Gravitational Redshift Observed from a Kerr Body with Spatial Resolution on the Surface

We analyze the extraction of the rotational energy of a Kerr black hole (BH) endowed with a test charge and surrounded by an external test magnetic field and ionized low-density matter. For a magnetic field parallel to the BH spin, electrons move outward(inward) and protons inward(outward) in a region around the BH poles(equator). For zero charge, the polar region comprises spherical polar angles -60∘≲θ≲60∘ and the equatorial region 60∘≲θ≲120∘. The polar region shrinks for positive charge, and the equatorial region enlarges. For an isotropic particle density, we argue the BH could experience a cyclic behavior: starting from a zero charge, it accretes more polar protons than equatorial electrons, gaining net positive charge, energy and angular momentum. Then, the shrinking(enlarging) of the polar(equatorial) region makes it accrete more equatorial electrons than polar protons, gaining net negative charge, energy, and angular momentum. In this phase, the BH rotational energy is extracted. The extraction process continues until the new enlargement of the polar region reverses the situation, and the cycle repeats. We show that this electrodynamical process produces a relatively limited increase of the BH irreducible mass compared to gravitational mechanisms like the Penrose process, hence being a more efficient and promising mechanism for extracting the BH rotational energy.

Gravitational redshift is generally calculated without considering the rotation of a body. Neglecting the rotation, the geometry of space time can be described by using the spherically symmetric Schwarzschild geometry. Rotation has great effect on general relativity, which gives new challenges on gravitational redshift. When rotation is taken into consideration spherical symmetry is lost and off diagonal terms appear in the metric. The geometry of space time can be then described by using the solutions of Kerr family. In the present paper we discuss the gravitational redshift for rotating body by using Kerr metric. The numerical calculations has been done under Newtonian approximation of angular momentum. It has been found that the value of gravitational redshift is influenced by the direction of spin of central body and also on the position (latitude) on the central body at which the photon is emitted. The variation of gravitational redshift from equatorial to non - equatorial region has been calculated and its implications are discussed in detail.

Interstellar is the first Hollywood movie to attempt depicting a black hole as it would actually be seen by somebody nearby. For this, our team at Double Negative Visual Effects, in collaboration with physicist Kip Thorne, developed a code called Double Negative Gravitational Renderer (DNGR) to solve the equations for ray-bundle (light-beam) propagation through the curved spacetime of a spinning (Kerr) black hole, and to render IMAX-quality, rapidly changing images. Our ray-bundle techniques were crucial for achieving IMAX-quality smoothness without flickering; and they differ from physicists’ image-generation techniques (which generally rely on individual light rays rather than ray bundles), and also differ from techniques previously used in the film industry’s CGI community. This paper has four purposes: (i) to describe DNGR for physicists and CGI practitioners, who may find interesting and useful some of our unconventional techniques. (ii) To present the equations we use, when the camera is in arbitrary motion at an arbitrary location near a Kerr black hole, for mapping light sources to camera images via elliptical ray bundles. (iii) To describe new insights, from DNGR, into gravitational lensing when the camera is near the spinning black hole, rather than far away as in almost all prior studies; we focus on the shapes, sizes and influence of caustics and critical curves, the creation and annihilation of stellar images, the pattern of multiple images, and the influence of almost-trapped light rays, and we find similar results to the more familiar case of a camera far from the hole. (iv) To describe how the images of the black hole Gargantua and its accretion disk, in the movie Interstellar, were generated with DNGR—including, especially, the influences of (a) colour changes due to doppler and gravitational frequency shifts, (b) intensity changes due to the frequency shifts, (c) simulated camera lens flare, and (d) decisions that the film makers made about these influences and about the Gargantua’s spin, with the goal of producing images understandable for a mass audience. There are no new astrophysical insights in this accretion-disk section of the paper, but disk novices may find it pedagogically interesting, and movie buffs may find its discussions of Interstellar interesting.

We present the first X-ray reflection model for testing the assumption that the metric of astrophysical black holes is described by the Kerr solution. We employ the formalism of the transfer function proposed by Cunningham. The calculations of the reflection spectrum of a thin accretion disk are split into two parts: the calculation of the transfer function and the calculation of the local spectrum at any emission point in the disk. The transfer function only depends on the background metric and takes into account all the relativistic effects (gravitational redshift, Doppler boosting, and light bending). Our code computes the transfer function for a spacetime described by the Johannsen metric and can easily be extended to any stationary, axisymmetric, and asymptotically flat spacetime. Transfer functions and single line shapes in the Kerr metric are compared to those calculated from existing codes to check that we reach the necessary accuracy. We also simulate some observations with NuSTAR and LAD/eXTP and fit the data with our new model to show the potential capabilities of current and future observations to constrain possible deviations from the Kerr metric.

Generally gravitational redshift has been calculated without consideration of rotation of a body. Neglecting the rotation, the geometry of space time can be described by using the well-known spherically symmetric Schwarzschild geometry. Rotation has great effect on general relativity, which gives new challenges on gravitational redshift. When rotation is taken into consideration spherical symmetry lost and off diagonal terms appears in the metric and the geometry of space time can be described by using the Kerr solution, which is the exact solution of the Einstein's field equations known at present. In this paper we will derive the expression for gravitational redshift for rotating source in Kerr field, and also apply the derived expression to calculate the gravitational redshift in case of Sun under Newtonian approximation of angular momentum.

Supermassive black holes (SMBHs) like M87* and Sgr A* have been observed with the Event Horizon Telescope (EHT), facilitating the investigation of null geodesics near a black hole. The null geodesic around black holes and relevant features such as photon ring, lensing ring and black hole shadow, have been modeled analytically. The emission and absorption of plasmas surrounding a black hole is highly related to the physical quantities such as mass density, momentum, internal energy, magnetic field, black hole spin and angular momentum. The images of accretion disks around black holes are affected by Doppler shift, gravitational redshift, and frequency-dependent emissivity and absorption coefficient, which can be assessed by numerical simulations. In this work, a numerical ray-tracing scheme and a covariant radiative transfer scheme are proposed to simulate the photon trajectories emitting from the accretion disks around Kerr black holes, hence to reconstruct their images. A set of magnetized, geometrically thick accretion disks are adopted to compute the emissivity and absorption coefficient, which are then used to reconstruct the images of the subject accretion disks. The effects of magnetization, black hole spin, angular momentum and geometry of the disks are studied. The effects of Doppler shift, gravitational lensing, and frequency-dependent emissivity and absorption coefficient are investigated by simulations.

A compatible theory of relativistic geodesy is quite necessary for future development of any formulation of the subject. In this paper, following recent studies which invoke an idea of relativistic geoid, i.e. the chronometric definition implied by the gravitational redshift, we revisit known calculations on some standard spacetimes. Among all, the notion of frame dragging in the Kerr spacetime and its relationship to the rotative observers needed for the consistent definition of the geoid is analyzed. A compatible theory of relativistic geodesy is quite necessary for future development of any formulation of the subject. In this paper, following recent studies which invoke an idea of relativistic geoid, i.e. the chronometric definition implied by the gravitational redshift, we revisit known calculations on some standard spacetimes. Among all, the notion of frame dragging in the Kerr spacetime and its relationship to the rotative observers needed for the consistent definition of the geoid is analyzed.

Using the Sasaki-Nakamura equation, we have computed the energy, linear and angular momentum of the gravitational radiation induced by a particle of mass µ and angular momentum µLz plunging in an equatorial plane into a Kerr black hole of mass M (≫µ) and angular momentum Ma. It is found that the total energy ΔE ≈(µ/ M) µc2 is emitted by the particle with sufficient large orbital angular momentum. For the same value of |Lz|, a corotating particle emits more energy than a counter-rotating one. We have also calculated the energy from a rotating ring plunging into a Kerr black hole. In this case, we have found that a corotating ring emits less gravitational energy than a counter-rotating one for the same |Lz|. The maximum of the linear momentum is 6 ×10-2(µ/ M) µc, which suggests the recoil velocity of the coalesced black hole is 160 km/s for µ= 0.1 M.

We present a general relativistic radiative transfer code, ARTIST (Authentic Radiative Transfer In Space–Time), that is a perfectly causal scheme to pursue the propagation of radiation with absorption and scattering around a Kerr black hole. The code explicitly solves the invariant radiation intensity along null geodesics in the Kerr–Schild coordinates, and therefore properly includes light bending, Doppler boosting, frame dragging, and gravitational redshifts. The notable aspect of ARTIST is that it conserves the radiative energy with high accuracy, and is not subject to the numerical diffusion, since the transfer is solved on long characteristics along null geodesics. We first solve the wavefront propagation around a Kerr black hole that was originally explored by Hanni. This demonstrates repeated wavefront collisions, light bending, and causal propagation of radiation with the speed of light. We show that the decay rate of the total energy of wavefronts near a black hole is determined solely by the black hole spin in late phases, in agreement with analytic expectations. As a result, the ARTIST turns out to correctly solve the general relativistic radiation fields until late phases as t ∼ 90 M. We also explore the effects of absorption and scattering, and apply this code for a photon wall problem and an orbiting hotspot problem. All the simulations in this study are performed in the equatorial plane around a Kerr black hole. The ARTIST is the first step to realize the general relativistic radiation hydrodynamics.

