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The proper orthochronous Lorentz group: boosts, rotations, unique factorization and invariants

Statement

Let $(V, η)$ be a $4$ -dimensional Lorentz vector space with time orientation $C^{+}$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space is the form; Time orientation, future and past timelike/null/causal vectors, and the closed cones fixes $C^{+}$ ), and let $(o; e_{0}, e_{1}, e_{2}, e_{3})$ be an inertial frame of the model (Inertial frames and inertial charts of the affine Minkowski model), with spatial three-space $S$ and Euclidean form $δ$ . Then:

(i) For every $β \in R^{3}$ with $∣ β ∣ < 1$ the passive boost $B (β)$ is a proper orthochronous Lorentz map with $B (β)^{- 1} = B (- β)$ , and for every unit spatial vector $e \in S$ and all $β_{1}, β_{2} \in (- 1, 1)$ one has $B (β_{1} e) B (β_{2} e) = B (\frac{β_{1} + β_{2}}{1 + β_{1} β_{2}} e)$ ; in particular collinear boosts commute. For each fixed unit spatial axis $e$ and scalar $b \in (- 1, 1)$ the collinear boost $B (b e)$ satisfies $b = \tanh (r)$ for a unique scalar rapidity $r \in R$ , and rapidities add for boosts along that same fixed axis. This signed scalar rapidity statement does not identify a vector $β \in R^{3}$ with the scalar $\tanh (r)$ .

(ii) $S O^{+} (V, η)$ is a subgroup of $O (V, η)$ , which is a subgroup of the general linear group $G L (V)$ ; every passive boost $B (β)$ and every spatial rotation of the frame belongs to $S O^{+} (V, η)$ .

(iii) Every $L \in S O^{+} (V, η)$ has a unique factorization $L = B (- β) \hat{R}$ with $∣ β ∣ < 1$ and $\hat{R}$ the extension of a spatial rotation of the frame; here $β = u_{v e c} / u^{0}$ where $(u^{0}, u_{v e c})$ are the coordinates of $u = L e_{0}$ . Consequently every proper orthochronous Lorentz map is the product of a passive boost and a spatial rotation, in that order, and $L$ is itself a passive boost if and only if the rotation $\hat{R}$ is the identity.

(iv) Every $L \in S O^{+} (V, η)$ preserves the interval $η (v, v)$ , the causal classification of every nonzero vector, the future and past closed cones, the orientation, and the oriented volume form of the model: $ϵ (L v_{0}, L v_{1}, L v_{2}, L v_{3}) = ϵ (v_{0}, v_{1}, v_{2}, v_{3})$ for all $v_{0}, \dots, v_{3} \in V$ (The oriented volume form epsilon and the permutation symbols of an inertial frame). An arbitrary element $L$ of $O (V, η)$ satisfies $\det L = \pm 1$ and either preserves or interchanges the future and past cones; moreover $\det L = 1$ alone does not force $L (C^{+}) = C^{+}$ .

Facts & Assumptions

Given: A $4$ -dimensional Lorentz vector space $(V, η)$ with time orientation $C^{+}$ , an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with spatial three-space $S$ and Euclidean form $δ$ , and the sets $O (V, η)$ , $S O^{+} (V, η)$ .

[L1]

A linear Lorentz map is a linear $L$ with $η (L v, L w) = η (v, w)$ for all $v, w$ ; it is proper when $\det L = + 1$ and orthochronous when $L (C^{+}) = C^{+}$ ; $G L (V)$ is the group of invertible linear maps (Linear and affine Lorentz maps, and the Lorentz group).

