# Surface Gravity of Dynamical Horizons: A Causal Perspective

Anamika Avinash Pathak<EMAIL_ADDRESS>Konka Raviteja
<EMAIL_ADDRESS>Swastik Bhattacharya<EMAIL_ADDRESS>pilani.ac.in Sashideep Gutti<EMAIL_ADDRESS>Department
of Physics, Birla Institute of Technology and Sciences-Pilani
Hyderabad, 500078, India

###### Abstract

We consider marginally trapped surfaces in a spherically symmetric spacetime
evolving due to the presence of a perfect fluid in D-dimensions and look at
the various definitions of the surface gravity for these marginally trapped
surfaces. We show that using Einstein equations it is possible to simplify and
obtain general formulae for the surface gravity in terms of invariant
quantities defined at these marginally trapped surfaces like area radius,
cosmological constant and principal values of the energy-momentum tensor
$\rho,p$. We then correlate these expressions of surface gravity to the cases
of dynamical horizons and timelike tubes and find which proposals of surface
gravity are causally sensitive as these surfaces undergo causal transitions
from spacelike to timelike and vice versa.

††preprint: APS/123-QED

##  I Introduction

Black holes are among the most mysterious objects that exist in our universe.
The formation of black holes, their evolution, and their mergers are fields of
intense study over many decades. One intriguing feature of black holes is the
relation between gravity and thermodynamics. The event horizon of the black
hole is found to possess entropy and temperature. The origin of the entropy of
the black hole and its description in terms of microstates is yet to be
properly understood. The connection between thermodynamics and black holes is
a well-established area of physics. A relatively less understood phenomenon is
the thermodynamics of a black hole that is in the process of evolution.

Black hole thermodynamics has been an area of intense study and analysis since
the discovery of black hole spacetimes. The connection between the surface
gravity of a stationary/static black hole event horizon and temperature is
very well established. The relation between black hole entropy and its area
has been firmly placed on a strong theoretical foundation due to the presence
of Hawking radiation. Though the non-evolving black hole thermodynamics is
very well understood, the realistic scenario involving an evolving black hole
is still at the initial stages of its formulation. The main reason for this is
due to the fact that the dynamical phenomenon describing the formation and
evolution of a black hole is extremely complicated, that is except for highly
special situations, obtaining analytically solvable solutions in general
relativity is difficult.

To capture the features of evolving horizons and trapped regions, Ashtekar et
al. [1, 2] defined dynamical horizons. The dynamical horizon is a spacelike
hypersurface foliated by marginally trapped regions. Using this definition,
they prove an important result stating that the area of the dynamical horizon
always increases. They also defined timelike membranes where the evolving
horizon is timelike. Hayward in his paper [3] has refined the concept of
trapping horizons based on a 2+2 decomposition framework which introduced
different types of horizons like future outer trapped horizon (FOTH), future
inner trapped horizon (FITH), past outer trapped horizon (POTH) and past inner
trapped horizon (PITH). There are many works [4, 5, 6, 7] where solutions are
found for dynamical horizons, timelike membranes, and situations where an
evolving horizon makes a transition from a dynamical horizon to a timelike
membrane were discussed. Bousso in [8] has introduced the construction of past
holographic screens which are to be defined in terms of marginally trapped
surfaces, and in works with Engelhardt [9, 10] they have proved a new area law
in general relativity where the area of holographic screens follow a monotonic
evolution even though the causal nature of these screens changes during its
dynamics.

The thermodynamics of evolving horizons is a work in progress, as the
evolution of these horizons is a non-equilibrium phenomenon. An evolving
horizon can be a dynamical horizon or a timelike tube depending on its causal
nature and dynamical horizons tend to increase in their area while time-like
tubes tend to decrease in their area. In general, the dynamical horizons are
outer horizons (FOTH) while the timelike tubes are inner (FITH). Dynamical
horizons are more generic while timelike tubes occur in special circumstances
like Friedmann-Robertson-Walker (FRW) spacetime. There are therefore
fundamental differences between the nature of dynamical horizons and timelike
tubes. It is reasonable to expect that some thermodynamic properties, too,
would carry over the distinction between dynamical horizons and timelike
membranes. There have been various definitions for the surface gravity for the
dynamically evolving marginally trapped regions. We think that a good
formulation of surface gravity is crucial when one wants to define non-
equilibrium thermodynamic state variables for various astrophysically
realistic cases of evolving horizons. A few of the proposals for surface
gravity that are well known are, Kodama-Hayward surface gravity [11],
Hayward’s trapping horizon [3], Fodor et al. surface gravity [12], Booth and
Fairhurst surface gravity for the evolving horizon [13], these are well
described in [14]. The surface gravity expressions of various proposals for
the case of the general spherically symmetric metric are in Eddington-
Finkelstein coordinates and are written in terms of Misner-Sharp mass, metric
function, and their derivatives with respect to these coordinates. In the
paper [15], they discuss a few of the proposals for surface gravity and obtain
expressions for surface gravity using Painleve-Gullstrand coordinates and
highlight the differences between the proposals in a dynamical setting.

It is found in various solutions that the evolving horizon may transition from
being spacelike to timelike, and vice versa [4, 5, 6, 7]. When one wants to
study the thermodynamic aspects of the evolving horizons, it would be useful
to understand the behavior of various surface gravity proposals with respect
to the nature of causal transitions of the evolving horizons. The first paper
that addresses the issue of surface gravity and causal description of evolving
horizons is by [16]. The causal nature of an evolving horizon in Friedmann-
Lemaitre-Robertson-Walker (FLRW) spacetime is well known. For the case of
FLRW, the paper [16], evaluate the Kodama-Hawyard surface gravity and shows
that the surface gravity is sensitive to the causal nature of the evolving
horizon. We want to address this question in more a general context since FLRW
is a specific case restricted to the cosmological type solutions.

Our goal in this article is twofold. Firstly we consider the definitions of
various surface gravity proposals in the D-dimension. We show that for the
case of $D$ dimensional evolving marginally trapped surface, it is possible to
simplify and obtain elementary formulae for the surface gravity ideas. We
describe these formulae in simple terms of area radius $R$, the cosmological
constant $\Lambda$, the dimension D and the principal values of the energy-
momentum tensor. These formulae indicate that the surface gravity estimation
can be done only using local information at the evolving horizon and does not
depend upon any non-local information and the global aspects of the solution.
These formulae are obtained directly from simplifying the expressions using
Einstein’s equations and do not require the solutions of these Einstein’s
equations to define the surface gravity. We obtain these formulae for the
proposals of surface gravity by Kodama-Hayward [11], Fodor et al. [12], Booth-
Fairhurst [13] and also obtain expressions for trapping gravity by Hawyard
[3]. Secondly, we want to find out which definitions of surface gravity are
causally sensitive to the transitions among spacelike and timelike surfaces.
Using the general formula for each proposal, we find the relation between the
expression for the causal nature of the evolving horizon with the signature
for surface gravity. We find that Kodama-Hayward surface gravity gives a
positive value for the case of dynamical horizons and gives a negative value
if the evolving horizon is timelike.

##  II Causal nature of evolving marginally trapped surface

In this section, we review some of the results describing the causal aspects
of marginally trapped regions. The results can be found in [16, 5, 6, 17, 4].
Some of these results generalized to a $D$ dimensional scenario for the case
of spherically symmetric perfect fluid are derived in [7]. In the above
references, the criteria for the marginally trapped surface to be
timelike/null/spacelike is derived. Interestingly the causal nature of the
marginally trapped surface can be obtained using Einstein equations without
explicitly solving for the metric.