A spinar is a collapsing object with quasi-equilibrium. Its equilibrium is maintained by the balance of centrifugal and gravitational forces and its evolution is determined by its magnetic field. A model of spinar quasi-equilibrium has recently been discussed in the context of an extralong X-ray plateau in a gamma-ray burst. In this paper, we propose a simple non-stationary three-parameter collapse model with the determining role of rotation and magnetic field. The input parameters of the theory are the mass, angular momentum and magnetic field of the collapsar. The model includes an approximate description of the following effects: the centrifugal force, the relativistic effects of the Kerr metrics, the pressure of nuclear matter, the dissipation of angular momentum as a result of the magnetic field, the decrease of the dipole magnetic moment as a result of compression and general-relativity effects (the black hole has no hair), neutrino cooling, time dilatation and gravitational redshift. The model describes the temporal behaviour of the central engine and demonstrates the qualitative variety of the types of such behaviour in nature. We apply our approach to an explanation of the observed features of all types of gamma-ray bursts. In particular, the model allows the unification of the phenomena of precursors, X-ray and optical flares, and the appearance of a plateau on the time-scale of several thousand seconds.

With the rapid development of atomic clocks, the potential for gravitational-wave detection through clock comparisons emerges as a viable prospect for the future. Given the profound signiffcance of gravitational redshift and time dilation in such comparisons, assessing its impact on gravitational wave detection with clock comparisons becomes imperative. We contemplate a two-satellite system tailored for gravitational wave detection via clock comparisons, and simulate the gravitational redshift and time dilation induced by major
celestial bodies within the Solar System. Leveraging this satellite system, we conduct numerical simulations to dissect the inffuence of gravitational redshift and time dilation on detection sensitivity. Our analysis shows that the Sun dominates the gravitational redshift
budget. In the band above 10^−4 Hz, the resulting fractional frequency shift (and time dilation) spectral density is two to three orders of magnitude below 10^−20/√Hz, which is the target sensitivity of clock-comparison searches for gravitational waves in the millihertz band.
Therefore, gravitational redshift and time dilation does not pose a notable obstacle to clock-comparison-based gravitational wave detection in frequencies above 10^−4 Hz. This conclusion is also directly applicable to the frequency shifts of laser links in laser
interferometric gravitational wave detection like TianQin, Taiji, and LISA.

We use a ray-tracing technique to compute the observed spectrum of a thin accretion disk around a Kerr black hole. We include all relativistic effects such as frame-dragging, Doppler boost, gravitational redshift, and bending of light by the gravity of the black hole. We also include self-irradiation of the disk as a result of light deflection. Assuming that the disk emission is locally blackbody, we show how the observed spectrum depends on the spin of the black hole, the inclination of the disk, and the torque at the inner edge of the disk. We find that the effect of a nonzero torque on the spectrum can, to a good approximation, be absorbed into a zero-torque model by adjusting the mass accretion rate and the normalization. We describe a computer model, called KERRBB, which we have developed for fitting the spectra of black hole X-ray binaries. Using KERRBB within the X-ray data reduction package XSPEC, and assuming a spectral hardening factor fcol = 1.7, we analyze the spectra of three black hole X-ray binaries: 4U 1543-47, XTE J1550-564, and GRO J1655-40. We estimate the spin parameters of the black holes in 4U 1543-47 and GRO J1655-40 to be a/M ∼ 0.6 and ∼0.6-0.7, respectively. If fcol ∼ 1.5-1.6, as in a recent study, then we find a/M ∼ 0.7-0.8 and ∼0.8-0.9, respectively. These estimates are subject to additional uncertainties in the assumed black hole masses, distances, and disk inclinations.

