Gravity Dual of Connes Cocycle Flow

Consider a quantum field theory on Minkowski space $\mathbb{R}^{d-1,1}$ in standard Cartesian coordinates $(t,x,y_{1},\ldots,y_{d-2})$. Consider a Cauchy surface $\mathcal{C}$ that is the disjoint union of the open regions $A_{0},A^{\prime}_{0}$ and their shared boundary $\partial A_{0}$. Let $\mathcal{A}_{0},\mathcal{A}^{\prime}_{0}$ denote the associated algebras of operators. Let $|\psi\rangle$ be a cyclic and separating state on $\mathcal{C}$, and denote by $|\Omega\rangle$ the global vacuum (the assumption of cyclic and separating could be relaxed for $|\psi\rangle$, at the cost of complicating the discussion below). The Tomita operator is defined by

$\displaystyle S_{\psi|\Omega;\mathcal{A}_{0}}\alpha|\psi\rangle=\alpha^{\dagger}|\Omega\rangle,\forall\alpha\in\mathcal{A}_{0}~{}.$

(1)

The relative modular operator is defined as

$\displaystyle\Delta_{\psi|\Omega}\equiv S^{\dagger}_{\psi|\Omega;\mathcal{A}_{0}}S_{\psi|\Omega;\mathcal{A}_{0}}~{},$

(2)

and the vacuum modular operator is

$$\Delta_{\Omega}\equiv\Delta_{\Omega|\Omega}~{}.$$

(3)

Note that we do not include the subscript $\mathcal{A}_{0}$ on $\Delta$; instead, for modular operators, we indicate whether they were constructed from $\mathcal{A}_{0}$ or $\mathcal{A}^{\prime}_{0}$ by writing $\Delta$ or $\Delta^{\prime}$.

Connes cocycle (CC) flow of $|\psi\rangle$ generates a one parameter family of states $|\psi_{s}\rangle$, $s\in\mathbb{R}$, defined by

$$|\psi_{s}\rangle=(\Delta^{\prime}_{\Omega})^{is}(\Delta^{\prime}_{\Omega|\psi})^{-is}|\psi\rangle~{}.$$

(4)

Thus far the definitions have been purely algebraic. In order to elucidate the intuition behind CC flow, let us write out the modular operators in terms of the left and right density operators, $\rho^{\psi}_{A_{0}}=\text{Tr}_{A^{\prime}_{0}}|\psi\rangle\langle\psi|$ and $\rho^{\psi}_{A^{\prime}_{0}}=\text{Tr}_{A_{0}}|\psi\rangle\langle\psi|$:¹¹1We follow the conventions in Witten:2018zxz where complement operators are written to the right of the tensor product.

$\displaystyle\Delta_{\psi|\Omega}$

$\displaystyle=\rho^{\Omega}_{A_{0}}\otimes(\rho^{\psi}_{A^{\prime}_{0}})^{-1}~{}.$

(5)

One finds that the CC operator acts only in $\mathcal{A}^{\prime}_{0}$:

$$(\Delta^{\prime}_{\Omega})^{is}(\Delta^{\prime}_{\Omega|\psi})^{-is}=(\rho^{\Omega}_{A^{\prime}_{0}})^{is}(\rho^{\psi}_{A^{\prime}_{0}})^{-is}\in\mathcal{A}^{\prime}_{0}~{}.$$

(6)

It follows that the reduced state on the right algebra satisfies

$$\rho^{\psi_{s}}_{A_{0}}=\rho^{\psi}_{A_{0}}~{}.$$

(7)

Therefore, expectation values of observables $\mathcal{O}\in\mathcal{A}_{0}$ remain invariant under CC flow. These heuristic arguments would be valid only for finite-dimensional Hilbert spaces Witten:2018zxz ; but Eq. (6) can be derived rigorously Ceyhan:2018zfg .

It can also be shown that $(\Delta^{\prime}_{\psi|\Omega})^{is}\Delta^{is}_{\Omega|\psi}=1$. Hence for operators $\mathcal{O}^{\prime}\in\mathcal{A}^{\prime}_{0}$, one finds that CC flow acts as $\Delta_{\Omega}^{is}$ inside of expectation values:

	$\displaystyle\langle\psi_{s}|\mathcal{O}^{\prime}|\psi_{s}\rangle$	$\displaystyle=\text{Tr}_{\mathcal{A}^{\prime}_{0}}\left[\rho^{\psi}_{A^{\prime}_{0}}(\Delta_{\psi|\Omega}^{-is}\Delta_{\Omega}^{is})\mathcal{O}^{\prime}(\Delta_{\Omega}^{-is}\Delta^{is}_{\psi|\Omega})\right]~{},$
		$\displaystyle=\text{Tr}_{\mathcal{A}^{\prime}_{0}}\left(\rho^{\psi}_{A^{\prime}_{0}}(\rho^{\Omega}_{A^{\prime}_{0}})^{-is}\mathcal{O}^{\prime}(\rho^{\Omega}_{A^{\prime}_{0}})^{is}\right)$		(8)
		$\displaystyle=\text{Tr}\left[|\psi\rangle\langle\psi|\Delta_{\Omega}^{is}({\mathbf{1}}\otimes\mathcal{O}^{\prime})\Delta_{\Omega}^{-is}\right]~{},$		(9)

where we have used the cyclicity of the trace.