We assume a general metric for a $(D=n+2)$ dimensional spherically symmetric
spacetime is of the form

$ds^{2}=-e^{\sigma(t,r)}dt^{2}+e^{\lambda(t,r)}dr^{2}+R^{2}(t,r)~{}d{\Omega}_{n}^{2}$
(1)

where $d{\Omega}_{n}^{2}$ is the metric on an $n$ dimensional sphere of unit
radius with angular coordinates defined by
$(\theta_{1},\theta_{2},...,\theta_{n})$. Here, $t$ is the time coordinate, r
is the comoving radial coordinate, and $R(t,r)$ is the areal radius (we will
also refer to this as “physical radius”) of the n-dimensional sphere. The
advantage of comoving coordinates is that the metric remains regular across
the apparent horizon and becomes singular when the curvature singularity forms
(this statement is generally true as long as the initial conditions are such
that there are no shell crossing singularities). The matter we consider here
is a perfect fluid whose energy-momentum tensor is

$T_{\mu\nu}=(\rho(t,r)+p(t,r))u_{\mu}u_{\nu}+p(t,r)g_{\mu\nu}$ (2)

and the four-velocity in comoving coordinates is

$u^{\mu}=(e^{-\frac{\sigma}{2}},0,0,...,0)$ (3)

with $u^{\mu}u_{\mu}=-1$. The relevant Einstein equations and the relations
obtained by the conservation of energy-momentum tensor are given in Appendix
A. It is easily seen that the energy-momentum tensor of the perfect fluid is
diagonal in this coordinate system. As shown in [7], the formula for the
causal nature of the marginally trapped region can be expressed completely in
terms of coordinate invariants and the principal values of the energy-momentum
tensor ($\rho$ and $p$ in this context). To define the marginally trapped
regions, for the assumed metric (1), we define the future outgoing radial null
vector as

$k^{a}=(e^{-\frac{\sigma}{2}},e^{-\frac{\lambda}{2}},0,0....,0)$ (4)

and the future incoming radial null vector as

$l^{a}=(e^{-\frac{\sigma}{2}},-e^{-\frac{\lambda}{2}},0,0....,0)$ (5)

These null vectors are normalized as

$g_{ab}k^{a}l^{b}=-2$

Using these two null vectors, the induced metric on a codimension $D-2$
hypersurface that is orthogonal to the two null vectors is given by,

$h_{ab}=g_{ab}+\frac{1}{2}(k_{a}l_{b}+l_{a}k_{b})$

so the expansion for the congruence of outgoing null rays is

$\Theta_{k}=h^{ab}\nabla_{a}k_{b}=\frac{n}{R}\bigg{(}e^{-(\frac{\sigma}{2})}\dot{R}+e^{-(\frac{\lambda}{2})}R^{\prime}\bigg{)}$
(6)

and for completeness, the expansion for the congruence of incoming null rays
is

$\Theta_{l}=h^{ab}\nabla_{a}l_{b}=\frac{n}{R}\bigg{(}e^{-(\frac{\sigma}{2})}\dot{R}-e^{-(\frac{\lambda}{2})}R^{\prime}\bigg{)}$
(7)

The hypersurface given by the equation $\Theta_{k}=c$ is a curve foliated by a
marginally trapped region if we set the constant $c=0$, so the curve is a
marginally trapped tube. To obtain the causal nature of the curve, we find the
norm of normal $\beta_{k}$ to this curve evaluated at $\Theta_{k}=0$ and is
given by [7],

$\beta_{k}=-\bigg{(}(\pounds_{k}\Theta_{k})(\pounds_{l}\Theta_{k})\bigg{)}\bigg{|}_{\Theta_{k}=0}$
(8)

The lie derivative of $\Theta_{k}$ with respect to the outgoing null vector
$k^{a}$ and the incoming null vector $l^{a}$ are

$\pounds_{k}\Theta_{k}=k^{a}\nabla_{a}\Theta_{k}=e^{\frac{-\sigma}{2}}\partial_{t}\Theta_{k}+e^{\frac{-\lambda}{2}}\partial_{r}\Theta_{k}$
(9)
$\pounds_{l}\Theta_{k}=l^{a}\nabla_{a}\Theta_{k}=e^{\frac{-\sigma}{2}}\partial_{t}\Theta_{k}-e^{\frac{-\lambda}{2}}\partial_{r}\Theta_{k}$
(10)

These lie derivatives have to be evaluated at $\Theta_{k}=0$

$\pounds_{k}\Theta_{k}\bigg{|}_{\Theta_{k}=0}=-\tilde{\kappa}(\rho+p)$ (11)

and

$\pounds_{l}\Theta_{k}\bigg{|}_{\Theta_{k}=0}=\tilde{\kappa}(\rho-p)+2\Lambda-\frac{n(n-1)}{R^{2}}$
(12)

which gives us

$\beta_{k}=\tilde{\kappa}(\rho+p)\bigg{(}\tilde{\kappa}(\rho-p)+2\Lambda-\frac{n(n-1)}{R^{2}}\bigg{)}$
(13)

Instead of the normal, one can also obtain the causal nature of the curve
$\Theta_{k}=0$ using the ratio of lie derivatives. The ratio represents the
causal nature of the tangent to the curves that are foliated by marginally
trapped region, the proof of which is shown in [3, 18]. The ratio of the Lie
derivatives evaluated at $\Theta_{k}=0$ determines the causal nature of the
marginally trapped tube, which is

$\alpha_{k}=\frac{\pounds_{k}\Theta_{k}}{\pounds_{l}\Theta_{k}}\bigg{|}_{\Theta_{k}=0}=\frac{-\tilde{\kappa}(\rho+p)}{\tilde{\kappa}(\rho-p)+2\Lambda-\frac{n(n-1)}{R^{2}}}$
(14)

The causal nature is described by the sign of the norm of the normal or
tangent. Marginally trapped curve is timelike if $\beta_{k}>0$ , is spacelike
if $\beta_{k}<0$ and is null if $\beta_{k}=0$. We can see that $\beta_{k}$ and
$\alpha_{k}$ are always of opposite signs. So the causal criteria are reversed
for $\alpha_{k}$.

We note the fact that the formula for $\beta_{k}$ is completely local and does
not need the solution of the Einstein equations. The formula is described in
terms of geometric invariants and system parameters like area radius $R$ of
the marginally trapped surface, cosmological constant $\Lambda$, dimension of
the spacetime ($n=D-2)$ and the principal values of the energy-momentum tensor
$(\rho,p)$ at the location of the marginally trapped region. One can easily
see that by adjusting the density and pressure, one can obtain transitions of
the marginally trapped tube from timelike to spacelike and vice versa. We also
note that the norm of the normal is a better tool to look at these causal
transitions since $\beta_{k}$ goes to zero and hence regular while
$\alpha_{k}$ goes to infinity and therefore is analytically cumbersome at the
transition points.

Now we will analytically describe these transitions in the FRW spacetime
setting.
FRW case: The metric (1) can be brought to the standard FRW form,

$ds^{2}=-dt^{2}+a^{2}(t)\bigg{(}\frac{dr^{2}}{1-kr^{2}}+r^{2}~{}d{\Omega}_{n}^{2}\bigg{)}$
(15)

which is the higher dimensional spherically symmetric metric whose source is a
homogeneous perfect fluid. The function $a(t)$ has the standard interpretation
as the scale factor and $k$ takes the values in $(1,0,-1)$. For the metric
(15), the future incoming radial null vector is given by

$l^{a}=(1,-\frac{\sqrt{1-kr^{2}}}{a(t)},0,0....,0)$ (16)

and the future outgoing radial null vector is given by

$k^{a}=(1,\frac{\sqrt{1-kr^{2}}}{a(t)},0,0....,0).$ (17)

These are normalized to,

$g_{ab}k^{a}l^{b}=-2.$

The expansion scalar for the outgoing bundle of null rays is

$\Theta_{k}=h^{ab}\nabla_{a}k_{b}=\frac{n}{a(t)r}\bigg{(}\dot{a}(t)r+\sqrt{1-kr^{2}}\bigg{)}$
(18)

We know that the norm of the normal to the $\Theta_{k}=0$ curve can be
expressed as product of Lie derivatives (8). The lie derivative of
$\Theta_{k}$ with respect to the outgoing radial null vector is

$\pounds_{k}\Theta_{k}\bigg{|}_{\Theta_{k}=0}=-\tilde{\kappa}\rho(1+\omega)$
(19)

and, that with respect to the ingoing radial null vector is

$\pounds_{l}\Theta_{k}\bigg{|}_{\Theta_{k}=0}=\frac{n}{2R^{2}}\left(3-n-\omega(n+1)+\frac{2\Lambda
R^{2}(1+\omega)}{n}\right)$ (20)

The norm of the normal to the curves $\Theta_{k}=0$ can be expressed as

$\beta_{k}=\frac{n\tilde{\kappa}\rho(1+\omega)}{2R^{2}}\left(3-n-\omega(n+1)+\frac{2\Lambda
R^{2}(1+\omega)}{n}\right)$ (21)

As described earlier, the causal nature of the marginally trapped tube can
also be found using the ratio of lie derivatives, which represents the causal
nature of the tangent of the marginally trapped tube. The ratio of the lie
derivatives evaluated at $\Theta_{k}=0$ and $\Theta_{l}=0$ gives us

$\alpha_{k}=\frac{-2R^{2}\tilde{\kappa}\rho(1+\omega)}{n\left(3-n-\omega(n+1)+\frac{2\Lambda
R^{2}(1+\omega)}{n}\right)}$ (22)

Note that we have expressed the formula in terms of physical radius $R$
instead of $a(t)r$. The formula for the cosmological case is even more
elementary than the general case. If we assume $1+\omega$ is positive, the
sign of the expressions for $\beta_{k}$ and $\alpha_{k}$ are completely
determined by the following expression,

$\left(3-n-\omega(n+1)+\frac{2\Lambda R^{2}(1+\omega)}{n}\right)$ (23)

We note that the formula does not contain any dynamical variables of the model
and is expressed completely in terms of spacetime dimension ($n$), equation of
the state parameter ($\omega$), cosmological constant ($\Lambda$), and
physical radius ($R$) in this sense it is a geometrical result.

