The idea of a Black Hole Temperature is fundamentally counterintuitive, considering their most basic properties. A temperature is conventionally described in terms of energy distributions of particles. I can ask you to imagine a hot teacup and picture it’s molecules bouncing around chaotically and energetically. Meanwhile, a black hole is understood to be a boundary called the event horizon. Typically in the shape of a perfect sphere, this boundary is a unique quirk of Einstein’s general theory of relativity as it exhibits a spacetime curvature so severe that no matter, information, or light can escape the boundary once it crosses over. If such a boundary disallows for any matter to escape, where do this particles that characterize a temperature come from?
According to the No-hair theorem developed during the later 1960’s, a black hole can be entirely described by it’s mass, angular momentum, and electric charge. Under a classical physic interpretation(meaning pre-quantum physics), a black hole as no microstates, and therefore no entropy. Entropy and microstates are thermodynamic concepts which predate the discovery of black holes by many years. But because of a black hole’s unique property of “hiding” it’s interior from the rest of the universe, we can violate a very fundamental thermodynamic law. If I were to take a teacup, which has microstates, entropy, and therefore a temperature, and I were to throw it into the black hole and ruin my afternoon teatime. I would effectively “delete” the teacup’s entropy from the universe, as the black hole must have zero entropy. The second law of thermodynamics states that entropy must be non-decreasing, and this law serves as the fundamental backbone of the entire very-well tested theory of thermodynamics. The only way to rectify this problem is for a black hole to have it’s own associated entropy proportional to its surface area, as theorized by Bekenstein. In 1974, Stephen Hawking further developed this notion by developing a theoretical argument describing how a black hole being causally disconnected from the universe allows for a quantum field theory in which the virtual particles in the vacuum field of the universe are brought forth into reality. By creating these particles just above its horizon, a black hole is able to quantum mechanically produce a thermal bath of particles, and therefore has a temperature. This temperature helps define the Bekenstein-Hawking entropy of a black hole and solves the entropy paradox, thereby giving a better description of the black hole phenomenon.
In Stephen Hawking’s original calculation, he developed the theory using a Schwarzschild black hole metric, one of the simplest descriptions of a black hole. As is the case for most black hole theories, predictions cannot be experimentally verified without traveling to measure a black hole itself, especially since the predicted temperature is on the scale of $10^{-12} \degree K$ Kelvin. Therefore, results must be verified by calculating through various independent methods and seeing of the end result expressions agree. What we have done is recalculated the temperature for a Kerr-Newman black hole, using a methodology similar to Hawking’s original calculation. Our results agree with the previously seen expressions for the Kerr-Newman black hole temperature, thereby further verifying them.
Here is a GitHub link to a sample of the project Mathematica code in which I derive the Near-Horizon Spacetime Metrics, which constitutes an important part of our overall calculation.
Here are some snapshots of the research poster!
