Gravitational Waves: Linearized Gravity and Wave Solutions
Ripples in spacetime propagating at the speed of light, predicted by Einstein in 1916 and directly detected a century later
Gravitational waves are ripples in the fabric of spacetime that propagate at the speed of light. They are produced by the acceleration of massive objects and carry energy and momentum away from the source. Predicted by Einstein in 1916 as a consequence of general relativity, they were first directly detected by LIGO in September 2015, opening an entirely new window on the universe.
Linearized Gravity
To derive gravitational waves, we consider small perturbations around flat Minkowski spacetime. We write the full metric as a background plus a small perturbation:
$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1$$
The metric perturbation hฮผฮฝ encodes the gravitational wave degrees of freedom
Here \(\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)\) is the Minkowski metric and\(h_{\mu\nu}\) is a symmetric tensor perturbation. Since the perturbation is small, we work to first order in \(h_{\mu\nu}\), discarding all terms quadratic or higher. At linear order, indices are raised and lowered with the background metric \(\eta_{\mu\nu}\).
The inverse metric to first order is:
$$g^{\mu\nu} = \eta^{\mu\nu} - h^{\mu\nu} + \mathcal{O}(h^2)$$
The linearized Christoffel symbols take the form:
$$\Gamma^{\alpha}{}_{\mu\nu} = \frac{1}{2}\eta^{\alpha\beta}\left(\partial_\mu h_{\beta\nu} + \partial_\nu h_{\beta\mu} - \partial_\beta h_{\mu\nu}\right)$$
Computing the linearized Riemann tensor and contracting, the linearized Einstein tensor becomes:
$$G^{(1)}_{\mu\nu} = \frac{1}{2}\left(\partial_\alpha\partial_\nu h^{\alpha}{}_\mu + \partial_\alpha\partial_\mu h^{\alpha}{}_\nu - \Box h_{\mu\nu} - \partial_\mu\partial_\nu h - \eta_{\mu\nu}\partial_\alpha\partial_\beta h^{\alpha\beta} + \eta_{\mu\nu}\Box h\right)$$
where \(h = h^{\alpha}{}_{\alpha}\) is the trace and \(\Box = -\partial_t^2 + \nabla^2\) is the d'Alembertian
Trace-Reversed Perturbation
The linearized Einstein tensor simplifies enormously if we introduce the trace-reversed perturbation:
$$\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$$
Note: \(\bar{h} = \bar{h}^{\alpha}{}_{\alpha} = h - 2h = -h\), hence the name "trace-reversed"
The inverse relation is \(h_{\mu\nu} = \bar{h}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\bar{h}\). In terms of the trace-reversed perturbation, the linearized Einstein equations become:
$$G^{(1)}_{\mu\nu} = -\frac{1}{2}\left(\Box \bar{h}_{\mu\nu} + \eta_{\mu\nu}\partial^\alpha\partial^\beta \bar{h}_{\alpha\beta} - \partial^\alpha\partial_\nu \bar{h}_{\mu\alpha} - \partial^\alpha\partial_\mu \bar{h}_{\nu\alpha}\right)$$
Gauge Freedom: The Lorenz Gauge
Under an infinitesimal coordinate transformation \(x^\mu \to x^\mu + \xi^\mu\), the metric perturbation transforms as:
$$h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu$$
This is the linearized diffeomorphism invariance, analogous to gauge transformations in electromagnetism
We exploit this freedom by choosing the Lorenz gauge (also called harmonic gauge or de Donder gauge):
$$\partial_\mu \bar{h}^{\mu\nu} = 0$$
Lorenz gauge condition: four equations that fix four of the ten components
One can always find a gauge transformation \(\xi^\mu\) satisfying\(\Box\xi^\nu = \partial_\mu\bar{h}^{\mu\nu}\) that brings us to Lorenz gauge. With this choice, the linearized Einstein equations simplify dramatically to:
$$\Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}$$
A wave equation sourced by the stress-energy tensor -- gravitational waves!
In vacuum (\(T_{\mu\nu} = 0\)), this reduces to the simple wave equation:
$$\Box \bar{h}_{\mu\nu} = 0$$
Transverse-Traceless (TT) Gauge
Even after imposing the Lorenz gauge, residual gauge freedom remains: we can still perform transformations with \(\Box\xi^\mu = 0\). This allows us to impose additional conditions. For a plane wave propagating in the z-direction, we can choose the transverse-traceless (TT) gauge:
$$h^{TT}_{0\mu} = 0 \quad \text{(temporal components vanish)}$$
$$h^{TT}{}^i{}_i = 0 \quad \text{(traceless)}$$
$$\partial_j h^{TT}_{ij} = 0 \quad \text{(transverse)}$$
For a wave propagating along the z-axis, the most general TT perturbation takes the form:
$$h^{TT}_{ij} = \begin{pmatrix} h_+ & h_\times & 0 \\ h_\times & -h_+ & 0 \\ 0 & 0 & 0 \end{pmatrix} \cos\left(\omega(t - z/c)\right)$$
Only two independent polarization states survive all gauge conditions
The Two Polarization States
Gravitational waves have exactly two physical polarization modes, corresponding to the two independent components of the TT perturbation:
Plus polarization (h+)
$$h^{TT}_{xx} = -h^{TT}_{yy} = h_+$$
Stretches spacetime along x while compressing along y, then vice versa. A ring of test particles oscillates as an ellipse aligned with the x and y axes.
Cross polarization (hร)
$$h^{TT}_{xy} = h^{TT}_{yx} = h_\times$$
Same pattern but rotated 45 degrees. The ellipse oscillates along the diagonal directions. The cross polarization is the plus polarization rotated by ฯ/4.
The two polarizations are related by a 45-degree rotation, in contrast to electromagnetic waves where the two polarizations differ by 90 degrees. This is because the graviton (the hypothetical quantum of the gravitational field) is a spin-2 particle, while the photon is spin-1.
$$h_{ij}^{TT}(t,z) = \left(A_+ e_{ij}^+ + A_\times e_{ij}^\times\right)\cos\bigl(\omega(t - z/c) + \phi\bigr)$$
General plane wave solution as a superposition of the two polarization tensors
Effect on Test Particles: The Geodesic Deviation
To understand what gravitational waves physically do, consider the geodesic deviation equation applied in the TT gauge. For two nearby freely falling particles separated by a displacement vector \(\xi^i\), the relative acceleration is:
$$\frac{d^2 \xi^i}{dt^2} = \frac{1}{2}\ddot{h}^{TT}_{ij}\,\xi^j$$
Tidal acceleration: the wave stretches and compresses space between particles
For a plus-polarized wave with amplitude h propagating in the z-direction, the fractional change in proper distance between two particles separated by L along the x-axis is:
$$\frac{\delta L}{L} = \frac{1}{2}h_+(t)$$
This is the strain -- the measurable quantity in gravitational wave detectors
This tidal effect is the physical observable that gravitational wave detectors measure. For a typical astrophysical source, \(h \sim 10^{-21}\), which for LIGO's 4 km arms corresponds to a displacement of \(\delta L \sim 10^{-18}\) meters -- one thousandth the diameter of a proton.
Interactive Simulation: Binary Inspiral Waveform
Run this Python code to simulate a gravitational wave signal from a binary black hole merger similar to GW150914, the first directly detected gravitational wave event. Try modifying the masses to see how the waveform changes!
GW150914-like Inspiral Waveform
PythonSimulate gravitational waves from binary black hole merger
Click Run to execute the Python code
Code will be executed with Python 3 on the server