here beyond the geometric data listed above.

Remarks

Choice convention. Smooth vector fields are tangent-bundle sections, whose canonical smooth structure assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) (A smooth vector field is a smooth section of the tangent bundle); this definition inherits that assumption and uses no further choice principle.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The change between two future-directed tetrads is a Lorentz matrix

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , and let $E = (E_{0}, E_{1}, E_{2}, E_{3})$ and $F = (F_{0}, F_{1}, F_{2}, F_{3})$ be two future-directed tetrads at $p$ (Local frame fields, coordinate frames and tetrad fields). Write $η = diag (- 1, 1, 1, 1)$ . Then there is a unique real $4 \times 4$ matrix $Λ = (Λ^{B}_{A})$ such that $F_{A} = \sum_{B} Λ^{B}_{A} E_{B}$ for all $A$ , and:

$\sum_{C, D} Λ^{C}_{A} η_{C D} Λ^{D}_{B} = η_{A B}$ for all $A, B$ ; equivalently $Λ^{T} η Λ = η$ .
$Λ$ is invertible and $\det Λ = \pm 1$ .
$Λ^{0}_{0} > 0$ .
For every covariant $2$ -tensor $T$ at $p$ , its components in the two tetrads satisfy $T (F_{A}, F_{B}) = \sum_{C, D} Λ^{C}_{A} Λ^{D}_{B} T (E_{C}, E_{D})$ . More generally, for a type $(r, s)$ tensor at $p$ each contravariant slot contributes a factor $(Λ^{- 1})^{A}_{B}$ and each covariant slot a factor $Λ^{A}_{B}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a point $p \in M$ , and two future-directed tetrads $E = (E_{0}, \dots, E_{3})$ , $F = (F_{0}, \dots, F_{3})$ at $p$ .

[L1]

A tetrad at $p$ is an ordered basis $(E_{A})$ of $T_{p} M$ with $g_{p} (E_{A}, E_{B}) = η_{A B}$ ; a future-directed tetrad has $E_{0}$ future-directed (Local frame fields, coordinate frames and tetrad fields).

[L2]

A nonzero timelike vector has $g_{p} (v, N_{p}) \neq 0$ , and $C_{p}^{+} = {v : g_{p} (v, v) < 0, g_{p} (v, N_{p}) < 0}$ is the future timelike component (Classification of future and past causal vectors by a timelike field).

[L3]

In an orthonormal ordered basis of a Lorentz vector space, the timelike set splits as $T = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ the two components, in some order, in every orthonormal basis (The timelike set has exactly two convex components, described in orthonormal coordinates).

[L4]

Every vector of $T_{p} M$ has a unique coordinate list in the ordered basis $E$ (A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis).

[L5]

Orthonormal bases of a Lorentz vector space exist, and the timelike vector $E_{0}$ of a tetrad is future-directed exactly when $g_{p} (E_{0}, N_{p}) < 0$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L6]

A type $(r, s)$ tensor at $p$ is a multilinear map $(T_{p}^{*} M)^{r} \times (T_{p} M)^{s} \to R$ : its first $r$ slots are covectors and its last $s$ slots are vectors; components in a basis are its evaluations on the corresponding dual covectors and basis vectors (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

Proof

technique · direct

1.1L1L4given

Since $E$ is a basis, [L4] gives unique coefficients $Λ^{B}_{A}$ with $F_{A} = \sum_{B} Λ^{B}_{A} E_{B}$ , which is the uniqueness assertion.

2.1step 1.1L1algebra

Evaluating $g_{p}$ and using bilinearity together with the tetrad property $g_{p} (E_{C}, E_{D}) = η_{C D}$ gives $η_{A B} = g_{p} (F_{A}, F_{B}) = \sum_{C, D} Λ^{C}_{A} Λ^{D}_{B} g_{p} (E_{C}, E_{D})$ , which is claim 1.

2.2step 1.1L1L2L3L5given

In the orthonormal ordered basis $E$ , [L3] describes the two timelike components as $T_{\pm}$ ; the vector $E_{0}$ has coordinates $(1, 0, 0, 0) \in T_{+}$ , and $E_{0}$ is future-directed by hypothesis, so $C_{p}^{+} = T_{+}$ and a vector is future timelike exactly when its $E$ -coordinates satisfy $v^{0} > ∣ v_{v e c} ∣$ . The vector $F_{0}$ has coordinates $(Λ^{0}_{0}, Λ^{1}_{0}, Λ^{2}_{0}, Λ^{3}_{0})$ by step 1.1 and is future-directed by hypothesis, so $Λ^{0}_{0} > ∣ (Λ^{1}_{0}, Λ^{2}_{0}, Λ^{3}_{0}) ∣ \geq 0$ ; this proves claim 3.

2.3step 1.1L1L6algebra

For a covariant $2$ -tensor $T$ , bilinearity of $T$ at $p$ and $F_{A} = \sum_{C} Λ^{C}_{A} E_{C}$ , $F_{B} = \sum_{D} Λ^{D}_{B} E_{D}$ give $T (F_{A}, F_{B}) = \sum_{C, D} Λ^{C}_{A} Λ^{D}_{B} T (E_{C}, E_{D})$ , the first part of claim 4.

3.1step 2.1L1algebra

Taking determinants in the matrix identity $Λ^{T} η Λ = η$ of claim 1 gives $(\det Λ)^{2} \det η = \det η$ ; since $\det η = - 1 \neq 0$ , we get $(\det Λ)^{2} = 1$ , so $\det Λ = \pm 1$ and $Λ$ is invertible. This proves claim 2.

4.1step 1.1step 3.1L1L6algebra

For the general type $(r, s)$ statement, let $(ε^{A})$ and $(θ^{A})$ be the dual bases of $(E_{A})$ and $(F_{A})$ . From $θ^{A} (F_{B}) = δ^{A}_{B}$ and the expansion of step 1.1 one gets $θ^{A} = \sum_{B} (Λ^{- 1})^{A}_{B} ε^{B}$ , where $Λ^{- 1}$ exists by claim 2; substituting this on each contravariant slot and the expansion of step 1.1 on each covariant slot into the multilinear evaluation gives the stated products of $(Λ^{- 1})^{A}_{B}$ and $Λ^{A}_{B}$ .

