 Solution

ID: past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2019/1/a/solution

Use coordinates

X^{μ} = (c t, x)

and the mostly-plus Minkowski metric

η_{μν} = diag (- 1, 1, 1, 1)

. Along a future-directed timelike world line, proper time satisfies

c^{2} d τ^{2} = - η_{μν} d X^{μ} d X^{ν} = c^{2} d t^{2} - d x^{2}, d τ = d t 1 - v^{2} / c^{2} .

U^{μ} = \frac{d X ^{μ}}{d τ} = γ (c, v), A^{μ} = \frac{d U ^{μ}}{d τ}, γ = (1 - v^{2} / c^{2})^{- 1/2} .

The Minkowski norm is fixed:

U^{μ} U_{μ} = - c^{2}

. Differentiating with respect to proper time, using the constant Minkowski metric, gives

0 = \frac{d}{d τ} (U^{μ} U_{μ}) = 2 U_{μ} A^{μ}, U \cdot A = 0.

In the instantaneous rest reference frame,

U^{μ} = (c, 0, 0, 0)

, so the orthogonality says

A^{0} = 0

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