If we consider the case where the cosmological constant is zero, we see from
the above expression that as the marginally trapped region evolves, there is
no change of causal nature. It is uniformly timelike, spacelike, or null. At a
value of $\omega$ called $\omega_{critical}$ given below, the evolving
marginally trapped region is uniformly null. If $\omega<\omega_{critical}$, it
is timeline, and if $\omega>\omega_{critical}$, the marginally trapped tube is
spacelike.

$\omega_{critical}=-\frac{(n-3)}{(n+1)}$ (24)

The case where the equation of state coincides with $\omega_{critical}$ is
very special. The evolving marginally trapped tube is uniformly null. These
are called the null evolving horizons that do not fit in Hayward’s
classification criteria as shown in [7].

If we consider the case, $\Lambda\neq 0$ there exists a critical radius
$R_{critical}$ at which marginally trapped tube is null
($\beta=0=\alpha^{-1}$) and it also marks a transition of these curves from
spacelike region to timelike region or vice versa.

$R_{critical}^{2}=\frac{n(n-3)+n\omega(n+1)}{2\Lambda(1+\omega)}$

Whenever the marginally trapped tube crosses this critical radius, it makes a
transition in terms of its causal nature.

##  III Surface gravity of marginally trapped surfaces

In this section, we study the various proposals that define surface gravity in
a dynamic setting. For almost all the proposals, we obtain elementary formulae
where the surface gravity is obtained in terms of invariants and local
information like area radius, the cosmological constant, the number of
dimensions, and principal values of the energy-momentum tensor. These
simplified formulae are intuitively appealing and have not been reported
before in literature. With the help of these formulae, it is easy to compare
the behavior of various proposals for surface gravity for marginally trapped
regions with the causal behavior of the same. We note a useful result that the
surface gravity of $D$ dimensional Schwarzschild black hole is,
$\kappa=(D-3)/2R=(n-1)/2R$. This helps in fixing the arbitrary constant that
is due to the freedom of normalization of null rays.

###  III.1 Kodama-Hayward Surface Gravity

The paper [16], has worked on Kodama-Hayward surface gravity for the case of
marginally trapped surfaces in FRW spacetimes for the case of $D=4$
dimensions. The paper finds that Kodama-Hayward’s definition of surface
gravity is sensitive to the causal description of the evolving marginally
trapped surface. It is shown in the paper that for the FRW case, if we define
the perfect fluid with the equation of state given by, $\rho=\omega P$, for
$\omega<1/3$, the marginally trapped surface is timelike. The Kodama-Hayward
surface gravity is shown in [16] to be negative in this range. For
$\omega>1/3$, the marginally trapped surface is spacelike, and Kodama-Hayward
surface gravity is shown to be positive. The expressions for Kodama-Hayward
surface gravity for the dynamical scenario using Painleve-Gullstrand
coordinates were obtained in [15] in terms of derivatives of the Schwarzschild
mass. We now obtain a formula for a spherically symmetric scenario in $D$
dimensions for a perfect fluid. The Kodama vector for the spherically
symmetric spacetime generalized to $D$ dimensions is defined as,

$K^{\mu}=\frac{1}{\sqrt{-h}}\epsilon^{\mu\nu}\partial_{\nu}R$ (25)

where $R$ is the areal radius and $-h$ is the determinant of the metric
induced on the horizon $h_{ab}$ (on the two dimensional hypersurface
orthogonal to the $D-2$ dimensional sphere).

$\begin{split}\kappa_{K-H}=\frac{C_{1}}{\sqrt{-h}}\epsilon^{\alpha}_{\mu}\partial_{\alpha}K^{\mu}=\\\
\frac{C_{1}}{\sqrt{-h}}\frac{\partial}{\partial
x^{\mu}}\bigg{(}\sqrt{-h}h^{\mu\nu}\frac{\partial}{\partial
x^{\nu}}R\bigg{)}\end{split}$ (26)

The constant $C_{1}$ is dependent on the normalization of the Kodama vector
and is fixed indirectly by matching the value of the surface gravity for the
known static case in D dimensions. For the metric (1), using $\sqrt{-h}$ =
eσ/2 eλ/2 and evaluating the above expression, we obtain,

$\kappa_{K-H}=C_{1}\bigg{(}-e^{-\sigma}(\ddot{R}+\dot{R}(\frac{\dot{\lambda}-\dot{\sigma}}{2}))+e^{-\lambda}(R^{\prime\prime}+R^{\prime}\frac{(\sigma^{\prime}-\lambda^{\prime})}{2})\bigg{)}$
(27)

where, $\dot{R}=\frac{\partial R}{\partial t}$ and $R^{\prime}=\frac{\partial
R}{\partial r}$. We show that the complicated expression can, surprisingly, be
simplified to a simple form using Einstein equations for the marginally
trapped regions.

For the apparent horizon, the outgoing null vector is given by (4). The
condition for the outgoing null ray to be marginally trapped is obtained by
setting $\Theta_{k}$ to zero.

$\Theta_{k}=\frac{n}{R}\bigg{(}e^{-(\frac{\sigma}{2})}\dot{R}+e^{-(\frac{\lambda}{2})}R^{\prime}\bigg{)}=0$
(28)

This gives

$\dot{R}e^{-\sigma/2}=-R^{\prime}e^{-\lambda/2}$ (29)

Using the above relation, we can write

$-e^{-\sigma}\frac{\dot{R}\dot{\lambda}}{2}=e^{-\frac{(\sigma+\lambda)}{2}}\frac{R^{\prime}\dot{\lambda}}{2}$
(30)

and

$e^{-\lambda}\frac{R^{\prime}\sigma^{\prime}}{2}=-e^{-\frac{(\sigma+\lambda)}{2}}\frac{\dot{R}\sigma^{\prime}}{2}$
(31)

We now simplify equation 27 using the Einstein equations given in Appendix A.
We calculate $G_{00}-G_{11}$ (using equations 67 and 68 to obtain the
expression 70. We simplify the equation 70, making use of the results 29,30,
31). The expression 27 simplifies to the equation below.

$\kappa_{K-H}=C_{1}\bigg{(}\frac{n-1}{R}-\frac{R}{n}(\tilde{\kappa}(\rho-p)+2\Lambda)\bigg{)}$
(32)

Comparing the result for zero density, pressure, and the cosmological
constant, with the surface gravity of D-dimensional Scwarzschild spacetime, we
get $C_{1}=1/2$. We observe that the Kodama-Hayward surface gravity is
completely determined by the local information available at the marginally
trapped region. We note that if the apparent horizon is isolated, then the
curve $\Theta_{k}=0$ is null. We set $(\rho=0,p=0)$ in the above equation and
see that the surface gravity in $D$ dimensions is proportional to $1/R$. We
recover the surface gravity of the static black hole horizon when the formula
is adopted for the non-evolving scenario. We also recover the surface gravity
of the de Sitter horizon or cosmological horizon by setting $\rho=0,p=0$. We
see that the surface gravity of the de Sitter cosmological horizon in a pure
de Sitter spacetime is $-\sqrt{2\Lambda}/\sqrt{n(n+1)}$. For FRW case, using
reference [19], we obtain the following formula for the perfect fluid for the
equation of state given below,

$p=\omega\rho.$ (33)
$\kappa_{K-H}=\frac{1}{4}\bigg{(}\frac{n-3+\omega(n+1)}{R}-\frac{2\Lambda
R}{n}(1+\omega)\bigg{)}$ (34)

We note that this formula is derived by simplifying the expressions for the
cosmological case. Setting $\omega=0$ gives us the formula for dust. This
formula is valid for FRW spacetime ($\rho\neq 0$).

####  III.1.1 Causal correlation of Kodama-Hayward surface gravity

In the previous section, we observed the possibilities of transitions of an
evolving horizon from spacelike to timelike and vice versa. We now examine the
value of surface gravity as the marginally trapped region changes from
timelike to spacelike. To find this relation, we divide the surface gravity in
32 with 13. We get,

$\frac{\kappa_{K-H}}{\beta_{k}}=\frac{\bigg{(}\frac{n-1}{R}-\frac{R}{n}(2\tilde{\kappa}(\rho-p)+2\Lambda)\bigg{)}}{\tilde{\kappa}(\rho+p)\bigg{(}\tilde{\kappa}(\rho-p)+2\Lambda-\frac{n(n-1)}{R^{2}}\bigg{)}}$
(35)

The ratio can be simplified to be $-R/(2n\tilde{\kappa}(\rho+p))$. If we
assume the energy condition that $\rho+p>0$, we see that the ratio is a
negative definite quantity. This implies that always the surface gravity is
negatively correlated with the norm of the normal to the marginally trapped
region. This implies that for the dynamical horizon where the marginally
trapped curve is spacelike, the Kodama-Hayward surface gravity is also
positive. For the case when the marginally trapped curve is timelike that is
for timelike tubes, the surface gravity is negative.

