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Worked motion of the primed frame's origin under a passive boost

Example

Worked computation. Let $B = B (β e_{1})$ be a passive boost with parameter $β \in (- 1, 1)$ along the first spatial direction of an inertial frame with coordinates $(x^{0}, x^{1})$ (The passive Lorentz boost with parameter beta); in matrix expressions write $B_{x} : = E^{- 1} \circ B \circ E$ and $x' = B_{x} x$ for the transformed coordinates. Then:

(i) the locus $x^{' 1} = 0$ of the primed coordinates (with $x^{' 2}, x^{' 3}$ fixed) is, in the unprimed chart, $x^{1} = β x^{0}$ with $x^{2}, x^{3}$ fixed constants. The primed spatial origin is the case $x^{2} = x^{3} = 0$ ; arbitrary fixed transverse coordinates give parallel loci: indeed $x^{' 1} = γ (x^{1} - β x^{0}) = 0$ forces $x^{1} = β x^{0}$ , and the transverse coordinates are unchanged because $B$ fixes them. With $x^{0} = c t$ this is uniform motion with coordinate velocity $β c$ along $e_{1}$ ;

(ii) conversely the unprimed origin $x^{1} = 0$ satisfies $x^{' 1} = γ (0 - β x^{0}) = - β γ x^{0} = - β x^{' 0}$ , since $x^{' 0} = γ x^{0}$ when $x^{1} = 0$ ; so the unprimed frame moves with parameter $- β$ in the primed chart, consistent with $B^{- 1} = B (- β)$ ;

(iii) for $β = 3 / 5$ the concrete values are $x^{1} = (3 / 5) x^{0}$ in the first chart and $x^{' 1} = - (3 / 5) x^{' 0}$ in the second; the locus $x^{' 1} = 0$ passes through the event $(x^{0}, x^{1}) = (0, 0)$ and through $(5, 3)$ (choosing $x^{0} = 5$ ), which the boost maps to $(x^{' 0}, x^{' 1}) = ((5 / 4) (5 - (3 / 5) 3), (5 / 4) (3 - (3 / 5) 5)) = (4, 0)$ , confirming $x^{' 1} = 0$ .

Facts & Assumptions

Given: An inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with coordinates $x = (x^{0}, x^{1}, x^{2}, x^{3})$ , a parameter $β \in (- 1, 1)$ , and the boost $B : = B (β e_{1})$ ; write $x' = B_{x} x$ .

[L1]

For $β = β e_{1}$ the boost acts by $x^{' 0} = γ (x^{0} - β x^{1})$ and $x^{' 1} = γ (x^{1} - β x^{0})$ , with $γ = (1 - ∣ β ∣^{2})^{- 1 / 2} = (1 - β^{2})^{- 1 / 2} > 0$ , and it fixes the transverse coordinates: $x^{' 2} = x^{2}$ , $x^{' 3} = x^{3}$ ; the Euclidean norm $∣ β ∣$ is that of the frame's spatial coordinates (The passive Lorentz boost with parameter beta, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L2]

$B (β e_{1})$ is a proper orthochronous Lorentz map and $B (β e_{1})^{- 1} = B (- β e_{1})$ ; the maps $B (β e_{1})$ are injective in $β$ (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).

[L3]

An inertial frame has $η$ -orthonormal basis with $η (e_{a}, e_{b}) = diag (- 1, 1, 1, 1)_{a b}$ , and its chart $x (p) = E^{- 1} (p - o)$ is affine (Inertial frames and inertial charts of the affine Minkowski model).

[L4]

Fix a real constant $c > 0$ and define the reparametrized temporal coordinate $t : = x^{0} / c$ , so $x^{0} = c t$ . Coordinate velocity means the derivative $d x^{1} / d t$ with respect to this coordinate. This rescaling makes no assertion about physical clock readings or the empirical value or SI units of $c$ .

