Gravitational Waves & LIGO
Gravitational Waves & LIGO
Linearized GR predicts transverse, traceless gravitational waves propagating at \(c\). Binary mergers radiate GW energy, causing orbital inspiral. LIGO's 2015 detection of GW150914 opened GW astronomy, confirming black hole mergers and testing GR in the strong-field, highly dynamical regime.
Linearized Gravity & TT Gauge
Write \(g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\), \(|h|\ll 1\). In Lorenz gauge \(\partial^\mu \bar h_{\mu\nu}=0\) (where \(\bar h_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h\)), vacuum EFE become \(\Box\bar h_{\mu\nu} = 0\). Plane wave solutions carry two polarizations \(h_+\) and \(h_\times\), with TT gauge conditions \(h^{0\mu}=0\), \(h^i{}_i=0\), \(\partial^j h_{ij}=0\).
Quadrupole Formula
GW power radiated: \(P = -\frac{G}{5c^5}\langle\dddot{I}_{ij}\dddot{I}^{ij}\rangle\), where \(I_{ij} = \int \rho x_i x_j d^3x\) is the quadrupole moment tensor. For a circular binary, the inspiral timescale scales as \(\tau \propto a^4/(m_1 m_2 (m_1+m_2))\).
Example 1: GW Strain Estimate
Estimate strain \(h\) from GW150914 (\(M\sim 30M_\odot\) BH merger at \(d\sim 410\,\text{Mpc}\)).
Solution: Near merger \(r \sim r_s \approx 90\,\text{km}\): \(h\sim r_s^2/(rd) \sim 10^{-21}\). LIGO measured \(h\approx 10^{-21}\), implying arm-length changes \(\sim 10^{-18}\,\text{m}\).
Example 2: Chirp Mass
GW frequency evolves as \(\dot f = \frac{96}{5}\pi^{8/3}\left(\frac{G\mathcal{M}}{c^3}\right)^{5/3}f^{11/3}\). Define and extract the chirp mass.
Solution: \(\mathcal{M} = (m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}\). Measuring \(f\) and \(\dot f\) from the waveform directly yields \(\mathcal{M}\). For GW150914: \(\mathcal{M}\approx 28.3\,M_\odot\).
Practice
- Show \(h_+\) and \(h_\times\) produce perpendicular quadrupolar strain patterns.
- Derive the GW luminosity for a circular binary using the quadrupole formula.
- Estimate the merger timescale for a NS binary \(m_1=m_2=1.4M_\odot\), \(a_0=2R_\odot\).
- Why does GW150914 constrain the speed of gravity and Lorentz invariance?
Show Answer Key
1. In the transverse-traceless gauge, the metric perturbation has two independent polarizations: $h_+$ stretches along $x$ while compressing $y$ (and vice versa), $h_\times$ does the same rotated by 45°. Both produce quadrupolar tidal deformations. A ring of test masses oscillates as an ellipse with axes aligned with the polarization direction.
2. Quadrupole formula: $P_{GW} = \frac{G}{5c^5}\langle\dddot{Q}_{ij}\dddot{Q}^{ij}\rangle$ where $Q_{ij}$ is the mass quadrupole moment. For equal masses $m$ in circular orbit (separation $a$, $\omega = \sqrt{GM/a^3}$): $P = \frac{32G^4m^5}{5c^5a^5}$ (using total mass $M=2m$: $P = \frac{32G^4 M^5}{5c^5(4a)^5}$... more precisely $P = \frac{32}{5}\frac{G^4 m^2 M^3}{c^5 a^5}$ for reduced mass $\mu = m/2$ orbiting total $M=2m$).
3. Orbital decay: $\frac{da}{dt} = -\frac{64G^3\mu M^2}{5c^5 a^3}$. Integrating: $a^4(t) = a_0^4 - \frac{256G^3\mu M^2}{5c^5}t$. Merger at $a=0$: $t_{\text{merge}} = \frac{5c^5 a_0^4}{256G^3\mu M^2}$. For $m_1=m_2=1.4M_\odot$, $a_0=2R_\odot$: $\mu = 0.7M_\odot$, $M=2.8M_\odot$. $t \sim 3\times10^8$ years.
4. GW150914: the gravitational wave arrived ~1.7 ms before the gamma-ray burst associated with the event... actually GW150914 was a BBH merger, not a BNS. The GW signal propagated at the speed of gravity. LIGO's detection placed an upper bound $|c_g/c - 1| < 10^{-15}$ (from GW170817/GRB170817A, the neutron star merger). This constrains Lorentz-violating theories of gravity and massive graviton theories ($m_g < 7.7 \times 10^{-23}$ eV).