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\begin{document}
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\hbox{}
\hbox{hep-th/0302052}}


\bigskip\bigskip\bigskip\bigskip


\centerline{\Large \bf Chronology Protection in String Theory}


\bigskip\bigskip
\bigskip\bigskip
\centerline{Lisa Dyson
\footnote[1]{\tt  ldyson@mit.edu} }
\bigskip
\bigskip
\centerline{\it Center
for Theoretical Physics}
\centerline{\it Massachusetts Institute of Technology}
\centerline{\it Cambridge, MA  02139}
\bigskip
\bigskip
\centerline{\it Institute for Theoretical Physics}
\centerline{\it Stanford University}
\centerline{\it Stanford CA 94305-4060}
\bigskip\bigskip
\begin{abstract}

\medskip

\medskip

\noindent
Many solutions of General Relativity appear to allow the possibility of time travel.  This was initially a fascinating discovery, but geometries of this type violate causality, a basic physical law which is believed to be fundamental.  Although string theory is a proposed fundamental theory of quantum gravity, geometries with closed time-like curves have resurfaced as solutions to its low energy equations of motion.  In this paper, we will study the class of solutions to low energy effective supergravity theories related to the BMPV black hole and the rotating wave--D1--D5--brane system.   Time travel appears to be possible in these geometries.  We will attempt to build the causality violating regions and propose that stringy effects prohibit their construction.  The proposed chronology protection agent for these geometries mirrors a mechanism string theory employs to resolve a class of naked time-like singularities.
 

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\section{Introduction}
General Relativity is believed to be an accurate description of our classical world.  While this led to a revolution in physical thought in the twentieth century, it also introduced a myriad of ambiguities and physical conundrums.  One such ambiguity arises when attempting to understand the quantum description of black holes.  Naive calculations would indicate that information may be lost and thereby destroy our notion of unitary time evolution \cite{loss}.  Physicists have battled over this issue for decades.  As a proposed ultimate theory of quantum gravity, string theory has intervened and provided some answers.  For a certain class of black hole geometries, a microscopic description exists in string theory which enables one to calculate macroscopic quantities such as entropy and Hawking radiation \cite{entropy, non}.  Resulting arguments seem to indicate that there is no loss of information and the unitarity of physics is preserved \cite{su, witten, eternal}.

Another class of solutions to General Relativity which have caused much debate are geometries which are not black holes, but nevertheless have regions of infinite curvature, that is, geometries with singularities unshielded by horizons.  Such geometries are ubiquitous in General Relativity, but believed to be physically unsound. Again, string theory has intervened and provided insight.  There exist mechanisms in string theory which resolve naked singularities, resulting in consistent, non-singular physics \cite{ks, ps, enhancon}.

General relativity also admits geometries which have closed time-like curves.  Examples include the Godel universe, Kerr black holes and Gott's time machines.  Naively this would indicate that time travel is possible.  This in turn gives birth to a whole host of physical inconsistencies such as the Grandmother paradox, where one is allowed to modify the initial conditions that lead to ones own existence, and the Bootstrap paradox, where an effect is its own cause \footnote{See, for example, \cite{cpa2} for a review}.

There has been much work trying to address the paradoxes posed by these geometries.  Some have accepted the possibility of time travel, while others have argued that physical processes come into play rendering time travel impossible.  In the case of Gott's time machine, for instance, it has been shown that any attempt to create a time machine would cause the space to collapse entirely \cite{gott}.  Many believe that quantum physics will ultimately prevent all geometries which naively allow time travel from forming.  This led to a proposal by Stephen Hawking who stated  ``It seems that there is a Chronology Protection Agency which prevents the appearance of closed time-like curves and so makes the universe safe for historians.'' \cite{cpa}

As of yet, the full consistency of geometries of this type has not been tested in string theory \footnote{See \cite{horava} for a recent discussion on holography in supersymmetric Godel universes.}.  

In this work, we will propose that a Chronology Protection Agency does indeed exist in string theory.  Specifically, we will study solutions to ten dimensional supergravity which can be dimensionally reduced to give five dimensional rotating black holes.  When the rotation parameter exceeds a certain value, closed time-like curves appear outside of the horizon.  By studying the full ten dimensional solution, we are able to track down the source of causality violations.  We will find that our solution breaks down long before we reach the causally sick region.  We conclude that the geometry must be altered at this point.
The causality violating region is never created and our usual notion of chronology is retained.

In the next section, we will explore the geometry in detail, first presenting the five dimensional black hole solution, then its ten dimensional counterpart.  We will see how closed time-like curves form in both the five dimensional  and ten dimensional geometries and show that the causality bound is associated with a unitarity bound in the dual conformal field theory.  This bound, however, does not prohibit us from forming the causally undesirable geometry.  In fact, supersymmetric solutions appear to exist for arbitrary values of angular momentum.

In order to determine if there is some form of an obstruction to creating the causality violating region which would have a non-unitary conformal field theory dual, in section three, we examine the objects associated with the charges that make up the geometry.  We propose that there is indeed an obstruction to forming the geometry and that chronology is protected.  The proposed agency of protection for the class of geometries that we present is analogous to the one presented in \cite{enhancon} to resolve a class of naked singularities.  We conclude in section four.


\section{The Geometry}

\subsection{Five Dimensional Black Hole}

Let us begin by presenting the five dimensional rotating black hole solution.  We will consider a black hole with three charges, call them $Q_1, Q_5$, and $Q_k$.  The metric for this geometry is

\begin{eqnarray}
ds^2  &=&  - (f_1 f_5 f_k)^{-{2 \over 3}}\Big[ dt^2 + {J \over 2 r^2} (\sin^2 \theta d \phi_1 - \cos^2 \theta d \phi_2)\Big]^2 \nonumber \\ 
 && \quad \quad \quad \quad + (f_1 f_5 f_k)^{1 \over 3} \Big[ dr^2 + r^2 d \theta^2 + r^2 (\sin^2 \theta d \phi_1^2 + \cos^2 \theta d \phi_2^2) \Big]
\end{eqnarray}
where, as usual, the $f_i$ are harmonic functions associated with the charges, $f_i = 1 + Q_i/r^2$.  In addition to the metric, this supergravity solution has the following moduli and gauge fields under which the black hole is charged,

\be
e^{-2\phi} = {f_5 \over f_1} \quad \quad \quad e^{2 \sigma} = { f_k \over f_5^{1/4} f^{3/4}} \quad \quad\quad e^{\sigma_i} = {f_1^{1/4} \over f_5^{1/4}} 
\label{bhmoduli}
\ee
\be
A = {1 \over f_1} \ \Big[ {Q_1 \over r^2} dt + { J\over 2 r^2} (\sin^2 \theta d \phi_1 - \cos^2 \theta d \phi_2)\ \Big] 
\ee
\be
A^k = {1\over f_k} \ \Big[{Q_k \over r^2} dt + { J\over 2 r^2} (\sin^2 \theta d \phi_1 - \cos^2 \theta d \phi_2)\ \Big]
\ee
\be
B =  Q_5 \cos^2\theta d\phi_1 \wedge d \phi_2 +  {1 \over f_1} { J\over 2 r^2 } dt \wedge (\sin^2 \theta d \phi_1 - \cos^2 \theta d \phi_2)
\label{blackhole}
\ee
with $i=1,...4$.

This geometry has been studied extensively by many authors \cite{bmpv, cvetic, myersbh, herdeiro2, herdeiro}.  In order to have a supersymmetric rotating black hole, the horizon must be stationary.  In \cite{myersbh} it was shown that the horizon is not rotating.  The nonzero rotation parameter affects the horizon by turning it into a squashed sphere.  Since the squashing parameter depends on the angular momentum, $J$, one can show that a sensible description of the horizon as a null hypersurface can breakdown if $J$ becomes too large.  

It was additionally shown that this geometry has closed time-like curves.  Luckily, if $J$ is small enough, all closed time-like curves are hidden behind the horizon.  However, when $J$ becomes too large, closed time-like curves appear outside of the horizon leading to gross causality violations.  The full sickness of this over-rotating black hole geometry was studied in detail in \cite{herdeiro2}.  

The causality constraint on $J$ can be seen explicitly by looking at the angular components of the metric.  The chronology horizon, $R_{ch}$, is the location where $g_{\phi_i \phi_i}$ vanishes.  Closed time-like curves appear when $g_{\phi_i \phi_i}$ becomes negative.  This occurs if $J^2 > 4 \,(r^2 + Q_1) (r^2 + Q_5) (r^2 + Q_k)$.  Since the horizon is located at the origin in these coordinates, an observer outside of the horizon will see closed time-like curves if $J^2 > 4\, Q_1Q_5Q_k$.  

This constraint on $J$ also becomes apparent when studying the thermodynamics of the black hole.  Its Bekenstein-Hawking entropy is

\be
S = { \pi^2 \over 2 G_5} \sqrt{ Q_1 Q_5 Q_k - { J^2 \over 4}}
\label{entropy}
\ee
This can become imaginary if $J$ is too large.  To be complete, let us also specify the mass of the black hole

\be
M = { \pi\over 4 G_5} ( Q_1 + Q_5 + Q_k)
\label{mass}
\ee

Following \cite{herdeiro}, we can attempt to understand the origin of the causally sick region by lifting this five dimensional solution to ten dimensions and studying the fundamental objects which create the black hole.  We will begin this process by presenting the ten dimensional geometry below.

\subsection{Ten Dimensional D--Brane Geometry}

Upon lifting to ten dimensions, we find a supergravity description of D1 and D5--branes with momentum running along the effective D--string.  The D5--branes are additionally wrapped on $T_4$.  The metric, RR field and the dilaton are \cite{herdeiro}

\begin{eqnarray}
ds^2 &=& {1 \over \sqrt{f_1 f_5}} \Big[ -dt^2 + {Q_k \over r^2} (dz - dt)^2 + dz^2 + { J \over r^2} (\sin^2 \theta d\phi_1 - \cos^2\theta d\phi_2)(dz - dt)\Big] \nonumber\\
& & + \sqrt{{f_1 \over f_5}}\, ds^2_{T^4} +  \sqrt{f_1 f_5}\, \Big[ dr^2 + r^2 d \theta^2 + r^2 (\sin^2 \theta d \phi_1^2 + \cos^2 \theta d \phi_2^2)\Big]
\label{fullgeo}
\end{eqnarray}
\be
C^{(2)} = {1 \over f_1} dt \wedge dz +  {1 \over f_1}{J \over 2 r^2 } (\sin^2 \theta d\phi_1 - \cos^2\theta d\phi_2) \wedge (dz - dt) + Q_5 \ \sin^2 \theta d \phi_1 \wedge d\phi_2
\label{rr}
\ee
\be
e^{-2 \Phi} = {f_5 \over f_1}
\ee
The charges $Q_1, Q_5$, $Q_k$ and $J$ are given by

\be
Q_1 = { g \alpha'^3 \over V} N_1, \quad \quad   Q_5 = g \alpha' N_5, \quad   \quad Q_k = { g^2 \alpha'^4 \over R^2 V} N_k, \quad \quad J = { g^2 \alpha'^4 \over R V} F_R
\label{charges}
\ee
where we have $N_1$ D1--branes, $N_5$ D5--branes, $N_k$ units of right moving momentum, and the angular momentum is quantized in terms of integers $F_R$.  $V$ is the volume of $T^4$ and $R$ is the radius of the $S^1$.
  
This is a IIB supergravity solution with four supercharges.  There are no obvious causality violating regions in this geometry, so it may appear as though the full ten dimensional geometry has resolved the causal problems associated with the black hole.  This conclusion, however, is misleading.  To see this explicitly, consider space-like curves which have tangent vectors given by  \cite{herdeiro}
\be
l^{\mu} \partial_{\mu} = \alpha \, \partial _{z} + \beta \,(\partial_{\phi_1} -  \partial_{\phi_2})\ .
\label{cyclech}
\ee
Since $z$ is a compact direction, for certain values of $\alpha$ and $\beta$, curves of this type can be closed.  Also, a quick calculation of the proper length shows that these curves can become time-like in the region $r < R_{ch}$. Closed time-like curves reappear in ten dimensions.
We have not been able to successfully rid ourselves of the sickness associated with the five dimensional geometry.  

In fact, we can do better than this.  We can T-dualize our solution to find a system which should have the same physics.  In the new geometry, the full sickness of this configuration becomes apparent.  Performing a T-duality along the $z$ direction, we find a solution to IIA supergravity with the following metric.

\bea
ds^2 &=& {1\over f_k \sqrt{f_1 f_5}} \ \Big[ -dt^2 + {J \over 2 r^2} ( - \sin^2\theta d\phi_1+ \cos^2\theta d\phi_2) \Big] + \sqrt{ f_1 \over f_5} \ ds^2_{T^4} 
 \nonumber \\
&& + \, \sqrt{f_1 f_5} \Bigg[ \left( 1 - { J^2 \over 4} {1 \over r^6 f_1 f_5 f_k} \sin^2 \theta\right) r^2 \sin^2\theta \, d\phi_1^2 + \left( 1 - { J^2 \over 4} {1 \over r^6 f_1 f_5 f_k}  \cos^2 \theta\right) r^2 \cos^2\theta \, d\phi_2^2 \Bigg] \nonumber \\
&& +
\,  { J^2 \over 2} {1 \over r^4 f_k \sqrt{f_1 f_5}} \, \cos^2 \theta \, \sin^2\theta \, d\phi_1 \, d\phi_2\ + \sqrt{f_1 f_5} \, \Big[ {dz^2\over f_k} + dr^2 + r^2 d\theta^2\Big]
\label{tdual}
\eea
\\
Since $g_{\phi_i \phi_i}$ can become negative, we see clearly that for large enough $J$ causality violations are an integral part of ten-dimensional physics.  The causally sick region in the higher dimensional geometry coincides precisely with the causally sick region of the lower dimensional black hole.

\subsection{The Dual Conformal Field Theory}

The geometry is not the only place where we find a breakdown in our physical description of this solution.  Recall that there is a dual conformal field theory \cite{entropy, bmpv}.  If we take the size of the $S^1$ to be much larger than the size of the $T^4$, the effective description is the 1+1 dimensional gauge theory living on the world volume of the D1--branes.  

The central charge of the conformal field theory is $c = 6 N_1 N_5$.  Turning off angular momentum for a moment, the supersymmetric states that we are interested in are in the left moving ground state with arbitrary right moving charge.  Since these states are BPS, the energy and momentum are related by $E \sim N_k/R$, where $R$ is the radius of the $S^1$.  

Adding rotation requires that we use up some of the energy of the oscillator modes.  
We can break up the $N=4$ superconformal algebra into left and right moving $N=2$ algebras.  There is a $U(1)$ subgroup associated with each and we can specify the charges as $(F_R, F_L)$. 

One can bosonise the $U(1)$ currents \cite{bmpv} to show that a state with charge $F_R$ is represented by an operator of the form 
\be
\O = \exp \left({i \sqrt{3} F_R\phi \over \sqrt{c}}\right) \Phi
\ee
where $\Phi$ is an operator from the rest of the conformal field theory.   
The eigenvalue of $L_0$ gives the conformal weight of primary operators.  Since $L_0(\O) = N_k$,
\be
L_0(\Phi) = N_k  - {F_R^2 \over 4 N_1 N_5} 
\label{cftlevel}
\ee
Unitarity requires that conformal weights be positive.  We find then that $N_k \ge F_R^2/(4N_1N_5)$, or equivalently, $Q_k \ge J^2/(4Q_1Q_5)$.  This is nothing more than the causality bound that we saw previously.  Let us also note that (\ref{cftlevel}) represents the effective level of the system after we have used some energy to add angular momentum.  The level determines the degeneracy of states.  The entropy can be easily calculated and agrees with the Bekenstein- Hawking entropy (\ref{entropy}) calculated above  \cite{bmpv}.


We have seen that the physics of this solution breaks down in three dual descriptions: the five dimensional black hole, the ten dimensional D--brane configuration and the two dimensional conformal field theory.  In an attempt to understand how we might resolve the causally sick regions, we will study the  ten dimensional supergravity solution a bit further below.

\section{The Proposed Resolution}

\subsection{The Geometry}
We can imagine building our geometry by slowly bringing in D--brane charge from infinity. A probe calculation will determine if this is a consistent thing to do.  
The action for a D1--brane is

\be
S = -\tau_1 \int d^2 \sigma \, e^{-\Phi} \sqrt{-det \, g} + \tau_1 \int C^{(2)}
\ee
Working in static gauge $ t = \tau , z = \sigma$, and choosing the ansatz $ x^i = x^i(\tau), y^a = y^a$,
where the $x^i$ run over the transverse directions and the $y^a$ run over the $T^4$, the slow velocity limit of the action is
\be
S =  \tau_1 \int d^2 \sigma  \ f_1 f_5 f_k  \ \Big[{\dot r}^2 + r^2 ({\dot \theta}^2 + \sin^2 \theta {\dot \phi_1}^2 + \cos^2 \theta {\dot \phi_2}^2)\Big]  
\label{action}
\ee
In this limit, the potential vanishes leaving behind a purely kinetic term in the action.  We can therefore consistently bring D1--branes to the origin.  A similar analysis holds for D5--branes (after dualizing the RR 2-form to get the appropriate RR 6-form under which the D5--branes are electrically charged).  So there is no obstacle here.  Where, then, is the problem? 

We have probed this geometry with objects associated with two of the charges which make up the geometry.  Recall that there is also charge associated with momentum modes running along the effective D--string, $Q_k$.  This is nothing more than a gravitational wave.  In a T-dual picture, this charge is associated with a fundamental string.  There is additional momentum, $J$, associated with rotation.  In order to determine where the breakdown in our geometry may be occurring, let us consider its microscopic description.

We can think of a bound state of one D1--brane and $N_5$ D5--branes as a system of $N_5$ fractional strings \cite{mathur1,fatblack, mathur2}.  If we have $N_1$ D1--branes, there are a total of $N_1 N_5$ fractional strings.  In this model, angular momentum can be added to the system by allowing each sub-string to be in a particular spin state of the $SU_R(2) \times SU_L(2)$ symmetry \cite{mathur2}.  We can join together any number of the sub-strings.  The spin of the string created, however, must be the same as that of a single string.  The maximum angular momentum of the system, then, occurs when all of the strings are split and aligned along the same direction.  The angular momentum of this state is ( ${N_1 N_5 \over 2}, {N_1 N_5 \over 2} $).

There are also massless open strings in our system which begin and end on the D--branes.  When we add $N_k$ units of momentum along these massless strings, the  entropically favored configuration is one in which all of the fractional strings join and form a single long string with an effective length of order $R' =N_1 N_5 R$, where $R$ is the radius of the $S^1$ along which the D-branes are wrapped \cite{fatblack}.  The effective level of the string state becomes $N' = N_1 N_5 N_k$.   If we want to add angular momentum to this system, it must be carried by the massless strings since the spin of the single long string formed by the D--branes is effectively zero \cite{mathur2}.  The effective level, once we add angular momentum, becomes 

\be
N' = N_1 N_5 \left( N_k  - {F_R^2 \over 4 N_1 N_5} \right).
\label{level}
\ee
The standard thermodynamic quantities can be easily calculated in this model.  Since open strings are attached to the horizon in this limit, they account for the degrees of freedom which give rise to the entropy of the black hole.
The entropy is $S = 2 \pi \sqrt{N'}$.   This agrees with (\ref{entropy}).

Returning to the supergravity picture, we want to form the three charge black hole.  We therefore assume that the D--branes have fractionalized and joined together to form one long string.  We have seen from the above probe calculation (\ref{action}) that we can consistently bring D1 and D5--branes in from infinity to the origin to form our geometry.    What about the momentum modes with non-zero rotation which, in the supergravity description, correspond to rotating gravitational waves?

We will attempt to add rotating gravitational waves to a D1-D5 system by doing the following.  Assume we have an interior geometry made up of only D1 and D5--brane charge.  The metric for this system is:

\be
ds^2 = {1 \over \sqrt{f_1 f_5}} \Big[ -dt'^2 + dz'^2 \Big] + \sqrt{{f_1 \over f_5}}\, ds^2_{T^4} +  \sqrt{f_1 f_5}\, \Big[ dr^2 + r^2 d \theta^2 + r^2 (\sin^2 \theta d \phi_1'^2 + \cos^2 \theta d \phi_2'^2)\Big]
\label{d1d5}
\ee
We will join this to the geometry (\ref{fullgeo}) which additionally has rotating gravitational waves.  We will join these two geometries together at some radius $R$.  Continuity of the metric requires the coordinates to be related by the following transformations:

\bea
\phi_1' & = & \phi_1 + {J \over 2 R^4 F_1 F_5} \ (z - t) \nonumber \\
\phi_2' & = & \phi_2 - {J \over 2 R^4 F_1 F_5} \ (z - t) \nonumber \\
z' & = & z + {1 \over 2 R^2}\left( Q_k - { J^2 \over 4 R^4 F_1 F_5}\right) (z - t) \nonumber \\
t' & = & t + {1 \over 2 R^2} \left( Q_k  - { J^2 \over 4 R^4 F_1 F_5}\right) (z - t)
\label{coords}
\eea
where $F_i = (1 + Q_i/R^2)$.

Our new geometry has D1 and D5--branes at the origin and rotating gravitational waves localized on the shell at radius $R$.  The standard Israel matching conditions \cite{israel} give the tension and pressures associated with the joining shell.  With this, we can determine if it is consistent to bring the gravitational waves in to the origin and hence create the geometry which is described by equation (\ref{fullgeo}) for all $r$.  Switching to Einstein frame, $G_{\mu\nu} = e^{-\Phi/2} g_{\mu\nu}$, the energy-momentum tensor for the shell is 
$S^{\mu}_{\nu} = 8 \pi G_{10} \sqrt{G_{rr}} \ T^{\mu}_{\nu}$ with

\begin{eqnarray}
T_0^0 ={2 \over R^3} \Big[Q_k - {J^2 \over 4 R^4 F_1 F_5} \Big]\quad\quad
&&
T_z^z = -  {2 \over R^3}\ \Big[Q_k - {J^2 \over 4 R^4 F_1 F_5} \Big]\nonumber \\
T_0^z =  {2 \over R^3}\ \Big[Q_k - {J^2  \over 4 R^4 F_1 F_5} \Big]\quad\quad
& &
T_z^0 =  - {2 \over R^3}\ \Big[Q_k - {J^2 \over 4 R^4 F_1 F_5} \Big] \nonumber \\
T_{\phi_1}^0 = T_{\phi_1}^z = - {J \over 2 R^3} \ g(R) \ \sin^2 \theta 
& & 
T_{\phi_2}^0 = T_{\phi_2}^z = {J \over 2 R^3} \ g(R)\ \cos^2 \theta \nonumber \\
T_0^{\phi_1} =   T_z^{\phi_2} = {J \over 2 R^5 F_1 F_5} \ g(R)\  \quad
 & &
   T_0^{\phi_2} = T_z^{\phi_1} = -{J \over 2 R^5 F_1 F_5} \ g(R)\nonumber \\
T_{\alpha,\beta} = 0   \, \ \ \quad\quad\quad\quad \quad  \quad \quad \quad\quad && \ \
\label{tensor}
\end{eqnarray}
where ${\alpha,\beta}$ runs over the transverse directions and the $T^4$, $G_{10} =  8 \pi \alpha'^4 g^2$ is the ten dimensional Newton's constant and $g(R) = [1 + (R^2 F_1 F_5)'/(2 R F_1 F_5)]$.   The energy-momentum tensor is all we need determine where the breakdown of our geometry occurs.  But let us stop here for a moment to confirm that it is indeed what we expect.

First, notice that the stresses and energy vanish in the transverse direction.  This tells us that there are no transverse forces acting on the shell to prohibit it from forming and it is consistent to bring it in from infinity.
Next,  the asymptotic tension of the shell, $T_0^0$, goes like the inverse of the area of a three sphere.  This gives precisely the charge-energy relationship that we expect for a gravitational wave, $E \sim Q_k$, and it contributes the appropriate amount to the asymptotic mass of the black hole (\ref{mass}).  Finally,
 assuming that $J^2 < 4 \ Q_1 Q_5 Q_k$ for the moment, we can consistently bring the shell to the horizon, $R=0$.  In this limit, the tension of the shell is

\be
\tau_{nh} ={2 \over R^3}\ \Big[Q_k - {J^2  \over 4 Q_1 Q_5} \Big]
\label{nhtension}
\ee
Since $E \sim N/R \sim N'/R'$, the effective level of the system agrees with our expected result (\ref{level}).

Now that we believe that the energy-momentum tensor accurately describes our system in the asymptotic  and the near horizon regions, we can conclude that the tension  
\be
\tau ={2 \over R^3} \Big[Q_k - {J^2 \over 4 R^4 F_1 F_5} \Big]
\label{fulltension}
\ee
represents the local energy density of the incoming gravitational waves from the supergravity perspective and massless strings from the microscopic D--brane point of view.  Placing the shell at a radius $R$, the effective local Kaluza Klein momentum $E \sim P'$ is
\be
P' \sim Q_k - {J^2 \over 4 R^4 F_1 F_5}
\label{momentum}
\ee

Notice that for $J> 4 \, Q_1 Q_5 Q_k$, the tension and momentum of the shell vanish before the chronology horizon is reached.  This occurs at a radius

\be
R_{cph}^2 = - {(Q_1 + Q_5) \over 2} + {1 \over 2} \sqrt{(Q_1 - Q_5)^2 + {J^2 \over  Q_k}}. 
\label{cph}
\ee
If we place the shell at this radius, the energy density of our constructed geometry is smooth.  If we try to bring the shell of gravitons in past $R_{cph}$, the energy density of the shell becomes negative.  This would be inconsistent.  We conclude that the gravitons are unable to travel past $R_{cph}$ and that the causally sick region can never be created.  

Let us stop for a moment to recap.  We have argued that the rotation in our geometry is carried by the gravitational waves.  Assuming this is true, we were able to bring D1 and D5--branes in to the origin.  When we tried to bring in rotating gravitational waves from infinity to complete the construction of our geometry, we found that negative energy densities appear before the chronology horizon is reached.  In order to avoid negative energy densities, we propose that the gravitons are not able to travel beyond a chronology protection radius, $R_{cph}$.  This radius is the location where the energy of the shell due to momentum along the compact direction exactly cancels the energy due to rotation.  The ten dimensional geometry, therefore, has a smooth energy density. The final corrected geometry has momentum charge localized on the shell at radius $R_{cph}$ outside of the would-be chronology horizon with D1 and D5--branes at the origin.  The potential causality violating region is never able to form and chronology is protected.

We should note that although we wrapped the D5--branes on $T^4$, we could have wrapped them on K3.  Depending on the value of the parameters, wrapping branes on K3 can have the undesirable effect of creating naked singularities of repulson type.  Luckily, these singularities can easily be resolved as shown in \cite{enhanconbh, enhanconrot}.  The extension of our result to this case is straight forward.  The resulting geometry will have a shell of D1 and D5--branes placed at the appropriate radius to resolve the repulson singularity and the shell presented here placed at a different radius to prevent closed time-like curves from forming.

\subsection{Repulsive Coulomb Force}
Let us return to the five dimensional geometry (\ref{bhmoduli}) - (\ref{blackhole}).  The ten dimensional rotating gravitational wave becomes a particle charged under the gauge field $A_k$ in five dimensions.  We can probe our geometry with a point particle with charge $q$ which couples to the gauge field $A_k$.  The Hamiltonian for such a particle is \cite{herdeiro}.
\begin{eqnarray}
H &=& g^{\mu\nu} ( p_{\mu} + q A^k_{\mu})(p_{\nu} + q A^k_{\nu})
\end{eqnarray}

Let us focus on the Coulomb potential that the particle feels moving radially in this background. 
\be
V = q \ g^{\mu\nu} ( p_{\mu} A^k_{\nu} + p_{\nu} A^k_{\mu}) 
 = {q Q_k \over r^2} \sqrt{f_1 f_5} \left(1 - {J^2 \over 4 Q_k  r^4 f_1 f_5} \right)
\ee
The Coulomb interaction changes sign precisely at $r=R_{cph}$, if the Coulomb force was attractive for $r>R_{cph}$, it becomes negative for $r<R_{cph}$ and vice versa \cite{herdeiro}.  So it appears as though there is some non-trivial charged object localized at the surface $r=R_{cph}$.  Our proposed resolved geometry simply places it there.  Again, this is analogous to the mechanism of \cite{enhancon, consistency} where a point particle probe interacting with the gravitational field felt nontrivial mass localized on a surface.  The proposed resolved geometry in that case simply placed the appropriate massive objects (D6-D2*--branes) at that location.  

In fact, as we saw previously, the asymptotic tension of the shell goes like $ Q_k/ A_3$.  This is precisely the relationship that we expect for a particle charged under the gauge field $A_k$ with its charge smeared over a three sphere.  Recall also that the shell is located at a safe distance away from the causally sick region, $R_{cph}> R_{ch}$.   So the end result is that a rotating shell with asymptotic charge $Q_k$ and local charge density given by (\ref{fulltension}) forms outside of the chronology horizon and thereby prevents any attempt to create closed time-like curves.  The interior region is no longer sick.  It is just a standard compactified D1 D5--brane geometry.

Up to this point, we have only used the microscopic description of our solution to determine which objects carry the charges.  A similar analysis was carried out for geometries with naked singularities of repuslon type in \cite{consistency}.  We believe that there is a general rule here.   The limits of physical consistency are exceeded before the sick regions can be created.  For a large class of geometries which are physically unsound, a simple geometric study of the laws of physics alone can reveal how one might correct a geometry to render it physically sound.

\subsection{Massless Strings and Shrinking Cycles}

Returning now to the microscopic description of our configuration,
recall that the naive geometry is a solution to Type IIB effective low energy supergravity with momentum modes and no winding modes.  In a T-dual picture, momentum modes and winding modes are interchanged.  A T-dual string, then, would have winding number and hence mass proportional to (\ref{momentum}).  If we try to bring a string in beyond $R_{cph}$, its mass becomes negative.  This would be unphysical.  The string becomes massless before this occurs at the chronology protection horizon, $R_{cph}$.  When massive modes become light, their contribution to the low energy effective theory can no longer be ignored.  The low energy theory must be corrected at this point to include the new massless degrees of freedom.  
Happily, massless modes appear at the chronology protection horizon, $R_{cph}$, so the breakdown in the supergravity theory coincides precisely with the breakdown we encountered when trying to construct the geometry above.  Once the new massless degrees of freedom have been taken into account, we propose that the above geometry is, to leading order, the solution to the corrected  supergravity theory. 

We saw previously that there are curves in the naive geometry which shrink to zero size at the chronology horizon (\ref{cyclech}).  As it turns out, there are also curves which shrink to zero size before the chronology horizon is reached.
To see this, we can perform the following coordinate transformation: $v= z+t\, $, $u = z-t$.  Plugging this back into the metric (\ref{fullgeo}), we find that space-like curves of the form  
\be
l^{\mu} \partial_{\mu} = \alpha \, \partial _{u} + \beta \,(\partial_{\phi_1} - \partial_{\phi_2})
\label{cyclecp}
\ee
can become null at the chronology protection radius, $r=R_{cph}$.  For example, this is the case if we let $\beta/\alpha = - g_{u \phi_1}/ g_{\phi_1 \phi_1}$.  In the region $r<R_{cph}$, these curves are time-like.  

We can wrap one of these non-trivial cycles with a test D--string.   The tension of the probe vanishes precisely when the cycle shrinks to zero size, at $R_{cph}$.  If we try to probe the interior region, $r < R_{cph}$, the tension of the brane becomes negative.  Since negative tension branes are considered unphysical,  the physics of such a probe supports the proposal that the low energy theory must be corrected beyond $R_{cph}$.


{\subsection{Near Horizon Limit}

We have proposed that the naive BMPV black hole and D1-D5-Wave geometries cannot be constructed when the angular momentum parameter exceeds a certain bound, $J^2 > 4 Q_1 Q_5 Q_k$.  Instead, a shell carrying $Q_k$ and $J$ charge forms outside of the would be causally sick region.  The charges needed to create the causality violating region are not permitted to travel past this shell.  Closed time-like curves, therefore, are not able to form and the resulting geometry is causally safe.  We saw previously that the causality bound of our spacetime coincided with a unitarity bound in the conformal field theory.  The conformal field theory is dual to the near horizon limit of our geometry.  We can take the limit of our proposed resolved geometry to see if it agrees with the expected result.

First, let us assume that the angular momentum exceeds the unitarity bound.  In this case, the shell of gravitons forms outside of the horizon at the chronology protection radius, $R_{cph}$.  Zooming in on the near horizon region gives the near horizon geometry of the standard D1 D5--brane configuration.  This is nothing more than the usual AdS$_3 \times S^3 \times T^4$.  The dual conformal field theory of this geometry is known.  There are no violations of unitarity here.

Next, let us assume that the unitarity bound is not exceeded, but let us maintain our construction as presented above (\ref{tensor}).  The near horizon geometry is a BTZ $\times S^3 \times T^4$ matched to an AdS$_3 \times S^3 \times T^4$ with a joining shell of tension

\be
\tau_{nh} ={2 \over R^3}\ \Big[Q_k - {J^2 \over 4 Q_1 Q_5} \Big]\ .
\ee
The bound on $J$ resurfaces as a bound on the energy of the shell required to create the geometry.  We can place the shell at any radius $R$ and the energy will be positive or zero since, by assumption, this near horizon solution can only be created if  $J^2 \leq 4 Q_1 Q_5 Q_k$.  Of course, the conformal field theory dual in this case is the standard unitary theory that we began with.  

So we see that our proposed construction ensures that unitarity is preserved.


\section{Conclusion}

As a fundamental theory of quantum gravity, string theory has been shown to successfully resolve many of the paradoxes inherent in Einstein's theory of General Relativity.  Examples include non-singular descriptions of geometries with regions of infinite curvature and evidence that information is not lost in black hole geometries.  We propose that string theory also has something to say about closed time-like curves.

We studied the class of geometries related to the BMPV rotating black hole and its higher dimensional gravitational wave-D1-D5--brane analogue.  Naively, closed time-like curves appear outside of the horizon if the rotation parameter exceeds a certain value.  When trying to construct the geometry, however, we saw that massless degrees of freedom arise before reaching the causally sick region.  These new degrees of freedom appear at precisely the location where the energy due to transverse momentum cancels the energy due to rotation of the gravitons that make up our geometry.  We propose that this is the location of a chronology protection horizon, beyond which the gravitons cannot travel.  Instead they expand to form a shell outside of the chronology violating region.  Since the presence of the gravitons beyond this point was crucial to create the causally sick region, our geometry is rendered safe and chronology is protected.  

The agent of chronology protection for the class of geometries studied here has a natural physical interpretation.  
Assume we begin with the standard low energy solution (\ref{fullgeo}) with $J < 4 \, Q_1 Q_5 Q_k$.  If we increase the angular momentum so that this bound is no longer satisfied, the gravitons in the ten dimensional language and the Kaluza Klein particles in the lower dimensional language begin to expand.  This expansion is due to the centrifugal repulsion and causes the gravitons (particles) to grow to just the right size to compensate for the repulsive force acting on them.  The end result is that giant gravitons (particles) form well outside of the chronology horizon to prevent the causally sick region from being created

Time travel gives rise to a host of physical inconsistencies.  One may conclude that any fundamental theory of quantum gravity would not admit solutions where time travel is possible.  The low energy effective equations of string theory have as solutions an abundance of such causality violating geometries.  We believe, however, that string theory is causally sound.
As we have argued for the class of geometries studied here, 
stringy corrections to the low energy effective theory resolve causality violations.  We propose that once all stringy effects are taken into account,
 our usual notion of chronology will emerge as a protected law of nature.


\section{Acknowledgments}

I would like to thank B. Freivogel, S. Ganguli, S. Hellerman, P. Horava, S. Kachru, M. Kleban, S. Shenker, E. Silverstein, L. Susskind, and U. Varadarajan for useful discussions.




\begin{thebibliography}{999}

\bibitem{loss} S. Hawking, ``Particle Creation by Black Holes,'' Commun.Math.Phys. 43 199-220 (1975).

\bibitem{entropy} A. Strominger, C. Vafa, 'Microscopic Origin of the Bekenstein-Hawking Entropy,'' Phys. Lett. B379 (1996) 99, [hep-th/9601029].

\bibitem{non} G. Horowitz, J. Maldacena, A. Strominger, ``Nonextremal Black Hole Microstates and U-duality,'' Phys.Lett. B383 (1996) 151-159, [hep-th/9603109].

\bibitem{su} L. Susskind, J. Uglum, ``String Physics and Black Holes,'' Nucl.Phys.Proc.Suppl. 45BC (1996) 115-134, [hep-th/9511227].

\bibitem{witten} E. Witten, ``Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories,'' Adv.Theor.Math.Phys. 2 (1998) 505-532, [hep-th/9803131].

\bibitem{eternal} J. Maldacena, ``Eternal Black Holes in AdS,'' [hep-th/0106112]

\bibitem{ks} I. Klebanov, M. Strassler, ``Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities,'' JHEP 0008 (2000) 052, [hep-th/0007191]

\bibitem{ps} J. Polchinski, M. Strassler, ``The String Dual of a Confining Four-Dimensional Gauge Theory,'' [hep-th/0003136].

\bibitem{enhancon} C. Johnson, A. Peet, J. Polchinski, `''9911161 Gauge theory and the excision of repulson singularities'', Phys. Rev. D61, 086001  (2000) [hep-th/9911161].

\bibitem{singres} M. Natsuume, ``The singularity problem in string theory,'' [gr-qc/0108059].

\bibitem{cpa2} M. Visser, ``The quantum physics of chronology protection,'' [gr-qc/0204022].

\bibitem{gott} S. Carroll, E. Farhi, A. Guth, ``An obstacle to building a time machine,'' Phys.Rev.Lett.68:263-266 (1992)

\bibitem{cpa} S. Hawking, ``Chronology protection conjecture,'' Phys. Rev. D 46, 603-611 (1992).

\bibitem{horava} E. Boyda, S. Ganguli, P. Horava, U. Varadarajan, ``Holographic Protection of Chronology in Universes of the Godel Type,'' [hep-th/0212087].

\bibitem{consistency} C. Johnson, R. Myers, A. Peet, S. Ross, ``The Enhan\c con and the Consistency of Excision,'' Phys.Rev. D64 (2001) 106001, [hep-th/0105077].

\bibitem{bmpv} J. Breckenridge, R. Myers, A. Peet, C. Vafa, ``D--branes and Spinning Black Holes,'' Phys.Lett. B391 (1997) 93-98, [hep-th/9602065].

\bibitem{cvetic} M. Cvetic, F. Larsen, ``Near Horizon Geometry of Rotating Black Holes in Five Dimensions,'' Nucl.Phys. B531 (1998) 239-255, [hep-th/9805097].

\bibitem{myersbh} J. Gauntlett, R. Myers, P. Townsend, ``Black Holes of D=5 Supergravity,'' Class.Quant.Grav. 16 (1999) 1-21, [hep-th/9810204].

\bibitem{herdeiro2} G. Gibbons, C. Herdeiro, ``Supersymmetric Rotating Black Holes and Causality Violation,'' Class.Quant.Grav. 16 (1999) 3619-3652, [hep-th/9906098].

\bibitem{herdeiro} C. Herdeiro, ``Special Properties of Five Dimensional BPS Rotating Black Holes,'' Nucl.Phys. B582 (2000) 363-392, [hep-th/0003063].

\bibitem{mathur1} S. Das, S. Mathur, ``Excitations of D-strings, Entropy and Duality,'' Phys.Lett. B375 (1996) 103-110, [hep-th/9601152].

\bibitem{fatblack} J. Maldacena, L. Susskind, ``D--branes and Fat Black Holes,'' Nucl. Phys. B475, 679 (1996) [hep-th/9604042].

\bibitem{mathur2} S. Mathur, ``Gravity on AdS$_3$ and flat connections in the boundary CFT,'' [hep-th/0101118].

\bibitem{israel} W. Israel, ``Singular hypersurfaces and thin shells in general relativity,'' Nuovo Cim. 44B 1 (1966)

\bibitem{enhanconbh} C. Johnson, R. Myers, ``The Enhancon, Black Holes, and the Second Law,'' Phys.Rev. D64 (2001) 106002, [hep-th/0105159].

\bibitem{enhanconrot} L. Jarv, C. Johnson, ``Rotating Black Holes, Closed Time-Like Curves, Thermodynamics, and the Enhancon Mechanism,'' [hep-th/0211097].

\bibitem{adscft} J. Maldacena, ``The Large N Limit of Superconformal Field Theories and Supergravity,'' Adv.Theor.Math.Phys. 2 (1998) 231-252; Int.J.Theor.Phys. 38 (1999) 1113-1133, [hep-th/9711200].

\bibitem{lectures} J. David, G. Mandal, S. Wadia,  ``Microscopic Formulation of Black Holes in String Theory,'' Phys.Rept. 369 (2002) 549-686, [hep-th/0203048].

\bibitem{peet} A. Peet, ``TASI lectures on black holes in string theory,'' [hep-th/0008241].

\bibitem{cks} M. Caldarelli, D. Klemm, W. Sabra, ``Causality Violation and Naked Time Machines in AdS$_5$,'' JHEP 0105 (2001) 014, [hep-th/0103133].

\bibitem{supertubes} R. Emparan, D. Mateos, P. Townsend, ``Supergravity Supertubes,'' JHEP 0107 (2001) 011, [hep-th/0106012].

\bibitem{5dsolns} J. Gauntlett, J. Gutowski, C. Hull, S. Pakis, H. Reall, ``All supersymmetric solutions of minimal supergravity in five dimensions,'' [hep-th/0209114].

\end{thebibliography}

\end{document}


