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Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics
Abstract
Algebraically special solutions of Einstein's empty-space field ; equations that are characterized by the existence of a geodesic and shear-free ; ray congruence are considered. A class of solutions is presented for which the ; congruence is diverging and is not necessarily hypersurface orthogonal. (C.E.S.);
... The Kerr metric [8] was constructed in 1963, soon after the discovery of Quasars. It has a singular source with angular momentum as well as mass, surrounded by two elliptical event horizons. ...
... They need to be replaced with functions, (f (u), g(v)), that are positive and at least three times differentiable at the horizon. The standard choice is to exponentiate them, 8 We will see later that the second PNV in Kerr, k * ± , is asymptotic on both sides to the inner horizon at t = +∞ and to the outer horizon at t = −∞. It is a FALL ray between the horizons. ...
... Each new coordinate is zero on one of the "perpendicular" event horizons of Kruskal. The metric coefficients in (8) are functions of r alone so we only need to calculate this as a function of U and V . From eq. (7), ...
... It is generally accepted that plasma particles in a nearby region of AGN are embedded in the magnetic field strong enough [8,9] to provide the frozen in-condition [10], which when combined with effects of rotation (always present in Kerr-type black holes [11,12]) leads to the relativistic magnetocentrifugal effects close to the light cylinder zone [13]-a hypothetical area where the linear velocity of rotation equals exactly the speed of light (see the sketch in Figure 1). In this figure, we show the straight field line, representing a certain channel, along which the particles are sliding, reaching the maximum energies on the light cylinder area. ...
... The Kerr-type black holes rotate with the angular velocity [11,12] ...
In this study, we examine the efficiency of pair creation by means of the centrifugal mechanism. The strong magnetic field and the effects of rotation, which always take place in Kerr-type black holes, guarantee the frozen-in condition, leading to the generation of an exponentially amplifying electrostatic field. This field, upon reaching the Schwinger threshold, leads to efficient pair production. Researchers have studied this process across a wide range of AGN luminosities and black hole masses, and found that the mechanism is highly efficient, indicating that for AGNs where centrifugal effects are significant, the annihilation lines in the MeV range will be very strong.
... We show now that our new metric does fall within the Weyl [12] class of metrics, which is a class of static and axisymmetric solutions to Einstein's field equations. The well known Schwarzschild, Reissner-Nordström, Kerr [13], and Kerr-Newman [14,15] metrics are all within the Weyl class of solutions to Einstein's field equations, which has the generic form given by, ds 2 = −e 2ψ(ρ,z) dt 2 + e 2γ (ρ,z)−2ψ(ρ,z) (dρ 2 + dz 2 ) + e −2ψ(ρ,z) ρ 2 dφ 2 (11) where ψ(ρ, z) and γ (ρ, z) are the two metric functions. Here, we are following Gautreau, Hoffman and Armenti [16] that do a similar derivation for the Reissner-Nordström metric (see also Stephani et al. [17]). ...
... Obviously, the potentials given in (13) lead to an exact solution to Einstein's field equations if the potentials given in (12) does. For convenience of the reader, we indicate schematically below the procedure leading to the solution. ...
In the context of gravitational objects with spherical symmetry, we derive a solution to Einstein’s field equations using two methods leading to the same result. The first is based on a stress-energy tensor that takes into account both the electric field energy of the charge and the gravitational field energy of the mass. The second is based on the mass-energy equivalence and has more general validity. We show that the metric falls within the Weyl class of metrics, representing a static and axisymmetric solution to Einstein’s field equations. The metric, which has a form similar to that of Reisser-Nordström, is used for predictions in strong fields and possibly shows better agreement with observation in high z quasars.
... Unlike other astrophysical objects, the ringdown spectrum of a black hole is remarkably simple. General relativity predicts that the frequencies and damping times of the entire spectrum of damped sinusoids, or "quasinormal modes," are fully determined by just two numbers: the black hole mass M and angular momentum J, as described by the Kerr solution [4]. This prediction, a consequence of the black hole "nohair theorem," does not hold in many alternate theories [5]. ...
