Math 2.0

Relativity and gravitational theory Gravitational waves Fact

Gravitational memory effect

Builds on Gravitational waveLinearized gravityAsymptotically flat spacetime

Statement

Work in units with G=c=1G = c = 1, in an asymptotically flat spacetime in the weak-field regime of linearized gravity. Let a burst of gravitational waves pass a pair of freely falling test masses at a large distance rr from the source. Let hjkTTh^{TT}_{jk} be the transverse-traceless metric perturbation at the masses. Then hjkTTh^{TT}_{jk} need not return to its initial value after the burst. There can be a permanent change ΔhjkTT=hjkTT(after)−hjkTT(before)≠0,\Delta h^{TT}_{jk} = h^{TT}_{jk}(\text{after}) - h^{TT}_{jk}(\text{before}) \neq 0, and the proper distance between the test masses changes permanently by an amount proportional to ΔhjkTT\Delta h^{TT}_{jk}. Suppose the source emits or absorbs unbound bodies of masses MAM_A and velocities 𝐯A\mathbf{v}_A. The linear part of the change is then ΔhjkTT=Δ∑A[4MAr1−∣𝐯A∣2 vAjvAk]TT.\Delta h^{TT}_{jk} = \Delta \sum_A \left[ \frac{4 M_A}{r \sqrt{1 - \abs{\mathbf{v}_A}^2}} \, v_A^j v_A^k \right]^{TT}. Here Δ\Delta is the difference between the final and the initial sums. The symbol [ ⋅ ]TT[\,\cdot\,]^{TT} is the transverse-traceless part with respect to the direction to the observer.

Remarks

The linear memory comes from a net change of the second time derivative of the moments of the source between early and late times. Zel'dovich and Polnarev found it in 1974. Christodoulou showed in 1991 that there is also a nonlinear memory, sourced by the energy carried off by the gravitational waves themselves. The nonlinear memory is present even for a bound system such as a binary that merges, because the radiated waves act as an unbound outgoing flux of energy. The memory is a zero-frequency effect: it grows during the burst and then stays. It is linked to the asymptotic symmetries of asymptotically flat spacetimes.

Going deeper

For a student

In linearized theory the far-field wave form is a time derivative of source moments. So a net change of those derivatives between early and late times leaves a step in hjkTTh^{TT}_{jk}. Such steps arise when masses fly apart, as in a hyperbolic encounter or an explosion. In full general relativity the radiated gravitational energy adds a further step, the nonlinear memory, which is of the same order for compact binary mergers. Detecting the memory in gravitational-wave data is an active goal of observation.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Asymptotically flat spacetime, concept, no evidence levelAsymptotically flat s…conceptGravitational wave, concept, no evidence levelGravitational waveconceptLinearized gravity, concept, no evidence levelLinearized gravityconceptGravitational memory effect, statement, no evidence levelGravitational memory …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Asymptotically flat spacetime concept
  2. Gravitational wave concept
  3. Linearized gravity concept
  4. Gravitational memory effect statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gravitational_memory_effect
About this page
TagZ8E24ZVQPS
URIm2://Z8E24ZVQPS
Kindstatement, fact
WritingW0 stub
MSC83C35, 83C30
StatusThis copy of the site has no build data. Run m2 status Z8E24ZVQPS.
AboutGravitational wave
UsesGravitational wave, Linearized gravity, Asymptotically flat spacetime
Same as
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.