In the FRW case, an interesting class of solutions where the evolving
marginally trapped region is null. These are degenerate cases that escape the
classification criteria of evolving marginally trapped regions into outer and
inner horizons [7]. For four dimensions, this corresponds to an equation of
state given by $p=\rho/3$ and for a general dimension $D=n+2$, we have
$p=-\rho(n-3)/(n+1)$. The surface gravity corresponding to these cases is
$\kappa_{K-H}=0$.

Also, in the presence of a cosmological horizon, the evolving marginally
trapped region transitions from spacelike to timelike when the curve passes
through the critical radius given by (II). So the surface gravity too
transitions from a positive value to a negative value passing through zero.
The nature of transitions needs to be explored further and is left for future
consideration.

###  III.2 Hayward’s trapping gravity

Another useful quantity defined by Hayward [3], is called trapping gravity. It
is defined below.

$\kappa_{H}=\frac{1}{2}\sqrt{-l^{\alpha}\Theta_{k;\alpha}}=\frac{1}{2}\sqrt{-\pounds_{l}\Theta_{k}}$
(36)

We now provide a formula for estimating trapping gravity using equation 12,
(from [19])

$\kappa_{H}=\frac{1}{2}\sqrt{\frac{n(n-1)}{R^{2}}-2\Lambda-K(\rho-p)}$ (37)

for a general spherically symmetric perfect fluid scenario.

For the FRW case, we obtain the formula below for a perfect fluid with the
equation of state given by, $p=\omega\rho$. Using (20), we obtain the formula
for trapping horizon in FRW gravity to be,

$\kappa_{H}=\frac{1}{2}\sqrt{\frac{\bigg{(}n-3+\omega(n+1)\bigg{)}n}{2R^{2}}-\Lambda(1+\omega)}$
(38)

As can be seen from both the expressions of Hayward gravity, these are defined
only for the case of dynamical horizons and not timelike tubes. This is
because of the square root. The expression inside the square root is positive
(assuming the null energy condition is preserved) only when the marginally
trapped region is a spacelike curve and is negative if the marginally trapped
region is described by a timelike curve. We note that the expressions inside
the square root are causally sensitive in the same sense as the Kodama-Hayward
surface gravity. The value of Hayward’s trapping horizon becomes imaginary for
timelike tubes.

###  III.3 Fodor’s method

We now consider the definition of surface gravity due to Fodor et al. [12]. We
again try to obtain a formula for Fodor’s surface gravity. Obtaining an
expression for the surface gravity of a perfect fluid is found to be
analytically difficult. We instead obtain a formula where the marginally
trapped region is evolving due to dust. We, therefore, set the pressure to
zero. We find that the expression simplifies to give a closed form. We then
use the obtained expression to find the causal correlation. We recall the
definition of Fodor’s surface gravity as,

$\kappa_{F}=-l^{\beta}k^{\alpha}k_{\beta;\alpha}$ (39)

with the same definitions of $k$ and $l$ as given in the earlier sections. The
normalization prescribed in Fodor et al. [12], is given by

$l^{\alpha}k_{\alpha}=-1$

However, we work with a different normalization and finally decide on the
normalization by matching the surface gravity that is defined up to a
multiplicative constant, with the static black hole case. For the metric
defined in (1), we get the surface gravity to be,

$\kappa_{F}=\frac{C_{2}}{2\sqrt{2}}\bigg{(}e^{-\lambda/2}\sigma^{\prime}+e^{-\sigma/2}\dot{\lambda}\bigg{)}$
(40)

where $C_{2}$ is fixed by comparing with the known static scenario. For the
case of dust, the metric 1 becomes that of a D-dimensional Lemaitre Toman
Bondi (LTB) model. Comparing with the D-dimensional LTB model, [6], we get the
metric coefficient $\sigma=0$ and,

$e^{\lambda}=R^{\prime 2}$ (41)

We simplify the expression (40) using the above relation. We get,

$\kappa_{F}=\frac{C_{2}}{2\sqrt{2}}\dot{\lambda}$ (42)

Now, from the expression(41) we have

$\dot{\lambda}=\frac{2\dot{R}^{\prime}}{R^{\prime}}$

Using the standard relation for the LTB models [6]

$\dot{R}^{2}=\frac{F(r)}{R^{n-1}}+\frac{2\Lambda R^{2}}{n(n+1)}$ (43)

where $F(r)$ is a function only of the comoving radius $r$ and has the
interpretation of Misner-Sharp mass. It is the total mass within a shell with
comoving label $r$. For marginally trapped and anti-trapped surfaces, it is
shown in [6] that $\dot{R}^{2}=+1$ holds true. Here $\dot{R}$ is the
derivative of the area radius with comoving time. For the marginally trapped
case, we choose, $\dot{R}=-1$. We also have from reference [6],

$\frac{F^{\prime}}{2R^{n}R^{\prime}}=\frac{\tilde{\kappa}\rho}{n}$ (44)

Now by differentiating 43 w.r.t $r$, we arrive at the expression

$2\dot{R}\dot{R}^{\prime}=\frac{F^{\prime}}{R^{(n-1)}}+\frac{4\Lambda
RR^{\prime}}{n(n+1)}-(n-1)\frac{FR^{\prime}}{R^{n}}$ (45)

Dividing both sides by $R^{\prime}$, we arrive at,

$\frac{\dot{R}^{\prime}}{R^{\prime}}=R\bigg{(}\frac{-\tilde{\kappa}\rho}{n}-\frac{2\Lambda}{n(n+1)}+\frac{(n-1)F}{2R^{(n+1)}}\bigg{)}$
(46)

For the marginally trapped region, we set $\Theta_{k}=0$. This implies [6],

$\frac{F}{R^{n+1}}=\frac{1}{R^{2}}-\frac{2\Lambda}{n(n+1)}$ (47)

Using the above expression, equation (46) gives

$\frac{\dot{R}^{\prime}}{R^{\prime}}=\frac{n-1}{2R}-R\bigg{(}\frac{\tilde{\kappa}\rho}{n}+\frac{\Lambda}{n}\bigg{)}$
(48)

Using this, we arrive at the expression for surface gravity to be,

$K_{F}=C_{2}\bigg{(}\frac{n-1}{2R}-\frac{R}{n}(\tilde{\kappa}\rho+\Lambda)\bigg{)}$
(49)

Compared with $D$ dimensional Schwarzschild case, we get that $C_{2}=1$. We
see that the equation (49) matches with the result in [12] if we set the
dimension $D=4(n=2)$ and equate the cosmological constant to zero. In this
sense, the result obtained is a generalization of the result in [12] to $D$
dimensions and with a cosmological constant term.

####  III.3.1 Causal correlation of $\kappa_{F}$

For correlating with the causal nature of the marginally trapped surface, we
compare the expression for the surface gravity with the formula for causal
nature. Since the formula obtained in the Fodor et al. case is only for dust,
we adopt the formula by setting the pressure to zero.

$\frac{\kappa_{F}}{\beta_{k}}=\frac{\bigg{(}\frac{n-1}{2R}-\frac{R}{n}(\tilde{\kappa}\rho+\Lambda)\bigg{)}}{\tilde{\kappa}\rho\bigg{(}\tilde{\kappa}\rho+2\Lambda-\frac{n(n-1)}{R^{2}}\bigg{)}}$
(50)

We can rearrange the denominator to arrive at the,

$\frac{\kappa_{F}}{\beta_{k}}=\frac{-R\bigg{(}\frac{n-1}{2R}-\frac{R}{n}(\tilde{\kappa}\rho+\Lambda)\bigg{)}}{2n\tilde{\kappa}\rho\bigg{(}\frac{n-1}{2R}-\frac{R}{n}(\frac{\tilde{\kappa}\rho}{2}+\Lambda)\bigg{)}}$
(51)

Again assuming that $\rho+p>0$, we observe that there is a factor of 2 as the
coefficient in the density term $\rho$ that makes a difference from
$\kappa_{F}$ being causally correlated. The ratio
$\frac{\kappa_{F}}{\beta_{k}}$ is therefore not negative definite and hence
the relation between the causal nature of marginally trapped region breaks
down. However, there is a match between $\kappa_{KH}$ and $\kappa_{F}$ when
the energy density is zero. Also, there is an agreement between $\kappa_{KH}$
and $\kappa_{F}$ when the energy densities are small or when the area radius
of the marginally trapped region is small.

