# Well-posedness for an hyperbolic-hyperbolic-elliptic system describing cold
plasmas

Diego Alonso-Orán<EMAIL_ADDRESS>Departamento de Análisis Matemático y
Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna C/.
Astrofísico Francisco Sánchez s/n, 38200 - La Laguna, Spain.  and  Rafael
Granero-Belinchón<EMAIL_ADDRESS>Departamento de Matemáticas,
Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n,
Santander, Spain.

###### Abstract.

In this short note, we provide the well-posedness for an hyperbolic-
hyperbolic-elliptic system of PDEs describing the motion of collision free-
plasma in magnetic fields. The proof combines a pointwise estimate together
with a bootstrap type of argument for the elliptic part of the system.

###### Key words and phrases:

Cold plasma, well-posedness, hyperbolic-hyperbolic-elliptic system

###### 1991 Mathematics Subject Classification:

35R35, 35Q35, 35S10, 76B03

##  1\. Introduction and main result

The motion of a cold plasma in a magnetic field consisting of singly-charged
particles can be described by the following system of PDEs [3, 9]

$\displaystyle n_{t}+(un)_{x}$ $\displaystyle=0,$ (1a) $\displaystyle
u_{t}+uu_{x}+\frac{BB_{x}}{n}$ $\displaystyle=0,$ (1b) $\displaystyle
B-n-\left(\frac{B_{x}}{n}\right)_{x}$ $\displaystyle=0,$ (1c)

where $n,u$ and $B$ are the ionic density, the ionic velocity and the magnetic
field, respectively. Moreover, it has also been used as a simplified model to
describe the motion of collission-free two fluid model where the electron
inertial, charge separation and displacement current are neglected and the
Poisson equation (1c) is initially satisfied, [3, 10]. In (1) the spatial
domain $\Omega$ is either $\Omega=\mathbb{R}$ or $\Omega=\mathbb{S}^{1}$
(_i.e._ $x\in\mathbb{R}$ or $x\in[-\pi,\pi]$ with periodic boundary
conditions) and the time variable satisfies $t\in[0,T]$ for certain
$0<T\leq\infty$. The corresponding initial-value problem consists of the
system (1) along with initial conditions

$n(x,0)=n_{0}(x),\;u(x,0)=u_{0}(x),$ (2)

which are assumed to be smooth enough for the purposes of the work.

System (1) was introduced by Gardner & Morikawa [9]. Furthermore, Gardner &
Morikawa formally showed that the solutions of (1) converge to solutions of
the Korteveg-de Vries equation (see also the paper by Su & Gardner [12]).
Berezin & Karpman extended this formal limit to the case where the wave
propagates at angles of certain size with respect to the magnetic field [3].
Later on, Kakutani, Ono, Taniuti & Wei [10] removed the hypothesis on the
angle. This formal KdV limit was recently justified by Pu & Li [11]. Very
recently in [2], by means of a multi-scale expansion (cf. [1, 4]), the authors
derived three asymptotic models of (1) and studied several analytical
properties of the models: the existence of conserved quantities, the
Hamiltonian structure, the well-posedness and the formation of singularities
in finite time. More precisely, for the uni-directional model which resembles
the well-known Fornberg-Whitham equation (cf. [8]), the authors showed that
wave-breaking occurs, that is, the formation of an infinite slope in the
solution. In [14], a new sufficient condition on the initial data is given
which exhibits wave breaking extending the previous work [2]. To the best of
the author’s knowledge, although system (1) has been introduced more than 50
year’s ago, the well-posedness of the system has not been studied elsewhere.
The result of this work is to give a positive answer to the previous problem
and the main theorem reads as follows

######  Theorem 1.

Let $n_{0}(x)>0$, $n_{0}(x)-1\in H^{2}$ and $u_{0}(x)\in H^{3}$. Then, there
exists a $T>0$ and unique solution of (1) such that

$(n-1,u)\in C([0,T],H^{2}\times H^{3}).$

### Notation

For $1\leq p\leq\infty$, let $L^{p}=L^{p}(\mathbb{R})$ be the usual normed
space of $L^{p}$-functions on $\mathbb{R}$ with $||\cdot||_{p}$ as the
associated norm. For $s\in\mathbb{R}$, the inhomogeneous Sobolev space
$H^{s}=H^{s}(\mathbb{R})$ is defined as

$\displaystyle H^{s}(\mathbb{R})\triangleq\left\\{f\in
L^{2}(\mathbb{R}):\|f\|_{H^{s}(\mathbb{R})}^{2}=\int_{\mathbb{R}}(1+\xi^{2})^{s}|\widehat{f}(\xi)|^{2}<+\infty\right\\},$

with norm

$\left\|f\right\|_{H^{s}}=\left\|f\right\|_{L^{2}}^{2}+\left\|f\right\|_{\dot{H}^{s}}.$

Moreover, throughout the paper $C=C(\cdot)$ will denote a positive constant
that may depend on fixed parameters and $x\lesssim y$ ($x\gtrsim y$) means
that $x\leq Cy$ ($x\geq Cy$) holds for some $C$.

##  2\. Proof of Theorem 1

The proof follows the classical a priori estimates approach which combines the
derivation of useful a priori energy estimates and the use of a suitable
approximation procedure via mollifiers (see for instance [2]). First, we write
system (1) in the new variables $n=1+\eta,B=1+b$. Then system (1) becomes

$\displaystyle\eta_{t}+(u\eta)_{x}+u_{x}$ $\displaystyle=0,$ (3a)
$\displaystyle u_{t}+uu_{x}+\frac{(1+b)b_{x}}{1+\eta}$ $\displaystyle=0,$ (3b)
$\displaystyle b-\eta-\left(\frac{b_{x}}{1+\eta}\right)_{x}$
$\displaystyle=0.$ (3c)

We are going to find the appropriate energy estimates for the following energy

$\mathcal{E}(t)=\left\|\eta(t)\right\|^{2}_{H^{2}}+\left\|u(t)\right\|^{2}_{H^{3}}+\displaystyle\max_{x\in\mathbb{R}}\frac{1}{1+\eta(x,t)}.$
(4)

In order to estimate the last term in the energy $\mathcal{E}(t)$, we need to
derive a pointwise estimate. To that purpose, following [6] and defining

$\displaystyle
m(t)=\displaystyle\min_{x\in\mathbb{R}}\eta(x,t)=\eta(\underline{x}_{t},t),\mbox{
for }t>0,$

it is easy to check that $m(t)$ is a Lipschitz functions and one has the
following bound

$\left|m(t)-m(s)\right|\leq\max_{y,z}|\partial_{t}\eta(y,z)||t-s|.$

From Rademacher’s theorem it holds that $m(t)$ is differentiable in $t$ almost
everywhere and furthermore

$m^{\prime}(t)=\partial_{t}\eta(\underline{x}_{t},t)\text{ a.e.}$ (5)

Then, using (1a) and noticing that $n_{x}(\underline{x}_{t},t)=$ we readily
see that

$\displaystyle
m^{\prime}(t)=-u_{x}(\underline{x}_{t},t)m(t)-u_{x}(\underline{x}_{t},t)=-u_{x}(\underline{x}_{t},t)(1+m(t))$
(6)

Moreover, since by assumption $m(0)>-1$ we also have that

$m(t)>-1,\quad\mbox{ for }0<t\ll 1.$ (7)

We remark that this is not a monotonicity statement relying on a sign
condition for $u_{x}(\underline{x}_{t},t)$, but just a small in time argument.
Hence, following the argument in [7] and using (6) we find that

$\displaystyle\frac{d}{dt}\left(\displaystyle\max_{x\in\mathbb{R}}\frac{1}{1+\eta(x,t)}\right)=-\frac{\partial_{t}\eta(\underline{x}_{t},t)}{(1+m(t))^{2}}=\frac{u_{x}(\underline{x}_{t},t)}{1+m(t)}\leq
C(\mathcal{E}(t))^{2}.$ (8)

The lower order $L^{2}$ norm of $\eta$ is bounded by

$\displaystyle\frac{1}{2}\frac{d}{dt}\left\|\eta\right\|_{L^{2}}^{2}$
$\displaystyle\lesssim\left\|\eta\right\|_{L^{2}}^{2}\left\|u_{x}\right\|_{L^{\infty}}+\left\|\eta\right\|_{L^{2}}\left\|u_{x}\right\|_{L^{2}}$
(9)

Similarly, we find that

$\frac{1}{2}\frac{d}{dt}\left\|u\right\|_{L^{2}}^{2}\lesssim\left(1+\left\|b\right\|_{L^{\infty}}\right)\left\|\frac{b_{x}}{1+\eta}\right\|_{L^{2}}\left\|u\right\|_{L^{2}}$
(10)

Testing equation (3a) and (3b) with $\partial_{x}^{4}\eta$ and
$\partial_{x}^{6}u$ respectively, and integrating by parts we have that

$\displaystyle\frac{1}{2}\frac{d}{dt}\left\|\partial_{x}^{2}\eta\right\|_{L^{2}}^{2}$
$\displaystyle\lesssim\left\|\eta\right\|_{H^{2}}^{2}\left\|u\right\|_{H^{3}}+\left\|\eta\right\|_{H^{2}}\left\|u\right\|_{H^{3}},$
(11)
$\displaystyle\frac{1}{2}\frac{d}{dt}\left\|\partial_{x}^{3}u\right\|_{L^{2}}^{2}$
$\displaystyle\lesssim\left\|u\right\|_{H^{3}}^{3}+\left(1+\left\|b\right\|_{L^{\infty}}\right)\left\|\frac{b_{x}}{1+\eta}\right\|_{H^{3}}\left\|u\right\|_{H^{3}}.$
(12)

Therefore, combining (9)-(12) and using Sobolev embedding and Young’s
inequality that

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\left\|\eta\right\|_{H^{2}}^{2}+\left\|u\right\|_{H^{3}}^{2}\right)$
$\displaystyle\lesssim\left\|\eta\right\|_{H^{2}}^{3}+\left\|u\right\|_{H^{3}}^{3}+\left(1+\left\|b\right\|_{H^{1}}\right)^{2}\left\|\frac{b_{x}}{1+\eta}\right\|^{2}_{H^{3}}+\left\|u\right\|_{H^{3}}^{2}.$
(13)

Moreover, using (3c) we find that

$\displaystyle\left\|\frac{b_{x}}{1+\eta}\right\|^{2}_{\dot{H}^{3}}=\int_{\mathbb{R}}\left|\left(\frac{b_{x}}{1+\eta}\right)_{xxx}\right|^{2}dx=\int_{\mathbb{R}}\left(\frac{b_{x}}{1+\eta}\right)_{xxx}\left(b-\eta\right)_{xx}\
dx\leq\left\|\frac{b_{x}}{1+\eta}\right\|_{\dot{H}^{3}}\left(\left\|\eta\right\|_{H^{2}}+\left\|b\right\|_{H^{2}}\right).$

