Formula for black hole
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Formula for Black Hole Mass and Entropy
Black Hole Mass Formula
Kerr Black Holes Mass Formula
The mass of a Kerr black hole, which is a rotating black hole, can be expressed through a specific formula. This formula is derived by integrating the mass differential term by term, resulting in three distinct components: surface energy, rotational energy, and electromagnetic energy . This approach is analogous to comparing a rotating black hole to a rotating liquid drop, suggesting potential instabilities in Kerr black holes .
Charged Black Holes Mass Formula
For charged black holes, the mass can be determined as a function of its irreducible mass, angular momentum, and charge. Notably, it has been shown that up to 50% of the mass of an extreme charged black hole can be converted into energy, which is significantly higher compared to 29% for an extreme rotating black hole .
General Mass Formula
A more generalized formula for the mass of a black hole, considering linear momentum ( p ) and angular momentum ( L ), is given by: [ E^2 = m_{\text{ir}}^2 + \left(\frac{L^2}{4m_{\text{ir}}^2}\right) + p^2 ] where ( m_{\text{ir}} ) represents the irreducible mass .
Black Hole Entropy Formula
Schwarzschild Black Hole Entropy
The entropy of a Schwarzschild black hole, which is a non-rotating and uncharged black hole, can be derived using a Cardy-like formula. This formula is effective for static black holes whose gravitational entropy scales with the temperature and is particularly robust for solutions with negative heat capacity . The entropy corrections for dimensions greater than four are logarithmic in nature .
Loop Quantum Gravity and Black Hole Entropy
In the context of loop quantum gravity, the black hole entropy is calculated as a function of the horizon area. This approach provides an exact formula for both the leading and sub-leading terms, aligning with the Bekenstein-Hawking formula and uniquely fixing the value of the 'quantum of area' in the theory .
Higher Curvature Interactions
For black holes in Lovelock higher-curvature gravity theories, the entropy is not merely one quarter of the horizon's surface area. Instead, it includes a sum of intrinsic curvature invariants integrated over a cross-section of the horizon . This comprehensive approach is derived by integrating the first law of black hole mechanics using Hamiltonian methods .
Smarr Formula and Thermodynamics
Smarr Formula for AdS Black Holes
The Smarr formula for static AdS black holes is derived geometrically and includes variations in the cosmological constant. This formula relates to the first law of black hole mechanics through a scaling argument based on Euler's theorem. The mass of an AdS black hole is interpreted as the enthalpy of the spacetime, incorporating the effective volume outside the black hole horizon .
Lovelock Black Holes
For Lovelock black holes, the mass formula can be expressed using a Noether charge surface integral and a suitable volume integral. This formalism, applied to static Lovelock black holes, also proposes a new prescription for Wald's entropy, accounting for topological contributions to the entropy functional .
Conclusion
The formulas for black hole mass and entropy vary significantly depending on the type of black hole and the theoretical framework applied. From Kerr and charged black holes to those described by loop quantum gravity and higher curvature interactions, each formula provides unique insights into the complex nature of black holes. Understanding these formulas is crucial for advancing our knowledge of black hole physics and the fundamental laws governing the universe.
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