###  III.4 Booth and Fairhurst method

The dynamical surface gravity is defined in the paper [13]. This definition is
tailored for the case of a slowly evolving horizon. The criteria for the same
is described in [13]. Using the idea in [13], we can define the tangent to the
marginally trapped curve to be $V^{\alpha}=k^{\alpha}-\alpha_{k}l^{\alpha}$
and normal to be $\tau_{\alpha}=k_{\alpha}+\alpha_{k}l_{\alpha}$. The norm for
each vector is evaluated with a normalization $k^{\alpha}l_{\alpha}=-2$ is
given by $\alpha_{k}$ (14) and $\beta_{k}$ (13) respectively. We define
surface gravity to be,

$K_{BF}=-l^{\alpha}k^{\beta}k_{\alpha;\beta}-\alpha_{k}k^{\alpha}l^{\beta}l_{\alpha;\beta}$
(52)

To evaluate the above expression, we have

$l^{\beta}k^{\alpha}k_{\beta;\alpha}=-C_{3}\bigg{(}e^{-\lambda/2}\sigma^{\prime}+e^{-\sigma/2}\dot{\lambda}\bigg{)}$
$k^{\alpha}l^{\beta}l_{\alpha;\beta}=C_{3}\bigg{(}e^{-\lambda/2}\sigma^{\prime}-e^{-\sigma/2}\dot{\lambda}\bigg{)}$

after putting the metric coefficient $\sigma$ = 0, we get

$l^{\beta}k^{\alpha}k_{\beta;\alpha}=-C_{3}\dot{\lambda}$ (53)

and

$k^{\alpha}l^{\beta}l_{\alpha;\beta}=-C_{3}\dot{\lambda}$ (54)

Hence equation (52) gives us

$K_{BF}=C_{3}\bigg{(}1+\alpha_{k}\bigg{)}\dot{\lambda}$ (55)

Just as in the previous section, we evaluate the surface gravity for the
analytically tractable case of dust. We set the pressure to zero. We then
obtain the equation for the scalar $\alpha_{k}$ to be

$\alpha_{k}=\frac{-\tilde{\kappa}\rho}{\tilde{\kappa}\rho+2\Lambda-\frac{n(n-1)}{R^{2}}}$
(56)

and $\dot{\lambda}$ is evaluated in Fodor’s method. This yields

$\kappa_{BF}=\frac{C_{3}\bigg{(}\frac{n-1}{2R}-\frac{R}{n}(\tilde{\kappa}\rho+\Lambda)\bigg{)}\bigg{(}2\Lambda-\frac{n(n-1)}{R^{2}}\bigg{)}}{\tilde{\kappa}\rho+2\Lambda-\frac{n(n-1)}{R^{2}}}$
(57)

We see again that the expressions match with the static case of $D$
dimensional Schwarzschild black hole surface gravity where we set density
$\rho=0$ and $\Lambda=0$. Comparison sets the value of $C_{3}=1$. The result
matches with D-dimensional Schwarzschild deSitter surface gravity.

####  III.4.1 Causal correlation of the Booth-Fairhurst surface gravity

An inspection of (57) indicates that the denominator of the expression is
positive definite with respect to $\beta_{k}$, but the numerator is not. This
makes the expression $\kappa_{BF}$ not causally correlated. However, for the
slowly evolving scenario for which the expression was the originally intended
purpose of defining the surface gravity [13].

##  IV Conclusions

In this article, we consider the marginally trapped surfaces which can be
spacelike, timelike, or null depending on the local information at the
horizon, i.e density $\rho$, pressure p, area radius $R$, and cosmological
constant $\Lambda$. We analyze the various ideas regarding the definition of
surface gravity when the horizon is evolving. We derived the following simple
formulae for the various proposals. The Kodama-Hayward surface gravity for the
general case is (32). For the case of FLRW, we obtain the formula (34). We
showed that both formulae are sensitive to causal nature. The formula yields
that surface gravity is positive for dynamical horizons and negative for
timelike tubes. The surface gravity smoothly transitions whenever the evolving
horizon makes a transition from timelike to spacelike or vice versa. The
formula obtained for Kodama-Hayward surface gravity holds for any spherically
symmetric situation with and without cosmological constant for any density and
pressure of the fluid. This definition of surface gravity seems better suited
to study thermodynamic aspects of evolving horizons since it is sensitive to
the causal nature of the horizon. This also holds promise for future analysis
between the nature of causal transitions and non-equilibrium thermodynamic
state variables that could be defined on the horizons.
We obtained the formula for Hayward’s trapping gravity (37), and similarly,
for the FLRW case the formula is derived in (38). Based on the expressions, it
is clear that this quantity is defined only for dynamical horizons and this
quantity becomes imaginary for timelike tubes. We obtained the formula for
Fodor et al. surface gravity in (49). This formula is obtained for the case of
zero pressure and with the cosmological constant. The case with pressure is
found to be analytically not tractable and a closed-form expression was not
possible. Nevertheless, we could obtain the causal correlation for the Fodor
et al case and find that there is a mismatch with the causal description. The
parameter space where Fodor’s surface gravity transitions in terms of its sign
is different from the transitions in evolving horizons. We similarly obtain
the formula for Booth and Fairhurst’s proposal for surface gravity in (52).
Again the expressions were obtainable in a closed form only for the case of
zero pressure. Just as in Fodor’s case, the Booth and Fairhurst proposal does
not correlate with the causal description of the evolving horizon in terms of
changing the sign of surface gravity when one goes from spacelike to timelike
evolving horizons. These findings in the article, are therefore crucial for
the first steps towards defining thermodynamic variables in evolving horizons.

##  V Appendix

The nonzero components of the energy-momentum tensor are listed below

$\displaystyle T_{00}=\rho e^{\sigma}$ $\displaystyle T_{11}=pe^{\lambda}$
$\displaystyle T_{22}=pR^{2}$ $\displaystyle
T_{(l+1~{}l+1)}=sin^{2}{\theta_{(l-1)}}T_{(ll)}$

where $l$ takes values from 2 to n. From the Bianchi identities

${T^{\mu\nu}}_{;\nu}=0$ (58)

we get the following relations

$\dot{\rho}+\frac{(\rho+p)}{2}\bigg{(}\frac{2n\dot{R}}{R}+\dot{\lambda}\bigg{)}=0$
(59) $p^{\prime}+\frac{\sigma^{\prime}}{2}(p+\rho)=0$ (60)

where the dot represents a derivative with time coordinate and the prime
represents a derivative with the comoving radial coordinate.

For a nonzero cosmological constant ($\Lambda\neq 0$), the Einstein equations
are

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\tilde{\kappa}T_{\mu\nu}$ (61)

here $\tilde{\kappa}$ is a constant and is related to gravitational constant
$G_{n}$, ($\tilde{\kappa}$ = 8$\pi G_{n}$). With these conditions, we evaluate
the left-hand side components of the Einstein equation $(G_{\mu\nu}+\Lambda
g_{\mu\nu})$ which are summarized below

$\displaystyle G_{00}+\Lambda
g_{00}=\frac{e^{-\lambda}}{R^{2}}\bigg{[}\frac{n(n-1)}{2}(e^{\lambda+\sigma}+e^{\lambda}\dot{R}^{2}-e^{\sigma}R^{\prime
2})$
$\displaystyle+\frac{n}{2}R(-2R^{\prime\prime}e^{\sigma}+e^{\sigma}R^{\prime}\lambda^{\prime}+e^{\lambda}\dot{R}\dot{\lambda})-\Lambda
e^{\lambda+\sigma}R^{2}\bigg{]}$ (62) $G_{01}+\Lambda
g_{01}=\frac{n}{2}\frac{(R^{\prime}\dot{\lambda}-2\dot{R}^{\prime}+\sigma^{\prime}\dot{R})}{R}$
(63) $\displaystyle G_{11}+\Lambda
g_{11}=\frac{e^{-\sigma}}{R^{2}}\bigg{[}-\frac{n(n-1)}{2}(e^{\lambda+\sigma}+e^{\lambda}\dot{R}^{2}-e^{\sigma}R^{\prime
2})$
$\displaystyle+\frac{n}{2}R(e^{\sigma}R^{\prime}\sigma^{\prime}+e^{\lambda}(\dot{R}\dot{\sigma}-{\color[rgb]{0,0,0}2\ddot{R}}))+\Lambda
e^{\lambda+\sigma}R^{2}\bigg{]}$ (64) $\displaystyle G_{22}+\Lambda
g_{22}=\frac{e^{-(\lambda+\sigma)}}{4}\bigg{[}2(n-1)(n-2)(e^{\sigma}R^{\prime
2}-e^{\lambda+\sigma}$
$\displaystyle-e^{\lambda}\dot{R}^{2})-2(n-1)R(e^{\sigma}R^{\prime}(\lambda^{\prime}-\sigma^{\prime})-2e^{\sigma}R^{\prime\prime}$
$\displaystyle+e^{\lambda}(\dot{R}(\dot{\lambda}-\dot{\sigma})+2\ddot{R}))+R^{2}(4e^{\lambda+\sigma}\Lambda-e^{\lambda}(2\ddot{\lambda}+\dot{\lambda}^{2}$
$\displaystyle-\dot{\lambda}\dot{\sigma})+e^{\lambda}(2\sigma^{\prime\prime}+{\sigma^{\prime}}^{2}-\lambda^{\prime}\sigma^{\prime}))\bigg{]}$