To summarize, expectation values of one-sided operators transform as follows:

	$\displaystyle\langle\psi_{s}|\mathcal{O}|\psi_{s}\rangle$	$\displaystyle=\langle\psi|\mathcal{O}|\psi\rangle~{},$		(10)
	$\displaystyle\langle\psi_{s}|\mathcal{O}^{\prime}|\psi_{s}\rangle$	$\displaystyle=\langle\psi|\Delta^{is}_{\Omega}\mathcal{O^{\prime}}\Delta^{-is}_{\Omega}|\psi\rangle~{}.$		(11)

There is no simple description of two-sided correlators in $|\psi_{s}\rangle$ such as $\langle\psi_{s}|\mathcal{O}\mathcal{O^{\prime}}|\psi_{s}\rangle$; we discuss such objects in Sec. 6.4.

Let us now specialize to the case where $\partial A_{0}$ corresponds to a cut $v=V_{0}(y)$ of the Rindler horizon $u=0$. We have introduced null coordinates $u=t-x$ and $v=t+x$ and denoted the transverse coordinates collectively by $y$. It can be shown that the modular operator $\Delta^{is}_{\Omega}$ acts locally on each null generator $y$ of $u=0$ as a boost about the cut $V_{0}(y)$ Casini:2017roe . More explicitly, one can define the full vacuum modular Hamiltonian $\widehat{K}_{V_{0}}$ by

$\displaystyle\widehat{K}_{V_{0}}=-\log\Delta_{\Omega;\mathcal{A}_{V_{0}}}~{}.$

(12)

We can write the full modular Hamiltonian as

$\displaystyle\widehat{K}_{V_{0}}=K_{V_{0}}\otimes\boldsymbol{1}^{\prime}-\boldsymbol{1}\otimes K^{\prime}_{V_{0}}~{}.$

(13)

Let $\Delta$ denote vacuum subtraction, $\Delta\langle\mathcal{O}\rangle=\langle\mathcal{O}\rangle_{\psi}-\langle\mathcal{O}\rangle_{\Omega}$. Then, for arbitrary cuts of the Rindler horizon, we have Casini:2017roe

$\displaystyle\Delta\langle K^{\prime}_{V_{0}}\rangle=-2\pi\int dy\int_{-\infty}^{V_{0}}dv[v-V_{0}(y)]\langle T_{vv}\rangle_{\psi}~{},$

(14)

and similarly for $K_{V_{0}}$. Thus $K^{\prime}_{V_{0}}$ is simply the boost generator about the cut $V_{0}(y)$ in the left Rindler wedge. That is, it generates a $y$-dependent dilation,

$$v\to V_{0}(y)+[v-V_{0}(y)]e^{2\pi s}~{}.$$

(15)

This allows us to evaluate Eq. (11) explicitly for local operators at $u=0$. For example, the CC flow of the stress tensor is

$\displaystyle\langle\psi_{s}|T_{vv}|\psi_{s}\rangle\lvert_{v<V_{0}}=e^{-4\pi s}\langle\psi|T_{vv}\left(V_{0}+e^{-2\pi s}(v-V_{0})\right)|\psi\rangle\lvert_{v<V_{0}}~{},$

(16)

and similarly for the other components of $T_{\mu\nu}$. There is a slight caveat here since $\Delta^{is}_{\Omega}$ only acts as a boost strictly at $u=0$. This would be sufficient for free theories, where $T_{vv}$ can be defined through null quantization on the Rindler horizon with a smearing that only needs support on $u=0$ Wall:2011hj . More generally, $T_{\mu\nu}$ must be smeared in an open neighborhood of $u=0$. However, if $V_{0}(y)$ is a perturbation of a flat cut then one can show that inside correlation functions $\Delta^{is}_{\Omega}$ approximately acts as a boost with subleading errors that vanish as $u\rightarrow 0$, to all orders in the perturbation Balakrishnan:2017bjg ; Balakrishnan:2020lbp . In the non-perturbative case, evidence comes from the fact that classically the vector field on the Rindler horizon which generates boosts about $V_{0}(y)$ can be extended to an approximate Killing vector field in a neighborhood of the horizon Koga:2001vq ; Ashtekar:2001jb . Therefore we expect Eq. (16) to hold on the null surface even after smearing.

Now consider a second cut $V(y)$ of the Rindler horizon which lies entirely below $V_{0}(y)$, so $V<V_{0}$ for all $y$. The cut defines a surface $\partial A_{V}$ that splits a Cauchy surface $\mathcal{C}_{V}=A^{\prime}_{V}\cup\partial A_{V}\cup A_{V}$; we take $A^{\prime}_{V}$ to be the “left” side ($v<V$), with operator algebra $\mathcal{A}^{\prime}_{V}$. The Araki definition of relative entropy is Witten:2018zxz

$\displaystyle S^{\prime}_{\text{rel}}(\psi|\Omega;V)=-\langle\psi|\log\Delta_{\psi|\Omega;\mathcal{A}^{\prime}_{V}}|\psi\rangle~{}.$

(17)

It has the following transformation properties Ceyhan:2018zfg :

	$\displaystyle S_{\text{rel}}(\psi_{s}|\Omega;V)$	$\displaystyle=S_{\text{rel}}(\psi|\Omega;V_{0}+e^{-2\pi s}(V-V_{0}))~{},$		(18)
	$\displaystyle\frac{\delta S_{\text{rel}}(\psi_{s}|\Omega;V)}{\delta V}$	$\displaystyle=e^{-2\pi s}\frac{\delta S_{\text{rel}}(\psi|\Omega;V_{0}+e^{-2\pi s}(V-V_{0}))}{\delta V}~{}.$		(19)

