New forms and thermodynamics of the neutral rotating squashed black hole in five-dimensional vacuum Einstein gravity theory

We initiated the program to look for new and simple forms for the five-dimensional rotating squashed black holes by solving directly the equation of motion. In a recent paper, the metric ansatz of dimensional reduction along the fifth spatial dimension was used to obtain a new but rather simple form for the five-dimensional rotating uncharged black hole solution with squashed horizons via solving the vacuum Einstein field equations. In this work, we continue to seek for another new but relatively simple form for the neutral rotating squashed black hole solution by using a different metric ansatz of time-like dimensional reduction. We then find its relation to our previous solution and investigate its thermodynamics by means of the counterterm method. Compared with the previous results given by the other author, both of our new metric forms and their associated thermodynamic expressions are very concise and elegant. Both of two new forms for the neutral rotating squashed black hole solution presented in this paper can be used as the seed to generate its charged generalization in D=5 minimal supergravity.