General Relativity: An Introduction for Physicists · Problem 17.23

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · gravitoelectric-gravitomagnetic-fields-rotating-body

Problem

Use your answer to Exercise 17.19 to show that, in the linear theory, a spherically symmetric body of mass MM rotating steadily with angular momentum J⃗\vec{J} produces gravitoelectric and gravitomagnetic fields given respectively by Eg(x⃗)=−GM∣x⃗∣2x^andBg(x⃗)=2Gc2∣x⃗∣3[J⃗−3(J⃗⋅x^)x^],E_\mathrm{g}(\vec{x}) = -\frac{GM}{|\vec{x}|^2}\hat{x} \qquad \text{and} \qquad B_\mathrm{g}(\vec{x}) = \frac{2G}{c^2|\vec{x}|^3}\left[\vec{J} - 3\left(\vec{J}\cdot\hat{x}\right)\hat{x}\right], where x^\hat{x} is a unit vector in the x⃗\vec{x}-direction. Hint: For any scalar field ϕ\phi and spatial vector fields a⃗\vec{a} and b⃗\vec{b} one has ∇⃗(ϕa⃗)=∇⃗ϕ×a⃗+ϕ(∇⃗×a⃗)\vec{\nabla}(\phi\vec{a}) = \vec{\nabla}\phi\times\vec{a} + \phi(\vec{\nabla}\times\vec{a}) and ∇⃗×(a⃗×b⃗)=a⃗(∇⃗⋅b⃗)−b⃗(∇⃗⋅a⃗)+(b⃗⋅∇⃗)a⃗−(a⃗⋅∇⃗)b⃗\vec{\nabla}\times(\vec{a}\times\vec{b}) = \vec{a}(\vec{\nabla}\cdot\vec{b}) - \vec{b}(\vec{\nabla}\cdot\vec{a}) + (\vec{b}\cdot\vec{\nabla})\vec{a} - (\vec{a}\cdot\vec{\nabla})\vec{b}. Also, ∇⃗(1/∣x⃗∣3)=−3x⃗/∣x⃗∣5\vec{\nabla}(1/|\vec{x}|^3) = -3\vec{x}/|\vec{x}|^5.

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