Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 311 / 3 / a / ii

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- Solution ii

Solution

Perform the Legendre transform in mechanics, using

\dot{h}_{ij} = 2 N K_{ij} + D_{i} N_{j} + D_{j} N_{i}

. The trace contraction

π^{ij} K_{ij} = h (K_{ij} K^{ij} - K^{2})

gives

H = \int d^{3} x [N h (K_{ij} K^{ij} - K^{2} -^{(3)} R) + 2 π^{ij} D_{i} N_{j}] .

To avoid differentiating a tensor density as though it were an ordinary tensor, set

p^{ij} = π^{ij} / h

. Integration by parts with the Levi-Civita connection of

h

gives

\int d^{3} x h 2 p^{ij} D_{i} N_{j} = - \int d^{3} x h 2 N^{j} D_{i} p^{i}_{j}

after dropping the surface term. Also

K_{ij} K^{ij} - K^{2} = \frac{1}{h} (π^{ij} π_{ij} - \frac{1}{2} π^{2}), π_{ij} = h_{ik} h_{j l} π^{k l} .

Therefore the requested Hamiltonian is

H = \int d^{3} x h (N H + N^{i} H_{i}), H = \frac{π ^{ij} π _{ij} - π ^{2} /2}{h} -^{(3)} R, H_{i} = - 2 D_{j} (\frac{π ^{j} _{i}}{h}) .

Varying the lapse function and shift vector imposes the Hamiltonian constraint

H = 0

and momentum constraint

H_{i} = 0

. The total canonical generator also contains multipliers for the vanishing lapse and shift canonical momenta. With an asymptotic boundary, surface terms must be restored to define the Arnowitt-Deser-Misner energy.

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