g(x) = \alpha + \beta x + \gamma x^2 + \int_{(0,\infty)}(\lambda+1) \frac {1}{\lambda+ x} {\rm d}\mu\ ,
\end{equation*}

where $\mu$ is a finite Borel measure on $(0,\infty)$. Therefore

f(x) = \alpha x^{-2} + \beta x ^{-1}+ \gamma  + \int_{(0,\infty)}(\lambda+1) \frac {1}{\lambda+ x} {\rm d}\mu\ ,

and since $f$ is operator convex and non-negative $\alpha =0$ and $\beta,\gamma \geq 0$.    It follows immediately that $f$ is operator monotone decreasing.

§ INEQUALITIES RELATING TO OPERATOR MONOTONICITY

The following fundamental theorem of Hiai and Petz [45] is a substantial generalization of the first three theorems in [60], and we explain in the example following the theorem how one recovers those theorems by making two special choices of the  function $f$ figuring in the theorem, namely $f(x) = x^t$ and $f(x) = \int_0^1 x^t {\rm d} t$. The proof is extremely simple, and the only inequality that is used is the Lieb-Ruskai Monotonicity  Theorem.

Let $f:(0,\infty) \to (0,\infty)$ be continuous, and let $\Phi$ be a unital and $2$-positive map from $M_n(\C)$ to $M_m(\C)$. For $X,Y\in M_n^{++}(\C)$,
$$G_f(X,Y) = f(R_X L_Y^{-1})L_Y\ ,$$

and note that this is a positive definite operator.  The following are equivalent:

(1)  The function $f$ is operator monotone increasing.

(2)  For all positive definite $X,Y\in M_n(\C)$,

\begin{equation}\label{ph2Q}
\Phi G_f(\Phi^\dagger(X), \Phi^\dagger(Y))^{-1}\Phi^\dagger     \leq G_f(X,Y)^{-1}\ .
\end{equation}

(3)  For all positive definite $X,Y\in M_n(\C)$,

\begin{equation}\label{ph1Q}
\Phi^\dagger  G_f(X,Y)  \Phi  \leq   G_f(\Phi^\dagger(X), \Phi^\dagger(Y))\ .
\end{equation}

(4)  The map

\begin{equation}\label{ph3Q}
(X,Y,Z)\mapsto  \langle Z, G_f(X,Y)^{-1} Z\rangle
\end{equation}
is jointly convex on $M_n^{++}(\C)\times M_n^{++}(\C) \times M_n(\C)$.

The first three theorems of [60] follow directly from Theorem <ref> and the elementary
fact that $f(x) = x^t$, $0 \leq t \leq 1$ is operator monotone.
Then for $X,Y > 0$, $G_f(X,Y) = R_X^tL_Y^{1-t}$.
Then $\langle K, \Phi^\dagger  G_f(X,Y)  \Phi      K \rangle \leq \langle K, G_f(\Phi^\dagger(X), \Phi^\dagger(Y))K\rangle$
is the same as (<ref>). Likewise, $\langle K, \Phi G_f(\Phi^\dagger(X),
\Phi^\dagger(Y))^{-1}\Phi^\dagger  K\rangle \leq \langle K, G_f(X,Y)^{-1}K\rangle$ is the same as (<ref>).

Next, define $f(x) = \int_0^1 x^t{\rm d}t = (x-1)/ \log x$, so that $f$ is operator monotone increasing.
Then for $X,Y > 0$,
$$G_f(X,Y)K = \int_0^1 Y^{1-t}KX^t{\rm d}t\ ,$$
and then a simple computation gives

$$G_f(X,Y)^{-1}K  = \int_0^\infty \frac{1}{sI + Y} K \frac{1}{sI + X}{\rm d}s \ .$$

One way to do this is to observe that it suffices to consider $K$ of the form
$|v\rangle\langle u|$ where $v$ is an eigenvector of $Y$ and $u$ is an eigenvector of $X$.
by (2) of Theorem <ref>,  for all unital $2$-positive $\Phi$, (<ref>) is valid. Note that this gives somewhat more than the Theorems <ref>, <ref> and <ref>:  Here we have only had to assume that $\Phi$ is $2$-positive, not completely positive.

We now give the impressively simple proof of the Hiai-Petz Theorem, beginning with a simple lemma that is abstracted from their paper [45].

For positive invertible $B\in M_m(\C)$ and $C \in M_n(\C)$ and any $m\times n$ matrix $A$,

\begin{equation}\label{ph3}
A^* B^{-1}A \leq C^{-1} \quad \iff \quad ACA^* \leq B\ ,
\end{equation}

Evidently $A^* B^{-1}A \leq C^{-1} \iff C^{1/2}A^*B^{-1}AC^{1/2} \leq I$. However,
$C^{1/2}A^*B^{-1}AC^{1/2}  =  (B^{-1/2}AC^{-1/2})^* (B^{-1/2}AC^{-1/2})$ has the same non-zero spectrum as

$$ (B^{-1/2}AC^{-1/2})(B^{-1/2}AC^{-1/2})^*  = B^{-1/2}ACA^*B^{-1/2}\ ,$$

and hence  $C^{1/2}A^*B^{-1}AC^{1/2} \leq I \iff B^{-1/2}ACA^*B^{-1/2} \leq I$, which yields (<ref>).

Let $f:(0,\infty) \to (0,\infty)$ be a continuous function. For positive definite $X,Y\in M_n(\C)$,  define $G_f(X,Y) = f(R_X L_Y^{-1})L_Y$ , which is then an invertible operator on $M_n(\C)$. Let $\Phi:M_m(\C) \to M_n(\C)$ be unital and $2$-positive.   Then

\begin{equation}\label{ph1}
\Phi G_f(\Phi^\dagger(X), \Phi^\dagger(Y))^{-1}\Phi^\dagger     \leq G_f(X,Y)^{-1}
\end{equation}

if and only if

\begin{equation}\label{ph2}
\Phi^\dagger  G_f(X,Y)  \Phi  \leq   G_f(\Phi^\dagger(X), \Phi^\dagger(Y))\ .
\end{equation}

Apply Lemma <ref> with $B =G_f(\Phi^\dagger(X), \Phi^\dagger(Y))$, $C = G_f(X,Y)$ and $A = \Phi^\dagger$.

We first show that (1) $\Rightarrow$ (3). That is, suppose that $f$ is operator monotone increasing. We will then show that  (<ref>)
is valid.
By (<ref>) and (<ref>)

$$f(x) :=  \beta + \gamma x + \int_{(0,\infty]}   \frac{x}{\lambda + x}  (1 + \lambda)   {\rm d}\mu\ ,$$

with $\beta,\gamma \geq 0$.
G_f(X,Y) =  \beta  L_Y  + \gamma R_X + \int_{(0,\infty]}   \frac{R_X}{\lambda + R_XL_Y^{-1}}  (1 + \lambda)   {\rm d}\mu}$.
Hence to prove (<ref>), it suffices to prove that

\begin{equation}\label{ph7}
\Phi^\dagger L_Y \Phi \leq L_{\Phi^\dagger(Y)}\ ,\qquad   \Phi^\dagger R_X \Phi \leq R_{\Phi^\dagger(X)}
\end{equation}

and that

\begin{equation}\label{ph8}
\Phi^\dagger   \frac{R_X}{\lambda + R_XL_Y^{-1}}  \Phi  \leq  \frac{R_{\Phi^\dagger(X)}}{\lambda + R_{\Phi^\dagger(X)}L_{\Phi^\dagger(Y)}^{-1}} \ .
\end{equation}

For any $K$,

\begin{eqnarray*}
\langle K, \Phi^\dagger L_Y \Phi K\rangle &=& \tr[\Phi(K^*) Y \Phi(K)] = \tr[\Phi(K)\Phi(K^*) Y ] \\
&\leq& \tr[\Phi(KK^*)Y] = \tr[KK^*\Phi^\dagger(Y)] = \langle K, L_{\Phi^\dagger(Y)} K\rangle\ ,
\end{eqnarray*}

where we have used the Kadison-Choi Schwarz Inequality. The proof of the second inequality in (<ref>) is entirely similar.

By Lemma <ref>, (<ref>) is equivalent to
\Phi \left(\frac{R_{\Phi^\dagger(X)}}{\lambda + R_{\Phi^\dagger(X)}L_{\Phi^\dagger(Y)}^{-1}}  \right)^{-1}\Phi^\dagger  \leq \left(   \frac{R_X}{\lambda + R_XL_Y^{-1}} \right)^{-1}\ ,
which is the same as
\Phi  \left(  \lambda  R_{\Phi^\dagger(X)}^{-1} + L_{\Phi^\dagger(Y)}^{-1}  \right)\Phi^\dagger   \leq \lambda R_X^{-1} + L_Y^{-1}
This is true if and only if for all $n\times n$ matrices $Z$,

\begin{multline*}
\lambda \tr[ \Phi^\dagger(Z) (\Phi^\dagger(X))^{-1} \Phi^\dagger(Z^*) ] +  \tr[ \Phi(Z^*) (\Phi^\dagger(Y))^{-1} \Phi^\dagger(Z)] \leq\\ \lambda \tr[ZX^{-1}Z^*] + \tr[Z^*Y^{-1}Z]\ .
\end{multline*}

However, by Corollary <ref> of the Lieb-Ruskai Monotonicity Theorem (with the observation that this is still valid for $2$-positive maps),  together with the fact that $\Phi^\dagger$ is trace preserving since $\Phi$ is unital,
\tr[ \Phi^\dagger(Z) (\Phi^\dagger(X))^{-1} \Phi^\dagger(Z^*) ]  \leq \tr[ZX^{-1}Z^*]$ and $\tr[ \Phi(Z^*) (\Phi^\dagger(Y))^{-1} \Phi^\dagger(Z)]  \leq  \tr[Z^*Y^{-1}Z]$.
This proves (1) $\Rightarrow$ (3) and by Lemma <ref>, (2) $\iff$(3).
Next, (2)  $\Rightarrow$ (4), i.e., that
the map $(X,Y,Z)\mapsto  \langle Z, G_f(X,Y)^{-1} Z\rangle$ is jointly convex,  by taking $\Phi^\dagger$ to be the partial trace as in the proof of Kiefer's inequality using Corollary <ref>.

We now show that (4) $\Rightarrow$ (1), i.e., that $f$ is operator monotone increasing.  Let $u$ be any unit vector in $\C^n$ and then for any $v\in \C^n$ define $Z = |v\rangle \langle u|$.  Note that for $Y =I$, then $G_f(X,I) = f(R_X) = R_{f(X)}$. Hence the joint convexity of $(X,Y,Z) \mapsto \langle Z, G_f(X,Y)^{-1} Z\rangle$ implies the joint convexity of the map
$(X,v) \mapsto \langle v, f^{-1}(X) v\rangle$.   By a Theorem <ref>, this means that $1/f$ is operator monotone decreasing, and then $f$ is operator monotone increasing.

§ TRACE  INEQUALITIES ASSOCIATED TO OPERATOR CONVEX FUNCTIONS

The Data Processing Inequality

D(\Phi^\dagger(X)||\Phi^\dagger(Y))  \leq D(X||Y)

lies  outside the direct scope of Theorem <ref> because $D(X||Y)$ cannot be written in terms of $G_f(X,Y)$ for any operator monotone function $f$ from $(0,\infty)$ to $(0,\infty)$. However, one does have $D(X||Y)= \langle I, G_f(X,Y)  I \rangle$  for $f(x)= x\log x$ and also
$D(Y||X)= \langle I, G_f(X,Y)  I \rangle$ for $f(x) = - \log x$.  Both of these functions are operator convex,  as may be seen from the integral representation

\begin{equation}\label{logrep}
\log (x) = \int_0^\infty \left(\frac{1}{\lambda+1} - \frac{1}{\lambda +x}\right){\rm d}\lambda\ .
\end{equation}

Alternatively, in 1986 Donald [36]  gave an entirely elementary proof of the joint convexity of $D(X||Y)$ taking as his starting point the integral formula

\begin{equation}\label{eq4}
\int_0^1 \frac{(y-x)^2 t}{(1-t)x + ty}{\rm d} t =  x(\log x - \log y) + y -x
\end{equation}

valid for all $x,y>0$.
Applying this with $R_X$ in place of $x$ and $L_Y$ in place of $y$, where $X$ and $Y$ are two non-degenerate density matrices, he obtained

\begin{equation}\label{eq5}
D(X||Y) = \int_0^1\left\langle \one, (L_Y-R_X) \frac{1}{(1-t)R_X + t L_Y}  (L_Y-R_X) \one \right\rangle  t{\rm d}t \ .
\end{equation}

Now the joint convexity is an immediate consequence of Kiefer's inequality. However, Donald did something even more elementary: He used this formula to deduce an expression

D(X||Y)  =  \sup \{  \tr[XC  + YD ]\ : \ (X,Y) \in \Omega \}\ ,

where $\Omega$ is specific convex set in $M_n(\C) \times M_n(\C)$ consisting of pairs of self-adjoint matrices.    This effectively displays the relative entropy as a Legendre transform, and specifies the Legendre transform of the relative entropy. His proof  built on ideas of Pusz and Woronowicz [78, 79] who also  gave a proof of the Lieb Concavity Theorem and the joint convexity of the relative entropy by giving explicit Legendre transform representations in <cit.>.  The variational formula of  Pusz and Woronowicz was rediscovered by Kosaki in 1986 [55].  (Kosaki wrote that his result is implicit in [78], but he does not cite [79] in which his formula is explicitly proved.)

