Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 1 / d

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- Solution d

Solution

A rigidly rotating stretched solution is

X^{0} = A τ, X^{1} = A cos τ cos σ, X^{2} = A sin τ cos σ,

with all other coordinates constant. Each coordinate obeys the wave equation and

X^{'} = 0

at

σ = 0, π

. Directly,

\dot{X} \cdot X^{'} = 0

and

\dot{X}^{2} + X^{'2} = - A^{2} + A^{2} cos^{2} σ + A^{2} sin^{2} σ = 0,

so both Virasoro constraints hold. Its energy and planar angular momentum are

M = P^{0} = π T A, J = J^{12} = T A^{2} \int_{0}^{π} cos^{2} σ, d σ = \frac{π T A ^{2}}{2} .

Consequently

J = M^{2} / (2 π T) = α^{'} M^{2}

, the classical leading open-string Regge trajectory.

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