Formula for a black hole
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Formula for a Black Hole
Mass Formula for Black Holes
The mass of a black hole can be expressed through various formulas depending on its characteristics such as charge, angular momentum, and irreducible mass. One significant formula derived for the mass of a black hole as a function of its irreducible mass, angular momentum, and charge is given by ( E^2 = m_{\mathrm{ir}}^2 + \left(\frac{L^2}{4m_{\mathrm{ir}}^2}\right) + p^2 ), where ( E ) is the energy, ( m_{\mathrm{ir}} ) is the irreducible mass, ( L ) is the angular momentum, and ( p ) is the linear momentum . This formula highlights the relationship between these physical quantities in determining the mass of a black hole.
Primordial Black Hole Formation
For primordial black holes, a new analytic formula has been derived to determine the amplitude of density perturbation at the threshold of their formation. This formula is given by ( \delta^{\rm UH}{H c} = \sin^2\left[\pi \sqrt{w}/(1+3w)\right] ) and ( \tilde{\delta}{c} = \left[3(1+w)/(5+3w)\right]\sin^2\left[\pi\sqrt{w}/(1+3w)\right] ), where ( w ) is the equation of state parameter. This formula provides a better agreement with numerical simulations compared to conventional formulas.
Smarr Formula and Enthalpy
The Smarr formula for black holes, particularly in the context of anti-de Sitter (AdS) black holes, relates the mass of the black hole to its enthalpy. The formula incorporates variations in the cosmological constant, which is interpreted as pressure, and includes a term for effective volume times change in pressure. This approach suggests that the mass of an AdS black hole should be viewed as the enthalpy of the spacetime.
Entropy and Higher Curvature Interactions
In Lovelock higher-curvature gravity theories, the entropy of stationary black holes is not simply proportional to the surface area of the horizon. Instead, it includes a sum of intrinsic curvature invariants integrated over a cross-section of the horizon. This general formula for entropy is derived by integrating the first law of black hole mechanics.
Kerr Black Holes
For Kerr black holes, a specific mass formula has been deduced, which includes terms for surface energy, rotational energy, and electromagnetic energy. This formula is obtained by integrating the mass differential term by term. The comparison of Kerr black holes to rotating liquid drops suggests potential instabilities in these black holes.
Area (or Entropy) Product Formula
The area (or entropy) product formula for regular black holes, such as those derived by Ayón-Beato and García, shows that the entropy product of two physical horizons depends on the ADM mass parameter, indicating it is not a universal quantity. However, a more complex function of the event horizon area and Cauchy horizon area is mass-independent.
Cardy-Verlinde Formula
The Cardy-Verlinde formula relates the entropy of a conformal field theory to its total energy and Casimir energy. This formula holds for conformal field theories with AdS duals, including AdS Schwarzschild and AdS Kerr black holes. For AdS Reissner-Nordström black holes, the formula also holds, provided a proper internal energy is subtracted from the total energy.
Conclusion
The formulas for black holes encompass a range of physical properties and theoretical frameworks, from mass and entropy to density perturbations and enthalpy. These formulas provide critical insights into the complex nature of black holes and their interactions with the surrounding spacetime. Understanding these relationships is essential for advancing our knowledge of black hole physics and the fundamental laws governing the universe.
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