5.1step 2.1step 3.1step 2.2step 2.3step 4.1∎

Claims 1-4 are steps 2.1, 3.1, 2.2, 2.3 and 4.1.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Existence of a smooth future-directed tetrad field near every event

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold and let $p \in M$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Then there is an open neighborhood $U \subseteq M$ of $p$ and a smooth future-directed tetrad field $(E_{0}, E_{1}, E_{2}, E_{3})$ on $U$ (Local frame fields, coordinate frames and tetrad fields), i.e. $g_{q} (E_{A} (q), E_{B} (q)) = diag (- 1, 1, 1, 1)_{A B}$ for all $q \in U$ with $E_{0} (q)$ future-directed for every $q \in U$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ and a point $p \in M$ .

[L1]

A tetrad field on an open set is a tuple of four smooth vector fields that is pointwise a $g$ -orthonormal ordered basis, and it is future-directed when its first member is future-directed at every point (Local frame fields, coordinate frames and tetrad fields).

[L2]

There is an ordered $g_{p}$ -orthonormal basis of $T_{p} M$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)).

[L3]

For a nonzero timelike $v \in T_{p} M$ one has $g_{p} (v, N_{p}) \neq 0$ , and $C_{p}^{+} = {v : g_{p} (v, v) < 0, g_{p} (v, N_{p}) < 0}$ is the future timelike component (Classification of future and past causal vectors by a timelike field, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L4]

The component functions $g_{a b} (q) = g_{q} (\partial_{a} ∣_{q}, \partial_{b} ∣_{q})$ of $g$ in a smooth chart are $C^{\infty}$ (Smooth covariant 2-tensor fields on a smooth manifold).

[L5]

The coordinate derivations $\partial_{a} ∣_{q}$ of a chart form a basis of $T_{q} M$ , and every vector of $T_{q} M$ has a unique coordinate list (Coordinate derivations form a basis of the tangent space, A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis).

[L6]

A vector field on a chart domain is smooth exactly when its coordinate component functions are smooth (Smoothness of a vector field is equivalent to smooth coordinate components).

[L7]

A smooth map between smooth manifolds is continuous ( $C^{r}$ and smooth maps between smooth manifolds).

[L8]

If $f : X \to R$ is continuous at $x$ and $f (x) > 0$ , then there is an open neighborhood $W$ of $x$ with $f > 0$ on $W$ ; the same holds with $>$ replaced by $<$ . (Continuity of a map of topological spaces at a point and globally)

[L9]

Finite sums, products and composites of $C^{\infty}$ maps on Euclidean domains are $C^{\infty}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[L10]

On $(0, \infty)$ the squaring map $u \mapsto u^{2}$ has derivative $2 u$ , and for powers and quotients the product and quotient rules hold (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L11]

If $U \subseteq R^{n}$ is open and $f : U \to R^{n}$ is injective, $C^{k}$ with $k \geq 1$ , and $D f$ is invertible everywhere, then $f [U]$ is open and $f : U \to f [U]$ is a $C^{k}$ diffeomorphism (An injective regular $C^{k}$ map is a $C^{k}$ diffeomorphism onto its image).

[L12]

The smooth structure of the tangent bundle used for smooth vector fields assumes the axiom ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct, by an explicit construction with successive shrinking of neighborhoods

1.1L2L3L5L6L12given

Choose a smooth chart $(U_{0}, x)$ around $p$ . By [L2] there is an ordered $g_{p}$ -orthonormal basis $(b_{0}, b_{1}, b_{2}, b_{3})$ of $T_{p} M$ ; by [L3], $g_{p} (b_{0}, N_{p}) \neq 0$ , so after replacing $b_{0}$ by $- b_{0}$ if necessary we may assume $b_{0}$ is future-directed, still an orthonormal basis. Expand each $b_{A}$ uniquely in the coordinate basis, $b_{A} = \sum_{a} b_{A}^{a} \partial_{a} ∣_{p}$ [L5], and define $B_{A} : = \sum_{a} b_{A}^{a} \partial_{a}$ on $U_{0}$ , with the constants $b_{A}^{a}$ as coefficients. Each $B_{A}$ is a smooth vector field by [L6] (so that the tangent-bundle smooth structure of [L12] is in force), and $B_{A} (p) = b_{A}$ .

2.1step 1.1L4L9given

Consider the smooth functions $G_{A B} (q) : = g_{q} (B_{A} (q), B_{B} (q))$ and $h (q) : = g_{q} (B_{0} (q), N_{q})$ on $U_{0}$ . They are smooth because in the chart $x$ they equal the finite sums $G_{A B} = \sum_{a, b} b_{A}^{a} b_{B}^{b} g_{a b}$ and $h = \sum_{a, b} b_{0}^{a} N^{b} g_{a b}$ of products of smooth functions, using [L4] and [L9]; and $G_{A B} (p) = η_{A B}$ , while $h (p) < 0$ by step 1.1 and the definition of future-directedness.

3.1step 2.1L7L8

By [L7] and [L8] applied to the continuous functions $- G_{00}$ (positive at $p$ ) and $- h$ (positive at $p$ ), there is an open neighborhood $U_{1} \subseteq U_{0}$ of $p$ on which $G_{00} < 0$ and $h < 0$ .

4.1step 3.1L1L3L9L10L11

Define $F_{0} : = B_{0} / \sqrt{- G_{00}}$ on $U_{1}$ . The function $u \mapsto u^{- 1 / 2}$ is $C^{\infty}$ on $(0, \infty)$ : the map $u \mapsto u^{1 / 2}$ is the inverse of the injective $C^{\infty}$ squaring map $u \mapsto u^{2}$ on $(0, \infty)$ whose derivative $2 u$ is invertible everywhere there, so it is $C^{\infty}$ by [L11], and $u \mapsto u^{- 1}$ has derivatives $(- 1)^{n} n! u^{- n - 1}$ for every $n$ by iterating the product and quotient rules of [L10]; composites and products of $C^{\infty}$ maps are $C^{\infty}$ by [L9]. Hence $F_{0}$ is smooth on $U_{1}$ , and $g_{q} (F_{0} (q), F_{0} (q)) = G_{00} (q) / (- G_{00} (q)) = - 1$ ; since $- G_{00} > 0$ , the field $F_{0}$ is a positive multiple of $B_{0}$ at every point, so $g_{q} (F_{0} (q), N_{q}) = h (q) / \sqrt{- G_{00} (q)} < 0$ and $F_{0} (q)$ is future-directed.