Therefore, we find that

$\left\|\frac{b_{x}}{1+\eta}\right\|_{H^{3}}\leq\left\|\eta\right\|_{H^{2}}+\left\|b\right\|_{H^{2}}.$

Plugging the previous estimate in (13) we infer that

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\left\|\eta\right\|_{H^{2}}^{2}+\left\|u\right\|_{H^{3}}^{2}\right)$
$\displaystyle\lesssim\left\|\eta\right\|_{H^{2}}^{3}+\left\|u\right\|_{H^{3}}^{3}+\left(1+\left\|b\right\|_{H^{1}}\right)^{2}(\left\|\eta\right\|_{H^{2}}+\left\|b\right\|_{H^{2}})+\left\|u\right\|_{H^{3}}^{2}$
$\displaystyle\lesssim
1+\left\|\eta\right\|_{H^{2}}^{3}+\left\|u\right\|_{H^{3}}^{3}+\left\|b\right\|_{H^{2}}^{3}.$
(14)

To close the energy estimate, we need to compute
$\left\|b\right\|_{H^{2}}^{3}$. To that purpose, we first find using the
elliptic equation (3c) and integrating by parts that

$\left\|b\right\|_{L^{2}}^{2}=\int_{\mathbb{R}}\eta b\
dx+\int_{\mathbb{R}}\left(\frac{b_{x}}{1+\eta}\right)_{x}b\
dx=\int_{\mathbb{R}}\eta b\ dx-\int_{\mathbb{R}}\frac{b_{x}^{2}}{1+\eta}\ dx$

Therefore, using the pointwise estimate (7) we find that the last term

$-\int_{\mathbb{R}}\frac{b_{x}^{2}}{1+\eta}\ dx\leq 0,$

and hence Young’s inequality yields

$\left\|b\right\|^{2}_{L^{2}}\leq\left\|\eta\right\|^{2}_{L^{2}}$ (15)

To compute the higher-order norm, let us first write

$\left\|b_{x}\right\|^{2}_{L^{2}}=\int_{\mathbb{R}}\frac{1+\eta}{1+\eta}(b_{x})^{2}\
dx=-\int_{\mathbb{R}}\frac{b_{x}}{1+\eta}(1+\eta)_{x}b\
dx-\int_{\mathbb{R}}\left(\frac{b_{x}}{1+\eta}\right)_{x}(1+\eta)\ b\
dx=I_{1}+I_{2}.$ (16)

Using Hölders and Young’s inequality we readily see that

$|I_{1}|\leq\left\|b_{x}\right\|_{L^{2}}\left\|\eta_{x}\right\|_{L^{\infty}}\left\|b\right\|_{L^{2}}\left\|\frac{1}{1+\eta}\right\|_{L^{\infty}}\leq\frac{1}{2}\left\|b_{x}\right\|_{L^{2}}^{2}+C\left\|\eta\right\|^{2}_{H^{2}}\left\|b\right\|^{2}_{L^{2}}\left\|\frac{1}{1+\eta}\right\|^{2}_{L^{\infty}}.$
(17)

On the other hand, using once again the elliptic equation (3c) we find that

$I_{2}=\int_{\mathbb{R}}(\eta-b)(1+\eta)b\ dx=\int_{\mathbb{R}}\left(\eta
b+\eta^{2}b-b^{2}(1+\eta)\right)\
dx\leq\left\|b\right\|_{L^{2}}\left\|\eta\right\|_{L^{2}}+\left\|b\right\|_{L^{2}}\left\|\eta\right\|_{L^{2}}\left\|\eta\right\|_{L^{\infty}}.$
(18)

Therefore, collecting (16) -(18) we infer that

$\left\|b_{x}\right\|^{2}_{L^{2}}\leq\frac{1}{2}\left\|b_{x}\right\|_{L^{2}}^{2}+C\left\|\eta\right\|^{2}_{H^{2}}\left\|b\right\|^{2}_{L^{2}}\left\|\frac{1}{1+\eta}\right\|^{2}_{L^{\infty}}+\left\|b\right\|_{L^{2}}\left\|\eta\right\|_{L^{2}}+\left\|b\right\|_{L^{2}}\left\|\eta\right\|_{L^{2}}\left\|\eta\right\|_{L^{\infty}}$
(19)

and hence using (15) we conclude that

$\displaystyle\left\|b_{x}\right\|^{2}_{L^{2}}$
$\displaystyle\lesssim\left\|\eta\right\|^{2}_{H^{2}}\left\|\eta\right\|^{2}_{L^{2}}\left\|\frac{1}{1+\eta}\right\|^{2}_{L^{\infty}}+\left\|\eta\right\|_{L^{2}}^{2}+\left\|\eta\right\|_{L^{2}}^{2}\left\|\eta\right\|_{L^{\infty}}.$
(20)

We iterate the previous idea, to provide an estimate
$\left\|b_{xx}\right\|_{L^{2}}$. To that purpose, we write

$\left\|b_{xx}\right\|^{2}_{L^{2}}=-\int_{\mathbb{R}}\frac{1+\eta}{1+\eta}b_{xxx}b_{x}\
dx=\int_{\mathbb{R}}(1+\eta)b_{xx}\left(\frac{b_{x}}{1+\eta}\right)_{x}\
dx+\int_{\mathbb{R}}(1+\eta)_{x}b_{xx}\frac{b_{x}}{1+\eta}\ dx=J_{1}+J_{2}.$

Using the elliptic equation (3c), we have that

$\displaystyle J_{1}=\int_{\mathbb{R}}(1+\eta)b_{xx}(b-\eta)\ dx$
$\displaystyle\leq\left\|b_{xx}\right\|_{L^{2}}\left\|(1+\eta)(b-\eta)\right\|_{L^{2}}$
$\displaystyle\leq\frac{1}{2\epsilon}\left\|b_{xx}\right\|^{2}_{L^{2}}+C_{\epsilon}\left(1+\left\|\eta\right\|_{H^{2}}^{4}+\left\|b\right\|_{L^{2}}^{4}\right)$
(21)

where in the second inequality we have used the Sobolev embedding and Young’s
ineqality. Similarly,

$\displaystyle J_{2}$
$\displaystyle\leq\left\|b_{xx}\right\|_{L^{2}}\left\|(1+\eta)_{x}\frac{b_{x}}{1+\eta}\right\|_{L^{2}}\leq\frac{1}{2\epsilon}\left\|b_{xx}\right\|^{2}_{L^{2}}+C_{\epsilon}\left(1+\left\|\eta\right\|_{H^{2}}^{8}+\left\|\frac{1}{1+\eta}\right\|_{L^{\infty}}^{8}+\left\|b_{x}\right\|_{L^{2}}^{8}\right).$
(22)

Therefore taking $\epsilon\ll 1$ (for instance $\epsilon=1/4$), we find that

$\frac{1}{2}\left\|b_{xx}\right\|_{L}^{2}\leq
C\left(1+\left\|\eta\right\|_{H^{2}}^{8}+\left\|\frac{1}{1+\eta}\right\|_{L^{\infty}}^{8}+\left\|b_{x}\right\|_{L^{2}}^{8}\right).$
(23)

Hence, estimate (23) combined with the previous estimates for
$\left\|b_{x}\right\|_{L^{2}}$ given in (20) and $\left\|b\right\|_{L^{2}}$ in
(15) we conclude that

$\left\|b\right\|^{3}_{H^{2}}\leq C\left(1+\mathcal{E}(t)\right)^{p},$ (24)

for some $C>0$ and $p>2$ large enough. The precise power of $p$ can be
computed though it is not essential to provide a local-in-time solution.
Hence, plugging the previous estimate into (2) and taking into account (8) we
conclude that

$\frac{d}{dt}\mathcal{E}(t)\leq C\left(1+\mathcal{E}(t)\right)^{p}$ (25)

for some $C>0$ and $p>2$ large enough which ensures a local time of existence
$T^{\star}>0$ such that

$\mathcal{E}(t)\leq 4\mathcal{E}(0),\quad\mbox{ for }0\leq t\leq T^{\star}.$

In order to construct the solution, we first define the approximate problems
using mollifiers, which reads

$\displaystyle\eta^{\epsilon}_{t}+\mathcal{J}_{\epsilon}(\mathcal{J}_{\epsilon}u\mathcal{J}_{\epsilon}\eta)_{x}+\mathcal{J}_{\epsilon}\mathcal{J}_{\epsilon}u^{\epsilon}_{x}$
$\displaystyle=0,$ (26a) $\displaystyle
u^{\epsilon}_{t}+\mathcal{J}_{\epsilon}\left(\mathcal{J}_{\epsilon}u\mathcal{J}_{\epsilon}u_{x}\right)+\frac{(1+b)b_{x}}{1+\eta^{\epsilon}}$
$\displaystyle=0,$ (26b) $\displaystyle
b-\eta^{\epsilon}-\left(\frac{b_{x}}{1+\eta^{\epsilon}}\right)_{x}$
$\displaystyle=0.$ (26c)

Repeating the previous estimates we find a time of existence $T^{\star}>0$ for
the sequence of regularized problems. Using compactness arguments and passing
to the limit we conclude the proof of existence. The time continuity for the
solution is obtained by classical arguments. On the one hand, the differential
equation (25) gives the strong right continuity at $t=0$. Using the change of
variables $\hat{t}=-t$, we get the strong left continuity at $t=0$, which
combined show the continuity in time of the solution.

## Acknowledgments

D.A-O is supported by the Spanish MINECO through Juan de la Cierva fellowship
FJC2020-046032-I. R.G-B is supported by the project “Mathematical Analysis of
Fluids and Applications” Grant PID2019-109348GA-I00 funded by MCIN/AEI/
10.13039/501100011033 and acronym “MAFyA”. This publication is part of the
project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. This
publication is also supported by a 2021 Leonardo Grant for Researchers and
Cultural Creators, BBVA Foundation. The BBVA Foundation accepts no
responsibility for the opinions, statements, and contents included in the
project and/or the results thereof, which are entirely the responsibility of
the authors. D.A-O and R. G-B are also supported by the project “Análisis
Matemático Aplicado y Ecuaciones Diferenciales” Grant PID2022-141187NB-I00
funded by MCIN/ AEIand acronym “AMAED”.