Moreover, the “left” von Neumann entropy is defined as

$\displaystyle S^{\prime}(\psi,V)=-\text{tr}_{A^{\prime}_{V}}{\rho^{\psi}_{A^{\prime}_{V}}\log\rho^{\psi}_{A^{\prime}_{V}}}~{}.$

(20)

With these definitions in hand, one can decompose the relative entropy as

$\displaystyle S^{\prime}_{\text{rel}}(\psi|\Omega;V)=\Delta\langle K^{\prime}_{V}\rangle-\Delta S^{\prime}(V)~{}.$

(21)

At this point we drop the explicit vacuum subtractions, as we will only be interested in shape derivatives of the vacuum subtracted quantities, which automatically annihilate the vacuum expectation values. In particular, one can directly compute shape derivatives of $K^{\prime}_{V}$:

$\displaystyle\frac{\delta\langle K^{\prime}_{V}\rangle_{\psi}}{\delta V}\Big{\lvert}_{V_{0}}=2\pi\int_{-\infty}^{V_{0}}dv\ \langle T_{vv}\rangle_{\psi}~{}.$

(22)

Hence the transformations of both $K^{\prime}_{V}$ and its derivative simply follow from Eq. (16).

Combining Eq. (18) and Eq. (14), as well as Eq. (19) and Eq. (22), we see that $S^{\prime}(\psi,V)$ and its derivative transform as

	$\displaystyle S^{\prime}(\psi_{s},V)$	$\displaystyle=S^{\prime}(\psi,V_{0}+e^{-2\pi s}(V-V_{0}))~{},$		(23)
	$\displaystyle\frac{\delta S^{\prime}}{\delta V}\Big{\lvert}_{\psi_{s},V}$	$\displaystyle=e^{-2\pi s}\frac{\delta S^{\prime}}{\delta V}\Big{\lvert}_{\psi,V_{0}+e^{-2\pi s}(V-V_{0})}~{}.$		(24)

The respective properties of the complement entropy follow from purity.

CC flow generates a stress tensor shock at the cut $V_{0}$, proportional to the jump in the variation of the one-sided von Neumann entropy under deformations, at the cut Bousso:2019dxk . To see this, let us start with the sum rule derived in Ceyhan:2018zfg for null variations of relative entropy:²²2For type I algebras, one can derive the analogous sum rule from simpler arguments Bousso:2014uxa .

$\displaystyle 2\pi(P_{s}-e^{-2\pi s}P_{0})=(e^{-2\pi s}-1)\frac{\delta S^{\prime}_{\text{rel}}(\psi|\Omega;V)}{\delta V}\Big{\lvert}_{V_{0}}~{},$

(25)

where

$\displaystyle P\equiv\int_{-\infty}^{\infty}dv\ T_{vv}$

(26)

is the averaged null energy operator at $u=0$, and $P_{s}\equiv\langle\psi_{s}|P|\psi_{s}\rangle$, so in particular $P_{0}\equiv\langle\psi|P|\psi\rangle$. (There is one such operator for every generator, i.e., for every $y$.)

Inserting Eq. (21) and Eq. (22) into Eq. (25), and making use of Eq. (16), we see that there must exist a shock at $v=V_{0}(y)$:

$\displaystyle\langle\psi_{s}|T_{vv}|\psi_{s}\rangle=(1-e^{-2\pi s})\frac{1}{2\pi}\frac{\delta S^{\prime}}{\delta V}\Big{\lvert}_{V_{0}}\delta(v-V_{0})+o(\delta)~{}.$

(27)

Here $o(\delta)$ designates the finite (non-distributional) terms. These are determined by Eq. (16), and by its trivial counterpart in the $v>V_{0}$ region.

This $s$-dependent shock is a detailed characteristic of the CC flowed state. As such, reproducing it through the holographic dictionary will be the key test of our proposal of the bulk dual of CC flow (see Sec. 4).

For the remainder of the paper we further specialize to flat cuts of the Rindler horizon, so that $\partial A_{0}$ corresponds to $u=v=0$. We therefore set $V_{0}=0$ in what follows. We take $\mathcal{C}$ to be the Cauchy surface $t=0$, so that $A_{0}$ ($t=0$, $x>0$) and $A_{0}^{\prime}$ ($t=0$, $x<0$) are partial Cauchy surfaces for the right and left Rindler wedges.

In this case $\Delta^{is}_{\Omega}$ is a global boost by rapidity $s$ about $\partial A_{0}$ Bisognano:1976za . Thus, it has a simple geometric action not only on the null plane $u=0$, but everywhere. CC flow transforms observables in $\mathcal{A}^{\prime}_{0}$ by $\Delta^{is}_{\Omega}$ and leaves invariant those in $\mathcal{A}_{0}$. For a flat cut, this action can be represented as a geometric boost in the entire left Rindler wedge. This allows us to characterize the CC flowed state $|\psi(s)\rangle$ on $\mathcal{C}$ very simply in terms of a different state on a different Cauchy surface which we call the “precursor slice”. This description will motivate the formulation of our bulk construction in Sec. 3.1.

