Displacement Memory Effect

Consider a test particle at large $r$ in Bondi coordinates $(u, r, x^A)$. The geodesic equation $\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$ for a slowly-moving particle at large $r$ reduces to:

For the Bondi-Sachs metric, expanding the Christoffel symbol $\Gamma^i_{uu}$ at large $r$:

Now consider the geodesic deviation between two nearby test particles with separation vector $\xi^\mu$. The equation of geodesic deviation is $\frac{D^2\xi^\mu}{d\tau^2} = R^\mu{}_{\alpha\beta\gamma}\frac{dx^\alpha}{d\tau}\xi^\beta\frac{dx^\gamma}{d\tau}$. For our slowly-moving particles with $dx^\alpha/d\tau \approx (1, 0, 0, 0)$ in the TT frame:

The Riemann tensor component $R^i{}_{uju}$ in the linearized regime $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ is computed from:

Computing each Christoffel symbol in the TT gauge where $h_{0\mu} = 0$ and $h^i{}_i = 0$:

Therefore:

This is the tidal acceleration due to the gravitational wave, directly proportional to the second time derivative of the metric perturbation.

Starting from $\ddot{\xi}^i = \frac{1}{2}\ddot{h}_{ij}^{TT}\xi^j$, we integrate once from $u = -\infty$ to $u$. At early times $\dot{\xi}^i(-\infty) = 0$ and $\dot{h}_{ij}(-\infty) = 0$ (no radiation before the burst):

The second term is of order $\mathcal{O}(h^2)$ since $\xi^j - \xi^j_0 = \mathcal{O}(h)$. At linear order:

Integrating again from $-\infty$ to $+\infty$, using $\xi^i(-\infty) = \xi^i_0$:

where $\Delta C_{AB} = C_{AB}(+\infty) - C_{AB}(-\infty)$. If $\Delta C_{AB} \neq 0$, the test particles are permanently displaced even after all radiation has passed. This is the displacement memory effect.

The $uu$-component of the Einstein equation at order $r^{-1}$ gives the evolution of the Bondi mass aspect. Starting from the Einstein tensor:

Rearranging, this gives the celebrated Bondi mass-loss formula:

Integrating over $S^2$, the $D^AD^BN_{AB}$ term vanishes by integration by parts (it is a total divergence). This yields the Bondi mass loss:

The total Bondi mass can only decrease: gravitational waves always carry positive energy to $\mathscr{I}^+$. This is the gravitational analogue of the Larmor formula in electromagnetism.

The key insight of Christodoulou (1991) is that the energy flux of gravitational waves itself acts as a source for a DC shift in the shear. Decompose the Bondi mass-loss equation into a constraint relating $\Delta C_{AB}$ to the energy flux. Start by integrating the mass-loss formula over all retarded time:

The total change in shear consists of a linear part (sourced by matter or unbound masses) and a nonlinear part (sourced by the gravitational wave energy flux). Separating $\Delta C_{AB} = \Delta C_{AB}^{\rm lin} + \Delta C_{AB}^{\rm NL}$ and using the constraint equation, the nonlinear memory satisfies:

Writing the STF tensor in terms of its scalar potential: $C_{AB} = (D_A D_B - \frac{1}{2}\gamma_{AB}D^2)\Phi$, the constraint becomes an elliptic equation on $S^2$ for $\Delta\Phi^{\rm NL}$:

The operator $(D^2 + 2)D^2$ has eigenvalues $-\ell(\ell+1)[\ell(\ell+1)-2]$ on $S^2$. The $\ell = 0$ and $\ell = 1$ modes are in the kernel (corresponding to supertranslation ambiguity), and the equation is invertible for $\ell \geq 2$.

For a binary merger, the dominant contribution is $\ell = 2, m = 0$. Expand the right-hand side in spherical harmonics: $\int N_{AB}N^{AB}\,du = \sum_{\ell m} \mathcal{E}_{\ell m}\,Y_{\ell m}(\theta, \phi)$. For the $\ell = 2$ mode, the eigenvalue of $(D^2+2)D^2$ is $-6 \times 4 = -24$:

The memory shear in the $\ell = 2, m = 0$ sector is then:

For a compact binary with total mass $M$, symmetric mass ratio $\eta = m_1 m_2 / M^2$, and total radiated energy $E_{\rm rad}$, the memory strain observed at distance $R$ along the polar axis ($\theta = 0$) is dominated by the $\ell = 2, m = 0$ mode:

For a face-on equal-mass binary ($\eta = 1/4$) with $E_{\rm rad} \approx 0.05\,Mc^2$ (from numerical relativity), the peak memory strain is:

This is roughly 10-20 times smaller than the peak oscillatory strain $h_{\rm osc} \sim \eta M/R$, making it challenging but potentially detectable by stacking multiple events in LIGO/Virgo/KAGRA or by next-generation detectors like LISA and the Einstein Telescope.

We model the inspiral-merger-ringdown news tensor for several mass ratios, integrate to obtain the shear $C_{AB}(u)$ showing the permanent offset, and compute the Christodoulou memory strain: