# Shockwave The shockwave solution in General Relativity is a non-linear solution that describes the gravitational backreaction of an infinitely boosted particle which travels along a null geodesic. Boosting a particle increases the backreaction, so naively taking the large-boost limit will not necessarily give a good geometry. But if one takes the limit by making the rest mass of the source scale inversely with the boost factor, the resulting geometry is geodesically complete. All geodesics are only shifted a finite distance away from the vacuum solution. The solution is still singular, but it is well-defined in a distributional sense. The black hole shockwave solution is a very useful object in understanding [[0008 Quantum chaos|chaos]] and [[0091 Boundary causality|causality]] in [[0001 AdS-CFT|holography]]. See below. There are also shockwaves on other backgrounds such as pure AdS. ## Refs - [[1971#Aichelburg, Sexl]] establishes that shock waves describe the gravitational field of a massless particle - [[1985#Dray, t'Hooft]]: vacuum Einstein equations - [[1994#Sfetsos]]: extension to non-vanishing matter fields and cosmological constant - AdS shockwaves - [[HottaTanaka1993]] - [[PodolskyGriffiths1998]] - [[1999#Horowitz, Itzhaki]] - 27-32 of [[2006#Cornalba, Costa, Penedones, Schiappa (a)]] - with higher derivatives (for shocks on the horizon) - [[2016#Roberts, Swingle]] - [[2018#Huang]] - [[2022#Dong, Wang, Weng, Wu]]: arbitrary higher derivative gravity - quantum extremal shock - [[2023#Parrikar, Singh]] ## Alternative formulations - effective action for shockwaves: [[2022#Verlinde, Zurek]] ## Applications - computing the boundary [[0482 Out-of-time-order correlator|OTOC]] - two point function in the presence of shock waves [[0129 Dual of shockwaves]] - in [[0248 Black hole microstates|BH microstates]]: [[ChakarabartyRawashTurton2021]][](https://arxiv.org/pdf/2112.08378.pdf) - [[0091 Boundary causality|boundary causality]]: [[2014#Camanho, Edelstein, Maldacena, Zhiboedov]] ## Relation to other concepts - relation to information exchange in dS: [[AalsmaColeMorvanVanDerSchaarShiu2021]][](https://arxiv.org/abs/2105.12737) - relation to [[0416 Modular Hamiltonian]]: [[2022#Verlinde, Zurek]] - realising wormhole in AdS with two uniform shock waves: [[Bzowski2020]] - use Noether charge method to understand shock waves for higher derivative gravity: [[LiuYoshida2021]] - quantum imprints (supertranslation shifts the vacuum state) - [[GrayKubiznakMayTimmermanTjoa2021]][](https://arxiv.org/pdf/2105.09337.pdf) - related to [[0287 Memory effect]] and supertranslations in [[0064 BMS group]] - quantum memory: [[Majhi2021]][](https://arxiv.org/abs/2108.01307) <!-- ## Shock wave equation in [[0006 Higher derivative gravity|higher derivative gravity]] from [[Huang2018]]: - $d s^{2}=2 A(U V) d U d V+\sum_{S} h^{(S)}(U V) \bar{g}_{i j}^{(S)}(x) d x_{(S)}^{i} d x_{(S)}^{j}$ - after adding a perturbation $T_{(\text{shock}) \hat{U} \hat{U}}=E e^{2 \pi t / \beta} a(\hat{x}) \delta(\hat{U})$, the spacetime time is still given as above but replace $V\rightarrow V+\alpha(x)$ for the patch of the spacetime to the future the shockwave - go to new coordinates $\hat{U}=U$, $\hat{V}=V+\Theta(U) \alpha(x)$ - $d s^{2}=2 \hat{A}(\hat{U}, \hat{V}) d \hat{U} d \hat{V}+\sum_{S} \hat{g}_{i j}^{(S)}(\hat{U}, \hat{V}, \hat{x}) d \hat{x}_{(S)}^{i} d \hat{x}_{(S)}^{j}-2 \hat{A} \hat{\alpha}(\hat{x}) \hat{\delta}(\hat{U}) d \hat{U}^{2}$ - the shock wave equation is then (dropping hats) ==$G_{U U}^{(1)}+2 G_{U V}^{(0)} \alpha(x) \delta(U)=E e^{2 \pi t / \beta} a(x) \delta(U)$== --> ## Formulas \[*see e.g. [[2016#Roberts, Swingle]]*\] Use perturbations: - $\delta T_{u u}=\frac{P}{\ell^{d+2}} e^{\frac{2 \pi}{\beta} t} \delta(u) \delta(x)-2 h(x, t) \delta(u) T_{u v}^{0}$ - and $\delta T_{u v}=-h(x, t) \delta(u) T_{v v}^{0}$ - n.b. $T^0$ can be made zero if unperturbed metric solved vacuum Einstein equation Write backreacted metric as $d s^{2}=2 A(u v) d u d v+B(u v) d x^{i} d x^{i}-2 A(u v) h(x, t) \delta(u) d u^{2},$then use Einstein equation on the perturbed metric $\delta(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R)=8 \pi G_{N}\delta T_{\mu \nu}.$ ![[A0001 Black tsunami.png]]