We return to the study of operators of the form

\begin{equation}\label{gfdef}
G_f(X,Y) := f(R_X L_Y^{-1})L_Y \ ,
\end{equation}

where now we shall take $f$ to be operator convex.   Some  useful monotonicity theorems in this setting can be obtained from  integral representations and the Lieb-Ruskai Monotony Theorem by making somewhat use of the perspective function construction [34] which associates the function

$$g_f(x,y) := f(x/y)y$$

to the function $f(x)$.
Note that for $f(x) = x^2$, the associated perspective function is $x^2/y$. The integral representation of a general operator convex function
allows it to be expressed in functions of this type, opening the way to the application of the  Lieb-Ruskai Monotony Theorem.

Let $f$ be an operator convex function on $[0,\infty)$ with $f'_+(0)> -\infty$. Let $g_f(x,y)$ be the perspective function of $f$.
Then for some finite Borel measure $\nu$ on $[0,\infty]$, and some $\alpha,\beta \in \R$,

\begin{equation}\label{low13}
=  \alpha y + \beta x  +  \int_{(0,\infty]} \left(   \frac{x^2}{x + \lambda y}   \right)  (1+\lambda) {\rm d} \nu \ .
\end{equation}

From the integral representation (<ref>) for $f$,  and the identity
{\displaystyle \frac{x^2}{\lambda+x} = x -  \frac{\lambda x}{\lambda +x}}

\begin{eqnarray*}
g_f(x,y) &=&  \alpha y + \beta x  + \int_{(0,\infty]} \left(  x - \lambda \frac{x}{\lambda + x/y}\right)(1+\lambda) {\rm d}\nu\nonumber\\
&=& \alpha y  + \beta x   + \int_{(0,\infty]} \left(  x - \lambda \frac{1}{(\lambda^{-1}x)^{-1} + y^{-1}}\right)(1+\lambda) {\rm d}\nu\ .
\end{eqnarray*}

Now using the identity, valid for all $a,b>0$, $(a^{-1}+b^{-1})^{-1}  = a - a(a+b)^{-1}a$,
which follows easily from $a^{-1} + b^{-1} = a^{-1}(a+b) b^{-1}$,  (<ref>) is proved.

The following  lemma can be found in Effros' paper [37].

For operator convex functions $f$ on $(0,\infty)$, let $G_f$ be feined by (<ref>). Then for all positive invertible $X_1,X_2,Y_1,Y_2$,

\begin{equation}\label{low35}
G_f(X_1+X_2,Y_1+Y_2)  \leq G_f(X_1,Y_1) +  G_f(X_2,Y_2)\ .
\end{equation}

and consequently,
$(X,Y) \mapsto G_f(X,Y)$
is jointly convex.

Replacing $f(x)$ by $f(x+\epsilon)$ for $\epsilon>0$ if need be, we have
the integral representation (<ref>), which is equivalent to

\begin{equation}\label{low33}
g_f(x,y) =  \alpha y + \beta x +\int_{(0,\infty]} \left(  x \frac{1}{x + \lambda y} x\right)  (1+\lambda) {\rm d} \nu\ ,
\end{equation}

\begin{equation}\label{low34}
G_f(X,Y) = \alpha L_Y + \beta R_X + \int_{(0,\infty]} \left( R_X \frac{1}{R_X + \lambda L_Y}R_X\right) (1+\lambda) {\rm d} \nu\ .
\end{equation}

Then (<ref>) follows directly from the corollary of the Lieb-Ruskai Theorem asserting that $(A,Z) \mapsto Z^*A^{-1}Z$ is sub-additive  so that for all positive invertible $X_1,X_2,Y_1,Y_2$,

\begin{equation}\label{low35B}
G_f(X_1+X_2,Y_1+Y_2)  \leq G_f(X_1,Y_1)+ G_f(X_2,Y_2)\ ,
\end{equation}

and then since $(X,Y) \mapsto  G_f(X,Y)$ is homogeneous of degree one, it is also jointly convex.

For all operator convex functions $f$ on $\R_+$, and all $K \in M_n(\C)$, the function

\begin{equation}\label{low56}
(X,Y) \mapsto \tr[K^* f(R_XL_Y^{-1})YK]
\end{equation}

is jointly convex on $M_n^{++}(\C)$.

Evidently $ \tr[K^* f(R_XL_Y^{-1})YK] = \langle K, G_f(X,Y) K\rangle$.

Taking $f(x) = -\log x$ and $K = I$, this yields the joint convexity of
$$(X,Y) \mapsto  \tr[Y( \log Y - \log X)]\ .$$

Likewise, taking $f(x) = x^q$, $q \in (1,2]$, we obtain Ando's Convexity Theorem: For all $K\in M_n(\C)$, and all $1 \leq q \leq 2$,

\begin{equation}\label{ando1}
(X,Y) \mapsto \tr[K^* Y^{1-q} K X^{q}]
\end{equation}

is jointly convex.  More generally, Ando proved that if $1 \leq q \leq 2$, and $0 \leq p \leq q-1$,

\begin{equation}\label{ando2}
(X,Y) \mapsto \tr[K^* Y^{-p} K X^{q}]
\end{equation}

is jointly convex. However, as with the Lieb Concavity Theorem, the general case (<ref>) follows from the special case
(<ref>) by an entirely analogous argument.

The next results make a somewhat different use of the perspective function construction which does not involve the GNS representation.

Let $f:[0,\infty) \to [0,\infty)$.  Then for all positive invertible operators $X,Y$ on a Hilbert space $\cH$,  the operator perspective function $g_f(X,Y)$ is defined by

\begin{equation}\label{opper}
g_f(X,Y)  := Y^{1/2} f(Y^{-1/2}X Y^{-1/2}) Y^{1/2}\ ,
\end{equation}

where $f(Y^{-1/2}X Y^{-1/2})$ is defined using the spectral theorem.

Let  $f$ be an operator convex function on $[0,\infty)$.  For all positive
invertible operators $X,Y$ on a Hilbert space $\cH$, let $g_f(X,Y)$ be defined by (<ref>).   Then for all $2$-positive maps $\Phi$,

\begin{equation}\label{low21}
\Phi(g_f(X,Y) ) \geq   g_f(\Phi(X),\Phi(Y)) \ .
\end{equation}

Suppose that  $f$ is an operator convex function on $[0,\infty)$. Then from the integral representation (<ref>),

\begin{eqnarray*}%\label{irep1B}
f(Y^{-1/2}X Y^{-1/2}) &=& \alpha + \beta Y^{-1/2}X Y^{-1/2}\nonumber\\
&+&  \int_{(0,\infty]} Y^{-1/2}X Y^{-1/2} \frac {1}{\lambda+ Y^{-1/2}X Y^{-1/2}} Y^{-1/2}X Y^{-1/2}{\rm d}(\lambda+1)\nu\ .
\end{eqnarray*}

It follows that

\begin{equation*}%\label{irep1B2}
g_f(X,Y) = \alpha Y + \beta X
+  \int_{(0,\infty]}X\frac {1}{\lambda Y+ X } X(\lambda+1)  {\rm d}\nu \ ,
\end{equation*}

and hence

\begin{equation*}%\label{irep1B3}
\Phi(g_f(X,Y)) = \alpha  \Phi(Y) + \beta \Phi(X)
+  \int_{(0,\infty]}\Phi\left(X\frac {1}{\lambda Y+ X } X\right) (\lambda+1) {\rm d} \nu \ .
\end{equation*}

By Corollary <ref> to the Lieb-Ruskai Monotonicity Theorem,

\Phi\left(X\frac {1}{\lambda Y+ X } X\right)  \geq \Phi(X) \frac{1}{ \lambda \Phi(Y) + \Phi(X)} \Phi(X)\ .

One can dispense with the condition that $\Phi$ be $2$-positive, and instead assume only that $\Phi$
is a non-zero positive map. In the case $\Phi(I)$ is invertible, all one need do is to apply Theorem <ref> in place of
Corollary <ref> in the proof of Theorem <ref>.   In the general case, one proceeds as in the proof of Corollary <ref>:
One first defines the positive  map $\widehat{\Phi}$  in terms of $\Phi$ as in (<ref>), and then since this map is positive with $\Phi(I)$ invertible,
one obtains

\widehat{\Phi}(g_f(X,Y) ) \geq   g_f(\widehat{\Phi}(X),\widehat{\Phi}(Y)) \ .

Now one uses (<ref>) to recover $\Phi$ from $\widehat{\Phi}$, and crucially, the $*$-homomorphism property (<ref>).   See <cit.> for a proof of Theorem <ref> for positive maps $\Phi$ under the assumption that $\Phi(I)$ is invertible that is enatailed in the use of Ando's more restrictive definition [2] of positivity of $\Phi$; i.e., $\Phi:M_n^{++}(\C) \to M_m^{++}(\C)$, and not simply
$\Phi:M_n^{+}(\C) \to M_m^{+}(\C)$.

The following corollary of Theorem <ref> is  central to the Kubo-Ando theory of operator means [57].

Let  $f$ be an operator convex function on $[0,\infty)$. For all positive
invertible operators $X,Y$ on a Hilbert space $\cH$, let $g_f(X,Y)$ be defined by (<ref>).

\begin{equation}\label{low22}
(X,Y) \mapsto g_f(X,Y)
\end{equation}

is jointly operator convex.

This follows from Theorem <ref> by taking $\Phi$ to be the partial trace, exactly as in the proof of Kiefer's inequality (<ref>). (Note that for functions that are homogeneous of degree one, convexity is the same as subadditivity.)

Let $f$ be an operator convex function on $[0,\infty)$. For all positive
invertible operators $X,Y$ on a Hilbert space $\cH$ and all positive unital maps,

\tr[ Yf(Y^{-1/2}XY^{-1/2}) ] \geq \tr[ (\Phi^\dagger(Y))f((\Phi^\dagger(Y))^{-1/2}\Phi^\dagger(X)(\Phi^\dagger(Y))^{-1/2})]\ ,

This follows directly from Theorem <ref>, together with Remark <ref>, the fact that $\Phi^\dagger$ is trace preserving, and the definition of $g_f(X,Y)$.

The function $D_{BS}(Y||X) := \tr[ Y \log( Y^{1/2}X^{-1}Y^{1/2})]$ is known as Belavkin-Stasewski relative entropy [13]. By Corollary <ref>, we have the corresponding version of the Data Processing Inequality:

D_{BS}(\Phi^\dagger(Y)||\Phi^\dagger(X)) \leq D_{BS}(Y||X)\

for all unital positive maps $\Phi$; see [13] for the completely positive case.

The coda to this paper is a timeline. We have discussed many theorems that have been proven to be equivalent; in some sense
these theorems are the many faces of SSA.
The discussion has not been chronological, but rather focused on the connections between the ideas that have emerged in the 60 years  since the
1962 paper of Wigner and Yanase [101]. The SSA conjecture of Lanford and Robinson  [58] was published in 1968.  The 1970 paper of Araki and
Lieb accelerated the progress. It identified the need for a generalization of the Golden-Thompson inequality to three matrices. At least Lieb and Uhlmann
were working  on this problem, completely independently of one another. They  both realized that proving that
$X\mapsto \tr[\exp(H+\log X)]$ is concave on $M_n^{++}(\C)$ would lead to a proof of SSA; as explained above, this is
equivalent through the simple Lemma <ref> to Lieb's Triple Matrix Theorem, which, as explained in Section 4, can be used to close the gap between the partial result of Araki and Lieb and SSA.

