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Worked velocity transformation: collinear, transverse and null cases

Example

Let $x$ and $x^{'}$ be inertial charts related by a boost of parameter $v = 3 c / 5$ along the first axis ( $γ = 5 / 4$ , The passive Lorentz boost with parameter beta, Inertial frames and inertial charts of the affine Minkowski model).

(a) Collinear case. A worldline with coordinate velocity $w = (4 c / 5, 0, 0)$ in $x$ has, by clause (iv) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation, $w_{1}^{'} = (4 c / 5 - 3 c / 5) / (1 - (3 / 5) (4 / 5)) = (c / 5) / (13 / 25) = 5 c / 13$ , so $∣ w^{'} ∣ = 5 c / 13 < c$ ; the Galilean difference $c / 5$ is not the transformed velocity.

(b) Transverse case. A worldline with $w = (0, c / 2, 0)$ has $w_{1}^{'} = (0 - 3 c / 5) / (1 - 0) = - 3 c / 5$ and $w_{2}^{'} = (c / 2) / ((5 / 4) (1 - 0)) = 2 c / 5$ , hence $∣ w^{'} ∣ = \sqrt{9 / 25 + 4 / 25} c = (\sqrt{13} / 5) c < c$ .

(c) Null limit. For $w = (c, 0, 0)$ the formula gives $w_{1}^{'} = (c - 3 c / 5) / (1 - 3 / 5) = c$ and $∣ w^{'} ∣ = c$ : a null worldline is null in every inertial chart.

In each case take the affine coordinate curve $t \mapsto (c t, w t)$ on $R$ with the stated constant $w$ ; the first two are future timelike and the third future null. The three cases verify the velocity transformation and the subluminality statement of clause (iv) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation.

Facts & Assumptions

Given: Two inertial charts $x$ , $x^{'}$ related by the boost of parameter $v = 3 c / 5$ along the first axis with $γ = 5 / 4$ , and the three coordinate velocities $w = (4 c / 5, 0, 0)$ , $w = (0, c / 2, 0)$ , $w = (c, 0, 0)$ (The passive Lorentz boost with parameter beta, Inertial frames and inertial charts of the affine Minkowski model).

[L1]

Clause (iv) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation: for a $C^{1}$ future causal curve with coordinate velocity $w$ , $∣ w ∣ \leq c$ , in the chart $x$ , the coordinate velocity in $x^{'}$ is $w_{1}^{'} = (w_{1} - v) / (1 - v w_{1} / c^{2})$ and $w_{⊥}^{'} = w_{⊥} / (γ (1 - v w_{1} / c^{2}))$ , with positive denominators, and $∣ w^{'} ∣ < c$ for $∣ w ∣ < c$ , with $∣ w^{'} ∣ = c$ exactly when $∣ w ∣ = c$ .

[L2]

The boost parameter is $β = v / c = 3 / 5$ and $γ = (1 - β^{2})^{- 1 / 2} = 5 / 4$ (The passive Lorentz boost with parameter beta).

Verification

technique · direct

1.1L1L2algebra

Collinear case: with $w_{1} = 4 c / 5$ , $w_{⊥} = 0$ , $v = 3 c / 5$ , the denominator of [L1] is $1 - \frac{3}{5} \cdot \frac{4}{5} = 1 - \frac{12}{25} = \frac{13}{25} > 0$ and $w_{1}^{'} = \frac{c / 5}{13 / 25} = \frac{5 c}{13}$ ; hence $w^{'} = (5 c / 13, 0, 0)$ and $∣ w^{'} ∣ = 5 c / 13 < c$ because $5 / 13 < 1$ . The Galilean difference for this passive change of frame is $4 c / 5 - 3 c / 5 = c / 5$ , which is not the transformed velocity.

1.2L1L2algebra

Transverse case: with $w_{1} = 0$ , $w_{⊥} = (c / 2, 0)$ , the denominator is $1$ and [L1] gives $w_{1}^{'} = (0 - 3 c / 5) / 1 = - 3 c / 5$ and $w_{2}^{'} = (c / 2) / (γ \cdot 1) = 2 c / 5$ , so $∣ w^{'} ∣^{2} = (9 / 25 + 4 / 25) c^{2} = 13 c^{2} / 25$ and $∣ w^{'} ∣ = (\sqrt{13} / 5) c < c$ because $\sqrt{13} < 5$ .

1.3L1L2algebra

Null limit: with $w_{1} = c$ the denominator of [L1] is $1 - 3 / 5 = 2 / 5 > 0$ and $w_{1}^{'} = (c - 3 c / 5) / (2 / 5) = (2 c / 5) / (2 / 5) = c$ , so $w^{'} = (c, 0, 0)$ and $∣ w^{'} ∣ = c$ ; this is the equality case of the subluminality statement, consistent with $∣ w ∣ = c$ .

2.1step 1.1step 1.2step 1.3∎

Conclusion: steps 1.1-1.3 evaluate the transformation formula of [L1] in the three configurations and confirm $∣ w^{'} ∣ < c$ for subluminal $w$ and $∣ w^{'} ∣ = c$ in the null case; in particular the collinear case deviates from the Galilean difference by the relativistic denominator $1 - v w_{1} / c^{2}$ , and the transverse case exhibits the factor $1 / γ$ in the perpendicular component.

Worked velocity transformation: collinear, transverse and null cases

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

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