5.1step 4.1L4L7L8L9algebra

For $i = 1, 2, 3$ define $P_{i} : = B_{i} + g_{q} (B_{i}, F_{0}) F_{0}$ on $U_{1}$ ; these are smooth fields by [L9], and $g_{q} (P_{i}, F_{0}) = g_{q} (B_{i}, F_{0}) + g_{q} (B_{i}, F_{0}) (- 1) = 0$ , so $P_{i} (q)$ lies in the $g_{q}$ -orthogonal complement of $F_{0} (q)$ . The function $φ_{1} : = g_{q} (P_{1}, P_{1})$ is smooth by [L4] and [L9], and $φ_{1} (p) = g_{p} (b_{1}, b_{1}) = 1 > 0$ because $F_{0} (p) = b_{0}$ and $g_{p} (b_{1}, b_{0}) = 0$ . By [L7] and [L8] there is an open $U_{2} \subseteq U_{1}$ containing $p$ with $φ_{1} > 0$ on $U_{2}$ . Define $F_{1} : = P_{1} / \sqrt{φ_{1}}$ on $U_{2}$ ; it is smooth by the argument of step 4.1, $g (F_{1}, F_{1}) = 1$ , and $g (F_{1}, F_{0}) = 0$ .

6.1step 5.1L4L7L8L9algebra

Define $P_{2}^{'} : = P_{2} - g (P_{2}, F_{1}) F_{1}$ on $U_{2}$ , a smooth field with $g (P_{2}^{'}, F_{1}) = g (P_{2}, F_{1}) - g (P_{2}, F_{1}) = 0$ and $g (P_{2}^{'}, F_{0}) = 0$ , and let $φ_{2} : = g (P_{2}^{'}, P_{2}^{'})$ , smooth with $φ_{2} (p) = g_{p} (b_{2}, b_{2}) = 1 > 0$ because $F_{1} (p) = b_{1}$ and $g_{p} (b_{2}, b_{1}) = 0$ . Shrink to an open $U_{3} \subseteq U_{2}$ containing $p$ with $φ_{2} > 0$ and define $F_{2} : = P_{2}^{'} / \sqrt{φ_{2}}$ ; then $F_{2}$ is smooth, $g (F_{2}, F_{2}) = 1$ and $g (F_{2}, F_{0}) = g (F_{2}, F_{1}) = 0$ .

7.1step 6.1L4L7L8L9algebra

Define $P_{3}^{'} : = P_{3} - g (P_{3}, F_{1}) F_{1} - g (P_{3}, F_{2}) F_{2}$ on $U_{3}$ , smooth with $g (P_{3}^{'}, F_{1}) = g (P_{3}^{'}, F_{2}) = g (P_{3}^{'}, F_{0}) = 0$ , and let $φ_{3} : = g (P_{3}^{'}, P_{3}^{'})$ , smooth with $φ_{3} (p) = g_{p} (b_{3}, b_{3}) = 1 > 0$ . Shrink to an open $U_{4} \subseteq U_{3}$ containing $p$ with $φ_{3} > 0$ and define $F_{3} : = P_{3}^{'} / \sqrt{φ_{3}}$ ; then $F_{3}$ is smooth, $g (F_{3}, F_{3}) = 1$ and $g (F_{3}, F_{A}) = 0$ for $A = 0, 1, 2$ .

8.1step 4.1step 5.1step 6.1step 7.1L1algebra

On $U_{4}$ the four smooth fields $F_{0}, F_{1}, F_{2}, F_{3}$ satisfy $g (F_{A}, F_{B}) = η_{A B}$ by steps 4.1-5.1, and $F_{0}$ is future-directed by step 4.1. The Gram matrix $η$ is invertible, so a linear relation $\sum_{A} λ^{A} F_{A} (q) = 0$ implies $\sum_{A} λ^{A} η_{A B} = 0$ for all $B$ after pairing with $F_{B} (q)$ , hence $λ = 0$ ; thus the tuple is pointwise an ordered basis, a future-directed tetrad field on $U_{4}$ by [L1].

9.1step 8.1∎

The open neighborhood $U_{4}$ of $p$ and the field $(F_{0}, F_{1}, F_{2}, F_{3})$ have the asserted properties; this proves the theorem.

Remarks

Choice convention. The construction uses smooth vector fields, hence tangent-bundle sections, whose canonical smooth structure assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) as recorded in [L12]; no further choice principle is used, and the construction selects a definite basis at $p$ by an explicit finite procedure.

TheoremStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Relative velocity decomposition of a future unit timelike vector

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , let $n$ be an observer at $p$ , let $c > 0$ be a real parameter, and let $u \in T_{p} M$ be future unit timelike, $g_{p} (u, u) = - 1$ (Observers, rest spaces and the spatial projection at an event). Put

$γ : = - g_{p} (n, u), w : = \frac{c h_{n} (u)}{γ},$

where $h_{n}$ is the spatial projection. Then:

$γ \geq 1$ , with $γ = 1$ if and only if $u = n$ ; in particular $w$ is well defined.
$w \in n^{⊥}$ , $g_{p} (w, w) = c^{2} (1 - γ^{- 2}) < c^{2}$ , and

$u = γ (n + \frac{w}{c}) .$
Uniqueness. If $u = λ (n + v / c)$ with $λ \in R$ and $v \in n^{⊥}$ , then $λ = γ$ and $v = w$ .
Converse. Conversely, if $λ \geq 1$ and $v \in n^{⊥}$ satisfy $g_{p} (v, v) = c^{2} (1 - λ^{- 2})$ , then $u^{'} : = λ (n + v / c)$ is future unit timelike, $- g_{p} (n, u^{'}) = λ$ and $h_{n} (u^{'}) = λ v / c$ ; the maps $u \mapsto (γ, w)$ and $(λ, v) \mapsto u^{'}$ are mutually inverse bijections between the future unit timelike vectors and the pairs $(λ, v)$ with $λ \geq 1$ , $v \in n^{⊥}$ and $g_{p} (v, v) = c^{2} (1 - λ^{- 2})$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a point $p$ , an observer $n$ at $p$ , a real $c > 0$ and a future unit timelike $u \in T_{p} M$ .