The parallel but partial insights of Uhlmann are recorded in the final section of
[97], in which he proved a mild generalization of Theorem 1 in  [6], and then he conjectured <cit.>  the concavity of
$X\mapsto \tr[\exp(H+\log X)]$. Presumably, by the time he made this conjecture, Lieb had already proved it, but in any case when Lieb proved the result, he was unaware of any such conjecture by anybody else, as he has told me. Uhlmann [97] did not explain how the concavity of $X\mapsto \tr[\exp(H+\log X)]$ would lead to the proof of SSA.  He also noted, again without explanation,  that his Conjecture III
(Lieb's Theorem <ref>) would follow from the generalization of the Wigner-Yanase result that became Epstein's Theorem. While he did not prove his
conjecture, he was following the trail blazed by Araki and Lieb [6] and had realized the connection with the work of  Wigner and Yanase [101, 102]. However,
Lieb [60] was  the first to seal the connection between the work of Wigner and Yanase and the SSA conjecture, and to prove the missing
Triple Matrix Theorem, Theorem <ref>.  All of the many deep results [60] are derived by simple arguments from the Lieb Concavity Theorem, Theorem <ref>, which had settled the Wigner Yanase Dyson conjecture.  In this sense, all of the deep theorems in [60] owe their  depth to the Lieb Concavity Theorem.

The 1973 paper of Lieb [60] opened the floodgates.  The 1973 proof of SSA by Lieb and Ruskai quickly followed. By what has been explained in the last paragraph, all of the results in their paper
ultimately depended on the Lieb Concavity Theorem. The paper [60] was written while Elliott Lieb was visiting I.H.E.S., and after
Henri Epstein, then a Professor at I.H.E.S., learned of the results, he provided not only a proof of Theorem <ref>,  the one  conjecture in [60],
but also a new proof of  the concavity of
$X\mapsto \tr[\exp(H+\log X)]$, Theorem <ref>, and his proof was independent of the Lieb Concavity Theorem.  Epstein does not discuss SSA,
but with his proof of Theorem <ref>, one has the first route to the proof of SSA,
building on [6], that does not depend upon the Lieb Concavity Theorem; Theorem <ref>.

The next important breakthrough came  in 1977  and  is due to Uhlmann [98]. He proved, building on work of  Pusz and Woronowicz [78],
that the DPI was valid not only for all completely positive trace preserving  maps, but  for all  adjoints of unital Schwarz maps. He also introduced the monotonicity version of the Lieb Concavity Theorem, and even proved it for the wider class of unital Schwarz maps <cit.>.

The next progress came in 1978 with the work of Pusz and Woronowicz [79] who gave a new proof of the Lieb Concavity Theorem
and the joint convexity of the relative entropy by displaying the relevant  functionals as explicit Legendre transforms.  They did not discuss SSA directly,
but their results provided an alternate route to SSA on account of the equivalence with joint convexity of the relative entropy which was well understood by then.  The approach of Pusz and Woronowicz
was greatly simplified in 1986  by Donald [36] who provided the first truly elementary proof of the joint convexity of the relative entropy,
using nothing but direct computation. Again, he didnot discuss SSA, but his results provided the first completely elementary   path to
SSA.  Finally, in [21], a proof of  classical SSA  that does not refer
to conditional probabilities was provided, and it was shown, using Epstein's Theorem, that this proof extended to the quantum case.

As noted above, since the work of Uhlmann, it has been known that  complete positivity was not required for the DPI; it is valid for all trace preserving Schwarz maps. In [77]  Petz asked an important  question: Is the DPI
valid for all positive maps trace preserving maps?  It was recently proved by Müller-Hermes and Reeb [69] that this is the case. They drew on work of Beigi [11] who proved the monotonicity under quantum operations  of certain sandwiched Rényi entropies that are discussed in Section 8 of this paper. His method relied on interpolation for certain  non-commutative $L^p$  spaces. Although he invoked complete positivity in his proof, Müller-Hermes and Reeb noted that positivity alone would have sufficed, and carried this through, and also in an infinite dimensional setting. Shortly afterwards, Jencova [50] extended the proof to a general von Neumann algebra setting.

I thank Elliott Lieb for many stimulating discussions of these topics and more over many years.
I thank Alexander Müller-Hermes for helpful discussion on the extension of monotonicity inequalities beyond the case of $2$-positive maps
to the case of Schwarz maps, or even merely positive maps.  I thank Frank Hansen for helpful discussions on operator monotonicity. I am very grateful to an anonymous referee who found many
typographical errors in the first draft, and who also made substantive suggestions for improvement that heve benefited the paper. Finally, I thank Rupert Frank for another very careful reading, and helpful suggestions after that.

This work was partially supported by  U.S.
National Science Foundation grant  DMS 2055282

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# Arithmetic Progressions of Integers that are
Relatively Prime to their Digital Sums

Ryan Blau<EMAIL_ADDRESS>Department of Mathematics &
Physical Sciences, The College of Idaho  Joshua Harrington
<EMAIL_ADDRESS>Department of Mathematics, Cedar Crest
College  Sarah Lohrey<EMAIL_ADDRESS>Department of Mathematics, Bryn
Mawr College  Eliel Sosis<EMAIL_ADDRESS>Department of Mathematics,
University of Michigan  Tony W. H. Wong<EMAIL_ADDRESS>Department of
Mathematics, Kutztown University of Pennsylvania

###### Abstract

For an integer $b\geq 2$, we call a positive integer $b$-anti-Niven if it is
relatively prime to the sum of the digits in its base-$b$ representation. In
this article, we investigate the maximum lengths of arithmetic progressions of
$b$-anti-Niven numbers.
MSC: 11A63, 11B25.
Keywords: Niven, anti-Niven, arithmetic progressions.

##  1 Introduction

Throughout this paper, let $b\geq 2$ be an integer. For all positive integers
$n$, let $s_{b}(n)$ denote the sum of the digits in the base-$b$ expansion of
$n$, i.e., if $n=\sum_{j=0}^{m}a_{j}b^{j}$, where $m$ is a nonnegative integer
and $0\leq a_{j}\leq b-1$ are integers for each $0\leq j\leq m$, then
$s_{b}(n)=\sum_{j=0}^{m}a_{j}$.

For positive integers $n$, $d$, and $t$, we call the sequence $\\{n+jd:0\leq
j\leq t-1\\}$ a _$d$ -AP of length $t$_ and we call the sequence
$\\{n+jd:j\geq 0\\}$ a _$d$ -AP of infinite length_. A positive integer $n$ is
$b$-_Niven_ if $s_{b}(n)\mid n$. If every term of a $d$-AP is $b$-Niven, we
call it a _$b$ -Niven $d$-AP_. We note that a $b$-Niven $1$-AP is a sequence
of consecutive Niven numbers.

In 1993, Cooper and Kennedy [1] showed that the maximum length of a $10$-Niven
$1$-AP is $20$. Grundman [3] generalized this result in 1994 by showing that
the maximum length of a $b$-Niven $1$-AP is $2b$. These maximum lengths were
shown to be attainable by Wilson [7]. More recently, Grundman, Harrington, and
Wong [4] investigated maximum length $b$-Niven $d$-APs for $d>1$ and
Harrington, Litman, and Wong [5] showed that every infinite $d$-AP contains
infinitely many $b$-Niven numbers.

In 1975, Olivier [6] studied sets
$S_{b}=\\{n\in\mathbb{Z}:\gcd(n,s_{b}(n))=1\\}$ and showed that the natural
density of these sets is $\frac{6}{\pi^{2}}\prod_{p\mid(b-1)}\frac{p}{p+1}$.
In 1997, Cooper and Kennedy [2] published a weaker result that established
Olivier’s density as an upper bound for the density of $S_{10}$.

In this paper, we define a positive integer $n$ to be _$b$ -anti-Niven_ if
$\gcd(s_{b}(n),n)=1$. If every term of a $d$-AP is $b$-anti-Niven, then we
call it a _$b$ -anti-Niven $d$-AP_. In Section 2 we give necessary and
sufficient conditions on $d$, $b$, and $n$ for which the $d$-AP $\\{n+jd:j\geq
0\\}$ contains at least one $b$-anti-Niven number. We also show that there is
no $b$-anti-Niven $d$-AP of infinite length, but for any $b$ and $t$, there
are infinitely many $b$-anti-Niven $d$-APs of length $t$. In Section 3 we
investigate the maximum length of $b$-anti-Niven $d$-APs when $b$ and $d$
satisfy various constraints.

##  2 $b$-anti-Niven Numbers in $d$-APs

In this section, we are going to give several general results on $b$-anti-
Niven numbers in $d$-APs.

######  Lemma 2.1.

Let $\delta$ be a positive integer such that $\delta\mid(b-1)$. Then for all
positive integers $n$, $\delta\mid n$ if and only if $\delta\mid s_{b}(n)$.

###### Proof.

Let $n=\sum_{j=0}^{m}a_{j}b^{j}$, where $m$ is a nonnegative integer and
$0\leq a_{j}\leq b-1$ are integers for each $0\leq j\leq m$. The proof follows
from the simple observation that $b\equiv 1\pmod{\delta}$ and thus
$\sum_{j=0}^{m}a_{j}b^{j}\equiv\sum_{j=0}^{m}a_{j}\pmod{\delta}$. ∎

######  Theorem 2.2.

The $d$-AP of infinite length $\\{n+jd:j\geq 0\\}$ contains a $b$-anti-Niven
number if and only if $\gcd(n,d,b-1)=1$.

###### Proof.

Assuming that $\gcd(n,d,b-1)=1$, we have $\gcd(s_{b}(n),s_{b}(d),b-1)=1$ by
Lemma 2.1. By Proposition 2.6 of Harrington, Litman, and Wong [5], there
exists a positive multiple $\overline{d}$ of $d$ such that
$\gcd(s_{b}(n),s_{b}(\overline{d}))=1$. Let $k$ be a positive integer such
that $s_{b}(n)+k\cdot s_{b}(\overline{d})=p$ is a prime with
$p>\max(b,\overline{d})$, and we further let
$m_{0}=\lfloor\log_{b}(n)\rfloor+1$ and
$m_{i}=m_{i-1}+\lfloor\log_{b}(\overline{d})\rfloor+1$ for all $1\leq i\leq
k$. Consider $n+j\overline{d}$ and $n+j^{\prime}\overline{d}$, where
$j=\sum_{i=0}^{k}b^{m_{i}}$ and $j^{\prime}=j-b^{m_{k}}+b^{m_{k}+1}$. Note
that both $s_{b}(n+j\overline{d})$ and $s_{b}(n+j^{\prime}\overline{d})$ are
equal to $s_{b}(n)+k\cdot s_{b}(\overline{d})=p$, and
$(n+j^{\prime}\overline{d})-(n+j\overline{d})=b^{m_{k}}(b-1)d$ is not
divisible by $p$ since $p>\max(b,\overline{d})$. Hence, at least one of
$n+j\overline{d}$ and $n+j^{\prime}\overline{d}$ is our desired $b$-anti-Niven
number in the given $d$-AP.

Conversely, if $\gcd(n,d,b-1)=\delta>1$, then
$\delta\mid\gcd(n+jd,s_{b}(n+jd))$ for all integers $j\geq 0$ by Lemma 2.1.
Therefore, $\\{n+jd:j\geq 0\\}$ does not contain any $b$-anti-Niven numbers. ∎

The following theorem is a consequence of a result of Harrington, Litman, and
Wong [5] who showed that every arithmetic progression of infinite length
contains at least one $b$-Niven number $n$ with $s_{b}(n)\neq 1$.

######  Theorem 2.3.

For any positive integer $d$, there is no $b$-anti-Niven $d$-AP of infinite
length.

Our next theorem shows that there exist arithmetic progressions of arbitrary
length containing only $b$-anti-Niven numbers.

######  Theorem 2.4.

For every positive integer $t$, there exist positive integers $n$ and $d$ such
that $\\{n+jd:0\leq j\leq t-1\\}$ is a $b$-anti-Niven $d$-AP of length $t$.

