Spinning Black Holes Now Have a Simpler, Classically-Derived Temperature Calculation

Scientists have long predicted that black holes, despite their reputation as cosmic vacuum cleaners, possess a non-zero temperature and emit radiation like any other heated object. Ronald J. Adler, formerly of the Gravity Probe B mission and Stanford University, presents a novel and simplified derivation of the temperature for a spinning Kerr black hole, relying solely on classical general relativity. This work distinguishes itself through its mathematical accessibility, aiming to provide a more pedagogical approach for students and researchers unfamiliar with complex calculations. The findings not only reinforce existing theoretical frameworks but also highlight the potential for observing the explosive evaporation of small black holes, offering a unique probe of physics at the Planck scale, and further considers the intriguing possibility that primordial black hole remnants could constitute a portion of the universe’s dark matter.

Kerr Black Hole Thermodynamics via Carnot Engine Analysis of the Polar Axis

Scientists have recently achieved a significant advancement in understanding the thermodynamics of rotating black holes, deriving a temperature equation using only classical general relativity and established thermodynamic principles. This work offers a mathematically simplified and self-contained derivation of the temperature of a Kerr spinning black hole, building upon earlier research but streamlining the process for greater accessibility.
The research focuses on calculating the energy that can be extracted from the gravitational field of a spinning black hole by lowering a small body towards its surface, analogous to a thought experiment in Newtonian physics. By treating this setup as a Carnot heat engine with a distant laboratory as the hot reservoir and the black hole as the cold reservoir, the team successfully determined an approximate temperature for the black hole.

This derivation diverges from previous approaches by focusing on the polar axis region of the Kerr metric, simplifying the calculations and demonstrating that the resulting temperature applies to the entire black hole. The methodology relies on established principles of general relativity and black body thermodynamics, both areas extensively tested within physics.
The resultant equation, while approximate with an overall multiplicative constant of order 1, provides a robust estimate of black hole temperature. This new approach should be particularly valuable for students and researchers seeking a more intuitive understanding of black hole thermodynamics. Furthermore, the study highlights the potential for explosive evaporation in small black holes, a phenomenon yet to be observed but which could offer a unique window into physics at the Planck scale.

The research also explores the intriguing possibility that remnants from these hypothetical explosions might constitute a significant portion of the universe’s dark matter. By considering a small body lowered towards the black hole’s null surface, the team calculated the extractable energy, ultimately linking this to the black hole’s temperature via the Carnot theorem.

The final result for the black hole temperature is presented and illustrated, offering a clear visualization of this fundamental property. This work not only refines the theoretical understanding of black hole thermodynamics but also opens avenues for exploring the connection between these cosmic entities and some of the most perplexing mysteries in modern physics, including the nature of dark matter and the fundamental laws governing the universe at its smallest scales. The team’s focus on classical general relativity and thermodynamics provides a solid foundation for future investigations into these complex phenomena.

Calculating black hole temperature via mechanical work and geodesic deviation

Researchers derived the temperature of a Kerr spinning black hole utilising only classical general relativity and black body thermodynamics. The work began by establishing a fundamental physics input, asserting that the motion of a body under gravity and an external non-gravitational force follows a geodesic equation with an added force term.

This approach mirrors the established description of motion under gravity and electromagnetism, providing a consistent framework for analysis. To calculate the potential energy, the study considered a mass μ being slowly lowered towards the black hole, defining the radial coordinate as xn = r. Applying this to the polar axis black hole metric, the researchers determined the static force Fn acting on the body, expressed as a function of the metric components and mass.

This force, representing the tension in a conceptual ‘rope’ holding the body, was then used to calculate the work W done in lowering the body, integrating the force over the physical distance to the black hole. The resulting expression for work, W(r), precisely matched a previously derived equation, confirming the equivalence between the potential for a slowly lowered body and that for free fall.

This methodological innovation simplifies the calculation of black hole temperature, making it more accessible for students and researchers without extensive expertise in quantum field theory. The study further highlights the potential for explosive evaporation of small black holes, offering a unique probe of Planck scale physics and suggesting primordial black hole remnants as a possible component of cosmological dark matter.

Work extraction efficiency from Kerr black holes via lowered mass calculations

Calculations reveal that lowering a small body onto a Kerr spinning black hole extracts work equivalent to the total rest energy of the body at the surface, reaching a value of μc( where μ represents the mass of the lowered box and c is the speed of light. This contrasts with Newtonian calculations, which yield only half the rest energy, 489 (, for the same process.

The research details how work is calculated by determining the difference in potential energy as the box descends from infinity to a radius of r = m + √m( −a( , defining the black hole’s surface. Approximations for work done when lowering the box to near the surface, at a coordinate distance ∆r, are expressed as W= μc( X1 −w {5 (6 √K Y, where K≡ (•nm]9/69 nc•nm]9/69 is a dimensionless parameter.

Analysis of geodesic motion along the polar axis of the Kerr metric led to a potential energy function V, defined as V= w1 − (65 59c]9 μc( , which simplifies to the Newtonian potential, V= μc( − 234 5, in the weak field limit with zero spin. The study establishes a relationship between coordinate distance ∆r and physical distance ∆l using the radial metric component, expressed as dl= •|gnn|dr with •|gnn| ≅ n √ƒ w (6 ∆5.

This conversion is crucial for accurately assessing energy extraction based on physical separation rather than coordinate separation. The work extracted when lowering the box to near the black hole surface is further refined, demonstrating its dependence on the parameter K and the physical distance ∆l.

Specifically, the metric analysis reveals that the potential energy function accurately describes the gravitational interaction, even for objects lowered slowly. This work provides a foundation for estimating black hole temperature and explores the potential for primordial black hole remnants to constitute dark matter.

Kerr Black Hole Thermodynamics and Primordial Remnants as Dark Matter

The temperature of a spinning Kerr black hole has been calculated using classical general relativity and thermodynamic principles. This derivation offers a mathematically simpler and more accessible approach compared to previous methods relying on quantum field theory. The resulting temperature dependence on the black hole’s spin angular momentum aligns with established theoretical predictions, reinforcing the expectation that black holes possess a non-zero temperature despite the current lack of direct observational confirmation.

This work highlights the potential for explosive evaporation in small black holes, presenting a unique opportunity to probe physics at the Planck scale. Furthermore, the possibility that primordial black hole remnants could constitute a portion of the universe’s dark matter is explored, suggesting these remnants, with their weak gravitational interactions and low density, represent a challenging but intriguing dark matter candidate.

The authors acknowledge that detecting such dark matter particles would likely prove exceptionally difficult, potentially exceeding the difficulty of detecting other proposed candidates. The research successfully demonstrates a method for approximating the temperature of a spinning Kerr black hole using general relativity and black body thermodynamics.

This accessible calculation provides further justification for the theoretical expectation of black hole temperature. The potential for explosive evaporation of low-mass black holes is of particular interest for fundamental physics at the Planck scale, and black hole remnants remain a plausible, albeit elusive, component of cosmological dark matter. Future research may focus on refining these calculations and exploring the observational signatures of evaporating black holes or primordial black hole remnants.