[L2]

Every linear Lorentz map is a linear isomorphism, satisfies $\det L = \pm 1$ , preserves $η (v, v)$ and the causal character of every nonzero vector, and maps $C^{+}$ onto one of the two components $C^{+}, C^{-}$ of the timelike set; $L (C^{+}) = C^{+}$ holds if and only if $L (C^{+}) \subseteq C^{+}$ ; $S O (V, η) = {L \in O (V, η) : \det L = 1}$ , $O^{↑} (V, η) = {L \in O (V, η) : L (C^{+}) = C^{+}}$ and $S O^{+} (V, η) = S O (V, η) \cap O^{↑} (V, η)$ are subgroups of $O (V, η)$ , which is a subgroup of $G L (V)$ ; if $L$ is proper then $ϵ (L v_{0}, \dots, L v_{3}) = ϵ (v_{0}, \dots, v_{3})$ and if $\det L = - 1$ the two sides differ by the sign $- 1$ (Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups, The oriented volume form epsilon and the permutation symbols of an inertial frame).

[L3]

For every $∣ β ∣ < 1$ the passive boost $B (β)$ is a proper orthochronous Lorentz map with $B (β)^{- 1} = B (- β)$ ; the map $β \mapsto B (β)$ is injective and $B (0) = {id}_{V}$ (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta), The passive Lorentz boost with parameter beta).

[L4]

For every unit spatial vector $e$ and all $β_{1}, β_{2} \in (- 1, 1)$ , $B (β_{1} e) B (β_{2} e) = B (\frac{β_{1} + β_{2}}{1 + β_{1} β_{2}} e)$ with $∣ \frac{β_{1} + β_{2}}{1 + β_{1} β_{2}} ∣ < 1$ , and the two collinear boosts commute (Collinear boost composition and the relativistic velocity-addition law).

[L5]

The map $r \mapsto \tanh (r)$ is a strictly increasing bijection of $R$ onto $(- 1, 1)$ , and for every unit spatial vector $e$ and all $r, s \in R$ one has $B (\tanh (r) e) B (\tanh (s) e) = B (\tanh (r + s) e)$ (Rapidity parametrization of collinear boosts).

[L6]

For a proper orthochronous Lorentz map $L$ with $u : = L e_{0}$ and coordinates $(u^{0}, u_{v e c})$ , the parameter $β : = u_{v e c} / u^{0}$ satisfies $∣ β ∣ < 1$ , and $L = B (- β) \hat{R}$ with $\hat{R} : = B (β) L$ the extension $diag (1, R)$ of a spatial rotation of the frame; this factorization is unique among parameters in the open unit ball and spatial rotations; conversely every such $B (- β) \hat{R}$ is proper orthochronous; and $L$ is a passive boost if and only if $\hat{R} = {id}_{V}$ (Unique boost-rotation factorization of proper orthochronous Lorentz maps).

[L7]

A spatial rotation of the frame is the extension by $e_{0} \mapsto e_{0}$ of a linear map of $S$ preserving $δ$ with determinant $+ 1$ on $S$ ; its extension is denoted $diag (1, R)$ in the frame's basis (Spatial rotations of an inertial frame).

[L8]

In the frame the timelike set is $T = T_{+} \cup T_{-}$ with $T_{+} = {v : v^{0} > ∣ v_{v e c} ∣}$ and $T_{-} = {v : v^{0} < - ∣ v_{v e c} ∣}$ , and $T_{+}, T_{-}$ are exactly the two components; since $e_{0} \in C^{+}$ has coordinates $(1, 0, 0, 0)$ (Inertial frames and inertial charts of the affine Minkowski model), one has $C^{+} = T_{+}$ and $C^{-} = T_{-}$ , with closed cones $cl (C^{+}) = {v : v^{0} \geq ∣ v_{v e c} ∣}$ and $cl (C^{-}) = {v : v^{0} \leq - ∣ v_{v e c} ∣}$ (The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones).

[L9]

A nonzero vector is timelike when $η (v, v) < 0$ , null when $η (v, v) = 0$ , spacelike when $η (v, v) > 0$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space); the oriented volume form satisfies $ϵ_{0123} = + 1$ (The oriented volume form epsilon and the permutation symbols of an inertial frame); $\det$ of a linear operator is the determinant of its matrix in any ordered basis and the Leibniz sum over a diagonal matrix is the product of its diagonal entries (The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L10]

For spatial coordinate vectors $a, b$ , $∣ a \cdot b ∣ \leq ∣ a ∣ ∣ b ∣$ by Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, applied in dimension $3$ .