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[->, line width=, rounded corners, color=] (W23)–(B24)–(W19)–(B19)–(W15)–(B14)–(W32)–(B33)–(W11)–(B9)–(W8)–(B10);
[->, line width=, rounded corners, color=] (B6)–(W5)–(B7)–(W6)–(B8)–(W7)–(B4)–(W3);

[orange] at (7,5) $x_2^\prime$;

at (11,0)

[->, line width=, rounded corners, color=] (W21)–(B22)–(W17)–(B17)–(W13)–(B12)–(W31)–(B32)–(W11)–(B33)–(W12)–(4.6,8);
[->, line width=, rounded corners, color=] (B10)–(W12)–(B9)–(W11)–(B8)–(W6)–(B3)–(W2);
[->, line width=, rounded corners, color=] (W22)–(B23)–(W18)–(B18)–(W14)–(B13)–(W32)–(B14)–(W33)–(B15);
[->, line width=, rounded corners, color=] (B11)–(W30)–(B31)–(W10)–(B7)–(W5)–(B2)–(W1);
[->, line width=, rounded corners, color=] (4.6,0)–(B30)–(W9)–(B6);

[green] at (9,3) $x_3^\prime$;

The zigzag paths $x_1^\prime, x_2^\prime, x_3^\prime$ of $\overline{\nu}^\zig_\calX(\Gamma, \{z_1,z_2,z_3\})$

By Lemma <ref>, some edges constituting the zigzag paths $x^\prime_1, x^\prime_2, x^\prime_3$ shown in Figure <ref> might be removed by the operation (zig-5).
Thus, paying attention to these zigzag paths, we apply (zig-5) to $\overline{\nu}^\zig_\calX(\Gamma)$ and make it consistent.
Namely, if we remove pairs of edges (1)–(5) shown in the type A (left) of Figure <ref>, which are the intersections of pairs of zigzag paths
on the universal cover that intersect with each other in the same direction more than once, from $\overline{\nu}^\zig_\calX(\Gamma)$ with this order.
Then, we have the dimer model shown in the left of Figure <ref>,
and applying (zig-6) we finally have the dimer model $\nu^\zig_\calX(\Gamma, \{z_1,z_2,z_3\})$ shown in the right of Figure <ref>.

Whereas there are other ways to remove edges. For example, if we remove pairs of edges (1)–(5) shown in the type B (right) of Figure <ref>,
then we have another dimer model shown in the left of Figure <ref>,
and applying (zig-6) we have the dimer model $\nu^\zig_\calX(\Gamma, \{z_1,z_2,z_3\})$ shown in the right of Figure <ref>.

at (0,0)

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at (8,0)

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[line width=] (W18)–(B23) node[fill=white,inner sep=0.5pt, circle, midway,xshift=0cm,yshift=0cm] $(3)$;
[line width=, fill=white] (W18) circle [radius=];
[line width=] (W33)–(B15) node[fill=white,inner sep=0.5pt, circle, midway,xshift=0cm,yshift=0cm] $(4)$;
[line width=, fill=white] (W33) circle [radius=];
[line width=] (W19)–(B24) node[fill=white,inner sep=0.5pt, circle, midway,xshift=0cm,yshift=0cm] $(4)$;
[line width=, fill=white] (W19) circle [radius=];
[line width=] (W8)–(B10) node[fill=white,inner sep=0.5pt, circle, midway,xshift=0cm,yshift=0cm] $(5)$;
[line width=, fill=white] (W8) circle [radius=];
[line width=] (W17)–(B22) node[fill=white,inner sep=0.5pt, circle, midway,xshift=0cm,yshift=0cm] $(5)$;
[line width=, fill=white] (W17) circle [radius=];

at (0,-2.5) Type A; at (8,-2.5) Type B;

The two ways to remove edges from $\overline{\nu}^\zig_\calX(\Gamma, \{z_1,z_2,z_3\})$ by (zig-5)

at (0,0)

(B) at (,);

13/8/1,14/8/3,15/8/5,16/8/7, 17/10/1,18/10/3,19/10/5,20/10/7,21/12/1,22/12/3,23/12/5,24/12/7
(W) at (,);

[line width=]  (0,0) rectangle (12,8);

/i̱n 1/1,2/2,3/3,4/4,5/1,5/2,5/6,6/2,6/3,6/7,7/3,7/4,7/8,8/4,8/5,8/9[line width=] (W)–(B)̱;
/i̱n 9/6,9/11,10/7,10/12,11/8,11/9,11/13,12/9,12/14[line width=] (W)–(B)̱;
/i̱n 13/11,13/12,13/16,13/17,14/12,14/13,14/17,14/18,15/13,15/14,15/18,15/19,16/14,16/15,16/19,16/20[line width=] (W)–(B)̱;
/i̱n 17/16,17/17,17/21,17/22,18/17,18/18,18/22,18/23,19/18,19/19,19/23,20/19,20/20,20/24,21/21,21/22,22/22,22/23,23/23,23/24,24/24,24/25[line width=] (W)–(B)̱;

(W9) ++(-18:1cm) coordinate (B30); (W9) ++(-18:2cm) coordinate (W30);
(W10) ++(-18:1cm) coordinate (B31); (W10) ++(-18:2cm) coordinate (W31);
(W11) ++(-18:1cm) coordinate (B32); (W11) ++(-18:2cm) coordinate (W32);
(W12) ++(-18:1cm) coordinate (B33); (W12) ++(-18:2cm) coordinate (W33);

[line width=] (W2)–(B3); [line width=] (W3)–(B4);

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 [line width=, fill=black] (B) circle [radius=];

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 [line width=, fill=white] (W) circle [radius=];

/i̱n 30/31,31/32,32/33[line width=] (W)–(B)̱;
[line width=] (B30)–(5.3,0); [line width=] (W33)–(5.3,8);

in 30,31,32,33[line width=, fill=black] (B) circle [radius=];
in 30,31,32,33[line width=, fill=white] (W) circle [radius=];

at (9,0)

(B) at (,);

13/6/1,14/6/3,15/6/5,16/6/7, 17/8/1,18/8/3,19/8/5,20/8/7,21/10/1,22/10/3,23/10/5,24/10/7
(W) at (,);

[line width=]  (0,0) rectangle (10,8);

/i̱n 1/1,2/2,2/3,3/3,3/4,4/4,5/1,5/2,6/2,6/3,7/3,7/4,8/4,8/5[line width=] (W)–(B)̱;
/i̱n 5/6,6/7,7/8,7/9,8/9,9/6,9/7,10/7,10/8,11/8,11/9,12/9,12/10[line width=] (W)–(B)̱;
/i̱n 9/11,10/12,11/13,12/14[line width=] (W)–(B)̱;
/i̱n 13/11,13/12,13/16,13/17,14/12,14/13,14/17,14/18,15/13,15/14,15/18,15/19,16/14,16/15,16/19,16/20[line width=] (W)–(B)̱;
/i̱n 17/16,17/17,17/21,17/22,18/17,18/18,18/22,18/23,19/18,19/19,19/23,20/19,20/20,20/24,21/21,21/22,22/22,22/23,23/23,23/24,24/24,24/25[line width=] (W)–(B)̱;

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 [line width=, fill=black] (B) circle [radius=];

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 [line width=, fill=white] (W) circle [radius=];

[->,line width=0.03cm] (3.5,0)–(6,0);
at (4.75,0.35) (zig-6);

The dimer model $\nu^\zig_\calX(\Gamma, \{z_1,z_2,z_3\})$ obtained from the type A of Figure <ref>

at (0,0)

(B) at (,);

13/8/1,14/8/3,15/8/5,16/8/7, 17/10/1,18/10/3,19/10/5,20/10/7,21/12/1,22/12/3,23/12/5,24/12/7
(W) at (,);

[line width=]  (0,0) rectangle (12,8);

/i̱n 1/1,2/2,3/3,4/4,5/1,5/2,5/6,6/2,6/3,6/7,7/3,7/4,7/8,8/4,8/5,8/9[line width=] (W)–(B)̱;
/i̱n 9/6,9/11,10/7,10/12,11/8,11/9,11/13,12/9,12/14[line width=] (W)–(B)̱;
/i̱n 13/11,13/12,13/16,13/17,14/12,14/13,14/17,14/18,15/13,15/14,15/18,15/19,16/14,16/15,16/19,16/20[line width=] (W)–(B)̱;
/i̱n 17/16,17/17,17/21,18/17,18/18,18/22,19/18,19/19,19/23,20/19,20/20,20/24,21/21,21/22,22/22,22/23,23/23,23/24,24/24,24/25[line width=] (W)–(B)̱;

(W9) ++(-18:1cm) coordinate (B30); (W9) ++(-18:2cm) coordinate (W30);
(W10) ++(-18:1cm) coordinate (B31); (W10) ++(-18:2cm) coordinate (W31);
(W11) ++(-18:1cm) coordinate (B32); (W11) ++(-18:2cm) coordinate (W32);
(W12) ++(-18:1cm) coordinate (B33); (W12) ++(-18:2cm) coordinate (W33);

[line width=] (W1)–(B2); [line width=] (W2)–(B3); [line width=] (W3)–(B4);
[line width=] (W12)–(B10);

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 [line width=, fill=black] (B) circle [radius=];

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 [line width=, fill=white] (W) circle [radius=];

/i̱n 30/31,31/32,32/33[line width=] (W)–(B)̱;
[line width=] (B30)–(5.3,0); [line width=] (W33)–(5.3,8);

in 30,31,32,33[line width=, fill=black] (B) circle [radius=];
in 30,31,32,33[line width=, fill=white] (W) circle [radius=];

at (9,0)

1/1/0,2/1/2,3/1/4,4/1/6,5/1/8,6/3/6, 7/5/0,8/5/2,9/5/4,10/5/6,11/5/8,
(B) at (,);

14/8/1,15/8/3,16/8/5,17/8/7, 18/10/1,19/10/3,20/10/5,21/10/7,22/12/1,23/12/3,24/12/5,25/12/7
(W) at (,);

[line width=]  (0,0) rectangle (12,8);

/i̱n 1/1,1/2,2/2,2/3,3/3,3/4,4/4,5/1,5/2,5/7,6/2,6/3,6/8,7/3,7/4,7/6,7/9,8/4,8/5,8/6,
22/22,22/23,23/23,23/24,24/24,24/25,25/25,25/26[line width=] (W)–(B)̱;

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 [line width=, fill=black] (B) circle [radius=];

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 [line width=, fill=white] (W) circle [radius=];

[->,line width=0.03cm] (3.5,0)–(5.5,0);
at (4.5,0.35) (zig-6);

The dimer model $\nu^\zig_\calX(\Gamma, \{z_1,z_2,z_3\})$ obtained from the type B of Figure <ref>

Let $\Gamma_A$ (resp. $\Gamma_B$) be the deformed dimer model shown in the right of Figure <ref> (resp. Figure <ref>).
We can check that $\Gamma_A$ and $\Gamma_B$ are not isomorphic, but they are mutation equivalent.
Indeed, by applying the mutations of $\Gamma_B$ at the faces $1,\cdots,10$ with this order (see Figure <ref>), we can recover $\Gamma_A$.
Here, we recall that the mutation of a dimer model can be defined at a quadrangle face.
Although some faces indexed by $\{1,\cdots,10\}$ are not quadrangle, such faces will be a quadrangle in the process of these series of mutations.

