Starting from the suitable action of four-dimensional Einstein gravity, which is coupled to a scalar dilaton field and logarithmic nonlinear electrodynamics, the related field equations have been solved in a spherically symmetric and energy-dependent spacetime. It has been found that the scalar potential, as the exact solution of the scalar field equations, must be written as the linear combination of two Liouville-type potentials. Three families of novel exact black holes have been introduced as the exact solutions to the gravitational equations, which are asymptotically non-flat and non-AdS. It has been illustrated that our solutions, recover their corresponding quantities in the Einstein–Maxwell-dilaton gravity theory when the nonlinearity parameter of electrodynamics is chosen very large. The thermodynamic and conserved quantities of the black holes have been calculated under the influence of the rainbow function. Then, through calculation of the Smarr mass formula, we have shown that the first law of black hole thermodynamics remain valid for all of the new dilatonic black holes. Thermal stability or phase transition of the black holes have been investigated by use of the canonical ensemble and thermodynamic geometry approaches, separately. By comparison of the results of these two alternative approaches, it has been found that they lead to the same results provided that the thermodynamic metric of HPEM is used.

中文翻译：

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引力彩虹中具有对数源的膨胀子黑洞

从与标量膨胀子场和对数非线性电动力学耦合的四维爱因斯坦引力的适当作用出发，已在球对称和能量相关的时空中求解相关场方程。已经发现，标量势作为标量场方程的精确解，必须写成两个刘维尔型势的线性组合。引入了三个新的精确黑洞族作为引力方程的精确解，它们是渐近非平面和非 AdS 的。已经说明，当电动力学的非线性参数选择得非常大时，我们的解决方案在 Einstein-Maxwell-dilaton 引力理论中恢复了它们的相应量。在彩虹函数的影响下计算了黑洞的热力学和守恒量。然后，通过 Smarr 质量公式的计算，我们证明了黑洞热力学第一定律对所有新的膨胀型黑洞仍然有效。黑洞的热稳定性或相变已经通过使用正则系综和热力学几何方法分别进行了研究。通过比较这两种替代方法的结果，发现如果使用 HPEM 的热力学度量，它们会产生相同的结果。我们已经证明，黑洞热力学第一定律对所有新的膨胀黑洞仍然有效。黑洞的热稳定性或相变已经通过使用正则系综和热力学几何方法分别进行了研究。通过比较这两种替代方法的结果，发现如果使用 HPEM 的热力学度量，它们会产生相同的结果。我们已经证明，黑洞热力学第一定律对所有新的膨胀黑洞仍然有效。黑洞的热稳定性或相变已经通过使用正则系综和热力学几何方法分别进行了研究。通过比较这两种替代方法的结果，发现如果使用 HPEM 的热力学度量，它们会产生相同的结果。