General Relativity: An Introduction for Physicists · Problem 17.20

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · kerr-metric-linearised-gravity-rotating-source

Problem

Use your answer to Exercise 17.19 to show that, in the linear theory, the line element outside a spherically symmetric matter distribution rotating about the zz-axis at a steady rate is given by ds2=c2(1−2GMc2r)dt2+4GJc2r3(x dy−y dx) dt−(1+2GMc2r)(dx2+dy2+dz2),ds^2 = c^2\left(1 - \frac{2GM}{c^2 r}\right)dt^2 + \frac{4GJ}{c^2 r^3}(x\,dy - y\,dx)\,dt - \left(1 + \frac{2GM}{c^2 r}\right)(dx^2 + dy^2 + dz^2), where r=∣x⃗∣r = |\vec{x}|. Show that this is equal to the Kerr line element to first order in MM and JJ. Hint: Ai dxi=−δijAj dxi=−A⃗⋅dx⃗A_i\,dx^i = -\delta_{ij}A^j\,dx^i = -\vec{A}\cdot d\vec{x} and (J⃗×x⃗)⋅dx⃗=J⃗⋅(x⃗×dx⃗)(\vec{J}\times\vec{x})\cdot d\vec{x} = \vec{J}\cdot(\vec{x}\times d\vec{x}).

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