Will Gravitational Wave ripple permanently alter space-time?

Question

When Gravitational Waves alter space-time, would that change be permanent? Will the space-time returns to its initial state once the ripple has stopped?

Answer 1

There is something called gravitational memory from gravitational radiation. A gravitational wave may pass a region with some test masses. The ripple in space will cause their relative distances to change. However, there is nothing which says the test masses must return to their initial positions. This is the result of BMS translations [1] which illustrates some deep connections between quantum mechanics and spacetime.

It is illustrative for physical understanding to consider a linearized form of gravitational memory. In gravitational wave detection this is most likely to be the form gravitational waves and any memory effect will be observed. Gravitational memory from a physical perspective is the change in the spatial metric of a surface according to $$ \Delta h_{+.\times}~=~ \lim_{t\rightarrow \infty}h_{+,\times}(t)~-~\lim_{t\rightarrow -\infty}h_{+,\times}(t). $$ Here $+$ and $x$ refer to the two polarization directions of the gravity wave. See [2] for a more complete treatment. Let us suppose we treat the gravity wave as a linear form of diphoton or colorless state with two gluons, where each photon (or gluon) has a generic state, $$ |\Psi_{+,\times}\rangle~=~ |h_{+}(t)~+~ih_\times(t)\rangle~=~\sum_{l=2}^\infty\sum_{m=-l}^l Y_{lm}[(\theta,\phi)\left(|\uparrow\rangle_+~+~ |\downarrow \rangle_+\right)~-~i(\theta,\phi)\left(|\uparrow\rangle_\times~+~|\downarrow \rangle_\times\right)], $$ where the arrows indicate the polarization directions according to their respective axes. Bern and Dixon work on gravity-QCD duality this way. The matrix element $H_{+,\times}~=~ |\Psi_{+,\times}\rangle\langle \Psi_{+,\times}|$ describes the interaction of the gravity wave with a quantum particle. This expanded out is $$ H_{+,\times}~=~ \sum_{l,l'=2}^\infty\sum_{m=-l}^l \sum_{m'=-l'}^{l'} Y_{lm}(\theta,\phi)Y_{l'm'}(\theta,\phi)\left[\left(|\uparrow\uparrow\rangle_{+\times}~+~|\downarrow \downarrow \rangle_{+\times}\right)~+~i\left(|\uparrow\downarrow \rangle_{+\times}~-~ |\downarrow \uparrow\rangle_{+\times}\right)\right] $$ This tensor operation sets to zero terms like $| \rangle_{++}$ and $| \rangle_{\times\times}$ as unphysical states. This matrix contains a gravity wave term $|\uparrow\uparrow\rangle_{+\times}~+~|\downarrow \downarrow \rangle_{+\times}$ plus a scalar term $|\uparrow\downarrow \rangle_{+\times}~-~ |\downarrow \uparrow\rangle_{+\times}$ that again we set to zero.

One might object to this in that there is an operator for the gravity wave, but we do not have states. However, in the Choy-Jamilkowsky isomorphism \cite{key-5}\cite{key-6} there is a correspondence between states and operators (really operations) $|\Psi\rangle\langle \Psi'|~\rightarrow~|\Psi\rangle\otimes|\Psi'\rangle$ which for the Hilbert space is $H~\rightarrow~ H\otimes H$. Consequently the matrix pertains directly to states in this scheme. We then have the matrix $$ H_{+,\times}~=~ \sum_{l,l'=2}^\infty\sum_{m=-l}^l\sum_{m'=-l'}^{l'} Y_{lm}(\theta,\phi)Y_{l'm'}(\theta,\phi)\left(|\uparrow\uparrow\rangle_{+\times}~+~|\downarrow \downarrow \rangle_{+\times}\right), $$ which corresponds to a state vector $|\psi_{+,\times}\rangle$. This is a linear form of gravitational memory and its quantum mechanical analogue.

[1] A. Strominger and A. Zhiboedov, "Gravitational Memory, BMS Supertranslations and Soft Theorems," http://arxiv.org/abs/1411.5745

[2] M. Favata "The gravitational-wave memory effect," $\it Class.~Quant.~Grav.$ $\bf 27$, 084036, (2010) http://arxiv.org/abs/1003.3486