Null Rays Research Articles

Mass fluctuations in non-rotating BTZ black holes

We investigate the impact of oscillations of a black hole's mass around its average value on the three-dimensional black hole geometry. Drawing on a classical framework that conceptualizes fluctuations near an event horizon as mass variations, we introduce a model where the metric of a black hole, formed from the collapse of a massive null shell, exhibits oscillatory behavior in spherical modes. This dynamic is encapsulated by a non-rotating BTZ-Vaidya solution, characterized by the black hole mass fluctuating at a resonant frequency ω and a small amplitude parameter μ0. Using a perturbative approach, solutions to the null geodesic equation are determined up to the second order in μ0. The temporal fluctuations of the event horizon's location induce alterations in the thermodynamic variables' values. Upon calculating the time-averaged values, the mean Hawking temperature experiences a slight decrease due to these fluctuations, while the mean entropy exhibits an increase. Furthermore, the study explores the influence of these fluctuations on the trajectories of null rays near the horizon, which ultimately reach the anti-de Sitter boundary at late times. The analytical computation of the general solution for the perturbed rays up to the second order underscores the novel approach of this study in examining the effects of mass oscillations on black hole thermodynamics and geometry, thereby contributing a unique perspective to the field.

Redshift drift in a universe with structure: Lemaître-Tolman-Bondi structures with arbitrary angle of entry of light

We consider the redshift drift and position drift associated with astrophysical sources in a formalism that is suitable for describing emitters and observers of light in an arbitrary spacetime geometry, while identifying emitters of a given null-geodesic bundle that arrives at the observer worldline. We then restrict the situation to the special case of a Lemaitre-Tolman-Bondi (LTB) geometrical structure, and solve for light rays propagating through the structure with arbitrary impact parameters, i.e., with arbitrary angles of entry into the LTB structure. The redshift drift signal emitted by comoving sources and viewed by a comoving observer turns out to be dominated by Ricci curvature and electric Weyl curvature contributions as integrated along the connecting light ray. This property simplifies the computations of the redshift drift signal tremendously, and we expect that the property extends to more complicated models including Swiss-cheese models. When considering several null rays with random impact parameters, the mean redshift drift signal is well approximated by a single Ricci focusing term. This suggests that the measurement of cosmological redshift drift can be used as a direct probe of the strong energy condition in a realistic universe where photons pass through many successive structures.

Deflection and gravitational lensing of null and timelike signals in general asymptotically (anti–)de Sitter spacetimes

The deflection and gravitational lensing of light and massive particles in arbitrary static, spherically symmetric and asymptotically (anti--)de Sitter spacetimes are considered in this work. We first proved that for spacetimes whose metric satisfies certain conditions, the deflection of null rays with fixed closest distance will not depend on the cosmological constant $\mathrm{\ensuremath{\Lambda}}$, while that of timelike signals and the apparent angle in gravitational lensing still depend on $\mathrm{\ensuremath{\Lambda}}$. A two-step perturbative method is then developed to compute the change of the angular coordinate and total travel time in the weak field limit. The results are quasipower series of two small quantities, with the finite distance effect of the source/detector naturally taken into account. These results are verified by applying to some known asymptotically de Sitter spacetimes. Using an exact gravitational lensing equation, we solved the apparent angles of the images and time delays between them and studied the effect of $\mathrm{\ensuremath{\Lambda}}$ on them. It is found that generally, a small positive $\mathrm{\ensuremath{\Lambda}}$ will decrease the apparent angle of images from both sides of the lens and increase the time delay between them. The time delay between signals from the same side of the lens but with different energy however, will be decreased by $\mathrm{\ensuremath{\Lambda}}$.

