Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 49 / 2

Use $G=c=1$, select the retarded source field with no added homogeneous radiation, and work to leading order in the weak-field approximation and long-wavelength source approximation. The retarded fundamental solution of the Linearized Einstein equations in Lorenz gauge in linearized gravity isFor $r\gg d$, the denominator is $r$ to leading order. The delay is $t-r+\mathbf{n}\cdot {\mathbf{x}}^{\prime }+O(d^2/r)$, where $\mathbf{n}=\mathbf{x}/r$. In the usual slowly evolving source regime, its characteristic time $\tau$ obeys $d/\tau \ll 1$, so the source-size part of the delay is negligible:The stress-energy conservation equation ${\partial }_{\mu }{T}^{\mu \nu }=0$ is required at the approximation order used; it follows from the divergence of the gauge-fixed field equation. Compact support removes the integration by parts surface terms. Since $T^{00}=T_{00}$ and $T^{ij}=T_{ij}$ in this signature,Combining these identities gives the retarded quadrupole fieldFor an ordinary nonrelativistic bound source, $\tau \sim d/v$ makes $d/\tau \sim v\ll 1$. More generally the size relative to the variation timescale must also be small; speed alone does not exclude a rapidly varying small-amplitude motion.

The radiation boundary condition for linearized gravity is essential for a statement about the full field. Without it, the displayed implication is false: take $T_{\mu \nu }=0$ and add ${\bar{h}}_{11}=A\cos (k(t-z))$, ${\bar{h}}_{22}=-{\bar{h}}_{11}$, all other components zero. This weak plane gravitational wave satisfies both the homogeneous wave equation and Lorenz gauge in linearized gravity, while every $I_{ij}$ is zero. The proved formula is for the retarded source contribution, with the standard slow-source qualification. Its transverse–traceless projection gives the physical radiative strain.

The separation of the stars is $2R$. Newton's law of universal gravitation and circular acceleration giveTo leading nonrelativistic order, the mass density is the sum of the two translated point-mass Dirac delta distributions. Thuswith all components involving $z$ zero. The trace is $2M{R}^2$, a constant. Define the trace-free mass quadrupole moment $Q_{ij}=I_{ij}-{\delta }_{ij}{I}_{kk}/3$; its third derivatives equal those of $I_{ij}$. If $K=8M{R}^2{\Omega }^3$, thenBoth off-diagonal entries count in the contraction. Therefore ${\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}_{ij}=2K^2=128M^2{R}^4{\Omega }^6$, independent of phase. The quadrupole formula gives the equal-mass circular-binary quadrupole luminosityRestoring units, ${\Omega }^2=GM/(4R^3)$ andHere $R$ is each star's radius about the centre of mass, not the separation. The rest-frame prescription supplies the leading mass density; a moving star does not still have zero momentum density and spatial stress. The calculation uses the assumed quadrupole formula and the Newtonian orbit rather than imposing those rest-frame zeros on the moving binary.

At fixed masses, the equal-mass circular-binary quadrupole luminosity grows as $R^{-5}$. The binary's Newtonian binding energy is $-G{M}^2/(4R)$, so reducing the separation increases both the binding and the radiated power. Equivalently, its luminosity scaling iswhich makes the importance of orbital compactness explicit.

Ordinary extended stars cannot remain separate at very small orbital radii: contact, mass transfer and tidal disruption intervene. A neutron star or black hole can remain a compact orbiting object down to separations of order a few gravitational radii, allowing high orbital speeds, rapidly changing mass quadrupole moments and strong gravitational waves. Thus compact, tightly bound binaries are especially efficient emitters. The Newtonian quadrupole formula explains the scaling; precision predictions near merger require relativistic dynamics, where that approximation itself ceases to be reliable.