A modified Hamilton–Jacobi equation with Lorentz-violating scalar field in the curved spacetime is used to research tunneling radiation from the horizons of Reissner–Nordstr $$\ddot{\mathrm{o}}$$ m de Sitter black hole. We investigate the influence of different etheric-like vector $$u^\alpha $$ on the solution of the modified Hamilton–Jacobi equation by using both the semi-classical approximation and beyond the semi-classical approximation. We find that the solution of the modified Hamilton–Jacobi equation strongly depends on the choices of $$u^\alpha $$ . As $$u^\alpha =\delta ^\alpha _t u^t$$ , the Lorentz-violating scalar field can enhances black holes’ Hawking temperature but reduces its entropy. However, the case will reverses as $$u^\alpha =\delta ^\alpha _r u^r$$ . Our conclusions are applicable to other spherically symmetric charged black holes.

中文翻译：

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具有洛伦兹违反标量场的修正 Hamilton-Jacobi 方程的解

在弯曲的时空中具有洛伦兹违反标量场的修正 Hamilton-Jacobi 方程用于研究来自 Reissner-Nordstr $$\ddot{\mathrm{o}}$$ m de Sitter 黑洞视界的隧道辐射。我们通过使用半经典近似和超越半经典近似来研究不同的类以太向量 $$u^\alpha $$ 对修正的 Hamilton-Jacobi 方程的解的影响。我们发现修正的 Hamilton-Jacobi 方程的解很大程度上取决于 $$u^\alpha $$ 的选择。由于 $$u^\alpha =\delta ^\alpha _t u^t$$ ，洛伦兹违反标量场可以提高黑洞的霍金温度，但会降低其熵。但是，情况将反转为 $$u^\alpha =\delta ^\alpha _r u^r$$ 。