###### Proof.

Let $m$ be a positive integer such that $b^{m}/(m(b-1)+1)\geq t$, and let
$d=b(b^{m}-1)(m(b-1)+1)$. Consider the $d$-AP $\\{(d+1)+jd:0\leq j\leq
t-1\\}$. For all $0\leq j\leq t-1$, note that
$\widetilde{j}=(j+1)(m(b-1)+1)\leq b^{m}$. Hence,

$\displaystyle s_{b}((d+1)+jd)$
$\displaystyle=s_{b}(\widetilde{j}(b^{m+1}-b)+1)$
$\displaystyle=s_{b}\big{(}(\widetilde{j}-1)b^{m+1}+b(b^{m}-1-(\widetilde{j}-1))+1\big{)}$
$\displaystyle=s_{b}(\widetilde{j}-1)+s_{b}\big{(}b^{m}-1-(\widetilde{j}-1)\big{)}+1$
$\displaystyle=s_{b}(\widetilde{j}-1)+s_{b}\left(\sum_{j=0}^{m-1}(b-1)b^{j}-(\widetilde{j}-1)\right)+1$
$\displaystyle=s_{b}(\widetilde{j}-1)+m(b-1)-s_{b}(\widetilde{j}-1)+1$
$\displaystyle=m(b-1)+1.$

Since $(d+1)+jd\equiv 1\pmod{m(b-1)+1}$, we conclude that $(d+1)+jd$ is
$b$-anti-Niven for all $0\leq j\leq t-1$. ∎

Although Theorem 2.4 shows that there are $b$-anti-Niven $d$-APs of arbitrary
length, the maximum length of a $b$-anti-Niven $d$-AP is bounded above by
$b-2$ for many values of $b$ and $d$, as shown in the following theorem.

######  Theorem 2.5.

For $b>2$ and a positive integer $d$, let $p$ be the smallest prime such that
$p\mid(b-1)$ and $p\nmid d$. Then every $b$-anti-Niven $d$-AP has length at
most $p-1$.

###### Proof.

Since $p\nmid d$, every $d$-AP of length $p$ contains a multiple of $p$. By
Lemma 2.1, this multiple of $p$ is not $b$-anti-Niven. Hence, the maximum
length of a $d$-AP that contains only $b$-anti-Niven numbers is at most $p-1$.
∎

##  3 Maximum Length $b$-anti-Niven $d$-APs

Theorem 2.5 in the previous section gives a bound on the maximum length of
certain $b$-anti-Niven $d$-APs. We begin this section by demonstrating that
there are instances when this bound is achieved. Theorems 3.2 and 3.3
investigate $1$-APs, i.e. sequences of consecutive $b$-anti-Niven numbers, and
$2$-APs, respectively. The following lemma will be a common tool in
establishing these two theorems.

######  Lemma 3.1.

For all finite collections of distinct primes $q_{1},q_{2},\dotsc,q_{t}$,
there exist infinitely many positive integers $m$ such that $b^{m}\equiv
b\pmod{q}$.

###### Proof.

Without loss of generality, assume that there exists $0\leq t^{\prime}\leq t$
such that $q_{i}\nmid b$ for all $1\leq i\leq t^{\prime}$ and $q_{i}\mid b$
for all $t^{\prime}+1\leq i\leq t$. By Euler’s theorem,
$b^{k\varphi(q_{1}q_{2}\dotsb q_{t^{\prime}})}\equiv 1\pmod{q_{1}q_{2}\dotsb
q_{t^{\prime}}}$ for every positive integer $k$. Hence,
$m=k\varphi(q_{1}q_{2}\dotsb q_{t^{\prime}})+1$ is our desired choice of
integer. ∎

######  Theorem 3.2.

For $b>2$, let $p$ be the smallest prime such that $p\mid(b-1)$. Then the
maximum length of a sequence of consecutive $b$-anti-Niven numbers is $p-1$.
Furthermore, there exist infinitely many such sequences of length $p-1$.

###### Proof.

By Theorem 2.5, the maximum length of a $b$-anti-Niven $1$-AP is at most
$p-1$. It remains to show that such sequences occur infinitely often. Let
$q_{1},q_{2},\dotsc,q_{t}$ be all primes less than $p$. By Lemma 3.1, there
exist infinitely many positive integers $m$ such that $b^{m}\equiv
b\pmod{q_{1}q_{2}\dotsb q_{t}}$. Now, for all $0\leq j\leq p-2$, we have
$s_{b}(b^{m}+j)=j+1$. Since $j+1<p$, for any prime divisor $q$ of $j+1$, we
have $b^{m}+j\equiv b+j\equiv b-1\not\equiv 0\pmod{q}$. Therefore,
$\gcd(b^{m}+j,s_{b}(b^{m}+j))=1$, implying that $\\{b^{m}+j:0\leq j\leq
p-2\\}$ forms a sequence of $p-1$ consecutive $b$-anti-Niven numbers. ∎

######  Theorem 3.3.

Let $b>2$ be such that $b\neq 2^{r}+1$ for any integer $r$, and let $p$ be the
smallest odd prime such that $p\mid(b-1)$. Then the maximum length of a
$b$-anti-Niven $2$-AP is $p-1$. Furthermore, there exist infinitely many such
sequences of length $p-1$.

###### Proof.

By Theorem 2.5, the maximum length of a $b$-anti-Niven $2$-AP is at most
$p-1$. It remains to show that such sequences occur infinitely often. Let
$q_{1},q_{2},\dotsc,q_{t}$ be all primes less than or equal to $b$. By Lemma
3.1, there exist infinitely many positive integers $m$ such that $b^{m}\equiv
b\pmod{q_{1}q_{2}\dotsb q_{t}}$.

Consider the case when $b$ is even. For all $0\leq j\leq p-2$, we have
$s_{b}(b^{m}+2j+1)=2(j+1)$. Since $j+1<p$, for any prime divisor $q$ of
$2(j+1)$, we have $b^{m}+2j+1\equiv b+2j+1\equiv b-1\not\equiv 0\pmod{q}$.
Therefore, $\gcd(b^{m}+2j+1,s_{b}(b^{m}+2j+1))=1$, implying that
$\\{b^{m}+2j+1:0\leq j\leq p-2\\}$ forms a $b$-anti-Niven $2$-AP of length
$p-1$.

Next, consider the case when $b$ is odd. For all $0\leq j\leq(p-1)/2$, we have
$s_{b}(b^{m}+b-p+2j)=1+b-p+2j$. Since $1+b-p+2j\leq b$, for any prime divisor
$q$ of $1+b-p+2j$, we have $b^{m}+b-p+2j\equiv 2b-p+2j\equiv
2b-p+2j-2(1+b-p+2j)\equiv p-2j-2\pmod{q}$. Note that $-1\leq p-2j-2\leq p-2$,
so none of these odd numbers share a common prime factor with $b-1$. Hence,
$\gcd(p-2j-2,1+b-p+2j)=\gcd(p-2j-2,b-1)=1$, implying that $p-2j-2\not\equiv
0\pmod{q}$. Thus, $\gcd(b^{m}+b-p+2j,s_{b}(b^{m}+b-p+2j))=1$ when $0\leq
j\leq(p-1)/2$.

Furthermore, for all $0\leq j\leq(p-5)/2$, we have $s_{b}(b^{m}+b+1+2j)=3+2j$.
Since $3+2j\leq p-2<b$, for any prime divisor $q$ of $3+2j$, we have
$b^{m}+b+1+2j\equiv 2b+1+2j\equiv 2b+1+2j-(3+2j)\equiv 2(b-1)\pmod{q}$. Note
that $2(b-1)\not\equiv 0\pmod{q}$ since $q$ is an odd prime less than $p$.
Thus, $\gcd(b^{m}+b+1+2j,s_{b}(b^{m}+b+1+2j))=1$ when $0\leq j\leq(p-5)/2$.
Therefore, $\\{b^{m}+b-p+2j:0\leq j\leq(p-1)/2\\}\cup\\{b^{m}+b+1+2j:0\leq
j\leq(p-5)/2\\}$ forms a $b$-anti-Niven $2$-AP of length $p-1$. ∎

So far, the theorems in this section have shown that the bound provided in
Theorem 2.5 is attainable. However, there are infinitely many instances when
the maximum length of $b$-anti-Niven $d$-APs does not attain this bound. The
following theorem illustrates one such instance.

######  Theorem 3.4.

Let $b\geq 6$ be even, and let $3\leq d\leq b/2$ be an odd integer. Then the
maximum length of a $b$-anti-Niven $d$-AP is at most $\lceil 2b/d\rceil+2$.

Note that when $b-1$ is an odd prime, the bound given by Theorem 2.5 is $b-2$,
while the bound given by Theorem 3.4 is strictly smaller when $b>15$.

###### Proof of Theorem 3.4.

Suppose there are two $b$-anti-Niven numbers from a $d$-AP in the interval
$[ab,ab+b-1]$. Let these two numbers be $ab+a_{0}$ and $ab+a_{0}+d$ for some
nonnegative integers $a$ and $a_{0}\leq b-1-d$. Recalling that $d$ is odd,
there exists $\chi\in\\{0,1\\}$ so that $a_{0}+\chi d$ is even. As a result,
$ab+a_{0}+\chi d$ is also even since $b$ is even. Since $ab+a_{0}+\chi d$ is
$b$-anti-Niven, $s_{b}(ab+a_{0}+\chi d)=s_{b}(a)+a_{0}+\chi d$ is odd,
implying that $s_{b}(a)$ is odd.

Note that there are at most $\lceil 2b/d\rceil$ terms from a $d$-AP in the
interval $[ab,(a+1)b+(b-1)]$. Hence, if there is a $b$-anti-Niven $d$-AP of
length $\lceil 2b/d\rceil+3$, then there exists a nonnegative integer $a$ such
that each of the intervals $[ab,ab+b-1]$, $[(a+1)b,(a+1)b+b-1]$, and
$[(a+2)b,(a+2)b+b-1]$ contains at least two terms from this $d$-AP. From the
above observation, we conclude that $s_{b}(a)$, $s_{b}(a+1)$, and $s_{b}(a+2)$
are all odd, which is a contradiction. This establishes an upper bound for the
maximum length of a $b$-anti-Niven $d$-AP as stated in the theorem. ∎

We now turn our attention to $d$-APs for which Theorem 2.5 does not apply.

######  Theorem 3.5.

Let $b$ be even. Then the maximum length of a $b$-anti-Niven $(b-1)$-AP is
$2b+1$. Furthermore, there exist infinitely many such sequences of length
$2b+1$.

###### Proof.

Suppose there is a $b$-anti-Niven $(b-1)$-AP $\mathcal{S}$ of length at least
$2b+1$. Since $\gcd(b-1,b)=1$, there exist two terms in $\mathcal{S}$ that are
multiples of $b$. Let these two terms be $ab$ and $ab+(b-1)b$ for some
positive integer $a$. Note that $ab$ is even, so $s_{b}(ab)=s_{b}(a)$ is odd.
Since $ab+2(b-1)=(a+1)b+b-2$ is an even term in $\mathcal{S}$, the digit sum
$s_{b}((a+1)b+b-2)=s_{b}(a+1)+b-2$ must be odd, implying that $s_{b}(a+1)$ is
also odd. Hence, $a=cb+b-1$ for some nonnegative integer $c$, where $s_{b}(c)$
is even. In other words, $ab=cb^{2}+(b-1)b$ and $ab+(b-1)b=(c+1)b^{2}+(b-2)b$.
Also, $s_{b}(c+1)$ is odd since $s_{b}((c+1)b^{2}+(b-2)b)=s_{b}(c+1)+b-2$ is
odd.

Now, note that $cb^{2}$ and $s_{b}(cb^{2})=s_{b}(c)$ are even, so $cb^{2}$ is
not in $\mathcal{S}$. Similarly, $(c+1)b^{2}+(b-1)b+b-2$ and
$s_{b}((c+1)b^{2}+(b-1)b+b-2)=s_{b}(c+1)+b-1+b-2$ are even, so
$(c+1)b^{2}+(b-1)b+b-2=cb^{2}+(2b+2)(b-1)$ is also not in $\mathcal{S}$.
Therefore, $\mathcal{S}$ is a subsequence of $\\{cb^{2}+j(b-1):1\leq j\leq
2b+1\\}$, thus the maximum length of a $b$-anti-Niven $(b-1)$-AP is at most is
$2b+1$.

It remains to show that such sequences occur infinitely often. Let $c$ be a
nonnegative integer such that $s_{b}(c+1)=s_{b}(c)+1$. Then it is not
difficult to observe that $s_{b}(cb^{2}+j(b-1))=s_{b}(c)+b-1$ for $1\leq j\leq
b$ and $b+2\leq j\leq 2b$, and $s_{b}(cb^{2}+j(b-1))=s_{b}(c)+2(b-1)$ for
$j\in\\{b+1,2b+1\\}$. Hence, it suffices to show that there exist infinitely
many positive integers $c$ such that

  * •

$b\nmid(c+1)$,

  * •

$\gcd(cb^{2}+j(b-1),s_{b}(c)+b-1)=1$ for $1\leq j\leq b$ and $b+2\leq j\leq
2b$, and

  * •

$\gcd(cb^{2}+j(b-1),s_{b}(c)+2(b-1))=1$ for $j\in\\{b+1,2b+1\\}$.