By Eq. (11), the CC flowed state on the slice $\mathcal{C}$,

$\displaystyle\ket{\psi_{s}(\mathcal{C})}=(\Delta^{\prime}_{\Omega})^{is}(\Delta^{\prime}_{\Omega|\psi})^{-is}\ket{\psi(\mathcal{C})}~{},$

(28)

satisfies

	$\displaystyle\bra{\psi_{s}(\mathcal{C})}\mathcal{O}_{A}\ket{\psi_{s}(\mathcal{C})}=\bra{\psi(\mathcal{C})}\mathcal{O}_{A}\ket{\psi(\mathcal{C})}~{},$		(29)
	$\displaystyle\bra{\psi_{s}(\mathcal{C})}\Delta_{\Omega}^{-is}\mathcal{O}_{A^{\prime}}\Delta_{\Omega}^{is}\ket{\psi_{s}(\mathcal{C})}=\bra{\psi(\mathcal{C})}\mathcal{O}_{A^{\prime}}\ket{\psi(\mathcal{C})}~{},$		(30)

where $\mathcal{O}_{A}$ and $\mathcal{O}_{A^{\prime}}$ denote an arbitrary collection of local operators that act on spacelike half-slices $A$ and $A^{\prime}$ of $\mathcal{C}$ respectively.³³3More precisely, one would have to smear the operator in a codimension 0 neighborhood of points on the slices. In the second equality above, we used the fact that $\Delta_{\Omega}^{is}$ acts as a global boost to move it to the other side of the equality, compared to Eq. (11).

We work in the Schrödinger picture where the argument $\mathcal{C}$ should be interpreted as the time variable. The fact that $\Delta_{\Omega}^{is}$ acts as a boost around $\partial A_{0}$ motivates us to consider the time slice

$$\mathcal{C}_{s}=A^{\prime}_{s}\cup\partial A_{0}\cup A_{0}~{},$$

(31)

where

$$A^{\prime}_{s}=\{t=(\tanh 2\pi s)x,~{}x<0\}~{}.$$

(32)

By Eqs. (29) and (30), each side of the CC-flowed state $\ket{\psi_{s}(\mathcal{C}_{s})}$ is simply related to the left and right restrictions of the original state on the original slice:

	$\displaystyle\bra{\psi_{s}(\mathcal{C}_{s})}\mathcal{O}_{A}\ket{\psi_{s}(\mathcal{C}_{s})}=\bra{\psi(\mathcal{C})}\mathcal{O}_{A}\ket{\psi(\mathcal{C})}~{},$		(33)
	$\displaystyle\bra{\psi_{s}(\mathcal{C}_{s})}\mathcal{O}_{A^{\prime}_{s}}\ket{\psi_{s}(\mathcal{C}_{s})}=\bra{\psi(\mathcal{C})}\mathcal{O}_{A^{\prime}}\ket{\psi(\mathcal{C})}~{}.$		(34)

In the second equation, $\mathcal{O}_{A^{\prime}_{s}}$ denotes local operators on $A^{\prime}_{s}$ which are analogous to $\mathcal{O}_{A^{\prime}}$ on $A^{\prime}$. More precisely, because the intrinsic metric of $A^{\prime}$ and $A^{\prime}_{s}$ are the same, there exists a natural map between local operators on $A^{\prime}$ and $A^{\prime}_{s}$.

In words, Eqs. (33) and (34) say that correlation functions in each half of $\mathcal{C}$ in the state $\ket{\psi(\mathcal{C})}$ are equal to the analogous correlation functions on each half of $\mathcal{C}_{s}$ in the state $\ket{\psi_{s}(\mathcal{C}_{s})}$. This justifies calling $\mathcal{C}_{s}$ the precursor slice since the CC flowed state on $\mathcal{C}$ arises from it by time evolution.

We find it instructive to repeat this point in the less rigorous language of density operators. In the density operator form of CC flow,

$\displaystyle\ket{\psi_{s}(\mathcal{C})}=(\rho^{\Omega}_{A^{\prime}_{0}})^{is}(\rho^{\psi}_{A^{\prime}_{0}})^{-is}\ket{\psi(\mathcal{C})}~{},$

(35)

it is evident that the action of $(\rho^{\Omega}_{A^{\prime}_{0}})^{is}$ can be absorbed into a change of time slice $\mathcal{C}\to\mathcal{C}_{s}$:

$\displaystyle\ket{\psi_{s}(\mathcal{C}_{s})}=(\rho^{\psi}_{A^{\prime}_{0}})^{-is}\ket{\psi(\mathcal{C})}~{}.$

(36)

Tracing out each side of $\partial A_{0}$ implies

	$\displaystyle\rho^{\psi_{s}}_{A_{0}}=\rho^{\psi}_{A_{0}}~{},$		(37)
	$\displaystyle\rho^{\psi_{s}}_{A^{\prime}_{s}}=\rho^{\psi}_{A^{\prime}_{0}}~{}.$		(38)

The first equality is trivial and was already discussed in Eq. (7). The second equality follows because $(\rho^{\psi}_{A^{\prime}_{0}})^{is}$ commutes with $(\rho^{\psi}_{A^{\prime}_{0}})$. This is the density operator version of Eqs. (33) and (34).

Eq. (36) should be contrasted with the one-sided modular-flowed state $\ket{\phi(\mathcal{C})}=(\rho^{\psi}_{A^{\prime}_{0}})^{-is}\ket{\psi(\mathcal{C})}$. The latter state would live on the original slice $\mathcal{C}$, but it is not well-defined since it would have infinite energy at the entangling surface.

