### Gravitational Waves & Ringing Teacups

Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide. Warning sign, described by Thomas Moore as a “geeky insider GR joke” [image from Moore, 2013].

Energy and angular momentum are carried away by the gravitational waves. These waves are oscillations in the metric quantities that determine the structure of space and time. They are tiny perturbations of flat spacetime, often described as “ripples in the fabric of spacetime”. They are transverse polarized waves that travel at the speed of light, that carry energy and that can be detected by their action in causing relative motion between neighbouring test masses.

Gravitational Wave Equation

The structure of spacetime is determined by the metric tensor ${g_{\mu\nu}}$, which gives the “distance” between two events $\displaystyle d s^2 = g_{\mu\nu} d x^{\mu}d x^{\nu}$

In the case of flat spacetime, far from any massive bodies, the metric tensor is constant, ${g_{\mu\nu} = \eta_{\mu\nu} \equiv \mathrm{diag}(-1, 1, 1, 1)}$. When the departure from flat spacetime is small (the weak field limit) we write the metric tensor as $\displaystyle g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} \qquad\mbox{where}\qquad | h_{\mu\nu} | << 1 \,.$

Einstein’s field equations are ${G^{\mu\nu} = (8\pi G/c^2) T^{\mu\nu}}$, where ${T^{\mu\nu}}$ is the energy-momentum tensor, describing the distribution of energy/matter in spacetime. Using the weak field limit and gauge freedom, they can be reduced in vacuo to the linear wave equation $\displaystyle \partial^\alpha\partial_\alpha h^{\mu\nu} = \frac{1}{c^2}\frac{\partial^2 h^{\mu\nu}}{\partial t^2} - \nabla^2 h^{\mu\nu} = 0$

(strictly, we should write ${H^{\mu\nu}}$, the `trace-reversed’ metric, in place of ${h^{\mu\nu}}$). For a particle at rest, the geodesic equations reduce to ${d^2 x^{\alpha}/{d t^2} = 0}$, so the particle remains at rest in the chosen coordinate system as the wave passes.

Plus Waves and Cross Waves

We can study the behaviour of neighbouring particles in a circle of radius ${R}$ in the ${xy}$-plane. We expect wave-like solutions, so we consider a wave travelling in the ${z}$-direction, $\displaystyle h^{\mu\nu} = A^{\mu\nu} \cos k(z-ct) \,.$

We find that only four components of the amplitude are non-zero: $\displaystyle A^{\mu\nu} = A_{+} \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 &-1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}\right] + A_{\times} \left[\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}\right]$

Thus, the gravitational wave solutions are linear combinations of two distinct components, called “plus” waves and “cross” waves. The physical separation for the “plus” waves is $\displaystyle \Delta s = R \left[ 1+\frac{1}{2} A_{+} \cos 2\theta\cos \omega_{+} t \right]$

and for the “cross” waves it is $\displaystyle \Delta s = R \left[ 1+\frac{1}{2} A_{\times} \sin 2\theta\cos \omega_{\times} t \right]$

In Fig.~0, the distortion of a ring of particles perpendicular to the direction of propagation is shown over one wave cycle. The top row shows the “plus” mode and the bottom row shows the “cross” mode. Distortion of a ring of particles perpendicular to the direction of propagation, over one wave cycle. Top row: Plus mode. Bottom row: Cross mode.

Ringing Teacups

The transverse structure of GWs is similar to the pattern of oscillations that results from tapping a teacup. Tapping the rim over the handle, or at points displaced from it by 90, 180 or 270 degrees, we hear a clear note. The tap distorts the circular rim to an ellipse, and elastic forces cause it to oscillate, generating a musical tone. But tapping at intermediate points, displaced by an odd multiple of 45 degrees from the handle, we hear a slightly higher tone, perhaps a semi-tone higher.

In the first case, the handle is at a point of maximum distortion (an anti-node) and increases the inertia, resulting in lower frequency oscillations. In the second case, the handle is at a node: it does not move and has no effect on the inertia, so the frequency of the oscillations is higher. Tapping at an arbitrary point on the rim, we generate a combination of the two modes of oscillation, and the sound is somewhat discordant? Distortion of a teacup tapped at the top, bottom or sides (North, South, East and West points).

Instead of a teacup, we could consider a long cylinder like that found in tubular bells. A tap will distort the circular cross-section causing oscillations. These will propagate rapidly in both directions along the tube. The transverse structure of the oscillations is reminiscent of GW oscillations. Distortion of a coffee mug. Left: Tap at blue points and handle moves (lower pitch). Right: Tap at red points and handle doesn’t move (higher pitch). [Image from YouTube video by Tadashi Tokieda]

Confirmation

Astronomical sources generate GWs with amplitudes around ${10^{-20}}$m as they pass Earth. Thus, particles separated by 1000km oscillate with amplitudes of about ${10^{-14}}$m, comparable to a large atomic nucleus. The Laser Interferometer Gravitational wave Observatory, LIGO, was designed to detect gravitational waves directly. The detection of GWs in 2015 by LIGO was a triumph of modern technology and a dramatic confirmation of general relativity.

Sources ${\bullet}$ Cervantes-Cota, Jorge L., Salvador Galindo-Uribarri, George F.~Smoot, 2016: A Brief History of Gravitational Waves. PDF. ${\bullet}$ Moore, Thomas A, 2013: A General Relativity Workbook. University Science Books, 476pp. ISBN: 978-1-891-38982-5. [See especially Chapter 31]. ${\bullet}$ Hartle, James B, 2003: Gravity: an Introduction to Einstein’s General Relativity. Addison-Wesley, 582pp. ISBN: 978-0-805-386622. ${\bullet}$ YouTube: Tadashi’s Toys. Video.