The PM polygon of the dimer models $\Gamma_A$ and $\Gamma_B$ are the same (see Proposition <ref>),
and it coincides with the lattice polygon shown in the right of Figure <ref> by Theorem <ref>.

at (0,0)

1/1/0,2/1/2,3/1/4,4/1/6,5/1/8,6/3/6, 7/5/0,8/5/2,9/5/4,10/5/6,11/5/8,
(B) at (,);

14/8/1,15/8/3,16/8/5,17/8/7, 18/10/1,19/10/3,20/10/5,21/10/7,22/12/1,23/12/3,24/12/5,25/12/7
(W) at (,);

[line width=]  (0,0) rectangle (12,8);

/i̱n 1/1,1/2,2/2,2/3,3/3,3/4,4/4,5/1,5/2,5/7,6/2,6/3,6/8,7/3,7/4,7/6,7/9,8/4,8/5,8/6,
22/22,22/23,23/23,23/24,24/24,24/25,25/25,25/26[line width=] (W)–(B)̱;

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 [line width=, fill=black] (B) circle [radius=];

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 [line width=, fill=white] (W) circle [radius=];

at (1,1) $1$; at (3,7.5) $2$; at (4,5.5) $3$; at (6.5,3.5) $4$;
at (8,2) $5$; at (9,3) $6$; at (5,7) $7$; at (6.5,5.5) $8$;
at (8,4) $9$; at (9,5) $10$;

at (9,0)

(B) at (,);

13/6/1,14/6/3,15/6/5,16/6/7, 17/8/1,18/8/3,19/8/5,20/8/7,21/10/1,22/10/3,23/10/5,24/10/7
(W) at (,);

[line width=]  (0,0) rectangle (10,8);

/i̱n 1/1,2/2,2/3,3/3,3/4,4/4,5/1,5/2,6/2,6/3,7/3,7/4,8/4,8/5[line width=] (W)–(B)̱;
/i̱n 5/6,6/7,7/8,7/9,8/9,9/6,9/7,10/7,10/8,11/8,11/9,12/9,12/10[line width=] (W)–(B)̱;
/i̱n 9/11,10/12,11/13,12/14[line width=] (W)–(B)̱;
/i̱n 13/11,13/12,13/16,13/17,14/12,14/13,14/17,14/18,15/13,15/14,15/18,15/19,16/14,16/15,16/19,16/20[line width=] (W)–(B)̱;
/i̱n 17/16,17/17,17/21,17/22,18/17,18/18,18/22,18/23,19/18,19/19,19/23,20/19,20/20,20/24,21/21,21/22,22/22,22/23,23/23,23/24,24/24,24/25[line width=] (W)–(B)̱;

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 [line width=, fill=black] (B) circle [radius=];

in 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 [line width=, fill=white] (W) circle [radius=];

[->,line width=0.03cm] (3.2,0)–(6.3,0);
at (4.75,0.35) The mutation $\mu_f$;
at (4.75,-0.35) where $f=1,\cdots,10$;

The mutations of $\Gamma_B$ (left) at the faces $1,\cdots,10$ induce $\Gamma_A$ (right)

§ REMARKS ON DEFORMATIONS OF HEXAGONAL AND SQUARE DIMER MODELS

As we mentioned in Remark <ref>, we can skip the operations (zig-4) and (zig-5) (resp. (zag-4) and (zag-5)) for the case of hexagonal and square dimer models as we will see below.
Here, we say that a dimer model $\Gamma$ is hexagonal (resp. square) if $\Gamma$ is homotopy equivalent to a dimer model whose faces are all regular hexagon (resp. square) dimer model.
We note that hexagonal and square dimer models are isoradial.
These dimer models have been studied in several papers, especially we have the followings.

Let $\Gamma$ be a consistent dimer model. Then, we have the followings.

* $\Gamma$ is a hexagonal dimer model if and only if the PM polygon $\Delta_\Gamma$ is a triangle.

* If $\Gamma$ is a square dimer model, then the PM polygon $\Delta_\Gamma$ is a parallelogram.

For these nice classes of dimer models, we may skip the operations (zig-4) and (zig-5) (or (zag-4) and (zag-5)) when we apply the deformation.

Let the notation be the same as in Definition <ref>, <ref>, and <ref>.
In addition, we assume that a dimer model $\Gamma$ is hexagonal or square, in which case $\Gamma$ is isoradial.
Then, the deformation $\nu_\calX^\zig(\Gamma,\{z_1,\cdots,z_r\})$ is defined by the operations (zig-1)–(zig-3) and (zig-6).
Similarly, the deformation $\nu_\calY^\zag(\Gamma,\{z_1,\cdots,z_r\})$ is defined by the operations (zag-1)–(zag-3) and (zag-6).

We prove the case of the deformation at zig, and the case of the deformation at zag is similar.

The zigzag paths $z_1,\cdots,z_r$ have the same slope and such a slope is the outer normal vector of an edge of the PM polygon $\Delta_\Gamma$ by Proposition <ref>.

* We assume that $\Gamma$ is a hexagonal dimer model.
Then, $\Delta_\Gamma$ is a triangle by Proposition <ref>(1).
Let $e_1,e_2,e_3$ be edges of $\Delta_\Gamma$ which are ordered cyclically in the anti-clockwise direction.
We may assume that the slopes of $z_1,\cdots,z_r$ are the outer normal vector of $e_1$.
We then consider the zigzag paths $x_1,\cdots,x_s$ (resp. $y_1,\cdots,y_t$) intersected with $z_i$ at zags (resp. zigs) of $z_i$.
Since $\Gamma$ is isoradial, it is properly ordered.
Thus, by Proposition <ref> we see that $[x_1]=\cdots=[x_s]$ (resp. $[y_1]=\cdots=[y_t]$), and $[x_j]$ (resp. $[y_k]$) is the outer normal vector of $e_2$ (resp. $e_3$).

* We assume that $\Gamma$ is a square dimer model.
Then, $\Delta_\Gamma$ is a parallelogram by Proposition <ref>(2).
Let $e_1,e_2,e_3,e_4$ be edges of $\Delta_\Gamma$ which are ordered cyclically in the anti-clockwise direction.
In particular, $\{e_1,e_3\},\{e_2,e_4\}$ are pairs of edges that are parallel.
We may assume that the slopes of $z_1,\cdots,z_r$ are the outer normal vector of $e_1$, in which case
the zigzag paths having the slope $-[z_i]$ correspond to $e_3$.
Then, in a similar way as above, we have the zigzag paths $x_1,\cdots,x_s$ (resp. $y_1,\cdots,y_t$) such that $[x_j]$ (resp. $[y_k]$) is the outer normal vector of $e_2$ (resp. $e_4$).

In both cases, we see that any pair of zigzag paths in $x_1,\cdots,x_s$ (resp. $y_1,\cdots,y_t$) does not have the intersection on the universal cover by Lemma <ref> because $\Gamma$ is isoradial.

Then, we consider $\overline{\nu}_\calX^\zig(\Gamma,\{z_1,\cdots,z_r\})$ for the case of $r\neq1$ (see Definition <ref> for the case of $r=1$).
By Lemma <ref> and the fact that there is no intersection between $x_1,\cdots,x_s$, we see that the edges removed by (zig-5) are bypasses added in (zig-4).
Furthermore, by Observation <ref> and <ref> a zigzag path passing through a bypass takes the form $\widetilde{x}_j^\prime$.
Since the slopes of $x_1,\cdots,x_s$ are all the same in our situation, those of $x_1^\prime, \cdots, x_s^\prime$ are all the same (see Proposition <ref>).
Thus, any bypass on the universal cover of $\overline{\nu}_\calX^\zig(\Gamma,\{z_1,\cdots,z_r\})$ is either

* a self-intersection of a zigzag path $\widetilde{x}_j^\prime$, or

* the intersection of a pair of zigzag paths $\widetilde{x}_j^\prime, \widetilde{x}_{j^\prime}^\prime$ with $[x_j^\prime]=[x_{j^\prime}^\prime]$.

We also see that an edge which is either (i) or (ii) is certainly a bypass, because of Observation <ref> and the fact that such an intersection can not appear in the irrelevant part.
Moreover, if a bypass is the intersection of zigzag paths $\widetilde{x}_j^\prime$ and $\widetilde{x}_{j^\prime}^\prime$, then they have another intersection because $[x_j]=[x_{j^\prime}]$, and such an intersection is also a bypass.
Thus, if there exists a bypass that can not be removed by (zig-5), then it prevents the consistency condition.
Therefore, we can remove all bypasses added in (zig-4) by using (zig-5), and hence we may skip these operations.

§.§ Acknowledgement

The authors thank Alexander Kasprzyk for valuable lectures and discussions on mutation of Fano polygons.
The first author is supported by JSPS Grant-in-Aid for Young Scientists (B) 17K14177.
The second author is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and JSPS Grant-in-Aid for Young Scientists (B) 17K14159.

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# Unified first law of thermodynamics in Gauss-Bonnet gravity on an FLRW
background

Joel Saavedra  0000-0002-1430-3008<EMAIL_ADDRESS>Instituto de Física,
Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaíso,
Chile.  Francisco Tello-Ortiz  0000-0002-7104-5746<EMAIL_ADDRESS>Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla
4950, Valparaíso, Chile.

###### Abstract

Employing the thermodynamics unified first law through the thermodynamics-
gravity conjecture, in this article, we derive for an FLRW universe the
Friedmann equations in the framework of Gauss-Bonnet gravity theory. To do
this, we project this generalized first law along the Kodama vector field and
along the direction of an orthogonal vector to the Kodama vector. The second
Friedmann equation is obtained by projecting on the Kodama vector, while the
first is obtained by projecting along the flux on the Cauchy hypersurfaces.
This result does not assume a priory temperature and an entropy, so the
Clausius relation is not used here. Nevertheless, it is used to obtain the
corresponding Gauss-Bonnet entropy. In this way, the validity of the
generalized second law of thermodynamics is proved for the Gauss-Bonnet
gravity theory.

##  I Introduction

The connection between gravity and thermodynamics, or in other words, the
thermodynamics description of gravity and vice versa, was put forward by
seminal articles by Bardeen, Carter and Hawking Bardeen:1973gs , Bekenstein
Bekenstein:1973ur , Hawking Hawking:1975vcx , Gibbons and Hawking
Gibbons:1976ue  and York York:1986it . In this series of works, the black hole
(BH) thermodynamics laws were developed, leading to an important
interpretation such as the BH temperature as the surface gravity
Hawking:1975vcx ; Gibbons:1976ue  and the BH entropy as the surface area
Bekenstein:1973ur . All the mentioned results were obtained within the
framework of General Relativity (GR); however, later on, this analysis was
extended to higher dimensional gravitational theories, specifically Lovelock’s
gravity Lovelock:1971yv , by Myers and Simon in Myers:1988ze . After all these
advances, Jacobson obtained Einstein’s field equations from a pure
thermodynamics description Jacobson:1995ab .