It will be useful to define new coordinates adapted to the precursor slice $\mathcal{C}_{s}$. Let

	$\displaystyle\tilde{v}=v\,\Theta(v)+e^{-2\pi s}\,v\,(1-\Theta(v))~{},$		(39)
	$\displaystyle\tilde{u}=e^{2\pi s}u\,\Theta(u)+\,u\,(1-\Theta(u))~{},$		(40)

where $\Theta(.)$ is the Heaviside step function. Let $\tilde{t}=\frac{1}{2}(\tilde{v}+\tilde{u})$ and $\tilde{x}=\frac{1}{2}(\tilde{v}-\tilde{u})$. In these coordinates, the Minkowski metric takes the form

	$\displaystyle ds^{2}$	$\displaystyle=\left[\Theta(\tilde{t}+\tilde{x})+e^{2\pi s}(1-\Theta(\tilde{t}+\tilde{x}))\right]\left[e^{-2\pi s}\Theta(\tilde{t}-\tilde{x})+(1-\Theta(\tilde{t}-\tilde{x}))\right](-d\tilde{t}^{2}+d\tilde{x}^{2})$
		$\displaystyle+d^{d-2}y~{},$		(41)

and the precursor slice corresponds to $\tilde{t}=0$.

In these “tilde” coordinates, the stress tensor shock of Eq. (27) takes the form⁴⁴4We remind the reader that $o(\delta)$ refers to any finite (non-distributional) terms.

$\displaystyle\langle\psi_{s}|T_{\tilde{v}\tilde{v}}|\psi_{s}\rangle=\frac{1}{2\pi}\left(\frac{\partial v}{\partial\tilde{v}}\right)^{2}(1-e^{-2\pi s})\frac{\delta S}{\delta V}\Big{\lvert}_{V=0}\delta(v)+o(\delta)~{}.$

(42)

Recall that the entropy variation is evaluated in the state $|\psi\rangle$. By Eq. (24),

$\displaystyle\frac{\delta S}{\delta V}\Big{\lvert}_{\psi}=\frac{\delta S}{\delta\widetilde{V}}\Big{\lvert}_{\psi_{s}}~{},$

(43)

where $\widetilde{V}(y)$ is a cut of the Rindler horizon in the $\tilde{v}$ coordinates. Thus we may instead evaluate the entropy variation in the state $|\psi_{s}\rangle$ on the precursor slice. This will be convenient when matching the bulk and boundary.

The Jacobian in Eq. (42) has a step function in it, as will the Jacobian coming from $\delta(v)$. A step function multiplying a delta function is well-defined if one averages the left and right derivatives:

$\displaystyle\left(\frac{\partial v}{\partial\tilde{v}}\right)^{2}\delta(v)=\frac{1}{2}\left(\frac{\partial v}{\partial\tilde{v}}\Big{\lvert}_{0^{-}}+\frac{\partial v}{\partial\tilde{v}}\Big{\lvert}_{0^{+}}\right)\delta(\tilde{v})~{}.$

(44)

Thus Eq. (42) becomes

$\displaystyle\langle\psi_{s}|T_{\tilde{v}\tilde{v}}(\tilde{v})|\psi_{s}\rangle=\frac{1}{2\pi}\sinh(2\pi s)\frac{\delta S}{\delta\widetilde{V}}\Big{\lvert}_{\psi_{s},\widetilde{V}=0}\delta(\tilde{v})+o(\delta).$

(45)

Since we are dealing with a flat cut, the symmetry $s\leftrightarrow-s,v\leftrightarrow u$ implies that CC flow also generates a $T_{uu}$ shock in the state $|\psi_{s}\rangle$ at $u=v=0$:

$\displaystyle\langle\psi_{s}|T_{\tilde{u}\tilde{u}}|\psi_{s}\rangle=\frac{1}{2\pi}\sinh(2\pi s)\frac{\delta S}{\delta\widetilde{U}}\Big{\lvert}_{\psi_{s},\widetilde{V}=0}\delta(\tilde{u})+o(\delta)~{}.$

(46)

(Note that $\delta/\delta V$ goes to $-\delta/\delta U$.) The linear combination

$\displaystyle\langle\psi_{s}|T_{\tilde{t}\tilde{x}}(\tilde{t},\tilde{x})|\psi_{s}\rangle$

$\displaystyle=\frac{1}{2\pi}\sinh(2\pi s)\frac{\delta S}{\delta\widetilde{X}}\Big{\lvert}_{\psi_{s},\widetilde{X}=0}\delta(\tilde{x})+\langle\psi|T_{tx}(t=\tilde{t},x=\tilde{x})|\psi\rangle~{}.$

(47)

will be useful in Sec. 4. The last term was obtained from Eqs. (37) and (38); it makes the finite piece explicit. Note that these equations are valid in the entire left and right wedges, not just on $\mathcal{C}_{s}$.

Consider a $d+1$ dimensional spacetime $\mathcal{M}$ with metric $g_{\mu\nu}$ satisfying the Einstein field equations. (We will discuss higher curvature gravity in Sec. 6.5.) Let $\Sigma$ be a Cauchy surface of $\mathcal{M}$ that contains an extremal surface $\mathcal{R}$ of codimension 1 in $\Sigma$. (That is, the expansion of both sets of null geodesics orthogonal to $\mathcal{R}$ vanishes.)