Proof

technique · direct

1.1L3

Clause (i), first sentence: by [L3], $B (β)$ is a proper orthochronous Lorentz map with inverse $B (- β)$ for every $∣ β ∣ < 1$ .

1.2L4

Clause (i), second sentence: by [L4] the composition of two collinear boosts is the boost with parameter $\frac{β_{1} + β_{2}}{1 + β_{1} β_{2}}$ in the same direction, this parameter lies in $(- 1, 1)$ , and the two factors commute; in particular the composition law is a collinear law and not an addition of parameters.

1.3L5

Clause (i), third and fourth sentences: fix a unit spatial axis $e$ . By [L5], $r \mapsto \tanh (r)$ is a strictly increasing bijection of $R$ onto $(- 1, 1)$ ; hence for each $b \in (- 1, 1)$ there is exactly one $r \in R$ with $b = \tanh (r)$ , so $B (b e) = B (\tanh (r) e)$ has the unique rapidity $r$ , and for $r, s \in R$ the product $B (\tanh (r) e) B (\tanh (s) e)$ equals $B (\tanh (r + s) e)$ , so rapidities add along the fixed axis. The rapidity $r$ is a scalar attached to the fixed axis $e$ : the displayed equations concern the scalar parameters $β_{i} = \tanh (r_{i})$ of boosts $B (β_{i} e)$ , and no identification of a vector $β \in R^{3}$ with $\tanh (r)$ is made.

1.4L1L2

Clause (ii), subgroup part: [L2] states that $S O (V, η)$ , $O^{↑} (V, η)$ and their intersection $S O^{+} (V, η)$ are subgroups of $O (V, η)$ , and that $O (V, η)$ is a subgroup of $G L (V)$ .

1.5L3L6L7

Clause (ii), membership part: every passive boost lies in $S O^{+} (V, η)$ by [L3]. For a linear $δ$ -isometry $R : S \to S$ of determinant $1$ , whose extension $\hat{R} = diag (1, R)$ is a spatial rotation of the frame: by [L6] the product $B (0) \hat{R} = \hat{R}$ of the boost with parameter $0$ and the spatial rotation is a proper orthochronous Lorentz map; here $B (0) = {id}_{V}$ by [L3], so $\hat{R} \in S O^{+} (V, η)$ .

1.6L6L7

Clause (iii): by [L6] every $L \in S O^{+} (V, η)$ has the factorization $L = B (- β) \hat{R}$ with $β = u_{v e c} / u^{0}$ and $\hat{R}$ the extension of a spatial rotation of the frame, and this factorization is unique; consequently $L$ is a product of a passive boost and a spatial rotation in that order, and [L6] also gives that $L$ is a passive boost if and only if $\hat{R} = {id}_{V}$ .