Frequency shift of light emitted from growing and shrinking black holes

In this paper we present a method to study the frequency shift of signals sent from near a Schwarzschild black hole that grows or shrinks through accretion. We construct the numerical solution of Einstein's equations sourced by a spherical shell of scalar field, with positive energy density to simulate the growth and with negative energy density to simulate the shrink of the black hole horizon. We launch a distribution of null rays at various time slices during the accretion and estimate their energy along their own trajectories. Spatially the bundles of photons are distributed according to the distribution of dust, whose dynamics obeys Euler equations in the test field limit during the evolution of the black hole. With these elements, we construct the frequency shift of photons during the accretion process of growth or contraction of the hole, which shows a variability that depends on the thickness of the scalar field shell or equivalently the time scale of the accretion.

Celestial fields on the string and the Schwarzian action

This paper describes the motion of a classical Nambu-Goto string in three-dimensional anti-de Sitter spacetime in terms of two ‘celestial’ fields on the worldsheet. The fields correspond to retarded and advanced boundary times at which null rays emanating from the string reach the boundary. The formalism allows for a simple derivation of the Schwarzian action for near-AdS2 embeddings.

Universal time delay in static spherically symmetric spacetimes for null and timelike signals * *Supported by the National Natural Science Foundation of China (11504276) and MOST China (2014GB109004)

A perturbative method of computing the total travel time of both null and lightlike rays in arbitrary static spherically symmetric spacetimes in the weak field limit is proposed. The resultant total time takes a quasi-series form of the impact parameter. The coefficient of this series at a certain order n is shown to be determined by the asymptotic expansion of the metric functions to the order . For the leading order(s), the time delay, as well as the difference between the time delays of two types of relativistic signals, is shown to take a universal form for all SSS spacetimes. This universal form depends on the mass M and a post-Newtonian parameter of the spacetime. The analytical result is numerically verified using the central black hole of galaxy M87 as the gravitational lensing center.

Causal description of marginally trapped surfaces in D dimensions

In this paper, we analyse the causal aspects of evolving marginally trapped surfaces in a D-dimensional spherically symmetric spacetime, sourced by perfect fluid with a cosmological constant. The norm of the normal to the marginally trapped tube is shown to be the product of lie derivatives of the expansion parameter of future outgoing null rays along the incoming and outgoing null directions. We obtain a closed form expression for this norm in terms of principal density, pressure, areal radius and cosmological constant. For the case of a homogeneous fluid distribution, we obtain a simple formula for determining the causal nature of the evolving horizons. We obtain the causal phase portraits and highlight the critical radius. We identify many solutions where the causal signature of the marginally trapped tube or marginally anti-trapped tube is always null despite having an evolving area. These solutions don't comply with the standard inner and outer horizon classification for degenerate horizons. we propose an alternate prescription for this classification of these degenerate horizons.

Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of (1-b_c/b) where b is the impact parameter and b_c is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter l and the asymptotic velocity v of the signal. The BH shadow size were found to decrease slightly as l increases to its critical value, and increase as v decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of l and decrease of v will increase their values.

On the radiation gauge for spin-1 perturbations in Kerr–Newman spacetime

We extend previous work (2020 Class. Quantum Grav. 37 075001) to the case of Maxwell’s equations with a source. Our work shows how to construct a vector potential for the Maxwell field on the Kerr–Newman background in a radiation gauge. The vector potential has a ‘reconstructed’ term obtained from a Hertz potential solving Teukolsky’s equation with a source, and a ‘correction’ term which is obtainable by a simple integration along outgoing principal null rays. The singularity structure of our vector potential is discussed in the case of a point particle source.

Null Wave Front and Ryu-Takayanagi Surface.

The Ryu–Takayanagi formula provides the entanglement entropy of quantum field theory as an area of the minimal surface (Ryu–Takayanagi surface) in a corresponding gravity theory. There are some attempts to understand the formula as a flow rather than as a surface. In this paper, we consider null rays emitted from the AdS boundary and construct a flow representing the causal holographic information. We present a sufficient and necessary condition that the causal information surface coincides with Ryu–Takayanagi surface. In particular, we show that, in spherical symmetric static spacetimes with a negative cosmological constant, wave fronts of null geodesics from a point on the AdS boundary become extremal surfaces and therefore they can be regarded as the Ryu–Takayanagi surfaces. In addition, from the viewpoint of flow, we propose a wave optical formula to calculate the causal holographic information.