The spectrum of X-rays produced by an accretion disk around a black hole is influenced markedly by Doppler shifls, gravitational red shifts, and the gravitational lens effect. These influences can be described by a "transfer function," which one folds into any assumed spectrum emitted on the disk's surface to get the spectrum observed at various locations far from the disk. This paper formulates such a transfer function and tabulates it for Kerr black holes of aIM = 0 and 0.9981. The transfer function depends strongly on the polar angle of the observer: An observer near the plane of the disk sees radiation from its hot inner regions to be less redshifted and to subtend a greater solid angle than does an observer near the disk's polar axis. Consequently, the equatorial observer sees a much harder spectrum at high energies than does the polar observer. This effect is more pronounced for black holes of larger angular momentum. Subject headings: black holes - gravitation - relativity - X-ray sources

In this new theory the interaction between gravity and light has been discussed at astronomical levels; Gravitational RedShift, Black Holes and Dark Matter and at sub-atomic levels; the absorption and emission of light at sub-atomic levels in concentric spheres by an atom at discrete energy levels. Differently than in general relativity, the interaction between gravity and light fundamentally has been based on the sum of the “Stress Energy Tensor” and the introduced “Gravitational Tensor”. The theory describes “Gravitational-Electromagnetic Interaction” resulting in a mathematical tensor presentation for Black Holes. (Gravitational Electromagnetic Confinements) the “Electromagnetic Energy Gradient” creates a second order effect “Lorentz Transformation” which results in the gravitational field of black holes which determines the interaction force density between the confinement of light (black hole’s) and the gravitational field. Einstein approached the interaction between gravity and light by the introduction of the “Einstein Gravitational Constant” in the 4-dimensional Energy-Stress Tensor In this alternative approach related to general relativity, the interaction between gravity and light has been presented by the sum of the Electromagnetic Tensor and the Gravitational Tensor The new theory describes the impact of "CURL" within the gravitational fields around black holes and the impact on gravitational lensing. Gravitational "CURL" (Equation 6) is an effect which cannot be explained and calculated by general relativity. The new approach presents mathematical solutions for the black holes (gravitational electromagnetic interaction) introduced in 1955 by John Archibald Wheeler in the publication in Physical Review Letters in 1955. The mathematical solutions for black holes are fundamental solutions for the relativistic quantum mechanical Dirac equation (Quantum Physics) in tensor presentation (41). Assuming a constant speed of light “c” and Planck’s constant ħ within the black hole, the radius “R” of the black hole with the energy of a proton, is about 1% of the radius of the hydrogen atom (14). The new theory has been tested in an experiment with 2 galileo satellites and a ground station by measuring the gravitational redshift in an by the ground station emitted stable MASER frequency. The difference between the calculation for gravitational redshift, within the gravitational field of the Earth, in “General Relativity” and the “New Theory” is smaller than 10-16 (12) and (13). In all “General Redshift Experiments” general relativity and the new theory predict a gravitational redshift with a difference smaller than 15 digits beyond the decimal point which is beyond the accuracy of modern “Gravitational Redshift” observations. Both values are always within the measured gravitational redshift in all observations being published since the first observation of the gravitational redshift in the spectral lines from the white dwarf which was the measurement of the shift of the star Sirius B, the white dwarf companion to the star Sirius, by W.S. Adams in 1925 at Mt. Wilson Observatory. Theories which unify quantum physics and general relativity, like “String Theory”, predict the non-constancy of natural constants. Accurate observations of the NASA messenger observe in time a value for the gravitational constant “G” which constrains until One of the characteristics of the new theory is the “Constant Value” in time for the Gravitational constant “G” in unifying general relativity and quantum physics. Keywords Quantum physics; General relativity; Gravitational redshift; Black holes; Dark matter

We study and quantify gravitational redshift by means of relativistic ray tracing simulations of emission lines. The emitter model is based on thin, Keplerian rotating rings in the equatorial plane of a rotating black hole. Emission lines are characterised by a generalized fully relativistic Doppler factor or redshift associated with the line core. Two modes of gravitational redshift, shift and distortion, become stronger with the emitting region closer to the Kerr black hole. Shifts of the line cores reveal an effect at levels of 0.0015 to 60% at gravitational radii ranging from 105 to 2. The corresponding Doppler factors range from 0.999985 to 0.4048. Line shape distortion by strong gravity, i.e. very skewed and asymmetric lines occur at radii smaller than roughly ten gravitational radii. Gravitational redshift decreases with distance to the black hole but remains finite due to the asymptotical flatness of black hole space–time. The onset of gravitational redshift can be tested observationally with sufficient spectral resolution. Assuming a resolving power of ∼100000, yielding a resolution of ≈0.1 Å for optical and near‐infrared broad emission lines, the gravitational redshift can be probed out to approximately 75000 gravitational radii. In general, gravitational redshift is an indicator of black hole mass and spin as well as for the inclination angle of the emitter, e.g. an accretion disk. We suggest to do multi‐wavelength observations because all redshifted features should point towards the same central mass. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)