Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 1 / a / i

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- Solution i

Solution

Expanding the spatial projection tensor gives

Θ_{μ} = ⊥^{ρ}_{μ} Z_{ρ} = Z_{μ} + n_{μ} n^{ρ} Z_{ρ} = Z_{μ} - n_{μ} Θ.

Rearrangement yields the normal-plus-spatial decomposition

Z_{μ} = Θ_{μ} + n_{μ} Θ.

It also makes

n^{μ} Θ_{μ} = 0

immediate.

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