Let $p_{1},p_{2},\dotsc,p_{t}$ be all primes less than or equal to $2b$. By
Lemma 3.1, there exist infinitely many positive integers $m$ such that
$b^{m+1}\equiv b\pmod{p_{1}p_{2}\dotsb p_{t}}$. In other words,
$p_{1}p_{2}\dotsb p_{t}\mid b(b^{m}-1)$. Let $P=b^{m}+1$. Since
$\gcd(b^{m}+1,b)=1$ and $\gcd(b^{m}+1,b^{m}-1)=1$, we have
$\gcd(P,p_{1}p_{2}\dotsb p_{t})=1$. Next, consider
$q_{1},q_{2},\dotsc,q_{\tau}$ be all prime factors of $b^{m-1}+1$. Hence,
$b^{m-1}\equiv-1\pmod{q_{1}q_{2}\dotsb q_{\tau}}$. Let
$r_{1},r_{2},\dotsc,r_{(P-b+1)/2}$ be positive integers, where
$r_{i+1}-r_{i}\geq m+1$ for all $1\leq i\leq(P-b-1)/2$, be defined as follows.

  * •

If $(P-b+1)/2$ is odd, then

$b^{r_{i}+2}\equiv\begin{cases}-1\pmod{q_{1}q_{2}\dotsb q_{\tau}}&\text{if
}1\leq i\leq(P-b-1)/4;\\\ 1\pmod{q_{1}q_{2}\dotsb
q_{\tau}}&\text{otherwise}.\end{cases}$

  * •

If $(P-b+1)/2$ is even, then

$b^{r_{i}+2}\equiv\begin{cases}-1\pmod{q_{1}q_{2}\dotsb q_{\tau}}&\text{if
}1\leq i\leq(P-b-3)/4;\\\ 1\pmod{q_{1}q_{2}\dotsb q_{\tau}}&\text{if
}(P-b+1)/4\leq i\leq(P-b-3)/2;\\\ b\pmod{q_{1}q_{2}\dotsb
q_{\tau}}&\text{otherwise}.\end{cases}$

Now, let $c=\sum_{i=1}^{(P-b+1)/2}b^{r_{i}}(b^{m}+1)$. Since $b\mid c$, we
have $b\nmid(c+1)$. Next, $s_{b}(c)=P-b+1$ from our construction, thus
$s_{b}(c)+b-1=P=b^{m}+1$, which is a factor of $c$. Recalling that $P$ is
relatively prime to all positive integers up to $2b$, we have
$\gcd(cb^{2}+j(b-1),P)=1$ for all $1\leq j\leq 2b$. It remains to prove that
$\gcd(cb^{2}+j(b-1),s_{b}(c)+2(b-1))=1$ for $j\in\\{b+1,2b+1\\}$. Note that
$s_{b}(c)+2(b-1)=P-b+1+2(b-1)=P+b-1=b(b^{m-1}+1)$. For any prime factor $q$ of
$b^{m-1}+1$, we clearly have $q\nmid b$. Moreover, $q\nmid(b-1)$, or
otherwise, $q\mid b(b^{m-1}+1)$ and $q\mid(b-1)$ imply $q\mid P$,
contradicting that $P$ is relatively prime to all positive integers up to
$2b$. If $(P-b+1)/2$ is odd, then

$\displaystyle cb^{2}+(b+1)(b-1)$
$\displaystyle=\left(\sum_{i=1}^{(P-b+1)/2}b^{r_{i}+2}\right)P+b^{2}-1$
$\displaystyle\equiv P+b^{2}-1$ $\displaystyle\equiv P+b^{2}-1-(P+b-1)$
$\displaystyle\equiv b(b-1)$ $\displaystyle\not\equiv 0\pmod{q}$

and $cb^{2}+(2b+1)(b-1)=cb^{2}+(b+1)(b-1)+b(b-1)\equiv 2b(b-1)\not\equiv
0\pmod{q}$. If $(P-b+1)/2$ is even, then

$\displaystyle cb^{2}+(b+1)(b-1)$
$\displaystyle=\left(\sum_{i=1}^{(P-b+1)/2}b^{r_{i}+2}\right)P+b^{2}-1$
$\displaystyle\equiv 2bP+b^{2}-1$ $\displaystyle\equiv 2bP+b^{2}-1-2b(P+b-1)$
$\displaystyle\equiv-(b-1)^{2}$ $\displaystyle\not\equiv 0\pmod{q}$

and $cb^{2}+(2b+1)(b-1)=cb^{2}+(b+1)(b-1)+b(b-1)\equiv-(b-1)^{2}+b(b-1)\equiv
b-1\not\equiv 0\pmod{q}$. Finally, our proof is completed by noticing that
$\gcd(cb^{2}+j(b-1),b)=1$ for $j\in\\{b+1,2b+1\\}$. ∎

To complete the investigation on $1$-APs, we provide the following corollary
by choosing $b=2$ in Theorem 3.5.

######  Corollary 3.6.

For $b=2$, the maximum length of a sequence of consecutive $2$-anti-Niven
numbers is $5$. Furthermore, there exist infinitely many such sequences of
length $5$.

##  4 Concluding Remarks

Theorem 3.3 establishes the that the maximum length of a $b$-anti-Niven $2$-AP
is at most $b-2$ when $b-1$ has an odd prime divisor. Although we have not
established an upper bound for the maximum length of a $b$-anti-Niven $2$-AP
when $b=2^{r}+1$ for some nonnegative integer $r$, the following theorem
establishes a lower bound.

######  Theorem 4.1.

Let $b=2^{r}+1$ for some nonnegative integer $r$. Then the maximum length of a
$b$-anti-Niven $2$-AP is at least $b$.

###### Proof.

If $r=0$, then $b=2$, and $\\{2,4\\}$ forms a $2$-anti-Niven $2$-AP of length
$2$. If $r>0$, then $b$ is odd. For all $0\leq j\leq(b-1)/2$, we have
$\gcd(b+2j,s_{b}(b+2j))=\gcd(b+2j,1+2j)=\gcd(b+2j,b-1)=\gcd(2^{r}+1+2j,2^{r})=1$.
Furthermore, for all $0\leq j\leq(b-3)/2$, we have
$\gcd(2b+1+2j,s_{b}(2b+1+2j))=\gcd(2b+1+2j,3+2j)=\gcd(2b+1+2j,2b-2)=\gcd(2^{r+1}+3+2j,2^{r+1})=1$.
Therefore, $\\{b+2j:0\leq j\leq(b-1)/2\\}\cup\\{2b+1+2j:0\leq j\leq(b-3)/2\\}$
forms a $b$-anti-Niven $2$-AP of length $b$. ∎

Theorem 3.5 establishes the that the maximum length of a $b$-anti-Niven
$(b-1)$-AP is at most $2b+1$ when $b$ is even. The following theorem
establishes a lower bound when $b$ is an odd prime.

######  Theorem 4.2.

Let $b$ be an odd prime. Then the maximum length of a $b$-anti-Niven
$(b-1)$-AP is at least $2b+1$.

###### Proof.

Clearly, $1$, $b$, and $b^{2}$ are $b$-anti-Niven. For all $1\leq j\leq b-1$,
we have

$\displaystyle\gcd\big{(}b+j(b-1),s_{b}(b+j(b-1))\big{)}$
$\displaystyle=\gcd\big{(}(j+1)b-j,s_{b}(jb+b-j)\big{)}$
$\displaystyle=\gcd((j+1)b-j,j+b-j)$ $\displaystyle=\gcd((j+1)b-j,b)=1$

since $b$ is a prime. Furthermore, for all $1\leq j\leq b-1$, we have

$\displaystyle\gcd\big{(}b^{2}+j(b-1),s_{b}(b^{2}+j(b-1))\big{)}$
$\displaystyle=\gcd\big{(}b^{2}+jb-j,s_{b}(b^{2}+(j-1)b+b-j)\big{)}$
$\displaystyle=\gcd(b^{2}+jb-j,1+j-1+b-j)$
$\displaystyle=\gcd(b^{2}+jb-j,b)=1.$

Therefore, $\\{1,b\\}\cup\\{b+j(b-1):1\leq j\leq
b-1\\}\cup\\{b^{2}\\}\cup\\{b^{2}+j(b-1):1\leq j\leq b-1\\}$ forms a $b$-anti-
Niven $(b-1)$-AP of length $2b+1$. ∎

Recall that Theorem 3.4 shows that the upper bound on the maximum length of a
$b$-anti-Niven $d$-AP given by Theorem 2.5 may not be achievable for even $b$.
However, when $b$ is odd, computational data suggests otherwise. Of course, if
$d$ is odd, then Theorem 2.5 implies that the maximum length of a $b$-anti-
Niven $d$-AP is at most $1$, which is clearly attainable. It is more
interesting to investigate if $d$ is even. The next conjecture addresses this
more interesting case and generalizes Theorem 3.3 to other even values of $d$.

######  Conjecture 4.3.

Let $b$ be odd such that $b\neq 2^{r}+1$ for any positive integer $r$, let $d$
be even, and let $p$ be the smallest prime such that $p\mid(b-1)$ and $p\nmid
d$. Then there exist infinitely many $b$-anti-Niven $d$-APs of length $p-1$.

Theorem 3.4 established an upper bound for the maximum length of a $b$-anti-
Niven $d$-AP when $b\geq 6$ is even and $3\leq d\leq b/2$ is an odd integer.
We conjecture that this bound is attainable for infinitely many pairs $(b,d)$.

######  Conjecture 4.4.

Let $b\geq 6$ be even, and let $3\leq d\leq b/2$ be an odd integer. There
exist infinitely many pairs $(b,d)$ for which there is a $b$-Niven $d$-AP of
length $\lceil 2b/d\rceil+2$.

##  5 Acknowledgements

These results are based on work supported by the National Science Foundation
under grant numbered MPS-2150299.

## References

  * [1] C. Cooper and R. E. Kennedy, On consecutive Niven numbers, Fibonacci Quart. 31 (1993), 146–151.
  * [2] C. Cooper and R. E. Kennedy, On the set of positive integers which are relatively prime to their digital sum and its complement, J. Inst. Math. Comput. Sci. Math. Ser. 10 (1997), 173–180.
  * [3] H. Grundman, Sequences of consecutive $n$-Niven numbers, Fibonacci Quart. 32 (1994), 174–175.
  * [4] H. Grundman, J. Harrington, and T. W. H. Wong, Arithmetic progressions of $b$-Niven numbers, Rocky Mountain J. Math., to appear (https://projecteuclid.org/journals/rmjm/rocky-mountain-journal-of-mathematics/DownloadAcceptedPapers/211029-Wong.pdf).
  * [5] J. Harrington, M. Litman, and T. W. H. Wong, Every arithmetic progression contains infinitely many b-Niven numbers, Bull. Aust. Math. Soc., to appear (https://doi.org/10.1017/S0004972723000758).
  * [6] M. Olivier, Sur la probabilite que $n$ soit premier a la somme de ses chiffres, C. R. Acad. Sci. Paris Ser. A-B, 280 (1975), A543–A545.
  * [7] B. Wilson, Construction of $2*n$ consecutive $n$-Niven numbers, Fibonacci Quart. 35 (1997), 122–128.


# Thermodynamics for the k-essence Emergent Reissner-Nordstrom-de Sitter
Spacetime

Goutam Manna<EMAIL_ADDRESS>Department of Physics, Prabhat Kumar
College, Contai, Purba Medinipur-721404, India  Bivash Majumder
<EMAIL_ADDRESS>Department of Mathematics, Prabhat Kumar College,
Contai, Purba Medinipur-721404, India  Ashoke Das<EMAIL_ADDRESS>Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134,
India.

###### Abstract

The k-essence emergent Reissner-Nordstrom-de Sitter spacetime has exactly
mapped on to the Robinson-Trautman (RT) type spacetime with cosmological
constant $\Lambda$ for certain configuration of k-essence scalar field.
Theoretically, we evaluated that the thermodynamical quantities for the RT
type emergent black hole is different from the usual one in the presence of
kinetic energy of the k-essence scalar field i.e., the dark energy density. We
restrict ourselves into the fact that the dark energy density (K) is to be
unity, then the effective temperature and pressure both are negative for the
RT type emergent black hole which implies that the system is thermodynamically
unstable when the charge $Q\neq 0$ and the emergent spacetime is only dark
energy dominated and it does not radiate when $Q=0$. The thermodynamically
unstable situation is physically plausible only when we consider spin degrees
of freedom of a system. We have made this analysis in the context of dark
energy in an emergent gravity scenario having k-essence scalar fields $\phi$
with a Dirac-Born-Infeld type lagrangian. The scalar field also satisfies the
emergent equation of motion at $r\rightarrow\infty$.

