Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 3 / d

Table of contents
- Solution d

Solution

Using the stated Christoffel symbols in the definition of the Riemann curvature tensor, the two nontrivial contractions are

R^{x}_{uxu} R^{y}_{u y u} = - \partial_{u} (\frac{f ^{'}}{f}) - (\frac{f ^{'}}{f})^{2} = - \frac{f ^{''}}{f}, = - \partial_{u} (\frac{g ^{'}}{g}) - (\frac{g ^{'}}{g})^{2} = - \frac{g ^{''}}{g} .

Hence

R_{uu} = R^{α}_{uαu} = - \frac{f ^{''}}{f} - \frac{g ^{''}}{g},

and the vacuum Einstein field equations reduce to

\frac{f ^{''}}{f} + \frac{g ^{''}}{g} = 0.

This is the diagonal Rosen coordinates for a plane gravitational wave equation.

For the plus-polarized wave, set

H (u) = H_{+} e^{- iω u} .

Comparison of the transverse metric components gives, to linear order,

f (u)^{2} = 1 + H (u), g (u)^{2} = 1 - H (u),

or

f (u) = 1 + \frac{1}{2} H (u) + O (H_{+}^{2}), g (u) = 1 - \frac{1}{2} H (u) + O (H_{+}^{2}) .

Therefore

\frac{f ^{''}}{f} + \frac{g ^{''}}{g} = \frac{1}{2} H^{''} - \frac{1}{2} H^{''} + O (H_{+}^{2}) = O (H_{+}^{2}) .

The Vacuum Einstein equations are consequently satisfied at linear order. A real gravitational wave is obtained by taking the real part of the complex plane-wave notation.

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