The aforementioned unexpected link gave rise to possible quantum gravitational
effects, taking a first step in unifying these two branches: gravity and
quantum mechanics. However, an additional crucial point was necessary to
proceed further in this direction. In doing that, Hayward, in a series of
articles Hayward:1993wb ; Hayward:1994bu ; Hayward:1997jp , derived the
thermodynamics unified first law (UFL) for dynamical BHs. In this context, the
Killing vector field is replaced by the Kodama vector field Kodama:1979vn  and
event horizons by apparent horizon Hayward:1997jp ; Kodama:1979vn ;
Faraoni:2011hf . Interestingly, the Kodama vector field establishes a
privileged time direction, and its contraction with Einstein tensor defines a
conserved charge Kodama:1979vn ; Hayward:1997jp .

This dynamical framework allowed us to spread the thermodynamics description
to scenarios beyond the BHs arena. Concretely, this methodology has been
applied to the cosmological scenario in several gravity theories Cai:2005ra ;
Akbar:2006kj ; Cai:2006rs ; Sheykhi:2010zz ; Sebastiani:2023brr , where one
does not have an event horizon, instead one an inner past apparent horizon
(for expanding cosmology) or future inner apparent horizon (for contracting
cosmology)Faraoni:2011hf ; Faraoni:2015ula ; Helou:2015zma .

In this concern, it has been argued that to obtain the dynamic of the FLRW
universe, i.e., Friedmann field equations, one needs to assume from the very
beginning the notion of temperature and entropy Cai:2005ra ; Cai:2006rs . This
is because the term describing the energy variation in the UFL becomes zero
when projecting it along the apparent horizon. For the GR case, in
Helou:2015yqa ; Helou:2015zma , it was proven that this happens for every
constant radius and not only for the apparent horizon. However, this
conclusion drastically changes when different vector fields are used to
project the UFL along the apparent horizon. For example, when contracting the
UFL with the Kodama vector, the Friedmann equations naturally arise without
further consideration.

On the other hand, if one follows the approach given in Cai:2005ra ;
Cai:2006rs  in the Gauss-Bonnet gravity theory (as was done in Cai:2005ra ),
it is necessary to assume that the apparent horizon is fixed when projecting
the UFL along it. This is because the Misner–sharp energy Misner:1964je
contains non-trivial contribution coming from the gravitational sector
Maeda:2006pm ; Maeda:2007uu ; Maeda:2008nz , and these do not vanish unless
the apparent horizon is taken to be constant, as previously mentioned.
Nevertheless, as the apparent horizon is a dynamic object Faraoni:2015ula ,
considering it a fixed entity is a very strong restriction. Then, in this
article, we do not follow the approach used in Cai:2005ra  to regain Friedmann
equations from the UFL. Instead of it, we project the UFL along the apparent
horizon using the Kodama vector field. This is possible because the Kodama
vector becomes a null vector on the apparent horizon Kodama:1979vn ;
Hawking:1975vcx ; Faraoni:2015ula ; in this case, the apparent horizon behaves
as a null surface. This means that the Kodama vector is orthogonal and tangent
to the apparent horizon. In this way, the second Friedmann equation naturally
arises without the assumption of Clausius thermodynamics relation. Then, the
first Friedmann equation is obtained by projecting the UFL along the direction
of a vector field orthogonal to the Kodama vector, consequently hypersurface
orthogonal to the apparent horizon surface. The Clausius relation becomes
relevant in testing the so-called generalized second law. The Gauss-Bonnet
entropy is also obtained by starting from the UFL and identifying the energy-
supply contribution with the Clausius relation.

The article is organized as follows: Sect. II provides a short revision about
the apparent horizon description for an FLRW universe and a short review of
the Kodama vector field in this background. Sect. III is devoted to the
presentation of the Gauss-Bonnet gravity theory and also presents the main
results of this research, that is, the Friedmann equations obtained from the
UFL (III.2) and the Gauss-Bonnet entropy expression (III.3). Finally, Sect. IV
provides some remarks about the article.

Throughout the article, the mostly positive signature $\\{-,+,+,+\\}$ and
units where the speed of light $c=1$ are used.

##  II Apparent horizon and Kodama vector field

In this section, we briefly review the main and elemental ingredients to study
the thermodynamics of the FLRW space-time. To do this, we must define the
apparent horizon in describing dynamical solutions Faraoni:2011hf ;
Faraoni:2015ula . Along with this information, determining the proper
dynamical vector field is necessary, providing the correct information on the
apparent horizon. This vector field is the well-known Kodama vector field
Kodama:1979vn ; Hayward:1993wb ; Hayward:1994bu ; Hayward:1997jp  which is
replacing the usual time–like Killing vector field employed in the static
situation.

###  II.1 The apparent horizon of the FLRW space–time

As stated before, the pertinent entity to describe the thermodynamics of the
Universe is the so-called apparent horizon ($AH$) Faraoni:2011hf ;
Faraoni:2015ula . The $AH$ is the surface satisfying

$h^{ij}\nabla_{i}R\nabla_{j}R=0,\quad i,j=t,r,$ (1)

where $R$ is the areal radius defined as $R(t,r)=a(t)r$, being $a(t)$ the
scale factor, $r$ the co–moving radial distance and $h^{ij}$ the inverse
metric of the two–dimensional metric $h_{ij}$ on the $t-r$ plane. These
definitions come from the FLRW line element

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega_{(n-2)}^{2}\right),$
(2)

re-expressed by means of the warped product as

$ds^{2}=h_{ij}dx^{i}dx^{j}+R^{2}d\Omega_{(n-2)}^{2},$ (3)

where in the above expressions $k=0,\pm 1$ is the spatial curvature and
$d\Omega_{(n-2)}^{2}$ is the line element of a $(n-2)$–dimensional unit
sphere. Therefore, in this way the metric on $t-r$ plane is given by
$h_{ij}=\text{diag}\\{-1,a^{2}/\left(1-kr^{2}\right)\\}$.

Now, the solution $R_{AH}$ of Eq. (1) is given by111The apparent horizon in
this context, is a dynamical marginally outer trapped surface (MOTS) with
vanishing expansion Faraoni:2015ula .Faraoni:2011hf ; Faraoni:2015ula

$R_{AH}=\frac{1}{\sqrt{H^{2}+\frac{k}{a^{2}}}},$ (4)

with $H=\dot{a}/a$ being the Hubble’s parameter. For a flat FLRW, when $k=0$,
the expression (4) coincides with the Hubble’s radius. From now on, we will
keep the spatial curvature $k$ arbitrary.

Although the co-moving patch222Greeks indexes run over the full space-time
components, Latin indexes only run over time and radial components.
$x^{\mu}=\\{t,r,\theta_{1}\ldots\theta_{n-2}\\}$, with $n-3$ angles ranging
over $[0,\pi]$ and $n-2$ ranging over $[0,2\pi)$, is the natural chart for the
FLRW metric, it is more instructive to express this model using the
Schwarzschild–like coordinate chart
$y^{\mu}=\\{T,R,\theta_{1}\ldots\theta_{n-2}\\}$. One must establish the
relation between the co-moving time $t$ and radius $r$ with the new time $T$
and radius $R$ coordinates to do this. These relations are given by
Faraoni:2011hf ; Faraoni:2015ula ; Nielsen:2005af

$dt=FdT-\beta dR,$ (5) $dr=-\frac{HRF}{a}dT+\frac{1}{a}dR,$ (6)

where $F=F(T,R)$ is an integration factor in order to assure that $dt$ being
an exact differential, $H$ and $a$ implicit functions of the time coordinate
$T$ and $\beta$ a function given by

$\beta(t,R)=\frac{HR}{1-H^{2}R^{2}-kr^{2}}.$ (7)

In this way, the line element (2) acquires the following form

$\begin{split}ds^{2}=-\left(1-\frac{H^{2}R^{2}}{1-kR^{2}/a^{2}}\right)F^{2}dT^{2}&\\\
+\frac{dR^{2}}{1-kR^{2}/a^{2}-H^{2}R^{2}}+R^{2}d\Omega_{(n-2)}^{2}.\end{split}$
(8)

From (8) it is easy to get the $AH$ by solving

$g^{RR}=0,$ (9)

for $R$ leading as before to (4).

###  II.2 Kodama vector field for FLRW space–time

As it is well-known, there is no preferred time coordinate in an evolving
time-dependent space-time. This is because there is no asymptotically
time–like a Killing vector field. Then, to bypass this issue, a divergence-
free vector field which is present in any time-dependent spherically
symmetric333For generalizations of the Kodama vector to symmetries other than
the spherical one see for example Tung:2007vq ; Kinoshita:2021qsv . space-time
was introduced in Kodama:1979vn  and is referred as the Kodama vector field.
The Kodama vector field identifies a natural time-like direction and is used
to define locally conserved currents Kodama:1979vn ; Hayward:1993wb . So, for
dynamical situations, it is necessary to introduce or replace the objects used
in a static/stationary context for those giving the proper description
Hayward:1997jp , that is

  1. 1.

Killing vector field $\mapsto$ Kodama vector field.

  2. 2.

Killing horizon $\mapsto$ apparent horizon.