###### pacs:

04.20.−q, 04.70.−s, 04.70.Bw

##  I Introduction

The differences between the k-essence theory with non-canonical kinetic terms
and the relativistic field theories with canonical kinetic terms are that
nontrivial dynamical solutions of the k-essence equation of motion not only
spontaneously break Lorentz invariance but also, change the metric for the
perturbations around these solutions. Thus the perturbations propagate in the
so-called emergent or analogue curved spacetime babi1 ; babi2 ; babi3 ; babi4
; babi5  with the metric which is different from the gravitational one. The
emergent gravity metric $\tilde{G}_{\mu\nu}$ is not conformally equivalent to
the gravitational metric $g_{\mu\nu}$. The general form of the lagrangian for
k-essence model is: $L=-V(\phi)F(X)$ where $\phi$ is k-essence scalar field,
$X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and does not depend
explicitly on $\phi$ to start with babi1 ; babi2 ; babi3 ; babi4 ; babi5 ; gm1
; gm2 ; gm3 ; scherrer1 ; scherrer2 ; scherrer3 . The Lagrangian for k-essence
scalar fields contain non-canonical kinetic terms. Relevant kinds of
literature gor1 -gor15  for such fields discuss issues on cosmology,
inflation, dark matter, dark energy and strings.

Research interest in the context of the Hawking radiation as well as the
thermodynamics of black holes has gained momentum in several years. Based on a
specific Dirac-Born-Infeld born1 ; born2 ; born3  model for the k-essence
scalar field and specific standard physical black hole spacetimes such as
Schwarzschild, Reissner-Nordstrom, the emergent spacetime is establishing
conformal invariance with the Barriola-Velenkin, Robinson-Trautman (RT) type
spacetimes gm1 ; gm2  and for the Kerr, Kerr-Newman and Kerr-Newman-AdS
background, the emergent metrics are satisfying Einstein equation for large
$r$ along $\theta=0$ gm2 ; gm3 . Also, in the above references, the authors
have established the modification of the Hawking temperature haw1 -haw15  in
the presence of dark energy in an emergent gravity scenario.

Researchers cai ; zhao1 ; zhao2 ; dolan ; seki ; gibbon1 ; gibbon2 ; ren ;
urano ; li ; kubiznak  focus on the thermodynamics of black holes with
cosmological constant $\Lambda$ and describe thermodynamical first law of
black hole horizon spacetimes with $\Lambda$, through the mechanical first law
(MFL). The MFL is a geometrical relation and it relates the variations of the
mass parameter, horizon area and other supplemental quantities. For example,
the MFL of the Schwarzschild black hole is equivalent to its thermodynamical
first law by using the mass as the only variable which describes the effect of
a single horizon on the MFL. The mass parameter $M$ and the cosmological
constant $\Lambda$ are regarded as two independent variables in the MFL for
spacetimes with a cosmological constant $\Lambda$ which possess the black hole
event horizon (BEH) and cosmological event horizon (CEH). Also, in buri ;
ghosh , the authors have studied the thermodynamics and phase structure of the
de Rham, Gabadadze, and Tolley (dRGT) massive gravity rham1 ; rham2
spacetime. They have evaluated the mass, temperature and entropy of the dRGT
black hole, and also performed thermodynamical stability analysis. They have
shown that the presence of the graviton mass completely changes the black hole
thermodynamics. They also have discussed how the linear term $\gamma r$ in the
metric, which is the unique character of the dRGT massive gravity, affected
the structure of the spacetime.

In this work, we investigate theoretically the thermodynamical behaviour of a
black hole in the presence of dark energy in an emergent gravity scenario for
the Reissner-Nordstrom-de Sitter background using the k-essence model. Mention
that one class of theoretical model to investigate the effects of the presence
of dark energy on cosmological scenarios are k-essence models where background
metrics are the usual gravitational metric. We find here the emergent
spacetime in the presence of dark energy can be mapped on to the Robinson-
Trautman(RT) type spacetime with cosmological constant $\Lambda$. Also, we
find when the charge $Q=0$ and the dark energy density is constrained to be
unity then Hawking temperature is zero and when the charge $Q\neq 0$, the
temperature is negative i.e. this RT type of emergent spacetime is
thermodynamically unstable. So we can say that the thermodynamics of this RT
type black hole is different from gm2 . Note that in the Ref. gm2 , when the
dark energy density is unity this black hole has zero Hawking temperature also
but the charge $Q$ is non-zero. Also, we clearly mention that in this work we
can not study the massive gravity in the context of the emergent gravity
scenario.

The paper is organized as follows: we are briefly describe the k-essence
theory and emergent gravity babi1 ; babi2 ; babi3 ; babi4 ; babi5  where
$\tilde{G}_{\mu\nu}$ contains the dark energy field $\phi$ and this should
satisfy the emergent gravity equations of motion in section 2. Again, for
$\tilde{G}_{\mu\nu}$ to be a blackhole metric, it has to satisfy the Einstein
field equations. In section 3, we have shown that the emergent gravity metric
having background metric is Reissner-Nordstrom-de Sitter (RNdS) mapping on to
Robinson-Trautman (RT) type metric with cosmological constant $\Lambda$. So
Einstein field equation is automatically satisfied. The k-essence scalar field
$\phi$ is also satisfied emergent gravity equation of motion at
$r\rightarrow\infty$. In section 4, we discuss the first law of thermodynamics
in the context of dark energy and calculate the various thermodynamical
quantities of RT type emergent spacetime with cosmological constant $\Lambda$.
The last section is the conclusion of our work.

##  II Brief review of the k-essence and Emergent Gravity

In the k-essence theory, the action is babi1 ; babi2 ; babi3 ; babi4 ; babi5 ;
gm1 ; gm2 ; gm3

$\displaystyle S_{k}[\phi,g_{\mu\nu}]=\int d^{4}x{\sqrt{-}g}L(X,\phi)$ (1)

where $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, $g_{\mu\nu}$
is the gravitational metric and $\phi$ is the k-essence scalar field. The
energy momentum tensor is

$\displaystyle T_{\mu\nu}\equiv{2\over\sqrt{-g}}{\delta S_{k}\over\delta
g^{\mu\nu}}=L_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}L$ (2)

$L_{\mathrm{X}}={dL\over dX},~{}~{}L_{\mathrm{X}X}={d^{2}L\over
dX^{2}},~{}~{}L_{\mathrm{\phi}}={dL\over d\phi}$ and $\nabla_{\mu}$ is the
covariant derivative defined with respect to the metric $g_{\mu\nu}$. The
equation of motion is

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\tilde{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+2XL_{X\phi}-L_{\phi}=0$
(3)

where

$\displaystyle\tilde{G}^{\mu\nu}\equiv
L_{X}g^{\mu\nu}+L_{XX}\nabla^{\mu}\phi\nabla^{\nu}\phi$ (4)

and $1+{2XL_{XX}\over L_{X}}>0$.

The emergent gravity metric $\tilde{G}_{\mu\nu}$ is not conformally equivalent
to the gravitational metric $g_{\mu\nu}$ for the non-trivial configurations of
the k-essence field $\phi$, $\partial_{\mu}\phi\neq 0$ (for a scalar
field,$\nabla_{\mu}\phi\equiv\partial_{\mu}\phi$ ), hence for the $k-$essence
theory, the characteristics are do not coincide with those ones for canonical
scalar field the Lagrangian of which depends linearly on the kinetic term $X$.
So this k-essence field $\phi$ has properties different from canonical scalar
fields defined with $g_{\mu\nu}$ and the local causal structure for this field
is different from those one defined by metric $g^{\mu\nu}$.

Carrying out the conformal transformation $G^{\mu\nu}\equiv{c_{s}\over
L_{x}^{2}}\tilde{G}^{\mu\nu}$, with $c_{s}^{2}(X,\phi)\equiv{(1+2X{L_{XX}\over
L_{X}})^{-1}}\equiv sound~{}speed$ we have the inverse metric of $G^{\mu\nu}$
is babi1 ; babi2 ; babi3 ; babi4 ; babi5

$\displaystyle G_{\mu\nu}={L_{X}\over
c_{s}}[g_{\mu\nu}-{c_{s}^{2}}{L_{XX}\over
L_{X}}\nabla_{\mu}\phi\nabla_{\nu}\phi]$ (5)

A further conformal transformation gm1 ; gm2
$\bar{G}_{\mu\nu}\equiv{c_{s}\over L_{X}}G_{\mu\nu}$ gives

$\displaystyle\bar{G}_{\mu\nu}={g_{\mu\nu}-{{L_{XX}}\over{L_{X}+2XL_{XX}}}\nabla_{\mu}\phi\nabla_{\nu}\phi}$
(6)

Note that here always have $L_{X}\neq 0$ for the sound speed $c_{s}^{2}$ to be
positive definite and only then equations $(1)-(4)$ will be physically
meaningful. This can be seen as follows. $L_{X}=0$ means that $L$ independent
of $X$ so that in equation (1), $L(X,\phi)\equiv L(\phi)$. Then $L$ becomes
pure potential but $k-$essence fields correspond to such type of lagrangians
where the kinetic energy dominates over the potential energy. Also the very
concept of minimal coupling of $\phi$ to $g_{\mu\nu}$ becomes redundant and
equation (1) meaningless and equations (4-6) ambiguous.

Again, if we consider $L$ is not an explicit function of $\phi$ then the
equation of motion $(3)$ is reduces to:

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\bar{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ (7)

We shall take the Lagrangian as $L=L(X)=1-V\sqrt{1-2X}$ with $V$ is a
constant. This is a particular type of the Dirac-Born-Infeld (DBI) born1 ;
born2 ; born3  lagrangian

$\displaystyle L(X,\phi)=1-V(\phi)\sqrt{1-2X}$ (8)

for $V(\phi)=V=constant$ and $kinetic~{}energy~{}of~{}\phi>>V$
i.e.$(\dot{\phi})^{2}>>V$. This is typical for the k-essence field where the
kinetic energy dominates over the potential energy. Then
$c_{s}^{2}(X,\phi)=1-2X$. For scalar fields
$\nabla_{\mu}\phi=\partial_{\mu}\phi$. Thus (6) becomes

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\partial_{\mu}\phi\partial_{\nu}\phi$
(9)

##  III Mapping on to the Robinson-Trautman type metric with cosmological
constant $\Lambda$ when background metric is Reissner-Nordstrom-de Sitter

We consider the gravitational metric $g_{\mu\nu}$ to be Reissner-Nordstrom- de
Sitter (RNdS) and denote $\partial_{0}\phi\equiv\dot{\phi}$,
$\partial_{r}\phi\equiv\phi^{\prime}$. The line element of RNdS metric cai ;
zhao1 ; zhao2 ; jg ; hs  is

$\displaystyle ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}d\Omega^{2}$ (10)

with

$\displaystyle
f(r)=(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})$ (11)

where $M$ is the mass of the black hole, $Q$ its charge and $\Lambda$ is the
cosmological constant ($\Lambda>0$) and
$d\Omega^{2}=d\theta^{2}+sin^{2}\theta~{}d\Phi^{2}$.