1.7L2L8L9L10algebra

Clause (iv), preservation part: let $L \in S O^{+} (V, η)$ , so $L$ is Lorentz, proper and orthochronous. By [L2], $L$ preserves $η (v, v)$ and the causal character of every nonzero vector, and $L (C^{+}) = C^{+}$ , so $L e_{0}$ is future timelike because $e_{0} \in C^{+}$ by [L8]. For the closed cones, first record the criterion: if $w$ is future timelike and $v \neq 0$ has $η (v, v) \leq 0$ , then $v \in cl (C^{+})$ if and only if $η (v, w) < 0$ , and $v \in cl (C^{-})$ if and only if $η (v, w) > 0$ . Indeed [L8] gives $cl (C^{+}) = {v^{0} \geq ∣ v_{v e c} ∣}$ and $cl (C^{-}) = {v^{0} \leq - ∣ v_{v e c} ∣}$ in frame coordinates, with $η (v, v) = - (v^{0})^{2} + ∣ v_{v e c} ∣^{2}$ and $w^{0} > ∣ w_{v e c} ∣$ ; for nonzero $v \in cl (C^{+})$ one has $v^{0} \geq ∣ v_{v e c} ∣ \geq 0$ and $v^{0} > 0$ , so $η (v, w) \leq - v^{0} w^{0} + ∣ v_{v e c} ∣ ∣ w_{v e c} ∣ \leq v^{0} (∣ w_{v e c} ∣ - w^{0}) < 0$ , while for nonzero $v \in cl (C^{-})$ one has $- v^{0} \geq ∣ v_{v e c} ∣ \geq 0$ and $- v^{0} > 0$ , so $η (v, w) \geq (- v^{0}) w^{0} - ∣ v_{v e c} ∣ ∣ w_{v e c} ∣ \geq (- v^{0}) (w^{0} - ∣ w_{v e c} ∣) > 0$ ; the nonzero causal vectors split into $cl (C^{+}) ∖ {0}$ and $cl (C^{-}) ∖ {0}$ by [L8], so the signs are exhaustive. Since $L$ preserves the form, is an isomorphism and satisfies $η (L v, L e_{0}) = η (v, e_{0})$ , applying the criterion with $w : = L e_{0}$ gives $L (cl (C^{+})) \subseteq cl (C^{+})$ and $L (cl (C^{-})) \subseteq cl (C^{-})$ ; applying the same argument to $L^{- 1} \in S O^{+} (V, η)$ (which exists by [L2]) gives the reverse inclusions, hence $L (cl (C^{+})) = cl (C^{+})$ and $L (cl (C^{-})) = cl (C^{-})$ . By [L2] again, properness gives $ϵ (L v_{0}, \dots, L v_{3}) = ϵ (v_{0}, \dots, v_{3})$ for all $v_{0}, \dots, v_{3}$ , which is the preservation of the oriented volume form and of the orientation.

1.8L1L2L8L9

Clause (iv), last sentence: for any $L \in O (V, η)$ , [L2] gives $\det L = \pm 1$ and $L (C^{+}) \in {C^{+}, C^{-}}$ , so $L$ either preserves or interchanges the future and past cones. For the failure of the converse, let $Λ : = - {id}_{V}$ , with matrix $M = - diag (1, 1, 1, 1)$ in the frame; its entries satisfy $M^{T} η_{0} M = η_{0}$ , because the temporal diagonal entry is $(- 1) (- 1) (- 1) = - 1$ , each spatial diagonal entry is $(- 1) (+ 1) (- 1) = + 1$ , and all off-diagonal entries vanish, so $Λ$ is a Lorentz map by [L1]; moreover $\det Λ = (- 1)^{4} = 1$ , because the Leibniz sum over the diagonal matrix $M$ is the product of its diagonal entries by [L9], so $Λ$ is proper. On coordinates, $Λ$ sends a vector with coordinates $(v^{0}, v_{v e c})$ to $(- v^{0}, - v_{v e c})$ , and $v^{0} > ∣ v_{v e c} ∣$ holds exactly when $- v^{0} < - ∣ v_{v e c} ∣$ ; hence $Λ (T_{+}) = T_{-}$ and $Λ (T_{-}) = T_{+}$ with the notation of [L8], so $Λ$ interchanges the two components of the timelike set and $Λ (C^{+}) = C^{-} \neq C^{+}$ : $Λ$ is a proper but non-orthochronous Lorentz map, and $\det L = 1$ alone does not force $L (C^{+}) = C^{+}$ .

2.1step 1.1step 1.2step 1.3step 1.4step 1.5step 1.6step 1.7step 1.8∎

Clause (i) is steps 1.1-1.3; clause (ii) is steps 1.4-1.5; clause (iii) is step 1.6; clause (iv) is steps 1.7-1.8. Every clause is either a direct restatement of the cited supplier with its hypotheses (frame fixed, $∣ β ∣ < 1$ , $L$ proper orthochronous) or the displayed finite computation of step 1.8.

The proper orthochronous Lorentz group: boosts, rotations, unique factorization and invariants

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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