The other nonzero relations are given by

$G_{(j+1~{}j+1)}=sin^{2}{\theta_{(j-1)}}G_{(jj)}$ (65)

where $j$ takes values from 2 to $n$. So from the $G_{01}=0$ Einstein equation
we get

$R^{\prime}\dot{\lambda}-2\dot{R}^{\prime}+\sigma^{\prime}\dot{R}=0$ (66)

from the $G_{00}=\tilde{\kappa}\rho e^{\sigma}$ equation we have

$\displaystyle\frac{n(n-1)}{2}(e^{\lambda+\sigma}+e^{\lambda}\dot{R}^{2}-e^{\sigma}R^{\prime
2})$ (67)
$\displaystyle+\frac{n}{2}R(-2R^{\prime\prime}e^{\sigma}+e^{\sigma}R^{\prime}\lambda^{\prime}+e^{\lambda}\dot{R}\dot{\lambda})=(\tilde{\kappa}\rho+\Lambda)R^{2}e^{\lambda+\sigma}$

and $G_{11}=\tilde{\kappa}pe^{\lambda}$ equation we get

$\displaystyle-\frac{n(n-1)}{2}(e^{\lambda+\sigma}+e^{\lambda}\dot{R}^{2}-e^{\sigma}R^{\prime
2})$ (68)
$\displaystyle+\frac{n}{2}R(e^{\sigma}R^{\prime}\sigma^{\prime}+e^{\lambda}(\dot{R}\dot{\sigma}-2\ddot{R}))=(\tilde{\kappa}p-\Lambda)R^{2}e^{\lambda+\sigma}$

The following two expressions will be useful in the subsequent calculations
done in the paper. The sum (67) + (68) gives,

$\displaystyle\frac{n}{2}\bigg{(}e^{\sigma}R^{\prime}(\sigma^{\prime}+\lambda^{\prime})+e^{\lambda}\dot{R}(\dot{\sigma}+\dot{\lambda})-2(e^{\lambda}\ddot{R}+e^{\sigma}R^{\prime\prime})\bigg{)}$
$\displaystyle=\tilde{\kappa}(\rho+p)e^{(\lambda+\sigma)}R$ (69)

similarly the difference (67) - (68) gives us

$\displaystyle
n(n-1)(e^{\lambda+\sigma}+e^{\lambda}\dot{R}^{2}-e^{\sigma}R^{\prime 2})$ (70)
$\displaystyle+\frac{nR}{2}(e^{\sigma}(R^{\prime}\lambda^{\prime}-2R^{\prime\prime}-R^{\prime}\sigma^{\prime})-e^{\lambda}(\dot{R}\dot{\sigma}-2\ddot{R}-\dot{R}\dot{\lambda}))$
$\displaystyle=(\tilde{\kappa}(\rho-p)+2\Lambda)R^{2}e^{\lambda+\sigma}$

## References

  * [1] Abhay Ashtekar and Badri Krishnan.  Dynamical horizons: energy, angular momentum, fluxes, and balance laws.  Physical review letters, 89(26):261101, 2002.
  * [2] Abhay Ashtekar and Badri Krishnan.  Isolated and dynamical horizons and their applications.  Living Reviews in Relativity, 7(1):10, 2004.
  * [3] Sean A Hayward.  General laws of black-hole dynamics.  Physical Review D, 49(12):6467, 1994.
  * [4] Ivan Booth, Lionel Brits, Jose A Gonzalez, and Chris Van Den Broeck.  Marginally trapped tubes and dynamical horizons.  Classical and Quantum Gravity, 23(2):413, 2005.
  * [5] Alexis Helou, Ilia Musco, and John C Miller.  Causal nature and dynamics of trapping horizons in black hole collapse.  Classical and Quantum Gravity, 34(13):135012, 2017.
  * [6] Konka Raviteja and Sashideep Gutti.  Aspects of marginally trapped and antitrapped surfaces in a d-dimensional evolving dust model.  Physical Review D, 102(2):024072, 2020.
  * [7] Konka Raviteja, Asrarul Haque, and Sashideep Gutti.  Causal description of marginally trapped surfaces in $d$ dimensions.  Phys. Rev. D, 103:124005, Jun 2021.
  * [8] Raphael Bousso.  Holography in general space-times.  Journal of High Energy Physics, 1999(06):028, 1999.
  * [9] Raphael Bousso and Netta Engelhardt.  New area law in general relativity.  Physical review letters, 115(8):081301, 2015.
  * [10] Raphael Bousso and Netta Engelhardt.  Proof of a new area law in general relativity.  Physical Review D, 92(4):044031, 2015.
  * [11] Sean A Hayward.  Energy and entropy conservation for dynamical black holes.  Physical Review D, 70(10):104027, 2004.
  * [12] Gyula Fodor, Kouji Nakamura, Yoshimi Oshiro, and Akira Tomimatsu.  Surface gravity in dynamical spherically symmetric spacetimes.  Phys. Rev. D, 54:3882–3891, Sep 1996.
  * [13] Ivan Booth and Stephen Fairhurst.  The first law for slowly evolving horizons.  Phys. Rev. Lett., 92:011102, Jan 2004.
  * [14] Alex B Nielsen and Jong Hyuk Yoon.  Dynamical surface gravity.  Classical and Quantum Gravity, 25(8):085010, apr 2008.
  * [15] Mathias Pielahn, Gabor Kunstatter, and Alex B. Nielsen.  Dynamical surface gravity in spherically symmetric black hole formation.  Phys. Rev. D, 84:104008, Nov 2011.
  * [16] Valerio Faraoni.  Cosmological apparent and trapping horizons.  Phys. Rev. D, 84:024003, Jul 2011.
  * [17] Valerio Faraoni.  Correspondence between a scalar field and an effective perfect fluid.  Physical Review D, 85(2):024040, 2012.
  * [18] Olaf Dreyer, Badri Krishnan, Deirdre Shoemaker, and Erik Schnetter.  Introduction to isolated horizons in numerical relativity.  Physical Review D, 67(2):024018, 2003.
  * [19] Konka Raviteja, Asrarul Haque, and Sashideep Gutti.  Causal description of marginally trapped surfaces in $d$ dimensions.  Phys. Rev. D, 103:124005, Jun 2021.


11institutetext: Faculty of Mathematics and Informatics
West University of Timişoara
11email<EMAIL_ADDRESS>
# Benchmarking Optimization Solvers and Symmetry Breakers for the Automated
Deployment of Component-based Applications
in the Cloud
EXTENDED ABSTRACT

Bogdan David  Mădălina Eraşcu

###### Abstract

Optimization solvers based on methods from constraint programming (OR-Tools,
Chuffed, Gecode), optimization modulo theory (Z3), and mathematical
programming (CPLEX) are successfully applied nowadays to solve many non-
trivial examples. However, for solving the problem of _automated deployment_
in the Cloud of _component-based applications_ , their computational
requirements are huge making automatic optimization practically impossible
with the current _general_ optimization techniques. To overcome the
difficulty, we exploited the sweet spots of the underlying problem in order to
identify search space reduction methods. We came up with $15$ symmetry
breaking strategies which we tested in a static symmetry breaking setting on
the solvers enumerated above and on $4$ classes of problems.

As a result, all symmetry breaking strategies led to significant improvement
of the computational time of all solvers, most notably, Z3 performed the best
compared to the others. As an observation, the symmetry breaking strategies
confirmed that, when applied in a static setting, they may interact badly with
the underlying techniques implemented by the solvers.

###### Keywords:

Cloud Computing resource provisioning Wordpress optimization modulo theory,
mathematical programming constraint programming symmetry breaking Minizinc OR-
Tools Chuffed Gecode Z3 IBM CPLEX.

##  1 Introduction

The problem of _automated deployment_ in the Cloud of component-based
applications received attention due to increased demand of digitalization of
businesses. It consists of the following steps:  (1) selection of the
computing resources,  (2) the distribution/assignment of the application
components over the available computing resources, and  (3) its dynamic
modification to cope with peaks of user requests.  In paper [3], we tackled
only the first two steps of the deployment problem. In particular, our
approach was used to synthesize the initial static optimal deployment of the
application which consists of an assignment of application components to VMs
such that the application functional requirements are fulfilled and costs are
minimized.