Assuming that the k-essence field $\phi(r,t)$ is spherically symmetric, one
has to obtain from (9)

$\displaystyle\bar{G}_{00}=g_{00}-(\partial_{0}\phi)^{2}=f(r)-\dot{\phi}^{2};$
$\displaystyle\bar{G}_{11}=g_{11}-(\partial_{r}\phi)^{2}=-\frac{1}{f(r)}-(\phi^{\prime})^{2};$
$\displaystyle\bar{G}_{22}=g_{22}=-r^{2};\bar{G}_{33}=g_{33}=-r^{2}sin^{2}\theta;$
$\displaystyle\bar{G}_{01}=\bar{G}_{10}=-\dot{\phi}\phi^{\prime}.$ (12)

The emergent line element is

$\displaystyle
ds^{2}_{emer}=(f(r)-\dot{\phi}^{2})dt^{2}-(\frac{1}{f(r)}+(\phi^{\prime})^{2})dr^{2}$
$\displaystyle-2\dot{\phi}\phi^{\prime}~{}dtdr-r^{2}d\Omega^{2}.$ (13)

Now making a coordinate transformation such as gm1 ; gm2 ; wein  :

$\displaystyle
d\omega=dt-({{\dot{\phi}\phi^{\prime}}\over{f(r)-\dot{\phi}^{2}}})dr.$ (14)

and we choose

$\displaystyle\dot{\phi}^{2}=(\phi^{\prime})^{2}(f(r))^{2}$ (15)

Then the line element (13) becomes

$\displaystyle
ds^{2}_{emer}=(f(r)-\dot{\phi}^{2})d\omega^{2}-\frac{dr^{2}}{(f(r)-\dot{\phi}^{2})}-r^{2}d\Omega^{2}~{}~{}$
(16)

Let us assume a solution to (15) of the form
$\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$. Then (15) reduces to

$\displaystyle\dot{\phi}_{2}^{2}=(\phi_{1}^{\prime})^{2}(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})^{2}=K$
(17)

$K(\neq 0)$ is a constant ($K\neq 0$ means k-essence field will have non-zero
kinetic energy). From (17) we get $\dot{\phi}_{2}(t)=\sqrt{K}$ and
$\phi_{1}^{\prime}(r)=\frac{\sqrt{K}}{f(r)}$. So the solution of (15) is

$\displaystyle\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$
$\displaystyle=-\frac{3\sqrt{K}}{2\Lambda}[A~{}ln(r^{2}+l_{1}r+m_{1})+B~{}ln(r^{2}-l_{1}r+n_{1})]$
$\displaystyle-\frac{3\sqrt{K}}{2\Lambda}[\frac{A(2C-l_{1})}{\sqrt{m_{1}-l_{1}^{2}/4}}~{}tan^{-1}(\frac{r+l_{1}/2}{\sqrt{m_{1}-l_{1}^{2}/4}})$
$\displaystyle+\frac{B(2D+l_{1})}{\sqrt{n_{1}-l_{1}^{2}/4}}~{}tan^{-1}(\frac{r-l_{1}/2}{\sqrt{n_{1}-l_{1}^{2}/4}})]+\sqrt{K}t$
(18)

where

$\displaystyle\phi_{1}(r)=-\frac{3\sqrt{K}}{\Lambda}\int{\frac{r^{2}dr}{r^{4}-\frac{3}{\Lambda}r^{2}+\frac{6M}{\Lambda}r-\frac{3Q^{2}}{\Lambda}}}$
$\displaystyle=-\frac{3\sqrt{K}}{\Lambda}\int{\frac{r^{2}dr}{(r^{2}+l_{1}r+m_{1})(r^{2}-l_{1}r+n_{1})}}$
$\displaystyle=-\frac{3\sqrt{K}}{\Lambda}[\int{\frac{Ar+C}{(r^{2}+l_{1}r+m_{1})}dr}+\int{\frac{Br+D}{(r^{2}-l_{1}r+n_{1})}dr}]$
$\displaystyle=-\frac{3\sqrt{K}}{2\Lambda}[A~{}ln(r^{2}+l_{1}r+m_{1})+B~{}ln(r^{2}-l_{1}r+n_{1})]$
$\displaystyle-\frac{3\sqrt{K}}{2\Lambda}[\frac{A(2C-l_{1})}{\sqrt{m_{1}-l_{1}^{2}/4}}~{}tan^{-1}(\frac{r+l_{1}/2}{\sqrt{m_{1}-l_{1}^{2}/4}})$
$\displaystyle+\frac{B(2D+l_{1})}{\sqrt{n_{1}-l_{1}^{2}/4}}~{}tan^{-1}(\frac{r-l_{1}/2}{\sqrt{n_{1}-l_{1}^{2}/4}})]$
(19)

and

$\displaystyle\phi_{2}(t)=\sqrt{K}t.$ (20)

Now we clarify the parameters of the above equation (18)
$A=-\frac{l_{1}(m_{1}+n_{1})}{N}$, $B=\frac{l_{1}(m_{1}+n_{1})}{N}$,
$C=\frac{n_{1}(n_{1}-m_{1})}{N}$, $D=-\frac{m_{1}(n_{1}-m_{1})}{N}$,
$N=2l_{1}^{2}(m_{1}+n_{1})+(m_{1}-n_{1})^{2}$,
$l_{1}=[(\frac{E_{1}+\sqrt{E_{1}^{2}-4D_{1}^{3}}}{2})^{1/3}+(\frac{E_{1}-\sqrt{E_{1}^{2}-4D_{1}^{3}}}{2})^{1/3}+\frac{2}{\Lambda}]^{1/2}$,
$m_{1}=\frac{1}{2\Lambda l_{1}}(\Lambda
l_{1}^{3}-3l_{1}-6M)~{},~{}n_{1}=\frac{1}{2\Lambda l_{1}}(\Lambda
l_{1}^{3}-3l_{1}+6M)$,
$E_{1}=\frac{1}{\Lambda^{3}}[(36M^{2}-24Q^{2})\Lambda-2]~{},~{}D_{1}=\frac{1}{\Lambda^{2}}(1-4Q^{2}\Lambda)$

Applying the equation (17) in equation (16) we get the emergent line element

$\displaystyle
ds^{2}_{emer}=(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}-K)d\omega^{2}$
$\displaystyle-\frac{dr^{2}}{(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}-K)}-r^{2}d\Omega^{2}$
$\displaystyle=(\epsilon-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})d\omega^{2}$
$\displaystyle-\frac{dr^{2}}{(\epsilon-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})}-r^{2}d\Omega^{2}$
(21)

with $\epsilon=(1-K)=constant$.

Now we are using the Eddington-Finkelstein coordinates $(v,r)$ or $(u,r)$
i.e., introducing advanced and retarded null coordinates

$\displaystyle v=\omega+r^{*}~{}~{};~{}~{}u=\omega-r^{*}~{};~{}$
$\displaystyle
dr^{*}=\frac{dr}{(\epsilon-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})}$
(22)

Then from (21) we get

$\displaystyle
ds^{2}_{emer}=(\epsilon-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})dv^{2}-2dvdr-r^{2}d\Omega^{2}$
(23)

For the solution (18) of $\phi(r,t)$, the line element (21) and (23) are
analogous to the Robinson-Trautman (RT) type metric with cosmological constant
$\Lambda$ (jg , Chapter-9, page no.-166, Chapter-19 and hs , Chapter-28). The
$\Lambda$ parameter comes from the background RN-de Sitter gravitational
metric. This RT type spacetime contains surfaces of constant positive, zero or
negative Gaussian curvature according to whether $\epsilon=+1,0,-1$
respectively.

In our case $\epsilon\neq+1$ since if $\epsilon=1$ then
$K=\dot{\phi}_{2}^{2}=0$ which implies dark energy is absent but in the
k-essence theory, dark energy density (in unit of critical density) i.e.
kinetic energy of the k-essence scalar field can not be zero. Also,
$\epsilon\neq-1$ since if $\epsilon=-1$ then $K=2$ but the total energy
density cannot exceed unity
($\Omega_{matter}+\Omega_{radiation}+\Omega_{darkenergy}=1$). Although
$\epsilon=1$ is a valid solution in every spherically symmetric mass
distribution starting from earth, star, etc. even in the presence of the
cosmological constant. So, for the above reasons, the RT type emergent
solution of the Einstein equation provided us $\epsilon=0$ i.e., $K=1$ which
implies $\Omega_{darkenergy}=1$ and the other energy densities are taken to be
zero. So we can consider this is the special case where the emergent spacetime
is only dark energy dominated. This is one possibility for satisfying the
Einstein equation in the context of emergent spacetime. Here we have shown
that the total energy density cannot exceed unity by mentioning the relation
$\Omega_{matter}+\Omega_{radiation}+\Omega_{darkenergy}=1$ gm2  only to get
the permissible values of $\epsilon$ i.e., $K$ and most importantly we are not
relating it with Friedmann-Lemaitre-Robertson-Walker (FLRW) metric since we
have not considered FLRW spacetime as a background.

Note that dark energy density (K) i.e., K.E. of the k-essence scalar field is
the essential property of the k-essence theory which can not be redefined with
the radial coordinate $r$ to absorb the $\epsilon$. Also, the case K=1 of (21)
does not seem to have a Newtonian limit schutz , which makes it unsuitable for
describing astrophysical objects. However, this may not be suitable as an
astrophysical object but still, it is a consistent solution to Einstein’s
equation.

Also, mention that the emergent line element (21) or (23) is not the dRGT type
massive gravity rham1 ; rham2  line element. In a massive-gravity model buri ;
ghosh , all the physical parameters depend on graviton mass $m_{g}$ and in the
line element, present the linear term of $r$ in addition to the cosmological
constant. But in our case, there is no linear term of $r$ is present in the
line element (23). Here we are mapping the emergent gravity spacetime in the
context of dark energy for a particular type of k-essence scalar field (18)
with the RT type spacetime only.

Note that the solution $\phi(r,t)$ (18) is satisfied emergent gravity equation
of motion (7) at $r\rightarrow\infty$:
$\bar{G}^{00}\partial_{0}^{2}\phi_{\mathrm{2}}+[\bar{G}^{11}(\partial_{1}^{2}\phi_{\mathrm{1}}-\Gamma_{11}^{1}\partial_{1}\phi_{\mathrm{1}})]+\bar{G}^{01}\nabla_{0}\nabla_{1}\phi+\bar{G}^{10}\nabla_{1}\nabla_{0}\phi+\bar{G}^{22}\nabla_{2}\nabla_{2}\phi+\bar{G}^{33}\nabla_{3}\nabla_{3}\phi=0$.
The first term vanishes since $\phi_{2}(t)$ linear in $t$ and second term
within third bracket vanish at $r\rightarrow\infty$ because
$2nd~{}term=-\frac{\sqrt{K}}{2}\frac{(3\epsilon r^{2}-6Mr+3Q^{2}-\Lambda
r^{4})(6\Lambda r^{4}-18Mr+18Q^{2})}{3r(\Lambda r^{4}-3r^{2}+6Mr-3Q^{2})^{2}}$
and the third and fourth terms vanish because $\bar{G}^{01}=\bar{G}^{10}=0$
and the last two terms vanish since $\phi$ is not dependent on $\theta$ and
$\Phi$ in this case.

##  IV Thermodynamics of Robinson-Trautman type metric with Cosmological
constant $\Lambda$

We rewrite the RT metric (21) with cosmological constant $\Lambda$ as

$\displaystyle
ds^{2}_{RT}=F(r)d\omega^{2}-\frac{dr^{2}}{F(r)}-r^{2}d\Omega^{2}$ (24)

where

$\displaystyle
F(r)=(\epsilon-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}).$ (25)

At the horizon $F(r)=0$, which admits the quartic equation of $r$. The roots
are

$\displaystyle
r=\frac{-l\pm\sqrt{l^{2}-4m}}{2}~{},~{}\frac{l\pm\sqrt{l^{2}-4n}}{2}$ (26)

where
$l=[(\frac{E+\sqrt{E^{2}-4W^{3}}}{2})^{1/3}+(\frac{E-\sqrt{E^{2}-4W^{3}}}{2})^{1/3}+\frac{2\epsilon}{\Lambda}]^{1/2}$
; $m=\frac{1}{2}(\frac{-3\epsilon}{\Lambda}+l^{2}-\frac{6M}{\Lambda
l})~{},~{}n=\frac{1}{2}(\frac{-3\epsilon}{\Lambda}+l^{2}+\frac{6M}{\Lambda
l})$ ;
$E=\frac{1}{\Lambda^{3}}[(36M^{2}-24Q^{2}\epsilon)\Lambda-2\epsilon^{3}]~{};~{}W=\frac{1}{\Lambda^{2}}(\epsilon^{2}-4Q^{2}\Lambda)$.

Depending upon the values of parameters $M,Q,\Lambda~{}and~{}\epsilon$ we can
denote the roots are $r_{cd},r_{+d},r_{-d},r_{4d}$ in the presence of dark
energy. The order of roots are $r_{cd}>r_{+d}>r_{-d}>r_{4d}$ where $r_{4d}<0$.
The root $r_{cd}$ denotes the location of the cosmological event horizon,
$r_{+d}$ denotes the location of the black hole event (outer) horizon and
$r_{-d}$ denotes the location of the Cauchy (inner) horizon of this black hole
spacetime (24) zhao1 ; zhao2 . Note that $r_{4d}$ does not correspond to a
physical horizon in this spacetime.