Maxwell equations in a curved spacetime: Spin optics approximation

We study propagation of high-frequency electromagnetic waves in a curved spacetime. We demonstrate how a modification of the standard geometric optics allows one to include the helicity dependent corrections into the equations of motion of circularly polarized beams of radiation. As a result, polarized light rays are still null but not geodesic curves. To achieve these results we construct null frames associated with a set of (non-geodesic) null rays and use these frames for description of the high-frequency wave propagation. We call this approach spin optics approximation. It is completely covariant and it can be used in an arbitrary time-dependent gravitational field.

News from Horizons in Binary Black Hole Mergers.

In a binary black hole merger, it is known that the inspiral portion of the waveform corresponds to two distinct horizons orbiting each other and that the merger and ringdown signals correspond to the final horizon being formed and settling down to equilibrium. However, we still lack a detailed understanding of the relation between the horizon geometry in these three regimes and the observed waveform. Here we show that the well-known inspiral chirp waveform has a clear counterpart on black hole horizons, namely, the shear of the outgoing null rays at the horizon. We demonstrate that the shear behaves very much like a compact binary coalescence waveform with increasing frequency and amplitude. Furthermore, the parameters of the system estimated from the horizon agree with those estimated from the waveform. This implies that even though black hole horizons are causally disconnected from us, assuming general relativity to be true, we can potentially infer some of their detailed properties from gravitational wave observations.

Radiation from an Inertial Mirror Horizon

The purpose of this study is to investigate radiation from asymptotic zero acceleration motion where a horizon is formed and subsequently detected by an outside witness. A perfectly reflecting moving mirror is used to model such a system and compute the energy and spectrum. The trajectory is asymptotically inertial (zero proper acceleration)—ensuring negative energy flux (NEF), yet approaches light-speed with a null ray horizon at a finite advanced time. We compute the spectrum and energy analytically.

Perturbative deflection angles of timelike rays

Geodesics of both lightrays and timelike particles with nonzero mass are deflected in a gravitational field. In this work we apply the perturbative method developed in reference [] to compute the deflection angle of both null and timelike rays in the weak field limit for four spacetimes. We obtained the deflection angles for the Bardeen spacetime to the eleventh order of m/b where m is the ADM mass and b is the impact parameter, and for the Hayward, Janis–Newman–Winicour and Einstein–Born–Infeld spacetimes to the thirteenth, seventh and eleventh order respectively. The effects of the impact parameter b, velocity v and spacetime parameters on the deflection angle are analyzed in each of the four spacetimes. It is found that in general, the perturbative deflection angle depends on and only on the asymptotic behavior of the metric functions, and in an order-correlated way. Moreover, it is shown that although these deflection angles are calculated in the large b/m limit, their minimal valid b can be as small as a few m’s as long as the order is high enough. At these impact parameters, the deflection angle itself is also found large. As velocity decreases, the deflection angle in all spacetime studied increases. For a given b, if the spacetime parameters allows a critical velocity vc, then the perturbative deflection angle will deviate from its true value as v decreases to vc. It is also found that if the variation of spacetime parameters can only change the spacetime qualitatively at small but not large radius, then these spacetime parameter will not cause a qualitative change of the deflection angle, although its value is still quantitatively affected. The application and possible extension of the work are discussed.

Evolution of the electric field along null rays for arbitrary observers and spacetimes

We derive, in a manifestly covariant and electromagnetic gauge independent way, the evolution law of the electric field $E^\alpha = Ee^\alpha\ (e^\alpha e_\alpha=1)$, relative to an arbitrary set of instantaneous observers along a null geodesic ray, for an arbitrary Lorentzian spacetime, in the geometrical optics limit of Maxwell's equations in vacuum. We show that, in general, neither the magnitude $E$ nor the direction $e^\alpha$ of the electric field (here interpreted as the observed polarization of light) are parallel transported along the ray. For an extended reference frame around the given light ray, we express the evolution of the direction $e^\alpha$ in terms of the frame's kinematics, proving thereby that its expansion never spoils parallel transport, which bears on the unbiased inference of intrinsic properties of cosmological sources, such as, for instance, the polarization field of the cosmic microwave background (CMB). As an application of the newly derived laws, we consider a simple setup for a gravitational wave (GW) interferometer, showing that, despite the (kinematic) shear induced by the GW, the change in the final interference pattern is negligible since it turns out to be of the order of the ratio of the GW and laser frequencies.