[L1]

An observer is a vector $n$ with $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) < 0$ ; the spatial projection is $h_{n} (v) = v + g_{p} (n, v) n$ , and a future unit timelike vector is a future-directed vector $u$ with $g_{p} (u, u) = - 1$ (Observers, rest spaces and the spatial projection at an event, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L2]

$h_{n}$ is linear, $\ker h_{n} = R n$ , $im h_{n} = n^{⊥}$ , $T_{p} M = R n \oplus n^{⊥}$ with $h_{n} (v) = v$ for $v \in n^{⊥}$ , the projection identity $g_{p} (h (v), h (v^{'})) = g_{p} (v, v^{'}) + g_{p} (v, n) g_{p} (n, v^{'})$ holds, and $g_{p}$ is positive definite on $n^{⊥}$ (The observer rest-space decomposition and the spatial projection).

[L3]

If $z$ is nonzero future causal then $- g_{p} (n, z) > 0$ ; if $z$ is unit timelike then $z$ is future-directed exactly when $g_{p} (n, z) < 0$ (Pairing of an observer with future causal vectors).

Proof

technique · direct

1.1L2L3given

The vector $u$ is nonzero and future causal, so [L3] gives $γ = - g_{p} (n, u) > 0$ ; thus $w = c h_{n} (u) / γ$ is well defined and belongs to $n^{⊥}$ because $h_{n} (u)$ does by [L2].

1.2L1L2L3algebra

Conversely let $λ \geq 1$ and $v \in n^{⊥}$ with $g_{p} (v, v) = c^{2} (1 - λ^{- 2})$ , and set $u^{'} : = λ (n + v / c)$ . Then $g_{p} (u^{'}, u^{'}) = λ^{2} (g_{p} (n, n) + 2 g_{p} (n, v) / c + g_{p} (v, v) / c^{2}) = λ^{2} (- 1 + 0 + 1 - λ^{- 2}) = - 1$ , so $u^{'}$ is unit timelike, and $g_{p} (n, u^{'}) = λ g_{p} (n, n) = - λ < 0$ , so $u^{'}$ is future-directed by [L3]. Moreover $- g_{p} (n, u^{'}) = λ$ and $h_{n} (u^{'}) = λ h_{n} (v) / c = λ v / c$ by [L2].

2.1step 1.1L1L2algebra

By the projection identity of [L2] applied to $v = v^{'} = u$ , and using $g_{p} (u, u) = - 1$ , we get $g_{p} (h_{n} (u), h_{n} (u)) = γ^{2} - 1$ ; the left-hand side is $\geq 0$ because it is the square of the $g_{p}$ -norm of $h_{n} (u) \in n^{⊥}$ and $g_{p}$ is positive definite on $n^{⊥}$ , with equality if and only if $h_{n} (u) = 0$ . Since $γ > 0$ by step 1.1, $γ^{2} \geq 1$ gives $γ \geq 1$ , and $γ = 1$ holds exactly when $h_{n} (u) = 0$ , i.e. $u \in \ker h_{n} = R n$ by [L2], which for future unit timelike $u$ means $u = n$ . This proves claim 1.

2.2step 1.1L2algebra

By the decomposition $u = - g_{p} (u, n) n + h_{n} (u)$ of [L2] and $γ = - g_{p} (n, u)$ , we have $u = γ n + h_{n} (u) = γ (n + h_{n} (u) / γ) = γ (n + w / c)$ , which is the formula of claim 2.

2.3step 1.1L2algebra

Suppose $u = λ (n + v / c)$ with $λ \in R$ and $v \in n^{⊥}$ . Pairing with $n$ gives $g_{p} (u, n) = λ (g_{p} (n, n) + g_{p} (n, v) / c) = - λ$ , hence $λ = - g_{p} (n, u) = γ$ . Applying $h_{n}$ and using linearity, $h_{n} (n) = 0$ and $h_{n} (v) = v$ [L2] gives $h_{n} (u) = λ h_{n} (v) / c = λ v / c$ , hence $v = c h_{n} (u) / λ = c h_{n} (u) / γ = w$ . This proves claim 3.

3.1step 1.1step 2.1algebra

From $w = c h_{n} (u) / γ$ and step 2.1, $g_{p} (w, w) = c^{2} γ^{- 2} g_{p} (h_{n} (u), h_{n} (u)) = c^{2} γ^{- 2} (γ^{2} - 1) = c^{2} (1 - γ^{- 2}) < c^{2}$ , because $γ$ is a finite real number with $γ \geq 1$ by step 2.1, so $γ^{- 2} > 0$ and $c^{2} > 0$ . This strict bound also holds when $u = n$ , in which case $γ = 1$ and $w = 0$ . This completes claim 2.

4.1step 3.1step 2.3step 1.2∎

Steps 1.1-2.2 prove claims 1 and 2; step 2.3 proves claim 3; and step 1.2 together with steps 1.1-2.2 shows that the two assignments are mutually inverse on the stated sets, completing claim 4.

CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Every congruence admits a local adapted chart

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $U \subseteq M$ be a nonempty open subset, let $n$ be an observer congruence on $U$ , and let $p \in U$ (Observer congruences, their observer curves and adapted charts). Then there is a chart $(u^{0}, u^{1}, u^{2}, u^{3})$ of $M$ with domain $U^{'} \subseteq U$ containing $p$ such that

$n = \frac{\partial}{\partial u^{0}} on U^{'},$

and this chart is $n$ -adapted: its spatial coordinates $u^{1}, u^{2}, u^{3}$ are constant along every integral curve of $n$ contained in $U^{'}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , an observer congruence $n$ on an open set $U$ , and a point $p \in U$ .