When two black holes merge, the late stage of gravitational wave emission is a superposition of exponentially damped sinusoids. According to the black hole no-hair theorem, this ringdown spectrum depends only on the mass and angular momentum of the final black hole. An observation of more than one ringdown mode can test this fundamental prediction of general relativity. Here, we provide strong observational evidence for a multimode black hole ringdown spectrum using the gravitational wave event GW190521, with a maximum Bayes factor of 56±1 (1σ uncertainty) preferring two fundamental modes over one. The dominant mode is the ℓ=m=2 harmonic, and the subdominant mode corresponds to the ℓ=m=3 harmonic. The amplitude of this mode relative to the dominant harmonic is estimated to be A330/A220=0.2−0.1+0.2. We estimate the redshifted mass and dimensionless spin of the final black hole as 330−40+30M⊙ and 0.86−0.11+0.06, respectively. We find that the final black hole is consistent with the no-hair theorem and constrain the fractional deviation from general relativity of the subdominant mode’s frequency to be −0.01−0.09+0.08.
... In all three cases, we begin with uniform electron fraction, Y e = 0.1, throughout the disk. We begin with an accretion disk surrounding a 2.58M ⊙ Kerr (1963) black hole with dimensionless spin parameter a = 0.69, as in Miller et al. (2019b), and a disk mass of 0.12M ⊙ . We use a radially logarithmic grid of dimensions N r x N θ x N ϕ = 192 x 128 x 66 and run the simulation out to 10 4 GM BH /c 3 , which corresponds to 127 ms of physical time given this geometry. ...
Magnetohydrodynamic turbulence drives the central engine of post-merger remnants, potentially powering both a nucleosynthetically active disk wind and the relativistic jet behind a short gamma ray burst. We explore the impact of the magnetic field on this engine by simulating three post-merger black hole accretion disks using general relativistic magnetohydrodynamics with Monte Carlo neutrino transport, in each case varying the initial magnetic field strength. We find increasing ejecta masses associated with increasing magnetic field strength. We find that a fairly robust main r-process pattern is produced in all three cases, scaled by the ejected mass. Changing the initial magnetic field strength has a considerable effect on the geometry of the outflow and hints at complex central engine dynamics influencing lanthanide outflows. We find that actinide production is especially sensitive to magnetic field strength, with overall actinide mass fraction calculated at 1 Gyr post-merger increasing by more than a factor of six with a tenfold increase in magnetic field strength. This hints at a possible connection to the variability in actinide enhancements exhibited by metal poor, r-process-enhanced stars.
... The generalization of the Schwarzschild black hole solution to the electrically charged one [7,8] was done immediately after the work, and the more general and realistic rotating black hole solutions [9,10] were found in the 1960s. However, all of these black hole solutions have a curvature singularity at r = 0 at which spacetime is geodesically incomplete. ...
After finding a solution for the Hayward regular black hole (HRBH) in massive gravity, we embed the (3+1)-dimensional HRBHs both in massless and in massive gravities into (5+2)- and (6+3)-dimensional Minkowski spacetimes, respectively. Here, massive gravity denotes that a graviton acquires a mass holographically by broken momentum conservation in the HRBH. The original HRBH has no holographically added gravitons, which we call ‘massless’. Making use of newly found embedding coordinates, we obtain desired Unruh temperatures and compare them with the Hawking and local fiducial temperatures, showing that the Unruh effect for a uniformly accelerated observer in a higher-dimensional flat spacetime is equal to the Hawking effect for a fiducial observer in a black hole spacetime. We also obtain freely falling temperatures of the HRBHs in massless and massive gravities seen by freely falling observers, which remain finite even at the event horizons while becoming the Hawking temperatures in asymptotic infinity.
... This radiation, which carries away energy and reduces the mass of the BH over time, implies that black holes have a temperature and a corresponding entropy linked to their surface area. The expression for the entropy of BH is given by [1][2][3][4][5][6][7][8] ...
