Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 3 / a / Solution

On an oriented Euclidean four-space with its metric volume form, the Hodge star operator is defined by

α \land^{⋆} β = ⟨ α, β ⟩ vol

for forms

α

and

β

of the same degree. In

n

Euclidean dimensions it satisfies

^{⋆} (^{⋆} α) = (- 1)^{p (n - p)} α (α \in Λ^{p}) .

Hence on two-forms in four dimensions,

^{⋆}^{⋆} = 1 on Λ^{2} (R^{4}) .

The normalization of

vol

as the metric volume form is essential; reversing its orientation reverses a single Hodge star but leaves its square unchanged.

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