[L1]

An observer congruence on the nonempty open set $U$ is a smooth vector field $n$ on $U$ with $g_{q} (n_{q}, n_{q}) = - 1$ and $g_{q} (n_{q}, N_{q}) < 0$ for every $q \in U$ ; its maximal integral curves are its observer curves. A chart with nonempty open domain $U^{'} \subseteq U$ is $n$ -adapted when its spatial coordinates are constant along every integral curve of $n ∣_{U^{'}}$ whose parameter domain is an interval; no agreement of spatial labels across exit and re-entry is required (Observer congruences, their observer curves and adapted charts, Definition (a)-(c)).

[L2]

A smooth vector field $X$ on a manifold with $X_{p} \neq 0$ at a point $p$ admits local coordinates $v^{1}, \dots, v^{4}$ near $p$ in which $X = \partial / \partial v^{1}$ (The flow-box theorem).

[L3]

The value $g_{q} (n_{q}, n_{q})$ is a smooth function of $q$ on $U$ , and $g_{q} (n_{q}, n_{q}) = - 1$ implies $n_{q} \neq 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L4]

A smooth curve $γ : J \to U^{'}$ on a real interval $J$ is an integral curve of a smooth vector field $X$ on $U^{'}$ when $γ^{'} (s) = X_{γ (s)}$ for every $s \in J$ (Integral curves of a vector field).

Proof

technique · direct

1.1L1L2L3given

The field $n$ is smooth on the open set $U$ and nowhere vanishing: by [L1] it satisfies $g_{q} (n_{q}, n_{q}) = - 1$ at every $q \in U$ , so $n_{q}$ is a nonzero vector by [L3]. Applying [L2] to $n$ at the point $p$ gives a chart $(v^{1}, v^{2}, v^{3}, v^{4})$ with domain $U^{'} \subseteq U$ containing $p$ and $n = \partial / \partial v^{1}$ on $U^{'}$ .

2.1step 1.1L1L4algebra

Define $u^{0} : = v^{1}$ , $u^{1} : = v^{2}$ , $u^{2} : = v^{3}$ , $u^{3} : = v^{4}$ . This is again a chart with domain $U^{'}$ , and $n = \partial / \partial u^{0}$ on $U^{'}$ . For any integral curve $γ : J \to U^{'}$ of $n ∣_{U^{'}}$ as in [L4], its coordinate components satisfy $(u^{0} \circ γ)^{'} = 1$ and $(u^{i} \circ γ)^{'} = 0$ for $i = 1, 2, 3$ , since $\partial / \partial u^{0}$ has components $(1, 0, 0, 0)$ in this chart. Thus each $u^{i} \circ γ$ is constant on the interval $J$ , so by [L1] the chart is $n$ -adapted.

3.1step 2.1∎

The chart produced in step 2.1 has all the stated properties, so a local adapted chart exists near every point of $U$ .

PropositionStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Smoothness of the rest-space family and its Frobenius integrability criterion

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $U \subseteq M$ be a nonempty open subset, and let $n$ be an observer congruence on $U$ (Observer congruences, their observer curves and adapted charts). Let $α : = n^{♭} \in Ω^{1} (U)$ be the smooth $1$ -form $α_{q} (v) = g_{q} (n_{q}, v)$ (The inverse metric, musical isomorphisms and index raising, A smooth differential $k$ -form), and let $D_{q} : = \ker α_{q} = {v \in T_{q} M : g_{q} (n_{q}, v) = 0} = n_{q}^{⊥}$ be the rest space of the congruence at $q$ (Observers, rest spaces and the spatial projection at an event). Then:

$α$ is nowhere zero on $U$ , and $q \mapsto D_{q}$ is a smooth rank- $3$ distribution on $U$ (Smooth distributions on a manifold).
$D$ is integrable (Integrable distributions) if and only if $α \land d α = 0$ on $U$ (The exterior derivative by the invariant vector-field formula, The wedge product of differential forms).
If $α \land d α = 0$ on $U$ , then every point of $U$ has a chart $(u^{0}, u^{1}, u^{2}, u^{3})$ in which $D = span (\partial_{1}, \partial_{2}, \partial_{3})$ ; equivalently, the rest spaces are tangent to the level hypersurfaces of $u^{0}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a nonempty open $U \subseteq M$ , an observer congruence $n$ on $U$ , the one-form $α = n^{♭}$ and the family $D_{q} = \ker α_{q}$ .

[L1]

The congruence $n$ is a smooth vector field on $U$ with $g_{q} (n_{q}, n_{q}) = - 1$ for all $q \in U$ , so $n_{q} \neq 0$ (Observer congruences, their observer curves and adapted charts).

[L2]

A vector field on a chart domain is smooth exactly when its coordinate components are smooth (Smoothness of a vector field is equivalent to smooth coordinate components).

[L3]

In a chart the components $g_{a b} (q) = g_{q} (\partial_{a} ∣_{q}, \partial_{b} ∣_{q})$ of $g$ are $C^{\infty}$ (Smooth covariant 2-tensor fields on a smooth manifold).

[L4]

The flat map is $♭_{q} (v) = g_{q} (v, \cdot)$ , so $α_{q} (v) = g_{q} (n_{q}, v)$ ; the components of $α$ in a chart are $α_{a} = \sum_{b} g_{a b} n^{b}$ (The inverse metric, musical isomorphisms and index raising).

[L5]

Finite sums and products of $C^{\infty}$ functions on chart domains are $C^{\infty}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[L6]

A family of tangent subspaces is a smooth rank- $k$ distribution exactly when every point has a neighborhood carrying smooth vector fields $X_{1}, \dots, X_{k}$ that are pointwise independent and span the subspaces (A smooth distribution is exactly a locally framed constant-rank family of tangent spaces); smooth distributions are defined as rank- $k$ smooth subbundles of $T M$ , under the axiom ${A C}_{ω}$ (Smooth distributions on a manifold, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L7]

For a nowhere-zero one-form $α$ , the hyperplane distribution $\ker α$ is integrable if and only if $α \land d α = 0$ (The codimension-one Frobenius criterion).