The perturbative approach for the weak deflection angle

Both null and timelike rays experience trajectory bending in a gravitational field. In this work, we systematically develop a perturbative method to compute the deflection angle of rays with general velocity v in arbitrary static and spherically symmetric spacetimes and in equatorial plane of arbitrary static and axisymmetric spacetimes. We show that the expansion in the large closest approach x_0 limit depends on the asymptotic behavior of the metric functions only, and the generated integrand is always integrable, resulting in a deflection angle in a series form of either x_0 or b, the impact parameter. Using this method, the deflection angles as series of both x_0 and b are found in Schwarzschild, Reissner–Nordström and Kerr–Newman spacetimes to 17-th, 15-th and 6-th orders respectively, for both lightrays and particles with general velocity. The effects of the impact parameter, velocity and other parameters of the spacatimes are briefly analyzed. Moreover, we show that for spacetimes whose metric functions are only asymptotically known, the deflection angle in the weak field limit can also be calculated. Furthermore, it is shown that the deflection angle in general static and spherically symmetric spacetime and equatorial plane of static and axisymmetric spacetime to the lowest non-trivial order, depends only on the impact parameter, velocity of the particle, and the effective ADM mass of the spacetime but not on other parameters such as charge or angular momentum. These deflection angles are used in an exact gravitational lensing equation and the corresponding apparent angles of the images of the source are also solved perturbatively.

New scalar compact objects in Ricci-based gravity theories

Taking advantage of a previously developed method, which allows to map solutions of General Relativity into a broad family of theories of gravity based on the Ricci tensor (Ricci-based gravities), we find new exact analytical scalar field solutions by mapping the free-field static, spherically symmetric solution of General Relativity (GR) into quadratic f(R) gravity and the Eddington-inspired Born-Infeld gravity. The obtained solutions have some distinctive feature below the would-be Schwarzschild radius of a configuration with the same mass, though in this case no horizon is present. The compact objects found include wormholes, compact balls, shells of energy with no interior, and a new kind of object which acts as a kind of wormhole membrane. The latter object has Euclidean topology but connects antipodal points of its surface by transferring particles and null rays across its interior in virtually zero affine time. We point out the relevance of these results regarding the existence of compact scalar field objects beyond General Relativity that may effectively act as black hole mimickers.

Affine-null formulation of the gravitational equations: Spherical case

A new evolution algorithm for the characteristic initial value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation is applied in the spherically symmetric case to the gravitational collapse of a massless scalar field. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to critical collapse. Unexpected quadratures lead to a simple evolution algorithm based upon ordinary differential equations which can be integrated along the null rays. For collapse to a black hole in a Penrose compactified spacetime, these equations are regularized throughout the exterior and interior of the horizon up to the final singularity. They are implemented as a global numerical evolution code based upon the Galerkin method. New results regarding the global properties of critical collapse are presented.

Constraining higher order gravities with subregion duality

In higher derivative theories, gravity can travel slower or faster than light. With this feature in mind, we revisit the construction of the causal and entanglement wedges in this type of theories, and argue that they must be constructed using the fastest mode instead of null rays. We show that the property of causal wedge inclusion, i.e., the fact that the causal wedge must be contained in the entanglement wedge, leads to more stringent constraints on the couplings than those imposed by hyperbolicity and boundary causality. Our results imply that the full power of subregion-subregion duality could lead to the same conclusions previously obtained based on high energy graviton scattering. We illustrate our findings with a systematic analysis in Gauss-Bonnet gravity.

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordstr\"om-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.