This paper delves the thermodynamics of 5-dimensional Schwarzschild $$AdS_5 \times S^5$$ A d S 5 × S 5 black hole by considering the recently proposed effective models of exponential entropies. The conventional understanding of cosmological constant $$\Lambda $$ Λ as thermodynamic pressure with volume being its counterpart cannot be directly applied in the framework of the AdS/CFT correspondence. In order to resolve this issue, we establish a connection between the cosmological constant $$\Lambda $$ Λ in the boundary gauge theory and number of colors N while chemical potential is considered as thermodynamic conjugate. In this study, we replace the geometric parameters L and r of the AdS black hole with two thermodynamic parameters $$N^2$$ N 2 and S in the micro-canonical ensemble. Additionally, we explore the various thermodynamic geometry models such as Ruppeiner, Quevedo and Weinhold to derive the associated scalar curvatures for the 5-D Schwarzschild AdS BH. We analyze that these geometries exhibit microscopic attraction/repulsion forces on the particles of black hole.
... We hope that other researchers will closely investigate our metric and compare carefully its predictions with observations from black holes and the Schwarzschild metric. The comparison can, hopefully, be extended to other metrics such as the Reissner-Nordström [20,21], Kerr [22] metric, the Kerr-Newman [23,24] metric, and even our new generalized rotation metric [19]. ...
Haug and Spavieri have recently presented a new exact solution to Einstein’s field equation. In this paper, we will list multiple implications of this model concerning predicted: gravitational time dilation, redshift, light bending, escape velocity, and more. All the new results contain a relativistic correction term added to the Schwarzschild metric. The Schwarzschild metric can almost be seen as a weak gravitational field approximation of our mass-charge metric. The difference in predictions of the two metrics, can be verified even in the weak gravitational field of the Sun and also in the gravitational field of the Earth by utlizing atomic or optical clocks on board of satellites. Yet, in the gravitational field of the Earth the effect is so small, that to be detected, we must have access to the most advanced state of the art in modern optical clocks. However in the Sun’s gravitational field it seems that even current cesium atomic clocks are good enough to test the differences in predictions between the two metrics.
Also, photons, sent out in a strong gravitational field close to the event horizon of black holes, can likely be used to verify what metric derived from general relativity theory is the best to explain observed phenomena in black holes, such as the lack of velocity time dilation in high-z quasars.
... The nonzero components of the metric tensor g µν , describing the geometry of the well-known classical neutral rotating Kerr BH can be written in the standard Boyer-Lindquist coordinates in the form [42,43] ...
... There it allows to observe with less dilution the direct primary radiation and its polarization properties, especially for low black-hole spin (Podgorn ý et al. 2023a ). For slowly rotating Kerr black holes, the co-rotating No viko v-Thorne disc (assumed to extend to ISCO) reaches only six gravitational radii from the black hole (Kerr 1963 ;No viko v & Thorne 1973 ). In the equatorial directions, the central emission is nearly parallelly polarized at all studied X-ray energies (see Fig. A9 ). ...
We present a broad analysis of X-ray polarimetric observational prospects for radio-quiet active galactic nuclei (AGN), focusing on the role of parsec-scale components. We provide a revision of self-consistent type-1 and type-2 generic AGN radiative transfer models that were obtained with a Monte Carlo code stokes, evaluating the effects of absorption and scattering. Our model consists of a central disc–corona emission obtained with the kynstokes code in the lamp-post geometry, an equatorial wedge-shaped dusty torus and two symmetric conical polar outflows. We argue that the information on the mutual orientation, shape, relative size, and composition of such components, usually obtained from spectroscopy or polarimetry in other wavelengths, is essential for the X-ray polarization analysis of the obscured type-2 AGNs. We provide general detectability prospects for AGNs with 2–8 keV polarimeters onboard of the currently flying Imaging X-ray Polarimetry Explorer (IXPE) satellite and the forthcoming enhanced X-ray Timing and Polarimetry mission. Finally, we assess the role of contemporary X-ray polarimetry in our understandings of the unified AGN model after the first year and a half of IXPE operation.
Einstein's equations for empty space are solved for the class of metrics which admit a family of hypersurface-orthogonal, non-shearing, diverging null curves. Some of these metrics may be considered as representing a simple kind of spherical, outgoing radiation. (Among them are solutions admitting no Killing field whatsoever.) Examples of solutions to the Maxwell-Einstein equations with a similar geometry are also given.