[L8]

A rank- $k$ smooth distribution is integrable if and only if it is involutive (Involutive distributions), and in that case every point has a coordinate neighborhood in which the distribution equals $span (\partial_{1}, \dots, \partial_{k})$ (Frobenius local coordinate theorem).

Proof

technique · direct

1.1L1L2L3L4L5given

In a chart $(U^{'}, x)$ contained in $U$ , the components of $α$ are $α_{a} = \sum_{b} g_{a b} n^{b}$ by [L4], where $g_{a b}$ are $C^{\infty}$ by [L3] and the components $n^{b}$ of the smooth field $n$ are $C^{\infty}$ by [L2]; hence by [L5] the functions $α_{a}$ are $C^{\infty}$ , so $α$ is a smooth $1$ -form. Moreover $α_{q} (n_{q}) = g_{q} (n_{q}, n_{q}) = - 1 \neq 0$ for every $q \in U$ by [L1] and [L4], so $α$ is nowhere zero.

2.1step 1.1L2L4algebra

Fix $q_{0} \in U$ and a chart $(U^{'}, x)$ around $q_{0}$ ; since $α_{q_{0}} \neq 0$ by step 1.1, some component $α_{a} (q_{0})$ is nonzero, and after relabelling the coordinates we may assume $α_{0} (q_{0}) \neq 0$ . On the open subset $V \subseteq U^{'}$ where $α_{0} \neq 0$ define, for $i = 1, 2, 3$ , the vector fields $X_{i} : = α_{0} \partial_{i} - α_{i} \partial_{0}$ ; their components are $C^{\infty}$ by step 1.1, so they are smooth by [L2]. They are tangent to $D$ , because $α (X_{i}) = α_{0} α_{i} - α_{i} α_{0} = 0$ . They are pointwise independent on $V$ , since the coefficient of $\partial_{i}$ in $X_{i}$ is $α_{0} \neq 0$ . Finally they span $D_{q}$ at every $q \in V$ : if $v = \sum_{a} v^{a} \partial_{a}$ satisfies $α (v) = α_{0} v^{0} + \sum_{i \geq 1} α_{i} v^{i} = 0$ , then $v^{0} = - (\sum_{i \geq 1} α_{i} v^{i}) / α_{0}$ , so $v = \sum_{i \geq 1} (v^{i} / α_{0}) X_{i}$ .

2.2step 1.1L7given

Applying [L7] to the nowhere-zero smooth one-form $α$ of step 1.1 gives: $\ker α = D$ is integrable if and only if $α \land d α = 0$ . This proves claim 2.

3.1step 2.1L6given

Step 2.1 exhibits, near each point of $U$ , three smooth pointwise-independent fields spanning the $D_{q}$ ; by [L6] the family $q \mapsto D_{q}$ is therefore a smooth rank- $3$ distribution on $U$ . This proves claim 1.

3.2step 2.2L8given

Assume $α \land d α = 0$ on $U$ ; by step 2.2 the distribution $D$ is integrable. Then [L8] gives, near every point of $U$ , a chart $(u^{0}, u^{1}, u^{2}, u^{3})$ in which $D = span (\partial_{1}, \partial_{2}, \partial_{3})$ ; the slices with $u^{0}$ constant are integral hypersurfaces tangent to the rest spaces. This proves claim 3.

4.1step 3.1step 2.2step 3.2∎

Steps 3.1, 2.2 and 3.2 prove claims 1, 2 and 3 respectively.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Observer-relative energy, spatial momentum and rest mass of a tangent vector

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , let $n$ be an observer at $p$ , and let $c > 0$ be a real parameter (Observers, rest spaces and the spatial projection at an event, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(a) Observer energy and spatial momentum. For every $P \in T_{p} M$ define the observer energy and the spatial momentum of $P$ relative to $n$ by

$E_{n} (P) : = - c g_{p} (P, n), q_{n} (P) : = h_{n} (P),$

where $h_{n}$ is the spatial projection of the observer $n$ . Both maps are $R$ -linear in $P$ : the first because $g_{p}$ is bilinear, the second by The observer rest-space decomposition and the spatial projection. The second takes values in the rest space $n^{⊥}$ , and by the rest-space decomposition of that lemma every $P$ has the unique decomposition

$P = \frac{E_{n} (P)}{c} n + q_{n} (P), \frac{E_{n} (P)}{c} \in R, q_{n} (P) \in n^{⊥} .$

Equivalently $E_{n} (P) = c λ$ and $q_{n} (P) = w$ for the unique pair $(λ, w)$ with $P = λ n + w$ , $λ \in R$ , $w \in n^{⊥}$ .

(b) Rest mass and four-velocity normalization. For $P \in T_{p} M$ with $g_{p} (P, P) \leq 0$ define the rest mass

$m (P) : = \frac{\sqrt{- g_{p} (P, P)}}{c} \geq 0,$

where the square root is the nonnegative root of Rational powers $a^{r}$ of a positive base. By strict positivity of the square root of a positive number, $m (P) = 0$ exactly when $P = 0$ or $P$ is null, and $m (P) > 0$ exactly when $P$ is timelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space). When $m (P) > 0$ , define the normalized four-velocity direction

$U (P) : = \frac{P}{m (P)},$

so that $g_{p} (U (P), U (P)) = g_{p} (P, P) / m (P)^{2} = - c^{2}$ . Conversely, if $m > 0$ and $U \in T_{p} M$ satisfy $g_{p} (U, U) = - c^{2}$ , then $P : = m U$ has $g_{p} (P, P) = - m^{2} c^{2}$ and $m (P) = m$ , $U (P) = U$ ; the pair $(m, U)$ with $U$ future-directed determines and is determined by its future timelike $P = m U$ .