In general, the Kodama vector field is defined as

$K^{i}\equiv E^{ij}\nabla_{j}R,$ (10)

where $E^{ij}$ is the skew-symmetric Levi–Civita tensor defines as

$E^{i_{1}\ldots
i_{n}}=\frac{\text{sgn}\left(|h|\right)}{\sqrt{|h|}}\varepsilon^{i_{1}\ldots
i_{n}},$ (11)

where $h$ is the determinant of the metric $h_{ij}$, the sign function
$\text{sgn}(h)=(-1)^{q}$, with $q$ the number of negatives in the metric
signature444This definition occurs when the metric has an odd number of
negative signs in the signature as in our case. and $\varepsilon^{i_{1}\ldots
i_{n}}$ the standard Levi–Civita symbol defined according our basis coordinate
orientation as

$\varepsilon_{i_{1}\ldots i_{n}}=-\varepsilon^{i_{1}\ldots
i_{n}}\left\\{\begin{aligned} +1&\text{ even permutation, }\\\ -1&\text{ odd
permutation, }\\\ 0&\text{ otherwise. }\end{aligned}\right.$ (12)

On the other hand, if the metric is expressed in Schwarzschild–like coordinate
gauge, the components of the Kodama vector (10) assume the simple form
Kodama:1979vn ; Racz:2005pm

$K^{i}=\frac{1}{\sqrt{g_{TT}g_{RR}}}\left(\frac{\partial}{\partial
t}\right)^{i}.$ (13)

Then, for the line element (8) one gets

$K=K^{T}\partial_{T}=F\sqrt{1-\frac{k}{a^{2}}R^{2}}\,\partial_{T}.$ (14)

Now, from the norm of the Kodama vector $K^{2}=K_{i}K^{i}$

$K^{2}=K_{i}K^{i}=-\left(1-H^{2}R^{2}-\frac{kR^{2}}{a^{2}}\right)=-\left(1-\frac{R^{2}}{R_{AH}^{2}}\right),$
(15)

we can infer the following valuable information: the Kodama vector is
time–like if $K^{2}<0$, thus $R<R_{{AH}}$, null $K^{2}=0$ if $R=R_{{AH}}$ and
space–like if $K^{2}>0$, that is, outside the $AH$ $R>R_{{AH}}$. Thus, the
$AH$ behaves as a null surface in this case. Fig. 1 shows a schematic
representation of the Kodama vector (14). As can be appreciated, in the chart
$y^{\mu}=\\{T,R,\theta_{1}\ldots\theta_{n-2}\\}$ it is hypersurface orthogonal
to the Cauchy hyper–surface
$\Sigma(R,\theta_{1}\ldots\theta_{n-2}):T-T_{0}=0$. This fact is clear from
the general hyper–orthogonal condition, which reads:
$X_{i}=\Lambda(x^{j})\nabla_{i}\Sigma$, being $\Lambda$ some proportionality
factor, which in general will vary from point to point and
$\nabla_{i}\Sigma=\partial_{T}$. Besides, the Kodama vector is orthogonal to
the two–spheres of symmetry. Interestingly, the modulus of the norm (15)
coincides with the space–like slice of marginally trapped tube (MTT) given
by555As we are performing a foliation of the space-time, as shown in Fig. 1,
the surface $\chi$ is just a two-sphere of radius $R_{AH}$ embedded in a
specific Cauchy hypersurface $\Sigma$. However, if we collect all the constant
time slices $\Sigma$ one gets the MTT (purple surface in Fig. 1), equivalently
if one considers $\chi(T,\theta_{1}\ldots\theta_{n-2})$ instead of only
$\chi(\theta_{1}\ldots\theta_{n-2})$ one obtains the full MTT.

$\chi(\theta_{1}\ldots\theta_{n-2}):1-\frac{R^{2}}{R^{2}_{AH}}=0.$ (16)

It is worth mentioning that each space–like slice (red circles) defines the
MOTS given rise to the $AH$ Faraoni:2011hf ; Faraoni:2015ula .

In analogy with Killing vectors, the Kodama vector satisfies

$\frac{1}{2}h^{ij}K^{k}\left(\nabla_{k}K_{i}-\nabla_{i}K_{k}\right)=\kappa_{\text{Kodama
}}K^{j},$ (17)

where $\nabla_{i}$ is covariant the derivative compatible with the metric
$h_{ij}$ ($\nabla_{k}h_{ij}=0$). The Eq. (17) defines the notion of surface
gravity for dynamical scenarios. Usually, this is referred to as the
Hayward–Kodama (HK) surface gravity Hayward:1997jp , which in general has the
following expression independent of the gravitational theory at hand

$\kappa_{\text{HK
}}=\frac{1}{2\sqrt{|h|}}\partial_{i}\left(\sqrt{|h|}h^{ij}\partial_{j}R\right).$
(18)

Now, we can discuss the thermodynamics of the FLRW space-time (2) through the
associated thermodynamics UFL in the context of the higher dimensional GB
theory.

Figure 1: The Kodama vector $K^{i}$ parallel to the time–like vector field
$\partial_{T}$ orthogonal to the space–like Cauchy hyper–surfaces
$\Sigma=\Sigma_{T_{0}}(R,\theta,\phi):T-T_{0}=0$. The marginally trapped tube
(MTT) corresponds to the purple surface. Each space-like surface (red circles)
corresponds to marginally outer trapped surfaces (MOTS), giving rise to the
apparent horizon ($AH$) Faraoni:2015ula , on which the norm of the Kodama
vector nullifies. This schematic representation is constructed using the
Schwarzschild–like coordinate patch
$y^{\mu}=\\{T,R,\theta_{1}\ldots\theta_{n-2}\\}$.

##  III thermodynamics UFL in GB gravity

###  III.1 Preliminaries of GB gravity

The whole action of the $n$–dimensional GB666This theory is also known as
Einstein-Gauss-Bonnet (EGB) gravity; however, we will refer to this theory as
GB only. couple to matter fields, is given by

$S_{GB}=\frac{1}{16\pi
G_{n}}\int_{\mathcal{M}}d^{n}x\sqrt{-g}\left[\mathcal{R}+\alpha\mathcal{G}_{\text{GB}}\right]+\text{B.T.}+S_{m},$
(19)

where $\mathcal{R}$ is the $n$–dimensional Ricci scalar, B.T. stands for
boundary terms in order to have a well–defined action principle, $G_{n}$ the
$n$–dimensional gravitational constant, $\alpha$ is a constant with units of
length squared and $S_{m}$ is the action of the matter fields. The GB term
$\mathcal{G}_{GB}$ reads as

$\mathcal{G}_{\text{GB}}=R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\sigma\gamma}R_{\mu\nu\sigma\gamma}.$
(20)

It is worth mentioning that, in the case of superstring theory, in the low
energy limit, $\alpha$ is related to the inverse string tension and is
positive definite Gross:1986iv ; Gross:1986mw . Besides, the GB action is a
natural extension of the Einstein theory in that no derivatives higher than
second order appear in the field equations. For this to happen, the numerical
coefficient of the $\mathcal{G}_{GB}$ must be $\\{1,-4,1\\}$ as shown in Eq.
(20).

Now, taking variations with respect to metric tensor $g^{\mu\nu}$, the field
equations coming from the action (19) are given by

$G^{\mu}_{\nu}+\alpha H^{\mu}_{\nu}=8\pi G_{n}T^{\mu}_{\nu},$ (21)

being

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathcal{R},$ (22)

the Einstein tensor and

$\displaystyle H_{\mu\nu}$
$\displaystyle=2\left(\mathcal{R}R_{\mu\nu}-2R_{\mu\lambda}R_{\nu}^{\lambda}-2R^{\gamma\delta}R_{\gamma\mu\delta\nu}+R_{\mu}^{\alpha\gamma\delta}R_{\alpha\nu\gamma\delta}\right)$
(23) $\displaystyle-\frac{1}{2}g_{\mu\nu}\mathcal{G}_{GB}$

the contribution coming from the GB term (20), the so–called Lanczos tensor.
As we know, for $n=4$, the tensor $H_{\mu\nu}$ is zero because the GB term
(20) becomes a topological invariant, not affecting the dynamic of the theory.
Besides, $T^{\mu}_{\nu}$ is the energy–momentum tensor for matter fields
obtained from $S_{m}$.

Next, assuming that the energy-momentum tensor is the one describing an
isotropic fluid

$T_{\mu\nu}=\left(\rho+p\right)X_{\mu}X_{\nu}+pg_{\mu\nu},$ (24)

where $\rho$ is the density, $p$ the isotropic pressure and
$X^{\mu}X_{\mu}=-1$ the four–velocity. The field Eqs. (21) subject to the line
element (2) lead to the following Friedmann equations in the GB scenario

$\frac{1}{R^{2}_{AH}}+\frac{\tilde{\alpha}}{R^{4}_{AH}}=\frac{16\pi
G_{n}}{(n-1)(n-2)}\rho,$ (25)
$\left(1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right)\left(\dot{H}-\frac{k}{a^{2}}\right)=-\frac{8\pi
G_{n}}{n-2}(\rho+p).$ (26)

As expected, the above expressions are modified by the presence of the GB
terms throughout the coupling constant $\tilde{\alpha}\equiv(n-3)(n-4)\alpha$
and also by the dimension $n$ of the space-time, recovering the usual
Friedmann equations for the GR scenario when $n=4$.

In obtaining the above expressions, we have used Ricci’s scalar

$\mathcal{R}=6\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}+\frac{k}{a^{2}}\right),$
(27)

and the GB term

$\mathcal{G}_{\text{GB}}=24\left(\frac{k}{a^{2}}+H^{2}\right)\left(\dot{H}+H^{2}\right).$
(28)

###  III.2  UFL in GB gravity and Friedmann equations

In general, to deal with the thermodynamics of $AH$s, one needs to modify the
usual approach based on the thermodynamics first law, employed in the
static/stationary BHs case Bardeen:1973gs ; Bekenstein:1973ur ;
Hawking:1975vcx ; Gibbons:1976ue ; York:1986it . Hayward developed this
approach in a series of articles Hayward:1993wb ; Hayward:1994bu ;
Hayward:1997jp . The main ingredients of this formulation are: i) the
Misner–Sharp (MS) mass $M_{AH}$ Misner:1964je , interpreted as the internal
energy, ii) the HK surface gravity (18) (related with the temperature horizon)
and iii) the areal volume $V_{AH}=\frac{4\pi}{3}R^{3}_{AH}$. Additionally, it
is necessary to introduce new quantities to reach the desired thermodynamics
description. These quantities are: iv) the density work $W$ and the v)
energy–flux $\psi_{i}$ across the $AH$ Hayward:1997jp . Using the metric in
the representation given by Eq. (3), the definition of these new pieces is
given by

$W\equiv-\frac{1}{2}h_{ij}T^{ij},$ (29)

and

$\psi_{i}\equiv T^{j}_{i}\nabla_{i}R+W\nabla_{i}R,$ (30)

respectively.

Putting together all these objects, the thermodynamics UFL acquires the
following form Hayward:1993wb ; Hayward:1994bu ; Hayward:1997jp

$\nabla_{i}M=A\psi_{i}+W\nabla_{i}V,$ (31)

or by identifying the mass $M$ with the internal energy $E$ one obtains

$\nabla_{i}E=A\psi_{i}+W\nabla_{i}V.$ (32)

It is worth mentioning that the term $A\psi_{i}$ is called the energy-supply
vector.

As stated by Hayward in Hayward:1997jp , the thermodynamics description comes
when (32) is projected along the direction of the $AH$, that is,

$z^{i}\nabla_{i}E=\frac{\kappa_{\text{HK}}}{8\pi}z^{i}\nabla_{i}A+Wz^{i}\nabla_{i}V,$
(33)

where the following identification has been made
$A\psi_{i}=\frac{\kappa_{\text{HK}}}{8\pi}z^{i}\nabla_{i}A$ (see
Hayward:1997jp ; Cai:2006rs  for further details about how to obtain this
identification) and $z^{i}$ is a vector tangent to the $AH$. Interestingly,
the Eq. (33) can be re-expressed as

$\frac{\kappa_{\text{HK}}}{8\pi}z^{i}\nabla_{i}A=R^{n-3}z^{i}\nabla_{i}\left(\frac{E}{R^{n-3}}\right)-Wz^{i}\nabla_{i}V,$
(34)

where it is argued that the first term on the right-hand member of the above
expression becomes zero along the $AH$ Cai:2005ra ; Helou:2015yqa .

