Erratum to: Entropy of self-gravitating anisotropic matter

1 Erratum to: Eur. Phys. J. C (2019) 79:679 https://doi.org/10.1140/epjc/s10052-019-7189-2

• 1. The Eq. (45) should read

  $$\begin{aligned} s=s_0\rho ^{1-\frac{1}{2w_2}}r^{-\frac{1+w_2}{w_2}}\sqrt{\frac{1-2m/r}{1-2\bar{m}/r}}, \end{aligned}$$

  (1)

  where $\bar{m}=m-M$.

• 2. Contrary to the Lagrangian (37) which is deduced from the first principle, there is no guarantee that the trial Lagrangian (39) represents the true entropy for $w_1=-1$ although it matches well for $w_1\ne -1$ cases.

  The Lagrangian corresponding to a, b in Eq. (44) is

  $$\begin{aligned} \mathcal {L}~\propto ~ (\bar{m}')^{1-\frac{1}{2w_2}}r^{-1}/\sqrt{1-2\bar{m}/r}. \end{aligned}$$

  (2)

  However, if the solutions are analytic around $w_1=-1$, the $w_1\rightarrow -1$ limit of the Lagrangian (37) gives

  $$\begin{aligned} \lim _{w_1\rightarrow -1}\mathcal {L}~\propto ~ (m')^{1-\frac{1}{2w_2}}r^{-1}, \end{aligned}$$

  (3)

  which is different from the above Lagrangian. Thus, one should be cautious in using the entropy density for $w_1=-1$.

  Therefore, the exact form of the entropy density for $w_1=-1$ should be derived by taking $w_1\rightarrow -1$ limit of the entropy function Eq. (47). The entropy density so obtained has the form

  $$\begin{aligned} s=s_0\rho ^{1-\frac{1}{2w_2}}r^{-\frac{1+w_2}{w_2}}\sqrt{1-2m/r}. \end{aligned}$$

  (4)