Initial data on $\Sigma$ consist of wald2010general the intrinsic metric $(h_{\Sigma})_{ab}$ and the extrinsic curvature,

$\displaystyle(K_{\Sigma})_{ab}=P^{\mu}_{a}P^{\nu}_{b}\nabla_{(\mu}t_{\nu)}~{}.$

(48)

Here $P^{\mu}_{a}$ is the projector from $\mathcal{M}$ onto $\Sigma$, and $t^{\mu}$ is the unit norm timelike vector field orthogonal to $\Sigma$. Indices $a,b,\ldots$ are reserved for directions tangent to $\Sigma$. For matter fields, initial data consist of the fields and normal derivatives, for example $\phi(w^{a})$ and $[t^{\mu}\nabla_{\mu}\phi](w^{a})$, where $\phi$ is a scalar field and $w^{a}$ are coordinates on $\Sigma$.

By the Einstein equations, the initial data on $\Sigma$ must satisfy the following constraints:

		$\displaystyle r_{\Sigma}+K_{\Sigma}^{2}-(K_{\Sigma})_{ab}(K_{\Sigma})^{ab}=16\pi G\,T_{\mu\nu}t^{\mu}t^{\nu}~{},$		(49)
		$\displaystyle D^{a}(K_{\Sigma})_{ab}-D_{b}K_{\Sigma}=8\pi G\,T_{b\nu}t^{\nu}~{},$		(50)

where $D_{a}=P^{\mu}_{a}\nabla_{\mu}$ is the covariant derivative that $\Sigma$ inherits from $(\mathcal{M},g_{\mu\nu})$; $r_{\Sigma}$ is the Ricci scalar intrinsic to $\Sigma$; and $K_{\Sigma}$ is the trace of the extrinsic curvature: $K_{\Sigma}=(h_{\Sigma})^{ab}(K_{\Sigma})_{ab}$.

Let $\Sigma$ be a Cauchy slice of $\mathcal{M}$ containing $\mathcal{R}$ and smooth in a neighborhood of $\mathcal{R}$. The kink transform is then a map of the initial data on $\Sigma$ to a new initial data set, parametrized by a real number $s$ analogous to boost rapidity. The transform acts as the identity on all data except for the extrinsic curvature, which is modified only at the location of the extremal surface $\mathcal{R}$, as follows:

$\displaystyle(K_{\Sigma})_{ab}\to(K_{\Sigma_{s}})_{ab}=(K_{\Sigma})_{ab}-\sinh{(2\pi s)}~{}x_{a}x_{b}~{}\delta(\mathcal{R})~{}.$

(51)

Here $x^{a}$ is a unit norm vector field orthogonal to $\mathcal{R}$ and tangent to $\Sigma$, and we define

$\displaystyle\delta(\mathcal{R})\equiv\delta(x)~{},$

(52)

where $x$ is the Gaussian normal coordinate to $\mathcal{R}$ in $\Sigma$ ($\partial_{x}=x^{a}$). Thus, the only change in the initial data is in the component of the extrinsic curvature normal to $\mathcal{R}$. An equivalent transformation exists for initial choices of $\Sigma$ that are not smooth around $\mathcal{R}$ though the transformation rule will be more complicated than Eq. (51). We will discuss this later in the section.

Let $\Sigma_{s}$ be a time slice with this new initial data, as in Fig. 1, and let $\mathcal{M}_{s}$ be the Cauchy development of $\Sigma_{s}$. That is, $\mathcal{M}_{s}$ is the new spacetime resulting from the evolution of the kink-transformed initial data. Since the intrinsic metric of $\Sigma_{s}$ and $\Sigma$ are the same, they can be identified as $d$-manifolds with metric; the subscript $s$ merely reminds us of the different extrinsic data they carry. In particular the surface $\mathcal{R}$ can be so identified; thus $\mathcal{R}_{s}$ has the same intrinsic metric as $\mathcal{R}$. It also trivially has identical extrinsic data with respect to $\Sigma_{s}$. In fact, we will find below that like $\mathcal{R}$ in $\mathcal{M}$, $\mathcal{R}_{s}$ is an extremal surface in $\mathcal{M}_{s}$. However, the trace-free part of the extrinsic curvature of $\mathcal{R}_{s}$ in $\mathcal{M}_{s}$ may have discontinuities.

We will now show that the constraint equations hold on $\Sigma_{s}$; that is, the kink transform generates valid initial data. This need only be verified at $\mathcal{R}$ since the transform acts as the identity elsewhere. Here we will make essential use of the extremality of $\mathcal{R}$ in $\mathcal{M}$, which we express as follows.