The contributions of [3] are:  (i) we formalized the Cloud deployment problem
by abstracting the particularities of four classes of real-world problems;
(ii) we proposed a methodology analyzing the particularities of the problem
with the aim of identifying search space reduction methods (these are methods
exploiting the symmetries of the general Cloud deployment problem,
respectively methods utilizing the graph representation of the interaction
between the components of each application);  (iii) we assessed and compared
the performance of two types of tools, namely mathematical programming (CPLEX
[5]) and computational logic (the optimization modulo theory solver Z3 [6]);
(iv) we identified limits in their scalability and applied six search space
reduction methods aiming to improve their performance.

This paper extends [3] in the following ways:

  1. 1.

a new formalization in the Minizinc [7] constraint modeling language;

  2. 2.

the performance comparison adds the constraint programming solvers OR-Tools
[8], Gecode [4] and Chuffed [2] which are available from the Minizinc IDE;

  3. 3.

the list of the symmetry breakers from [3] is enriched with composition of all
possible combinations of single symmetry breakers. These symmetry breakers are
tested on the constraint programming solvers OR-Tools [8], Gecode [4] and
Chuffed [2], optimization modulo theory solver Z3 [1], and mathematical
programming solver CPLEX [5].

The rest of the paper is organized as follows. Section 2 briefly introduces
the problem and the particularities of each formalization. Section 3 shows the
need for strategies to reduce the search space and, to this aim, introduces
symmetry breaking techniques as well as the principles to combine them.
Conclusions of the experimental analysis are presented in Section 4.

##  2 Setting the Scene

##### Problem Definition

The description of the problem first appeared in [3]. We have $N$ interacting
components, $C=\\{C_{1},\ldots,C_{N}\\}$, to be assigned to a set of $M$
virtual machines, $V=\\{V_{1},\ldots,V_{M}\\}$. Each component $C_{i}$ is
characterized by a set of requirements concerning the hardware resources. Each
virtual machine, $V_{k}$, is characterized by a _type_ , which is comprised by
hardware/software characteristics and leasing price. There are also
_structural constraints_ describing the interactions between components. The
problem is to find:

  1. 1.

an assignment matrix $a$ with binary entries $a_{ik}\in\\{0,1\\}$ for
$i=\overline{1,N}$, $k=\overline{1,M}$, which are interpreted as follows:
$a_{ik}=1\hbox{ if }C_{i}\hbox{ is assigned to }V_{k},\text{ and }0$,
otherwise; and

  2. 2.

the type selection vector t with integer entries $\emph{t}_{k}$ for
$k=\overline{1,M}$, representing the type (from a predefined set) of each VM
leased;

such that:  (i) the structural constraints, and  (ii) the hardware
requirements (capacity constraints) of all components are satisfied; and
(iii) the purchasing/ leasing price is minimized.

The _structural constraints_ are _application-specific_ and derived in
accordance with the analysis of the case studies. These are:

  * •

_Conflict:_ components in conflict cannot be deployed on the same VM.

  * •

_Co-location:_ components in co-location must be deployed on the same VM.

  * •

_Exclusive deployment:_ Only one of the components in exclusive deployment
must be deployed in the same deployment plan.

  * •

_Require-Provide:_ one component requires or provides some functionalities
offered, respectively provides, of another. Such an interaction induces
constraints on the number of instances corresponding to the interacting
components as follows.

  * •

_Full deployment:_ components in this relationship must be deployed on all VMs
leased, except those which would induce conflicts between components.

  * •

_Deployment with bounded number of instances_ occur when the number of
instances of deployed components must be equal, greater or less than some
values.

_General constraints_ are always considered in the formalization and are
related to the:  (i) _basic allocation_ rules,  (ii) _occupancy_ criteria,
(iii) hardware _capacity_ of the VM offers, and  (iv) _link_ between the VM
offers and the components hardware/software requirements.

We stated the problems as a linear constraint optimization problem (COP). We
redirect the reader to [3] for a full description of it.

##### Problem Formalization

The formalization for all three types of solvers, that is constraint
programming solvers, SMT solvers, and mathematical programming solver, has
almost a one-to-one correspondence between linear constraints present in the
definition of COP and the implementation. We did not apply optimizations
exploiting the particularities of the modelling languages because we wanted to
have a fair comparison of the different formalisms. However, in the future
work we plan to take advantage of their sweet spots.

##### Minizinc models

The Minizinc models are the ones newly introduced in this paper. They are
organized as follows:  (i) there are surrogate models for each problem in
which the maximum number of needed VMs is computed;  (ii) there is a model
gathering together all constraints, both general and application specific (the
model corresponding to each application instantiates the constraints needed
for its modeling)  (iii) there is a model gathering together all symmetry
breakers developed which are then instantiated based on the tests which want
to be performed.

##  3 Experimental Analysis

The principles of the experimental analysis were introduced in [3]: on one
hand, we want to assess the _scalability_ of state-of-the-art general CP
(Chuffed [2], Gecode [4], OR-Tools [8]), MP (CPLEX [5]) and OMT (Z3 [1]) tools
in solving COPs corresponding to realistic case studies. On the other hand, we
evaluate the _effectiveness_ of various static symmetry breaking techniques in
improving the computational time of solving these problems (see Section 3.1).
This is because tests (see Tables 1-2) revealed that the naive application of
general CP, MP and OMT techniques is not sufficient to solve realistic Cloud
deployment applications.

We consider four case studies (Secure Web Container, Secure Billing Email
Service, Oryx2, and Wordpress) which exhibit:  (i) different hardware
characteristics of components and the rich interactions type in between
(structural constraints);  (ii) the kind of linear constraints used to
formalize the problem; and  (iii) the kind of solution we are searching for.
A full description of these case studies is in [3].

The scalability and effectiveness are evaluated from two perspectives: number
of VM offers, respectively number of deployed instances of components. For
_Secure Web Container_ , _Secure Billing Email_ and _Oryx2_ applications, we
considered up to 500 VM offers. Additionally, for the _Wordpress_ application,
we considered up to $12$ instances of the Wordpress component to be deployed.
The set of offers was crawled from the Amazon CPs offers list.

###  3.1 Experimental Settings

####  3.1.1 Selected Symmetry Breaking Strategies

Aiming to reduce the search space size, a set of strategies have been selected
in order to exploit the particularities of the problem:  (i) the VMs needed
for application deployment might have different characteristics;  (ii)
applications components might be in conflict hence conflict-type constraints
can be exploited;  (iii) the number of instances to be deployed is unknown.

Our approach is incremental and experimental: we start with traditional
symmetry breakers that have been used for other problems related to bin-
packing and combine them with the aim of further search space reduction.

#####  3.1.1.1 Simple symmetry breakers

_Price-based ordering_ (PR). This strategy aims to break symmetry by ordering
the vector containing the types of used VMs decreasingly by price, i.e.
$p_{k}\geq p_{k+1},\ k=\overline{1,M-1}.$ This means that the solution will be
characterized by the fact that the columns of the assignment matrix will be
ordered decreasingly by the price of the corresponding VMs.

_Lexicographic ordering_ (LX). This corresponds to the traditional strategy
aiming to break column-wise symmetries. The constraints to be added aiming to
ensure that two columns, $k$ and $(k+1)$ are in a decreasing lexicographic
order, i.e. $a_{*k}\succ_{lex}a_{*(k+1)}$, are
$\bigwedge\limits_{l=1}^{i-1}(a_{lk}=a_{l(k+1)})\Longrightarrow a_{ik}\geq
a_{i(k+1)},\ \forall i=\overline{1,N}$.

_Load-based ordering (L)_. This is a column-wise symmetry breaker which orders
decreasingly the columns by the number of the component instances they
accommodate: $\sum_{i=1}^{N}a_{ik}\geq\sum_{i=1}^{N}a_{i(k+1)},\
k=\overline{1,M-1}$.

_Fixed values (FV)._ The search space can be reduced also by fixing the values
of some variables starting from the application specific constraints, in
particular conflict constrains. More precisely, the graph composed by the
components being in conflict is used to identify components which must be
placed on different machines and hence the values of the corresponding
decision variables are fixed. The identification of these components is done
by constructing the clique with maximum deployment size.

#####  3.1.1.2 Composed symmetry breakers

The symmetry breakers above can be composed leading to the following symmetry
breakers:

  * •

FV-PR, FV-L, FV-LX, PR-L, PR-LX, L-PR, L-LX,

  * •

FV-PR-L, FV-PR-LX, FV-L-PR, FV-L-LX, PR-L-LX, L-PR-LX,

  * •

FV-PR-L-LX, FV-L-PR-LX

These symmetry breakers are so the subsequent breaks ties of the former. For
example, FV-PR fixes on separate machines the decision variables corresponding
to the component instances being in the clique with maximum deployment size
and the machines left unoccupied are ordered decreasingly by price. In the
case of PR-L-LX, the machines are ordered decreasingly by price, those with
the same price are ordered decreasingly by the number of components they host
and those with the same number of instances are ordered lexicographically.