(c) Scope and reading. $E_{n}$ and $q_{n}$ depend on the observer $n$ : for the same nonzero $P$ , different observers $n, n^{'}$ generally give different values, and no observer-independent splitting of $P$ into "energy" and "momentum" is defined by this item. The identities above are algebraic; this definition asserts no positivity of $E_{n}$ (that is proved separately), no conservation, no dynamics and no measurement. The names observer energy, spatial momentum and rest mass, and their physical units and interpretation, belong to the physical reading of the framework's unit conventions and postulates and are not asserted here; this item fixes only the mathematical dependence on the objects $P$ , $n$ , $g_{p}$ and the parameter $c$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Positivity, relative-velocity identities and the mass shell of observer-relative energy and momentum

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $p \in M$ , let $n$ be an observer at $p$ , let $c > 0$ , and for $P \in T_{p} M$ write $E : = E_{n} (P)$ and $q : = q_{n} (P)$ as in Observer-relative energy, spatial momentum and rest mass of a tangent vector. When $g_{p} (P, P) \leq 0$ , also write $m : = m (P) = \sqrt{- g_{p} (P, P)} / c$ ; $m (P)$ is not defined for spacelike $P$ . Then:

Master identity. $E^{2} = c^{2} (g_{p} (q, q) - g_{p} (P, P))$ .
Positivity. If $P$ is nonzero future causal, then $E > 0$ .
Mass shell. If $P$ is future timelike, then $m > 0$ and, with $U : = P / m$ , $u : = U / c$ and $γ : = - g_{p} (n, u)$ , one has $γ \geq 1$ , $E = m c^{2} γ$ , $q = m γ w$ where $w : = c h_{n} (u) / γ$ , $g_{p} (q, q) = m^{2} c^{2} (γ^{2} - 1)$ and

$E^{2} = m^{2} c^{4} + c^{2} g_{p} (q, q) .$
Null case. If $P$ is future null and nonzero, then $E > 0$ , $E = c \sqrt{g_{p} (q, q)}$ and $E^{2} = c^{2} g_{p} (q, q)$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , a point $p$ , an observer $n$ at $p$ , a real $c > 0$ , and $P \in T_{p} M$ with $E = E_{n} (P)$ , $q = q_{n} (P)$ , and $m = m (P)$ only when $g_{p} (P, P) \leq 0$ .

[L1]

$E = - c g_{p} (P, n)$ , $q = h_{n} (P) \in n^{⊥}$ , and $P = (E / c) n + q$ (Observer-relative energy, spatial momentum and rest mass of a tangent vector).

[L2]

$T_{p} M = R n \oplus n^{⊥}$ is an orthogonal direct sum with $g_{p} (n, n) = - 1$ , and $g_{p}$ is positive definite on $n^{⊥}$ (The observer rest-space decomposition and the spatial projection).

[L3]

If $z$ is nonzero future causal then $- g_{p} (n, z) > 0$ ; if $z$ is future unit timelike then $γ_{z} : = - g_{p} (n, z) \geq 1$ (Pairing of an observer with future causal vectors).

[L4]

For future unit timelike $u$ , with $γ = - g_{p} (n, u)$ and $w = c h_{n} (u) / γ$ , one has $u = γ (n + w / c)$ and $g_{p} (w, w) = c^{2} (1 - γ^{- 2})$ (Relative velocity decomposition of a future unit timelike vector).

[L5]

A vector is timelike, null or spacelike according to the sign of $g_{p} (P, P)$ , with the null case excluding $P = 0$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space), and a positive multiple of a future-directed vector is future-directed (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L6]

For $a \geq 0$ the nonnegative real $\sqrt{a}$ satisfies $\sqrt{a} \geq 0$ and $(\sqrt{a})^{2} = a$ , and $\sqrt{a} > 0$ for $a > 0$ (Rational powers $a^{r}$ of a positive base).

Proof

technique · direct

1.1L1L2algebra

By [L1], $P = (E / c) n + q$ with $q \in n^{⊥}$ ; since $g_{p} (n, q) = 0$ and $g_{p} (n, n) = - 1$ by [L2], bilinearity gives $g_{p} (P, P) = (E / c)^{2} g_{p} (n, n) + 2 (E / c) g_{p} (n, q) + g_{p} (q, q) = - (E / c)^{2} + g_{p} (q, q)$ . Multiplying by $c^{2}$ gives $E^{2} = c^{2} (g_{p} (q, q) - g_{p} (P, P))$ , the master identity of claim 1.

1.2L1L3algebra

If $P$ is nonzero future causal, then $- g_{p} (n, P) > 0$ by [L3], so $E = - c g_{p} (P, n) = c (- g_{p} (n, P)) > 0$ because $c > 0$ . This proves claim 2.

1.3L1L3L4L5algebra

Let $P$ be future timelike; then $g_{p} (P, P) = - m^{2} c^{2}$ with $m > 0$ . The vector $u = U / c = P / (m c)$ satisfies $g_{p} (u, u) = g_{p} (P, P) / (m^{2} c^{2}) = - 1$ , so $u$ is unit timelike, and it is future-directed as a positive multiple of $P$ by [L5]; hence $γ : = - g_{p} (n, u) \geq 1$ by [L3] and $u = γ (n + w / c)$ with $w = c h_{n} (u) / γ$ and $g_{p} (w, w) = c^{2} (1 - γ^{- 2})$ by [L4].

2.1step 1.3L1L2L4algebra

With $u = P / (m c)$ , the definitions give $E = - c g_{p} (P, n) = - c g_{p} (m c u, n) = - m c^{2} g_{p} (u, n) = m c^{2} γ$ . Moreover $q = h_{n} (P) = h_{n} (m c u) = m c h_{n} (u) = m c (γ w / c) = m γ w$ by linearity of $h_{n}$ [L2] and [L4], so $g_{p} (q, q) = m^{2} γ^{2} g_{p} (w, w) = m^{2} γ^{2} c^{2} (1 - γ^{- 2}) = m^{2} c^{2} (γ^{2} - 1)$ .

2.2step 1.1step 1.2L2L5L6algebra

Let $P$ be future null and nonzero; then $g_{p} (P, P) = 0$ by [L5], so the master identity gives $E^{2} = c^{2} g_{p} (q, q)$ with $g_{p} (q, q) \geq 0$ by [L2]. By claim 2, $E > 0$ ; hence $E = c \sqrt{g_{p} (q, q)}$ by [L6] and $c > 0$ . This proves claim 4.