In Cai:2005ra ; Akbar:2006kj ; Cai:2006rs , the above statement was employed
(in the GR framework) to recognize the left-hand member as the Clausius
relation. In this way, the second Friedmann equation is obtained using the
corresponding entropy and the temperature identification with the HK surface
gravity. Nevertheless, in Binetruy:2014ela ; Helou:2015yqa ; Helou:2015zma ,
it was shown that these assumptions are not necessary to regain the second
Friedmann equation. This is because the term claimed to be zero along the $AH$
is zero for any constant $R$ (including the particular case $R=R_{A}$).
Moreover, projecting the remaining terms leads to the desired result without
further consideration. The same argument was used in the context of GB gravity
Cai:2005ra . Nevertheless, the MS energy (mass) in the GB theory contains
additional terms (corrections) Maeda:2006pm ; Maeda:2007uu ; Maeda:2008nz ,
contributing to the equation of motion (see Eqs. (25)–(26)). Therefore, in the
present case, it is not correct to argue that the first term in the right-hand
side of Eq. (34) is vanishing along the $AH$ because both the left-hand side
and the second term in the right-hand side, are not containing any information
about the corrections introduced by the GB term. The only way to assume the
above is to set $AH$ to be constant and then assume the Clausius relation.
However, this is a strong restriction because the $AH$ is dynamic. Then, to
recast Friedmann Eqs. (25)–(26) from the UFL (32) we are going to project it
along the Kodama vector direction (leading to Eq. (26)) and along the
direction orthogonal to the Kodama vector (obtaining Eq. (25)). It is worth
mentioning that the information about the field equations is already contained
in (32). Therefore, by projecting it along different directions, we only
isolate a particular information about the FLRW dynamic.

To proceed further, we must introduce an essential ingredient: the MS energy
for GB theory. In Maeda:2006pm ; Maeda:2007uu ; Maeda:2008nz , the spherically
symmetric gravitational collapse of a dust cloud in the GB scenario, where the
MS energy Misner:1964je  associated with this process was found to be

$\displaystyle E$ $\displaystyle\equiv\frac{(n-2)\pi^{(n-1)/2}}{8\pi
G_{n}\Gamma\left[\frac{(n-1)}{2}\right]}\bigg{[}R^{n-3}\left(1-h^{ij}\nabla_{i}R\nabla_{j}R\right)$
(35)
$\displaystyle+\tilde{\alpha}R^{n-5}\left(1-h^{ij}\nabla_{i}R\nabla_{j}R\right)^{2}\bigg{]}.$

The MS energy (35) evaluated at the FLRW metric (2) reads

$E_{\text{FLRW}}=\frac{(n-2)\pi^{(n-1)/2}}{8\pi
G_{n}\Gamma\left[\frac{(n-1)}{2}\right]}\frac{R^{n-1}}{R^{2}_{AH}}\left[1+\frac{\tilde{\alpha}}{R^{2}_{AH}}\right].$
(36)

Of course, the above expressions reduce to the GR for $n=4$.

It should be noted that the UFL (32) was derived using the warped chart (3).
As we are working in the Schwarzschild patch (8) where the time $T$ and radial
$R$ coordinates are independent coordinates from each one, we need to
transform every object of (32) to get the desired result, taking into account
that the area $A$, the work density $W$ and the volume $V$ do not transform
since they are invariant (scalars). Therefore, for the component of the
covariant derivative, one has

$\displaystyle\nabla_{T}$ $\displaystyle=$ $\displaystyle\frac{\partial
t}{\partial T}\nabla_{t}+\frac{\partial r}{\partial
T}\nabla_{r}=F\nabla_{t}-\frac{HRF}{a}\nabla_{r},$ (37)
$\displaystyle\nabla_{R}$ $\displaystyle=$ $\displaystyle\frac{\partial
t}{\partial R}\nabla_{t}+\frac{\partial r}{\partial
R}\nabla_{r}=-\beta\nabla_{t}+\frac{1}{a}\nabla_{r},$ (38)

while for the energy–flux vector

$\displaystyle\psi_{T}$ $\displaystyle=$ $\displaystyle\frac{\partial
t}{\partial T}\psi_{t}+\frac{\partial r}{\partial
T}\psi_{r}=F\psi_{t}-\frac{HRF}{a}\psi_{r},$ (39) $\displaystyle\psi_{R}$
$\displaystyle=$ $\displaystyle\frac{\partial t}{\partial
R}\psi_{t}+\frac{\partial r}{\partial
R}\psi_{r}=-\beta\psi_{t}+\frac{1}{a}\psi_{r},$ (40)

obtaining

$\displaystyle\psi_{T}$ $\displaystyle=$ $\displaystyle-
HRF\left(p+\rho\right),$ (41) $\displaystyle\psi_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-k\frac{R^{2}}{a^{2}}\right)\left(1-\frac{R^{2}}{R^{2}_{AH}}\right)^{-1}\left(p+\rho\right).$
(42)

To get (41) and (42) we used $\psi_{t}=-HR\left(p+\rho\right)/2$ and
$\psi_{r}=a\left(p+\rho\right)/2$.

Now, the UFL (32) projected on the Kodama (14) vector yields to

$K^{T}\nabla_{T}E=AK^{T}\psi_{T}+WK^{T}\nabla_{T}V,$ (43)

giving

$\begin{split}K^{T}\nabla_{T}E=-\frac{(n-2)\pi^{(n-1)/2}}{4\pi
G_{n}\Gamma\left[\frac{(n-1)}{2}\right]}F^{2}\sqrt{1-\frac{k}{a^{2}}R^{2}}\frac{R^{n-1}\dot{R}_{AH}}{R^{3}_{AH}}&\\\
\times\left(1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right),\end{split}$ (44)
$WK^{T}\nabla_{T}V=0,$ (45)
$AK^{T}\psi_{T}=-\frac{2\pi^{(n-1)/2}}{\Gamma\left[\frac{(n-1)}{2}\right]}F^{2}HR^{n-1}\left(p+\rho\right).$
(46)

Putting together (44), (45) and (46) and after some algebra we obtain

$\left(1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right)\left(\dot{H}-\frac{k}{a^{2}}\right)=-\frac{8\pi
G_{n}}{n-2}(\rho+p),$ (47)

where the following expression has been used in obtaining the above result

$\dot{R}_{AH}=-HR^{3}_{AH}\left(\dot{H}-\frac{k}{a^{2}}\right),$ (48)

recalling that every object is an implicit function of the $T$ time
coordinate. Thus, the over-dot means differentiation with respect to $T$.

Next, the orthogonal vector to the Kodama vector is given by

$M=M^{R}\partial_{R}=-\frac{1}{a}\sqrt{1-k\frac{R^{2}}{a^{2}}}\left(1-\frac{R^{2}}{R^{2}_{AH}}-H^{2}R^{2}\right)\partial_{R}.$
(49)

As can be seen, the vector (49) is tangent to the hypersurface $\Sigma$, thus
orthogonal to the Kodama vector (see Fig. 1), and hypersurface orthogonal to
the two spheres defining the $AH$ (equivalently hypersurface orthogonal to the
(n–2)–dimensional hypersurface (16)). However, it is not a null vector on (16)
as the Kodama vector does.

Projecting (32) along the direction of (49) we have

$\begin{split}M^{R}\nabla_{R}E=-\frac{(n-2)\pi^{(n-1)/2}}{8\pi
aG_{n}\Gamma\left[\frac{(n-1)}{2}\right]}\sqrt{1-k\frac{R^{2}}{a^{2}}}&\\\
\times\left[1-\frac{R^{2}}{R^{2}_{AH}}-H^{2}R^{2}\right]R^{n-2}&\\\
\times\left[1-\frac{R^{2}}{R^{2}_{AH}}\right]^{-1}\bigg{[}-\left(n-1\right)\frac{H^{2}R^{n+2}}{R^{2}_{AH}}&\\\
\times\left(1+\frac{\tilde{\alpha}}{R^{2}_{AH}}\right)+\frac{2H\dot{R}_{AH}R^{n+2}}{R^{3}_{AH}}&\\\
\times\left(1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right)+\frac{\left(n-1\right)}{R^{2}_{AH}}\left(1+\frac{\tilde{\alpha}}{R^{2}_{AH}}\right)&\\\
\times\left(1-\frac{R^{2}}{R^{2}_{AH}}\right)\bigg{]},\end{split}$ (50)
$\begin{split}WM^{R}\nabla_{R}V=\frac{\pi^{(n-1)/2}}{\Gamma\left[(n-1)/2\right]}\left(1-\frac{R^{2}}{R^{2}_{AH}}\right)^{-1}&\\\
\times\left(1-\frac{R^{2}}{R^{2}_{AH}}-H^{2}R^{2}\right)\left(\rho-p\right)R^{n-2},\end{split}$
(51)
$\begin{split}AM^{R}\psi_{R}=\frac{\pi^{(n-1)/2}}{\Gamma\left[(n-1)/2\right]}\left(1-\frac{R^{2}}{R^{2}_{AH}}\right)^{-1}&\\\
\times\left(1-k\frac{R^{2}}{R^{2}_{AH}}\right)\left(\rho+p\right)R^{n-2}.\end{split}$
(52)

After some algebra, from expressions (50), (51) and (52) the output is

$\frac{1}{R^{2}_{AH}}+\frac{\tilde{\alpha}}{R^{4}_{AH}}=\frac{16\pi
G_{n}}{(n-1)(n-2)}\rho.$ (53)

To obtain the first Friedmann Eq. (53) we have used $W=(\rho-p)/2$,
$V=\pi^{(n-1)/2}R^{n-1}/\Gamma[(n+1)/2]$,
$A=2\pi^{(n-1)/2}R^{n-2}/\Gamma[(n-1)/2]$ and the previous result (51) to
cancel out the pressure $p$.