The extrinsic curvature of $\mathcal{R}$ with respect to $\mathcal{M}$ has two independent components. Often these are chosen to be the two orthogonal null directions, but we find it useful to consider

	$\displaystyle(B_{\mathcal{R}}^{(t)})_{ij}=P^{\mu}_{i}P^{\nu}_{j}\nabla_{(\mu}t_{\nu)}~{},$		(53)
	$\displaystyle(B_{\mathcal{R}}^{(x)})_{ij}=P^{\mu}_{i}P^{\nu}_{j}\nabla_{(\mu}x_{\nu)}~{}.$		(54)

Here $i,j$ represent directions tangent to $\mathcal{R}$, and $P^{\mu}_{i}$ is the projector from $\mathcal{M}$ to $\mathcal{R}$. Extremality of $\mathcal{R}$ in $\mathcal{M}$ is the statement that the trace of each extrinsic curvature component vanishes:

	$\displaystyle B_{\mathcal{R}}^{(t)}=(\gamma_{\mathcal{R}})^{ij}(B_{\mathcal{R}}^{(t)})_{ij}=0~{},$		(55)
	$\displaystyle B_{\mathcal{R}}^{(x)}=(\gamma_{\mathcal{R}})^{ij}(B_{\mathcal{R}}^{(x)})_{ij}=0~{},$		(56)

where $(\gamma_{\mathcal{R}})_{ij}=P^{a}_{i}P^{b}_{j}(h_{\Sigma})_{ab}$ is the intrinsic metric on $\mathcal{R}$.

Orthogonality of $t^{\mu}$ and $x^{\mu}$ implies that $P^{\mu}_{i}=P^{a}_{i}P^{\mu}_{a}$, and hence

$$(B_{\mathcal{R}}^{(t)})_{ij}=P^{a}_{i}P^{b}_{j}(K_{\Sigma})_{ab}~{}.$$

(57)

Since $x^{a}$ is the unit norm orthogonal vector field at $\mathcal{R}$, the trace of $(K_{\Sigma})_{ab}$ at $\mathcal{R}$ can be written as:

$\displaystyle\left.K_{\Sigma}\right|_{\mathcal{R}}=x^{a}x^{b}(K_{\Sigma})_{ab}+(\gamma_{\mathcal{R}})^{ij}(B_{\mathcal{R}}^{(t)})_{ij}=x^{a}x^{b}(K_{\Sigma})_{ab}~{}.$

(58)

A little algebra then implies

$$(K_{\Sigma_{s}})^{2}-(K_{\Sigma_{s}})_{ab}(K_{\Sigma_{s}})^{ab}=(K_{\Sigma})^{2}-(K_{\Sigma})_{ab}(K_{\Sigma})^{ab}~{}.$$

(59)

Moreover, we have $r_{\Sigma}=r_{\Sigma_{s}}$ since the two initial data slices have the same intrinsic metric. Thus Eq. (49) implies that the kink-transformed slice satisfies the scalar constraint equation:

$\displaystyle r_{\Sigma_{s}}+(K_{\Sigma_{s}})^{2}-(K_{\Sigma_{s}})_{ab}(K_{\Sigma_{s}})^{ab}=16\pi GT_{\mu\nu}t^{\mu}t^{\nu}~{}.$

(60)

To check the vector constraint Eq. (50), we separately consider the two cases of $b=x$ and $b=i$ where $i,j$ represent directions tangent to $\mathcal{R}$:

	$\displaystyle D_{a}(K_{\Sigma_{s}})^{a}_{x}-D_{x}K_{\Sigma_{s}}$	$\displaystyle=D_{a}(K_{\Sigma})^{a}_{x}-D_{x}K_{\Sigma}+B_{\mathcal{R}}^{(x)}\sinh{(2\pi s)}\delta(x)$
		$\displaystyle=D_{a}(K_{\Sigma})^{a}_{x}-D_{x}K_{\Sigma}=8\pi G\,T_{x\nu}t^{\nu}~{},$		(61)
	$\displaystyle D_{a}(K_{\Sigma_{s}})^{a}_{i}-D_{i}K_{\Sigma_{s}}$	$\displaystyle=D_{a}(K_{\Sigma})^{a}_{i}-D_{i}K_{\Sigma}=8\pi G\,T_{i\nu}t^{\nu}~{},$		(62)

where the second line of the first equation follows from the extremality of $\mathcal{R}$.

We conclude that the kink transform is a valid modification to the initial data. For both constraints to be satisfied after the kink, it was essential that $\mathcal{R}$ is an extremal surface. Thus the kink transform is only well-defined across an extremal surface. Note also that $\mathcal{R}_{s}\subset\Sigma_{s}$ is an extremal surface in $\mathcal{M}_{s}$. By Eq. (57),

$$(B_{\mathcal{R}_{s}}^{(t)})_{ij}=P^{a}_{i}P^{b}_{j}\left.(K_{\Sigma_{s}})_{ab}\right|_{\mathcal{R}_{s}}=(B_{\mathcal{R}}^{(t)})_{ij}\implies B_{\mathcal{R}_{s}}^{(t)}=0~{}.$$

(63)

In the second equality we used Eq. (51) as well as the fact that all relevant quantities are intrinsic to $\Sigma_{s}$, so $\mathcal{R}_{s}$ can be identified with $\mathcal{R}$. Moreover,

$$(B_{\mathcal{R}_{s}}^{(x)})_{ij}=(B_{\mathcal{R}}^{(x)})_{ij}\implies B_{\mathcal{R}_{s}}^{(x)}=0~{},$$

(64)

since this quantity depends only on the intrinsic metrics of $\Sigma$ and $\Sigma_{s}$, which are identical.

We will now establish important properties and an equivalent formulation of the kink transform.

