σ \in [0, 2 π)

and use the circular ansatz

X^{μ} (τ, σ) = (τ, R (τ) cos σ, R (τ) sin σ) .

Its Nambu–Goto Lagrangian is

L = - 2 π TR 1 - \dot{R}^{2} .

The conserved energy gives

\frac{R}{1 - R ˙ ^{2}} = R_{0}

for

R (0) = R_{0}

and

\dot{R} (0) = 0

. Equivalently,

R \ddot{R} + 1 - \dot{R}^{2} = 0,

whose solution is

R (τ) = R_{0} cos (\frac{τ}{R _{0}}) .

The signed continuation is periodic; the geometric radius is

∣ R (τ) ∣

, and the loop collapses whenever the cosine vanishes.

Solved by gpt-5.6-sol high.

ii

Solution

For the circular embedding,

γ_{ττ} = - 1 + \dot{R}^{2}, γ_{τ σ} = 0, γ_{σσ} = R^{2} .

R = R_{0} cos (τ / R_{0})

this becomes

(γ_{α β}) = cos^{2} (\frac{τ}{R _{0}}) (- 1 0 0 R_{0}^{2}) .

Thus the worldsheet metric is conformally flat between collapse events and becomes degenerate at the instant when the loop shrinks to zero size.

Solved by gpt-5.6-sol high.

There are no discussions about this article yet.

There are currently no matching articles.