At this stage, some comments are pertinent. First of all, to obtain the
Friedmann field equations (51) and (50), it is not necessary to assume from
the very beginning the Clausius relation or include the associated entropy and
temperature. Secondly, the complete information about the dynamic of the GB
theory (in this case) in an FLRW background is already contained in the UFL
(32), and its projection on different directions isolates a specific part of
it. As the Kodama vector defines a privileged time direction, it leads to a
dynamical equation as (51), while along the orthogonal direction with respect
to the Kodama flow, the vector (49) provides the equation for the density
$\rho$. Although we have not used the Clausius relation and the corresponding
GB entropy and temperature to obtain the above result, it does not imply that
this relation and associated quantities are not fundamental in the
thermodynamical description of gravity Jacobson:1995ab . In what follows, we
are to get the GB entropy by identifying the surface gravity (18) with the
temperature Bardeen:1973gs ; Hawking:1975vcx ; Bekenstein:1973ur ;
Hayward:1993wb ; Hayward:1994bu ; Hayward:1997jp ; Jacobson:1995ab  and those
terms accompanying it as the entropy variation. After integrating these terms,
we will obtain the desired expression for the GB entropy. Of course, to
achieve it, one needs to employ (32) as the starting point in conjunction with
the on-shell equations of motion.

###  III.3 The GB entropy

In the previous subsection III.2, we obtained the dynamic description
(equation of motion) for the FLRW space-time in the GB gravity theory. It was
reached starting from the thermodynamics UFL (32) and projecting it along
different directions concerning the $AH$. In doing this, there was no
assumption about temperature and entropy. Notwithstanding, as was proven and
established at the starting works Bardeen:1973gs ; Hawking:1975vcx ;
Bekenstein:1973ur ; Gibbons:1976ue ; York:1986it ; Jacobson:1995ab , there is
a strong connection between the temperature and the surface gravity and also
between the entropy and the area of the horizon (whatever the horizon, event
or apparent horizons). Therefore, the link _gravity_ $\iff$ _thermodynamics_
becomes evident (see Fig. 2 for a schematic representation of this
conjecture).

Figure 2: A schematic representation of the thermodynamics–gravity conjecture.
Basically, this diagram reflects that, from the action principle, the field
equations and thermodynamics description for a gravity theory are obtained.
Furthermore, given a thermodynamics description, the field equations can also
be obtained, and given the field equations, the thermodynamics description is
reached, too.

In this subsection, we want to obtain the entropy for the GB theory in the
FLRW background. Supported by the bi-directional relationship between gravity
and thermodynamics, we will start from the MS energy at the $AH$ together with
the equations of motion. The differential form of the MS energy together with
the UFL (32) and the surface gravity (18) shall allow us to recognize the
temperature and, consequently, the associated entropy. The result must be
consistent with those expressions obtained for the GB entropy using the
Euclidean path integral Gibbons:1976ue ; York:1986it ; Myers:1988ze  and also
the so-called Wald entropy formula based on Noether charges Wald:1993nt ;
Clunan:2004tb . It is worth mentioning that this approach has also been used
in the context of BHs. These three ways to obtain the horizon entropy lead to
the same expression. It can not be the exception for the cosmological
scenario, independently of the gravitational theory.

The first assumption is to consider that the whole MS energy of the system is
confined within an areal volume $V_{R_{AH}}$ along with a density $\rho_{AH}$,
that is,

$E_{AH}=\rho_{AH}V_{AH}.$ (54)

The time variation of the above expression is

$\frac{dE_{AH}}{dt}=\rho_{AH}\frac{dV_{AH}}{dt}+V_{AH}\frac{d\rho_{AH}}{dt}.$
(55)

Now, making use of the work density $W$, the above equation can be recast as

$\frac{dE_{AH}}{dt}=W\frac{dV_{AH}}{dt}+V_{AH}\frac{d\rho_{AH}}{dt}+\frac{\left(\rho_{AH}+p_{AH}\right)}{2}\frac{dV_{AH}}{dt}.$
(56)

The last two-term in the right-hand member of the previous expression provide

$\begin{split}V_{AH}\frac{d\rho_{AH}}{dt}+\frac{\left(\rho_{AH}+p_{AH}\right)}{2}\frac{dV_{AH}}{dt}=\frac{(n-2)\pi^{(n-1)/2}}{2G_{n}\Gamma[(n-1)/2]}&\\\
\times\left[1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right]R^{n-3}_{AH}\frac{dR_{AH}}{dt}&\\\
\times\left[\frac{1}{2\pi
R_{AH}}\left(\frac{1}{2HR_{AH}}\frac{dR_{AH}}{dt}-1\right)\right].\end{split}$
(57)

Now, from (18), it is not hard to show that for the FLRW at the $AH$ one gets

$\kappa_{\text{HK}}=\frac{1}{R_{AH}}\left(\frac{1}{2HR_{AH}}\frac{dR_{AH}}{dt}-1\right),$
(58)

and following Hawking:1975vcx ; Hayward:1997jp  we know that

$T\equiv\frac{\kappa_{\text{HK}}}{2\pi},$ (59)

from (57), it is clear that the last term corresponds to the temperature of
the system evaluated at the $AH$, so (57) can be recast as

$\begin{split}V_{AH}\frac{d\rho_{AH}}{dt}+\frac{\left(\rho_{AH}+p_{AH}\right)}{2}\frac{dV_{AH}}{dt}=\frac{(n-2)\pi^{(n-1)/2}}{2G_{n}\Gamma[(n-1)/2]}&\\\
\times\left[1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right]R^{n-3}_{AH}\frac{dR_{AH}}{dt}T_{AH}.\end{split}$
(60)

Turning back to (56) and plugging (60) we obtain

$\begin{split}\frac{dE_{AH}}{dt}=W\frac{dV_{AH}}{dt}+\frac{(n-2)\pi^{(n-1)/2}}{2G_{n}\Gamma[(n-1)/2]}&\\\
\times\left[1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right]R^{n-3}_{AH}\frac{dR_{AH}}{dt}T_{AH}.\end{split}$
(61)

So, as stated by Hayward in Hayward:1997jp , once the UFL (32) is projected
(33) and the surface gravity related to the temperature, the variation of the
area can be identified with the variation of the entropy, that is, $dA\sim dS$
or equivalently the energy–supply term is matching the Clausius relation. So,
from (61) we extract the following information for the GB entropy
$S_{\text{GB}}$

$dS_{\text{GB}}=\frac{(n-2)\pi^{(n-1)/2}}{2G_{n}\Gamma\left[(n-1)/2\right]}\left[1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\right]{R}^{n-3}_{AH}dR_{AH},$
(62)

integrating the above expression it provides

$S_{\text{GB}}=\frac{A_{AH}}{4G_{n}}\left[1+2\frac{\tilde{\alpha}}{R^{2}_{AH}}\frac{(n-2)}{(n-4)}\right],$
(63)

or for a general areal radius $R$

$S_{\text{GB}}=\frac{A}{4G_{n}}\left[1+2\frac{\tilde{\alpha}}{R^{2}}\frac{(n-2)}{(n-4)}\right].$
(64)

This result (63) was previously obtained using the Euclidean Path integral in
Myers:1988ze  and also using the Wald entropy formula in Clunan:2004tb . As
can be seen, all these approaches are consistent, yielding the same result.
Despite the entropy (63) obtained for BHs solutions, this is also compatible
with a cosmological scenario. The same happens in the GR framework for the
FLRW cosmological model, where the entropy is $S=A/4$ as in the BH case
Faraoni:2011hf . Nevertheless, it should be considered that for the
cosmological scenario, the area of the $AH$ is changing, leading to a
dynamical entropy compared with its static/stationary BH counterpart where the
area is constant.

##  IV Concluding remarks

The equations of motion, i.e., first and second Friedmann equations, for an
FLRW model in the framework of GB gravity theory have been obtained, Eqs. (47)
and (53). These equations were obtained by projecting the thermodynamics UFL
(32) along different directions with respect to the $AH$. For the second
Friedmann equation (47), the UFL is projected along the direction of the
Kodama vector field (14). This Kodama vector establishes a privileged time
direction (see Fig. 1). In the present context, this vector becomes a null
vector on the $AH$. Then, the $AH$ behaves as a null surface. The Kodama
vector is tangent to the hypersurface (16) describing the $AH$. So, projecting
the UFL (32) along the Kodama vector field, thus along the $AH$, the
acceleration equation is recovered. On the other hand, the projection of the
UFL (32) along the orthogonal direction of the Kodama vector, as (49) does,
the first Friedmann equation (53) is recast. Interestingly, the vector field
(49) is hypersurface orthogonal to the $AH$ (16), then given the equation
moderating the rate expansion of the Universe.

It is worth mentioning that, in deriving these equations, it is not necessary
to assume the notion of temperature and entropy and, therefore, the Clausius
relation. Nevertheless, it does not imply that Clausius’s relation is not
fundamental in the thermodynamics description of the actual scenario. Of
course, as was pointed out by Hayward in Hayward:1997jp , after projecting the
UFL (32), the energy–supply term can be interpreted as the heat flux (see the
first term in the right member of Eq. (33)). Thus, there is a correlation with
the Clausius thermodynamics relation. The point is that, in projecting the UFL
(32) along the Kodama vector (14), this term is vanishing, and the
modifications to the Friedmann equations coming from the GB sector are
introduced from the gradient of the MS energy member. Moreover, as we are
dealing with a dynamical horizon, the $AH$, it is more natural to use the
Kodama vector field (14) to project the UFL (32) along the $AH$ to isolated
each Friedmann equation, already contained in the UFL (32).

So, the previous result was obtained by applying the thermodynamics
conjecture, starting from the thermodynamics description, to reach the dynamic
equations of motion of the GB gravity theory in an FLRW background. Then,
inversely applying the same methodology, from gravity to thermodynamics, we
obtained the corrected or associated entropy to the GB theory (63). This
result coincides with previous ones obtained using different (but at the same
time equivalent) approaches, such as the Path integral Myers:1988ze  and Wald
entropy formula Clunan:2004tb . It should be noted that, in deriving (63), we
have not considered any particular cosmological era, leading to an expanding,
static, or contracting cosmology. This is so because, once a concrete era is
considered, one needs to take into account the signature or global sign of the
surface gravity (18) and temperature (59). This is dictated by the $AH$’s
future (past) inner (outer) feature of the $AH$. Notwithstanding, the
temperature expression appears on both sides and cancels out, leading without
any extra information to the correct entropy (63).

So, in general, to apply the gravity-thermodynamics conjecture to cosmological
scenarios, the relevant geometric object to do thermodynamics over the $AH$ is
the Kodama vector field. This is so because this object isolated the
corresponding dynamic information without assuming strong restrictions on the
entropy, temperature, and Universe evolution. Besides, the inverse Path to
regain thermodynamics potentials does not need to assume anything about the
causal structure of the $AH$ to reach the desired and correct result.

## ACKNOWLEDGEMENTS

J. Saavedra and F. Tello-Ortiz acknowledge to grant FONDECYT N°1220065.. F.
Tello-Ortiz acknowledges VRIEA-PUCV for financial support through Proyecto
Postdoctorado 2023 VRIEA-PUCV.

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