Let us write $\Sigma$ as the disjoint union

$$\Sigma=a^{\prime}\cup\mathcal{R}\cup a~{}.$$

(65)

The spacetime $\mathcal{M}$ contains $D(a)$ and $D(a^{\prime})$ where $D(.)$ denotes the domain of dependence. The kink transformed slice $\Sigma_{s}$ contains regions $a$ and $a^{\prime}$ with identical initial data, so $\mathcal{M}_{s}$ also contains $D(a)$ and $D(a^{\prime})$. Because $\Sigma_{s}$ has different extrinsic curvature at $\mathcal{R}$, the two domains of dependence will be glued to each other differently in $\mathcal{M}_{s}$, so the full spacetime will differ from $\mathcal{M}$ in the future and past of $\mathcal{R}$. This is depicted in Fig. 2.

We will now derive an alternative formulation of the kink transform as a one-sided local Lorentz boost at $\mathcal{R}$. The unit vector field $t_{\Sigma_{s}}^{\mu}$ normal to $\Sigma_{s}$ is discontinuous at $\mathcal{R}$ due to the kink. Let

	$\displaystyle({t_{\Sigma_{s}}^{\mu}})_{R}=\lim_{x\to 0^{+}}t_{\Sigma_{s}}^{\mu}~{},$		(66)
	$\displaystyle({t_{\Sigma_{s}}^{\mu}})_{L}=\lim_{x\to 0^{-}}t_{\Sigma_{s}}^{\mu}$		(67)

be the left and right limits to $\mathcal{R}$. The metric of $\mathcal{M}_{s}$ is continuous since it arises from valid initial data on $\Sigma_{s}$. Therefore, the normal bundle of 1+1 dimensional normal spacetimes to points in $\mathcal{R}$ is well-defined. The above vector fields $({t_{\Sigma_{s}}^{\mu}})_{R}$ and $({t_{\Sigma_{s}}^{\mu}})_{L}$ belong to this normal bundle. Therefore at each point on $\mathcal{R}$, the two vectors can only differ by a Lorentz boost acting in 1+1 dimensional Minkowski space. The kink transform, Eq. (51), implies:

$\displaystyle({t_{\Sigma_{s}}^{\mu}})_{R}={(\Lambda_{2\pi s})}^{\mu}_{\nu}({t_{\Sigma_{s}}^{\nu}})_{L}~{},$

(68)

where ${(\Lambda_{2\pi s})}^{\mu}_{\nu}$ is a Lorentz boost of rapidity $2\pi s$. In this sense, the kink transform resembles a local boost around $\mathcal{R}$. Alternatively, we can view Eq. (68) as the definition of the kink transform. This definition can be applied to Cauchy slices that are not smooth around $\mathcal{R}$, but it reduces to Eq. (51) in the smooth case.

This observation applies equally to any other vector field $\xi^{\mu}$ in the normal bundle to $\mathcal{R}$, if $\xi^{\mu}$ has a smooth extension into $D(a^{\prime})$ and $D(a)$ in $\mathcal{M}$. The norm of $\xi^{\mu}$ and its inner products with $({t_{\Sigma_{s}}^{\mu}})_{L}$ and $({t_{\Sigma_{s}}^{\mu}})_{R}$ are unchanged by the kink transform. Hence, in $\mathcal{M}_{s}$, the left and right limits of $\xi^{\mu}$ to $\mathcal{R}$ will satisfy

$$\xi^{\mu}_{R}={(\Lambda_{2\pi s})}^{\mu}_{\nu}\xi^{\nu}_{L}~{}.$$

(69)

Now let $\Xi\supset\mathcal{R}$ be another Cauchy slice of $D(\Sigma)$. Since $\Xi$ contains $\mathcal{R}$, its timelike normal vector field $\xi^{\mu}$ (at $\mathcal{R}$) lies in the normal bundle to $\mathcal{R}$. We have shown that Eq. (68) is equivalent to the kink transform of $\Sigma$; that Eq. (69) is equivalent to the kink transform of $\Xi$; and that Eq. (68) is equivalent to Eq. (69). Hence the kink transform of $\Sigma$ is equivalent to the kink transform of $\Xi$. In other words, the spacetime resulting from a kink transform about $\mathcal{R}$ does not depend on which Cauchy surface containing $\mathcal{R}$ we apply the kink transform to.

The kink transform (with $s\neq 0$) always generates physically inequivalent initial data. However $\mathcal{M}_{s}$ need not differ from $\mathcal{M}$. They will be the same if and only if $\Sigma_{s}$ is an initial data set in $\mathcal{M}$. There is an interesting special case where this holds for all values of $s$. Namely, suppose $\mathcal{M}$ has a Killing vector field that reduces to a boost in the normal bundle to $\mathcal{R}$. Then $\Sigma_{s}\subset\mathcal{M}$ (as a full initial data set), for all $s$. For example, the kink transform maps straight to kinked slices in the Rindler or maximally extended Schwarzschild spacetimes (see Fig. 3).

We can also consider the kink transform of matter fields on a fixed background spacetime with the above symmetry. Geometrically, $\mathcal{M}=\mathcal{M}_{s}$ for all $s$, but the matter fields will differ in $\mathcal{M}_{s}$ by a one-sided action of the Killing vector field. For example, let $\mathcal{M}$ be Minkowski space, with two balls at rest at $x=\pm 1$, $y=z=0$ (see Fig. 4); and let $\mathcal{R}$ given by $x=t=0$. In the spacetime $\mathcal{M}_{s}$ obtained by a kink transform, the two balls will approach with velocity $\tanh 2\pi s$ and so will collide. The right and left Rindler wedge, $D(a)$ and $D(a^{\prime})$, are separately preserved; the collision happens in the past or future of $\mathcal{R}$.