The equations

$\displaystyle
F(r_{+d})=0=Q^{2}-2Mr_{+d}+r_{+d}^{2}(\epsilon-\frac{\Lambda}{3}r_{+d}^{2})$
(27)

and

$\displaystyle
F(r_{cd})=0=Q^{2}-2Mr_{cd}+r_{cd}^{2}(\epsilon-\frac{\Lambda}{3}r_{cd}^{2})$
(28)

are rearranged in terms of $r_{cd}~{}and~{}r_{+d}$ as

$\displaystyle
Q^{2}=r_{cd}r_{+d}[\epsilon-\frac{\Lambda}{3}(r_{cd}^{2}+r_{cd}r_{+d}+r_{+d}^{2})]$
(29)

and

$\displaystyle
2M=(r_{+d}+r_{cd})[\epsilon-\frac{\Lambda}{3}(r_{cd}^{2}+r_{+d}^{2})].$ (30)

The surface gravity on the black hole and the cosmological horizons are

$\displaystyle\kappa_{+d}=\frac{1}{2}\frac{dF}{dr}\arrowvert_{r=r_{+d}}=\frac{1}{2r_{+d}^{3}}[2Mr_{+d}-2Q^{2}-2\frac{\Lambda}{3}r_{+d}^{4}]$
$\displaystyle=\frac{1}{2r_{+d}^{3}}[r_{+d}^{2}\epsilon-Q^{2}-\Lambda
r_{+d}^{4}]$
$\displaystyle=\frac{r_{+d}-r_{cd}}{2r_{+d}^{2}}[\epsilon-\frac{(r_{cd}^{2}+2r_{cd}r_{+d}+3r_{+d}^{2})(\epsilon
r_{cd}r_{+d}-Q^{2})}{r_{cd}r_{+d}(r_{+d}^{2}+r_{cd}r_{+d}+r_{cd}^{2})}]$
$\displaystyle=\frac{r_{+d}-r_{cd}}{2r_{+d}^{2}}[\epsilon-\frac{\Lambda}{3}(r_{cd}^{2}+2r_{+d}r_{cd}+3r_{+d}^{2})]$
(31)

and

$\displaystyle\kappa_{cd}=\frac{1}{2}\frac{dF}{dr}\arrowvert_{r=r_{cd}}=\frac{1}{2r_{cd}^{3}}[2Mr_{cd}-2Q^{2}-2\frac{\Lambda}{3}r_{cd}^{4}]$
$\displaystyle=\frac{1}{2r_{cd}^{3}}[r_{cd}^{2}\epsilon-Q^{2}-\Lambda
r_{cd}^{4}]$
$\displaystyle=\frac{r_{cd}-r_{+d}}{2r_{cd}^{2}}[\epsilon-\frac{(r_{+d}^{2}+2r_{cd}r_{+d}+3r_{cd}^{2})(\epsilon
r_{cd}r_{+d}-Q^{2})}{r_{cd}r_{+d}(r_{+d}^{2}+r_{cd}r_{+d}+r_{cd}^{2})}]$
$\displaystyle=\frac{r_{cd}-r_{+d}}{2r_{cd}^{2}}[\epsilon-\frac{\Lambda}{3}(r_{+d}^{2}+2r_{+d}r_{cd}+3r_{cd}^{2})]$
(32)

The first law of thermodynamics for the black hole horizon and the
cosmological horizon are expressed as zhao1 ; dolan ; seki ; gibbon1 ; gibbon2
; li ; kubiznak :

$\displaystyle\delta M=\frac{\kappa_{+d}}{2\pi}\delta
S_{+d}+\varphi_{+d}\delta Q+V_{+d}\delta P$ (33)

and

$\displaystyle\delta M=\frac{\kappa_{cd}}{2\pi}\delta
S_{cd}+\varphi_{cd}\delta Q+V_{cd}\delta P$ (34)

where $S_{+d}=\pi r_{+d}^{2}$, $S_{cd}=\pi r_{cd}^{2}$,
$\varphi_{+d}=Q/r_{+d}$, $\varphi_{cd}=-(Q/r_{cd})$,
$V_{+d}=\frac{4\pi}{3}r_{+d}^{3}$, $V_{cd}=\frac{4\pi}{3}r_{cd}^{3}$ and
$P=-\frac{\Lambda}{8\pi}$. This law means that total energy of the black hole
system is conserved.

In zhao1 ; zhao2 ; ren  obtained that the outgoing rate of the charged de
Sitter spacetime which radiates particles with energy $\omega$ is
$\Gamma=e^{(\Delta S_{+d}+\Delta S_{cd})}$ where $\Delta S_{+d}$ and $\Delta
S_{cd}$ are Bekenstein-Hawking entropy difference in the presence of dark
energy corresponding to the black hole horizon and the cosmological horizon
after the charged de Sitter spacetime radiates particles with energy $\omega$.
Therefore, in this context, we can say that the thermodynamic entropy of the
RT type spacetime in the presence of dark energy is the sum of the black hole
horizon entropy and the cosmological horizon entropy (as in zhao1 ; zhao2 ):

$\displaystyle S=S_{+d}+S_{cd}.$ (35)

substituting the values of $\delta S_{+d}$ and $\delta S_{cd}$ from equations
(33 and 34) in equation (35) we get

$\displaystyle
dS=2\pi(\frac{1}{\kappa_{+d}}+\frac{1}{\kappa_{cd}})dM-2\pi(\frac{\varphi_{+d}}{\kappa_{+d}}+\frac{\varphi_{cd}}{\kappa_{cd}})dQ$
$\displaystyle-2\pi(\frac{V_{+d}}{\kappa_{+d}}+\frac{V_{cd}}{\kappa_{cd}})dP=2\pi(\frac{1}{\kappa_{+d}}+\frac{1}{\kappa_{cd}})dM$
$\displaystyle-2\pi(\frac{\varphi_{+d}}{\kappa_{+d}}+\frac{\varphi_{cd}}{\kappa_{cd}})dQ+\frac{\pi}{3}(\frac{r_{+d}^{3}}{\kappa_{+d}}+\frac{r_{cd}^{3}}{\kappa_{cd}})d\Lambda.$
(36)

For the sake of simplicity, we consider the uncharged case, then the above
equation (36) become

$\displaystyle
dS=2\pi(\frac{1}{\kappa_{+d}}+\frac{1}{\kappa_{cd}})dM+\frac{\pi}{3}(\frac{r_{+d}^{3}}{\kappa_{+d}}+\frac{r_{cd}^{3}}{\kappa_{cd}})d\Lambda.$
(37)

Differentiating equations (27) and (28) and using equations (31) and (32) we
get

$\displaystyle
dr_{+d}=\frac{dM}{r_{+d}\kappa_{+d}}+\frac{(r_{+d}^{3}/3)}{2r_{+d}\kappa_{+d}}d\Lambda$
(38)

and

$\displaystyle
dr_{cd}=\frac{dM}{r_{cd}\kappa_{cd}}+\frac{(r_{cd}^{3}/3)}{2r_{cd}\kappa_{cd}}d\Lambda.$
(39)

The thermodynamic volume zhao1 ; dolan ; urano  of RT type spacetime (24) is
given as

$\displaystyle V=\frac{4\pi}{3}(r_{cd}^{3}-r_{+d}^{3}).$ (40)

Now substituting the values of $dr_{+d}$ and $dr_{cd}$ in (40) we get

$\displaystyle
dV=4\pi(\frac{r_{cd}}{\kappa_{cd}}-\frac{r_{+d}}{\kappa_{+d}})dM+\frac{2\pi}{3}(\frac{r_{cd}^{4}}{\kappa_{cd}}-\frac{r_{+d}^{4}}{\kappa_{+d}})d\Lambda.$
(41)

Using equations (37) and (41), one can derive the thermodynamic equation zhao1
; urano

$\displaystyle dM=T_{eff}^{d}dS-P_{eff}^{d}dV.$ (42)

The effective temperature $T_{eff}^{d}$ is

$\displaystyle T_{eff}^{d}=[\frac{(y^{4}+y^{3}-2y^{2}+y+1)}{4\pi
r_{cd}y(y+1)(y^{2}+y+1)}]\epsilon$
$\displaystyle-\frac{Q^{2}}{4\pi}[\frac{(y^{6}+y^{5}+y^{4}-2y^{3}+y^{2}+y+1)}{r_{cd}^{3}y^{3}(y+1)(y^{2}+y+1)}]$
(43)

and the effective pressure $P_{eff}^{d}$ is

$\displaystyle P_{eff}^{d}=[\frac{(1-y)(1+3y+3y^{2}+3y^{3}+y^{4})}{8\pi
r_{cd}^{2}y(1+y)(1+y+y^{2})^{2}}]\epsilon$
$\displaystyle-\frac{Q^{2}}{8\pi}[\frac{(1-y)(1+2y+3y^{2}-3y^{5}-2y^{6}-y^{7})}{r_{cd}^{4}y^{3}(1+y)(1-y^{3})(1+y+y^{2})}]$
(44)

where $y=\frac{r_{+d}}{r_{cd}}$ and $0<y<1$.

Because of the temperature and pressure are positive definite for ordinary
laboratory systems, it is natural that the thermodynamical stability should
express appropriately and it should satisfy the following requirement:
$T_{eff}^{d}>0$ and $P_{eff}^{d}>0$.

But in our case, we are restricting the value of $\epsilon$ is
$\epsilon=0~{}i.e.~{}K=1$, which implies the following situations:

a) For $Q\neq 0$, it is evident from (43) and (44) that the effective
temperature and pressure are negative. Negativity of pressure indicates that
the universe is dark energy dominated whereas negative temperature stands for
a thermodynamically unstable system. So in this situation, for the restriction
of the dark energy density limiting to unity, RT type spacetime (24) is
thermodynamically unstable though this type of spacetime is a valid solutions
of Einstein’s field equation. We can describe this type of situation as per
Ref. reif : thermodynamically unstable situation is physically plausible only
when we consider spin degrees of freedom of a system without taking
translational degrees of freedom (i.e., coordinates and momenta of the system)
into account. Based on this consideration the number of possible spin states,
at first, increase with energy leading to a maximum and finally decreases
again. This feature indicates that the absolute spin temperatures may be
negative as well as positive. So we can infer that the RT type black hole (24)
can be projected as a physical reality only by considering spin degrees of
freedom an exceptional condition.

b) Another condition: if $\epsilon=0$ and $Q=0$ then we see that from (43) and
(44) the effective temperature and pressure are both zero. Zero temperature
means that the RT type black hole (24) does not radiate and zero pressure
means the volume-thermodynamic system becomes area-thermodynamic since
horizons have nonzero area.

Now we are considering the problem when the black hole event horizon and the
cosmological event horizon are independent of each other i.e., the two
horizons are coincide zhao1 , then,

$\displaystyle
r_{+d/cd}^{2}=\frac{\epsilon\pm\sqrt{\epsilon^{2}-4Q^{2}\Lambda}}{2\Lambda}\equiv
r_{0d}^{2}.$ (45)

From above equation (45) we can see that
$r_{0d}^{2}=\frac{iQ}{\sqrt{\Lambda}}$ i.e., imaginary horizon since in our
case the dark energy density is to be unity ($K=1$) i.e., $\epsilon=0$. Again
from (31) and (32) we see that the surface gravity of the two horizons are
zero as two horizons are coinciding. Thus the temperatures of both horizons
are zero in the presence of dark energy when two horizons are coincided.
However, both horizons have the nonzero area, which means that the entropy for
the two horizons should not be zero.

##  V Conclusion

We have investigated the thermodynamics of a black hole in the context of dark
energy in an emergent gravity scenario using k-essence model for Reissner
-Nordstrom - de Sitter background. The emergent gravity spacetime has to be
conformally invariant with the Robinson -Trautman type spacetime with
cosmological constant $\Lambda$. The evaluated thermodynamical quantities like
surface gravity, area law, temperature, pressure etc. are different from the
usual ones. As our restriction of the emergent spacetime (23), taken to be the
dark energy density i.e., the kinetic energy of the k-essence scalar field is
unity, the RT type emergent black hole does not radiate when $Q=0$ and when
$Q\neq 0$ the system possess thermodynamically unstable equilibrium.So we can
say that the RT type emergent black hole (23) is thermodynamically unstable in
the presence of dark energy and which can be projected as a physical reality
only by considering spin degrees of freedom an exceptional condition. Also,
for this restriction in the RT type emergent metric (23), we conclude that
this type of spacetime is only dark energy dominated as a special
consideration since it is a valid solution of the Einstein equation.

Acknowledgement: The authors thanks to Dr. Pradipta Panchadhyayee, Associate
Professor, Department of Physics, Prabhat Kumar College, Contai for his
valuable suggestions. The authors would like to thank the referees for
illuminating suggestions to improve the manuscript.

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