[L5]

For every proper orthochronous linear Lorentz map $L : V \to V$ and every $c_{0} \in A$ the tuple $(c_{0}; L e_{0}, L e_{1}, L e_{2}, L e_{3})$ is an inertial frame whose chart $x_{L}$ satisfies $x_{L} (p) = M_{L}^{- 1} (x (p) - x (c_{0}))$ , where $M_{L} = E^{- 1} \circ L \circ E$ (Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition).

Proof

technique · direct

1.1L1L4algebra

By [L1], $x^{' 1} = γ (x^{1} - β x^{0})$ and $γ > 0$ , so $x^{' 1} = 0$ holds exactly when $x^{1} = β x^{0}$ ; the equations $x^{' 2} = x^{2}$ , $x^{' 3} = x^{3}$ of [L1] show that the transverse coordinates are arbitrary constants, unchanged by the boost. Under the convention $x^{0} = c t$ of [L4], the locus is $x^{1} = β c t$ , that is, uniform motion along $e_{1}$ with coordinate velocity $d x^{1} / d t = β c$ .

1.2L1L2algebra

For the unprimed origin set $x^{1} = 0$ (with $x^{2} = x^{3} = 0$ ). [L1] gives $x^{' 0} = γ x^{0}$ and $x^{' 1} = - γ β x^{0} = - β x^{' 0}$ , so in the primed coordinates the unprimed origin moves along the locus $x^{' 1} = - β x^{' 0}$ . This is consistent with [L2]: applying the inverse boost $B (- β e_{1})$ to a point with $x^{' 1} = - β x^{' 0}$ gives $x^{1} = γ (x^{' 1} + β x^{' 0}) = γ (- β x^{' 0} + β x^{' 0}) = 0$ , so the inverse map sends that locus to the unprimed origin, as $B^{- 1} = B (- β)$ requires.

1.3L1L2L3L5algebra

Finally, the transformed coordinates $x' = B_{x} x$ are the coordinates of an inertial chart of the model: by [L2] the map $L : = B (- β e_{1})$ is proper orthochronous, so [L5] applied with $c_{0} : = o$ provides an inertial frame whose chart is $x_{L} (p) = M_{L}^{- 1} (x (p) - x (o)) = B_{x} (β e_{1}) x (p) = x' (p)$ , where $B_{x}$ denotes the boost matrix and $M_{L} = B_{x} (- β e_{1})$ for every event $p$ , using $x (o) = 0$ and $L^{- 1} = B (β e_{1})$ . Thus the worked motion is the motion of the primed inertial chart's spatial origin in the unprimed chart, and conversely; the computation uses only the $1 + 1$ boost formulas and the affine chart structure.

2.1L1step 1.1step 1.2algebra

For $β = 3 / 5$ one has $γ = (1 - 9 / 25)^{- 1 / 2} = (16 / 25)^{- 1 / 2} = 5 / 4$ ; step 1.1 gives the locus $x^{1} = (3 / 5) x^{0}$ and step 1.2 gives $x^{' 1} = - (3 / 5) x^{' 0}$ . The event $(x^{0}, x^{1}) = (5, 3)$ satisfies $3 = (3 / 5) \cdot 5$ , so it lies on the locus, and [L1] gives $x^{' 0} = (5 / 4) (5 - (3 / 5) 3) = (5 / 4) (5 - 9 / 5) = (5 / 4) (16 / 5) = 4$ and $x^{' 1} = (5 / 4) (3 - (3 / 5) 5) = (5 / 4) (3 - 3) = 0$ ; the origin $(0, 0)$ is fixed because the boost is linear. Hence the displayed values of clause (iii) are correct.

3.1step 1.1step 1.2step 2.1step 1.3∎

Steps 1.1-1.2 and 2.1 verify clauses (i)-(iii) of the Example and step 1.3 identifies the primed coordinates with an inertial chart; every assertion is a finite computation with the displayed rational values.

Worked motion of the primed frame's origin under a passive boost

Example

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources