302. Dynamical black holes & expanding plasmas

Pau Figueras, research associate at the Department of Mathematical Sciences in Durham University, works on black hole solutions (and their relatives) in string theory. Dmitry.

The AdS/CFT correspondence has over the years played an invaluable role in providing insight into the dynamics of strongly coupled gauge theories. An important application of the correspondence has been to understand the holographic description of hydrodynamic properties of field theories. This can be used to understand qualitative features of the Quark-Gluon plasma (QGP) produced in heavy ion collisions. Current theoretical understanding of this system is that subsequent to rapid thermalization, the system evolves as an almost ideal fluid, expanding rapidly away from the central collision region. The evolution in this regime has been well described by the so called Bjorken flow [1].

The first step in understanding the holographic dual of the Bjorken flow was taken in the seminal works [2,3], where the spacetime dual to the Bjorken flow in Super-Yang Mills was constructed as a perturbation expansion at late times. This geometry models the early (but post-thermalization) stages of the expanding quark gluon plasma.

More generally, the fluid/gravity correspondence proposed in [4] establishes the dictionary between the hydrodynamic regime of any interacting quantum field theory and its gravitational dual. This provides a useful relation between the dynamics of strongly coupled systems in the long-wavelength regime and the corresponding asymptotically AdS black hole geometries.  More interestingly, it allows one to calculate properties such as transport coefficients of the field theory fluid.

In general, the flow of a viscous fluid, which involves dissipation, necessarily leads to entropy production.   An important ingredient in the fluid-gravity correspondence was the identification of the global event horizon in the bulk spacetime, which turned out to provide a simple geometric construction for a Boltzmann H-function in the bulk geometry. In spacetimes dual to non-linear fluid flows,  the event horizon can be determined locally despite its teleological nature assuming slow temporal variations as well as that the geometry will settle down to a stationary configuration at late times . Indeed, the location of the event horizon is  given by a perturbation around this final equilibrium position.  The area-form of this event horizon, when pulled back to the boundary, was shown to lead to a natural local entropy current with non-negative divergence as required by the second law.

Our main aim in this work is to determine the location of the horizons in certain time-dependent geometries that do not settle down to stationary finite-temperature solutions at late times. Our interest in this question originally arises from the Bjorken flow (BF) geometry, but we also consider the conformal soliton (CS) geometry, which provides a simpler example with stronger time dependence. The latter  corresponds to a black hole entering through the past Poincare horizon and exiting through the future Poincare horizon, with its closest approach to the boundary occurring at (Poincare time).  In the boundary CFT, this describes a finite energy lump which collapses and re-expands in a time-symmetric fashion. Here the hydrodynamic approximation is not valid at all times, but because this fluid flow is conformal to a stationary fluid on the Einstein static universe, the stress tensor is shear-free; that is, there is no dissipation in this fluid flow.

The Bjorken flow

To understand the hydrodynamics in the BF, consider Minksowski spacetime  written in Milne-type coordinates which respect boost invariance in a subspace, i.e.,

The coordinates and measure the proper time and rapidity in the longitudinal direction respectively, and collectively denotes the transverse directions. For a conformally invariant fluid, the equations of motion of hydrodynamics, viz., energy momentum conservation and tracelessness of stress tensor, can be shown to constrain the dynamics to be derivable from a single function which is conveniently taken to be the energy density.  For an ideal conformal fluid, the equations of motion lead to a power law fall-off for the energy density and temperature

with the entropy per unit rapidity remaining constant. As the fluid expands the energy density diffuses throughout the forward light-cone and at late times, , the ideal hydrodynamic description becomes more and more accurate. In fact, one can go beyond the ideal fluid description of the BF and argue that an expansion in powers of corresponds to the derivative expansion in the fluid dynamics.

Gravity dual to Bjorken flow

To construct the gravity dual to the BF, we start with a metric ansatz

where . The idea is solve Einstein’s equations iteratively in a late time expansion, , keeping fixed. In order to do so, one assumes that the functions , and can be expanded as

To solve the Einstein equations, one imposes the boundary conditions,

which ensure that the spacetime has boundary metric consistent with the Bjorken flow.

In order to check that the spacetime dual to the BF is regular, in [5] the apparent horizon of the spacetime was determined explicitly up to second order in the expansion. The idea in this paper was to argue that this apparent horizon must be enclosed by a global event horizon, concealing the singularities from the asymptotic region.   However, the presence of an apparent horizon implies, by virtue of the singularity theorems, that the spacetime will evolve into a singularity in the future. In order to prove that the spacetime is regular, one has to show that the spacetime has a well behaved global event horizon. This is defined as the boundary of the past of future infinity , and it is generated by null geodesics.  We can therefore determine the location of the horizon directly, by studying the geodesic motion on this spacetime  and determining which points cannot send signals to infinity.

Since one has three Killing fields and the location of the horizon is simply given by a curve . The null geodesic equation reduces to

The event horizon is the outermost solution of this equation which does not reach at finite .  At late times, for the event horizon can be shown to admit an expansion in   of the form

where the ‘s are some yet to be determined constants. These constants can be found by solving the null geodesic equation order by order using the previously determined expansion for    at the same order.  Using the metric  quoted in [5] we find that

This gives the location of the event horizon up to second order in the late time expansion.

The Conformal Soliton flow

The Conformal Soliton (CS) spacetime is extremely simple — it is just the global AdS black hole sliced in Poincare coordinates. To obtain the CS spacetime we take the global Schwarzschild-AdS black hole, which is dual to a fluid in global thermal equilibrium in the Einstein Static Universe , and consider it in a `Poincare patch’. From the dual field theory point of view, we are making a conformal transformation to map the field theory on to the field theory on Minkowski space, . This maps the stationary fluid on the Einstein Static Universe to a time-dependent fluid configuration on Minkowski space. From the bulk spacetime point of view, this corresponds to considering only the portion of the null infinity of global Schwarzschild-AdS restricted to this Minkowski patch in the Einstein Static Universe.  The corresponding Poincare patch in the bulk contains not only the region outside the black hole, which is simply the portion of global Schwarzschild-AdS visible from this portion of null infinity, but also a finite region inside the black hole. The former is bounded by past and future Poincare horizons, as in the description of global AdS in Poincare coordinates, whereas the latter covers a larger  region. For simplicity we will concentrate on the BTZ black hole, since it is simpler but it contains the essential features of interest.

The coordinate transformation between the global coordinates in which pure has metric

and the Poincare coordinates in which the metric is

can be written as

The idea now is to apply this coordinate transformation to the BTZ spacetime,

The resulting geometry is what we call the CS spacetime. Note that the metric in these new coordinates is time-dependent.

Event horizon for the CS spacetime

The event horizon will correspond to the boundary of the region visible from . The latter  corresponds to the portion of null infinity, , of the global BTZ spacetime restricted to the Minkowski patch in the Einstein Static Universe. In the global BTZ spacetime, can be taken to be the region at containing with . The CS event horizon is then the boundary of the causal past of this region. Since all points in lie in the causal past of the boundary point , the problem of finding the CS event horizon reduces to the problem of finding null geodesics ending on this point. We find,

where parametrises the angular momentum per unit energy of each geodesic. These equations give the event horizon as a surface parametrised by and . Here .

Event horizon area, apparent horizon area & field theory entropy

One of the most important and physically interesting attributes of the event horizon is the area of its cross-sections, which we would usually take to give the entropy of the corresponding field theory state. We have seen that the horizon is dynamical, so we expect the area to be varying. To translate the variation of the area into a statement of the boundary field theory entropy, we compute the area of the cross-sections of constant Poincare time since this is the natural time coordinate of the boundary field theory. We find:

which grows logarithmically at large .

Thus, we have seen that the area of the cross-sections of the global event horizon in the CS spacetime increases with time, with a logarithmic divergence. This implies that we cannot identify it with the entropy in the dual field theory. In the field theory, passing from the global BTZ black hole to the CS spacetime is just a conformal transformation, and the entropy of the fluid is invariant under conformal transformations. Thus, we expect the total entropy of the fluid flow corresponding to the CS spacetime to be the same as in the static fluid dual to the global BTZ black hole, independent of the spatial slice on the boundary we choose to measure it on. The point is that although the dual fluid flow is time-dependent, it is an ideal fluid, and in the absence of any viscous or dissipative effects we cannot have any entropy production. As a result we should have the field theory entropy being constant even in the CS spacetime. We are thus led to propose that in dynamical spacetimes with `significant’ time variations, one should not associate the area of the event horizon to the entropy of the dual field theory. Instead,  we will argue that in the CS spacetime the relevant object is the apparent horizon.

The notion of apparent horizon is intimately tied to the notion of trapped surfaces.  Recall that a closed, co-dimension two spacelike surface   (which for the CS geometry is just a closed curve) is trapped if both (ingoing and outgoing) future-directed null geodesic congruences emanating normal to the surface have negative expansions, i.e. the areas of the `wavefronts’ for these null congruences decrease in time. A surface is marginally trapped if the outgoing null congruence has zero expansion, while the ingoing congruence has negative expansion. Then, the apparent horizon can be defined as the outermost marginally trapped surface on a given spacelike slice.

An important subtlety to note about both of these definitions is that a given spacetime geometry does not by itself specify the location of the apparent horizon; we first need to specify a foliation of the spacetime, with respect to which we can then define the apparent horizon. In the present case, the physically relevant foliation is one corresponding to constant Poincare time slices.

Now we will argue that the apparent horizon in fact coincides with the event horizon of the global BTZ black hole. Since the BTZ spacetime satisfies the energy conditions and has a complete , the apparent horizon on any spacelike slice must lie inside or on the event horizon.  The position of the apparent horizon on any given slice does not depend on the rest of the foliation, so we can view a given constant slice as part of a -foliation of the CS, or as part of a foliation of the global BTZ spacetime (obtained by translating the given slice by rather than ).  In the latter case, the relevant event horizon is not the CS event horizon which `flares out’, but rather the global event horizon, which stays at constant radius for all times.  This means that the apparent horizon cannot lie outside the surface. Since the slice in Poincare coordinates coincides with the slice in BTZ coordinates, where the apparent horizon of the static black hole coincides with its event horizon, we know the apparent horizon on this slice coincides with the global horizon at .  Furthermore, by the area theorem pertaining to the apparent horizon, this apparent horizon cannot recede in the future since its area cannot decrease.  This, combined with the previous argument that the apparent horizon cannot lie outside the global event horizon, forces the apparent horizon to coincide with the global event horizon for all times, which immediately implies that  its area is constant.

Since the area of the apparent horizon is constant, it can be identified with the entropy of the dual field theory. This example thus provides strong evidence that the entropy of the field theory fluid should in general be identified with the area of the apparent horizon rather than that of the event horizon.

Discussion

For the boost invariant Bjorken flow, we have shown that it has a regular event horizon and is a genuine regular black hole spacetime. This provides a final consistency check of the late time expansion as a gradient expansion used in the hydrodynamic context.

For the Confromal Soliton flow, we have found that the event horizon is a dynamical null hypersurface, whose spatial cross-section area diverges logarithmically for positive Poincare times. We then argued that for sensible foliations of the spacetime, including the constant Poincare time slices, the apparent horizon on the slices will coincide with the global BTZ event horizon at . This example shows that in strongly time-dependent settings, it is the apparent horizon, not the event horizon, which encodes the field theory entropy in the gravitational dual. On the other hand, the event horizon is teleological as its determination requires knowledge of the entire future evolution of the spacetime. Thus by using its area as a measure of entropy we would be predicting a drastic non-locality in the field theory dynamics.

References

[1] J. D. Bjorken, “Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region.” Phys. Rev. D27 (1983) 140-151.

[2] R. A. Janik and R. B. Peschanski, “Asymptotic perfect fluid dynamics as a consequence of AdS/CFT,” Phys. Rev. D73 (2006) 045013, arXiv:hep-th/0512162.

[3] R. A. Janik and R. B. Peschanski, “Gauge/gravity duality and thermalization of a boost-invariant perfect fluid,” Phys. Rev D74 (2006) 046007, arXiv:hep-th/0606149.

[4] S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, “Nonlinear Fluid Dynamics from Gravity,” JHEP 02 (2008) 045, arXiv:0712.2456 [hep-th].

[5] S. Kinoshita, S. Mukohyama, S. Nakamura, and K.-y. Oda, “A Holographic Dual of Bjorken Flow,” arXiv:0807.3797 [hep-th].