Using the definition of a black hole’s volume introduced by Christodoulou and Rovelli, we calculate the interior volume of a (2 + 1)-dimensional charged Banados Teitelboim Zanelli ( BTZ ) black hole, and find that the volume increases linearly with time. Afterwards, the entropy of a massless scalar field inside the black hole is calculated and the result indicates that the entropy will be also increasing with time infinitely. Moreover, thinking about Hawking radiation, the ratio of variation of the scalar field’s entropy to the variation of Bekenstein–Hawking entropy is approximately a linear function of m , which is quite different from a RN black hole while m is relatively large. In the end, we extend the calculation above to a massive BTZ black hole and find that the relationship with different M ~ $\tilde {M}$ between two kinds of entropy is similar to the previous result. But the difference is that the relationship function F ( m , q , M ~ ) $F(m,q,\tilde {M})$ will tend to be a constant when the mass parameter M ~ $\tilde {M}$ becomes big enough.

中文翻译：

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霍金辐射下带电(2+1)维BTZ黑洞的熵变

使用Christodoulou 和Rovelli 引入的黑洞体积定义，我们计算了(2 + 1) 维带电Banados Teitelboim Zanelli (BTZ) 黑洞的内部体积，发现体积随时间线性增加。随后，计算了黑洞内部一个无质量标量场的熵，结果表明熵也将随时间无限增加。而且，考虑到霍金辐射，标量场熵的变化与贝肯斯坦-霍金熵的变化之比近似为 m 的线性函数，这与 RN 黑洞有很大不同，而 m 相对较大。到底，我们将上面的计算扩展到一个巨大的 BTZ 黑洞，发现两种熵之间不同 M ~ $\tilde {M}$ 的关系与之前的结果相似。但不同的是关系函数 F ( m , q , M ~ ) $F(m,q,\tilde {M})$ 当质量参数 M ~ $\tilde {M}$ 会趋向于一个常数变得足够大。