3.1step 1.1step 1.3step 2.1L1algebra

Substituting $g_{p} (P, P) = - m^{2} c^{2}$ and $g_{p} (q, q) = m^{2} c^{2} (γ^{2} - 1)$ of step 2.1 into the master identity gives $E^{2} = c^{2} (m^{2} c^{2} (γ^{2} - 1) + m^{2} c^{2}) = m^{2} c^{4} γ^{2} = m^{2} c^{4} + c^{2} g_{p} (q, q)$ , the mass shell. This proves claim 3.

4.1step 1.1step 1.2step 3.1step 2.2∎

Claims 1-4 are steps 1.1, 1.2, 3.1 and 2.2.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

Charts, frames, tetrads, congruences and observers are distinct objects

Remark

This page introduces several objects that are related but not interchangeable; the distinctions below are part of their definitions and are used throughout the general-relativity track.

Observer. An observer at $p$ is one future-directed unit timelike vector $n \in T_{p} M$ , $g_{p} (n, n) = - 1$ (Observers, rest spaces and the spatial projection at an event). It carries no parametrized curve and no clock reading by itself.
Congruence. An observer congruence on an open set $U$ is a smooth field $n$ whose value at every point is an observer; its maximal integral curves are its observer curves, and its flow parameter is not a physical time (Observer congruences, their observer curves and adapted charts).
Frame and coordinate frame. A local frame field is a tuple of four pointwise independent smooth vector fields; the coordinate frame of a smooth chart is the special case of the fields $\partial / \partial x^{a}$ , whose members pairwise commute. A frame whose members do not commute is therefore not the coordinate frame of any chart (Local frame fields, coordinate frames and tetrad fields).
Tetrad. A tetrad field is a pointwise $g$ -orthonormal frame field; it is future-directed when its first member is future-directed, and it is adapted to a congruence $n$ when that first member equals $n$ . A tetrad field need not be a coordinate frame. Local existence near every point is proved in Existence of a smooth future-directed tetrad field near every event; global existence on $M$ is not asserted and would be a nontrivial parallelizability property of the manifold.
Adapted chart. A chart is $n$ -adapted when its spatial coordinates are constant along each integral curve of $n$ contained in its domain, with an interval as parameter domain (Observer congruences, their observer curves and adapted charts, Definition (c)). Near every point of a congruence there is such a chart with $n = \partial / \partial u^{0}$ (Every congruence admits a local adapted chart). The definition does not require this normalization: replacing $u^{0}$ in that chart by $v^{0} = 2 u^{0}$ and keeping the spatial coordinates gives another adapted chart with $n = 2 \partial / \partial v^{0}$ . Thus adapted charts are not unique, and their zeroth coordinate field need not equal $n$ .
Rest spaces and their integrability. The rest spaces $n_{q}^{⊥}$ of a congruence always form a smooth rank- $3$ distribution, and they are tangent to local hypersurfaces exactly when $α \land d α = 0$ for $α = n^{♭}$ . Thus an adapted chart always exists, while an orthogonal rest-space foliation need not (Smoothness of the rest-space family and its Frobenius integrability criterion).
Components. Coordinate components of a tensor field transform by products of chart Jacobians (The component transformation law of a smooth tensor field), while its components in two future-directed tetrads are related by a matrix $Λ$ with $Λ^{T} η Λ = η$ and $Λ^{0}_{0} > 0$ (The change between two future-directed tetrads is a Lorentz matrix). The two component systems are different data (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).
Observer-relative quantities. For $P \in T_{p} M$ and a fixed real parameter $c > 0$ , observer energy $E_{n} (P) = - c g_{p} (P, n)$ and spatial momentum $q_{n} (P) = h_{n} (P)$ depend on the chosen observer $n$ , where $h_{n} (P) = P + g_{p} (n, P) n$ is the spatial projection of Observers, rest spaces and the spatial projection at an event, Definition (c). Different observers generally assign different energies to the same tangent vector, and no observer-independent energy-momentum split is asserted. In contrast, rest mass $m (P) = \sqrt{- g_{p} (P, P)} / c$ , defined when $g_{p} (P, P) \leq 0$ , depends only on $P$ , $g_{p}$ and $c$ , and is independent of $n$ (Observer-relative energy, spatial momentum and rest mass of a tangent vector, Definition (a)-(c)).
Physical readings. The interpretation of these geometric objects as observers, reference frames, clocks, particles and measured energies belongs to the physical postulates and unit conventions of the framework, recorded elsewhere on this track; the items on this page assert no measurement, dynamics or claim about nature.

5 · Examples, experiments and counterexamples

None yet.

Tensors And Observers

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The tensor bundle on a smooth four-manifold from coordinate evaluations

Statement

Facts & Assumptions

Proof

Tensor fields of type (r,s) on a Lorentzian four-manifold and their components

Definition

Observers, rest spaces and the spatial projection at an event

Definition

Remarks

Observer congruences, their observer curves and adapted charts

Definition

Remarks

The component transformation law of a smooth tensor field

Statement

Facts & Assumptions

Proof

Contraction of a smooth tensor field is a smooth tensor field

Statement

Facts & Assumptions

Proof

The observer rest-space decomposition and the spatial projection

Statement

Facts & Assumptions

Proof

Pairing of an observer with future causal vectors

Statement

Facts & Assumptions

Proof

Local frame fields, coordinate frames and tetrad fields

Definition

Remarks

The change between two future-directed tetrads is a Lorentz matrix

Statement

Facts & Assumptions

Proof

Existence of a smooth future-directed tetrad field near every event

Statement

Facts & Assumptions

Proof

Remarks

Relative velocity decomposition of a future unit timelike vector

Statement

Facts & Assumptions

Proof

Every congruence admits a local adapted chart

Statement

Facts & Assumptions

Proof

Smoothness of the rest-space family and its Frobenius integrability criterion

Statement

Facts & Assumptions

Proof

Observer-relative energy, spatial momentum and rest mass of a tangent vector

Definition

Positivity, relative-velocity identities and the mass shell of observer-relative energy and momentum

Statement

Facts & Assumptions

Proof

Charts, frames, tetrads, congruences and observers are distinct objects

Remark

5 · Examples, experiments and counterexamples

Sources