It is worth noticing that the symmetry breakers involving FV must apply FV the
very first. This is because FV is used as a preprocessing step which has a
positive impact on the solvers as it introduces equalities.

####  3.1.2 Software and Hardware Settings

We used Minizinc v0.7.0 as the constraint modeling language. We mention that
the Minizinc models follow the formalization and no optimizations were
performed because we wanted to be as close as possible to the OMT and CPLEX
formalizations in order to have a fair computational comparison between the
newly considered solvers and the existing results. The CP solvers used
(Chuffed, Gecode, OR-Tools) are called from Minizinc IDE with the default
values for parameters. The OMT formalization is done using the Z3 Python API
and uses quantifier-free linear integer arithmetic. Z3 was used with the
default values of the parameters. In the case of the mathematical programming
solver CPLEX, we used the Python API with the no symmetry breaking option
manually activated.

The source code and the experimental results are available online at
https://github.com/BogdanD02/Cloud-Resource-Provisioning, release v1.0.0. All
reported timings are in seconds. They only include the actual solving time of
the optimization problem and not the pre-processing steps.

All tests in this paper were performed on an Intel(R) Core (TM) i5-9400F CPU @
3.90GHz using Chuffed v0.10.4, Gecode v6.3.0, OR-Tools v9.0.0, CPLEX v12.9.0
and Z3 v4.10.2.

###  3.2 Results

Tables 1-4 include the results obtained without using symmetry breaking
strategies. The list of offers (columns #o) was crawled from the Amazon
site111https://aws.amazon.com/. Each list of VM offers covers the main
instance types, for example, small, medium, large. The list of offers can be
viewed as a containment hierarchy (i.e. the list of 20 offers is included in
the list of 40 offers etc.).

Table 1: Scalability tests for Wordpress with different instances (#i). Time values are expressed in seconds. #i | #o=20 | #o=40 | #o=250 | #o=500 | #o=20 | #o=40 | #o=250 | #o=500
---|---|---|---|---|---|---|---|---
OR-Tools | CPLEX
3 | 3.49 | 8.38 | 96.05 | 191.04 | 9.66 | - | - | -
4 | 23.25 | 56.07 | 501.43 | 987.91 | 121.96 | - | - | -
5 | 149.47 | 425.03 | - | - | 446.02 | - | - | -
6 | 493.39 | 1173.46 | - | - | 664.68 | - | - | -
Gecode | Chuffed
3 | 2.13 | 2.19 | - | - | 2.05 | 3.7 | 45.88 | 447.56
4 | 14.84 | 23.83 | - | - | 23.73 | 114.18 | 1866.61 | -
5 | 162.13 | - | - | - | 531.19 | 2278.76 | - | -
Z3 |  |  |  |
3 | 2.82 | 4.13 | 103.19 | 391.87 |  |  |  |
4 | 46.46 | 275.81 | - | - |  |  |  |
Table 2: Scalability tests for Oryx2. Time values are expressed in seconds. #o=20 | #o=40 | #o=250 | #o=500 | #o=20 | #o=40 | #o=250 | #o=500
---|---|---|---|---|---|---|---
OR-Tools | CPLEX
0.95 | 1.12 | 4.25 | 5.79 | 0.16 | 0.54 | - | -
Gecode | Chuffed
70.99 | 104.54 | 234.27 | 465.72 | 128.67 | 154.11 | 294.25 | 396.58
Z3 |  |  |  |
13.35 | 15.36 | 453.2 | 717.99 |  |  |  |
Table 3: Scalability tests for Secure Billing Email Service. Time values are expressed in seconds. #o=20 | #o=40 | #o=250 | #o=500 | #o=20 | #o=40 | #o=250 | #o=500
---|---|---|---|---|---|---|---
OR-Tools | CPLEX
0.85 | 1.22 | 8.04 | 18.68 | 0.73 | 4.73 | 378.47 | -
Gecode | Chuffed
0.73 | 1.42 | 76.63 | 119.32 | 0.79 | 0.82 | 1.35 | 3.11
Z3 |  |  |  |
0.34 | 0.70 | 3.37 | 8.29 |  |  |  |
Table 4: Scalability tests for Secure Web Container. Time values are expressed in seconds. #o=20 | #o=40 | #o=250 | #o=500 | #o=20 | #o=40 | #o=250 | #o=500
---|---|---|---|---|---|---|---
OR-Tools | CPLEX
1.50 | 4.85 | 28.79 | 57.33 | 2.68 | 49.64 | - | -
Gecode | Chuffed
10.15 | 43.84 | 76.63 | 779.44 | 0.89 | 1.01 | 2.64 | 4.54
Z3 |  |  |  |
0.45 | 1.14 | 8.28 | 17.48 |  |  |  |

The tables include only those cases for which we obtained a result in a 40
minutes timeframe. The missing values ($-$) mean that no solution is returned
in this timeframe.

One can observe that CPLEX scales the worst for all case studies while OR-
Tools the best. However, none of the tools scale for Wordpress with more than
6 instances and several dozens of offers.

To overcome the lack of scalability issue, we applied the symmetry breaking
strategies described in Section 3.1.1.

##  4 Discussion of the Results

We conducted various tests involving $5$ solvers and $15$ symmetry breakers.
Because of lack of space, we can not include all of them here and we direct
the reader to check them in the folder PlotData from release v1.0.0 at
https://github.com/BogdanD02/Cloud-Resource-Provisioning.

We draw the following remarks:

  1. 1.

Using the $15$ symmetry breakers, out of the $5$ solvers, the best one from
the computational time point of view is Z3.

  2. 2.

For the virtual best solver, that is Z3, the best symmetry breaker is FVPR.
This is because we have many Wordpress files to be analyzed ($40$ files
corresponding to Wordpress with $3$ up to $12$ instances and $20$, $40$,
$250$, $500$ offers), compared to the other applications ($4$ files
corresponding to $20$, $40$, $250$, $500$ offers for each of the other
applications) for which reduction methods which exploit the graph
representation, that is FV, is of benefit.

  3. 3.

We also considered the best symmetry breaker for Z3 for each of the case
studies. In case of Secure Web Container and Secure Billing Email Service
applications the best is FVL, while for Oryx2 is FVLX. There is no surprise
that symmetry breakers involving FV give best results, however for more
reliable results we should consider more test cases for Secure Billing, Secure
Web and Oryx2, since now there are only 4 files analyzed corresponding to
different number of offers. We plan to run more tests for each of the case
studies for a more accurate analysis.

  4. 4.

One would expect that the best symmetry breaker is one composing a higher
number of individual symmetry breakers as more symmetries are broken so the
search space is significantly reduced. However, this is not true: FVPR,
composing $2$ symmetry breakers, is better than those composing $3$ or $4$. An
explanation for this is, on one hand the number of added constraints which
influence the solving time, on the other hand, when using static symmetry
breaking, the symmetry breakers can interact badly with the SMT solvers which
we used as black box.

## References

  * [1] Bjørner, N., Phan, A., Fleckenstein, L.: $\nu$Z - An Optimizing SMT Solver. In: Tools and Algorithms for the Construction and Analysis of Systems - 21st International Conference, TACAS 2015, London, UK, April 11-18, 2015. Proceedings. pp. 194–199 (2015)
  * [2] Chu, G., Stuckey, P.J., Schutt, A., Ehlers, T., Graeme Gange, K.F.: Chuffed, a lazy clause generation solver (2022), https://github.com/chuffed/chuffed, last accessed 20 August 2022
  * [3] Erascu, M., Micota, F., Zaharie, D.: Scalable optimal deployment in the cloud of component-based applications using optimization modulo theory, mathematical programming and symmetry breaking. J. Log. Algebraic Methods Program. 121, 100664 (2021)
  * [4] Gecode Team: Gecode: Generic Constraint Development Environment (2006), available from http://www.gecode.org
  * [5] IBM: IBM ILOG CPLEX Optimization Studio CPLEX User’s Manual. Version 12 (2016)
  * [6] de Moura, L., Bjørner, N.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) Tools and Algorithms for the Construction and Analysis of Systems. pp. 337–340. Springer Berlin Heidelberg, Berlin, Heidelberg (2008)
  * [7] Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., Tack, G.: Minizinc: Towards a standard CP modelling language. In: Bessiere, C. (ed.) Principles and Practice of Constraint Programming - CP 2007, 13th International Conference, CP 2007, Providence, RI, USA, September 23-27, 2007, Proceedings. Lecture Notes in Computer Science, vol. 4741, pp. 529–543. Springer (2007)
  * [8] Perron, L., Furnon, V.: OR-Tools, https://developers.google.com/optimization/