Observers in Kerr spacetimes: the ergoregion on the equatorial plane

We perform a detailed analysis of the properties of stationary observers located on the equatorial plane of the ergosphere in a Kerr spacetime, including light-surfaces. This study highlights crucial differences between black hole and the super-spinner sources. In the case of Kerr naked singularities, the results allow us to distinguish between “weak” and “strong ” singularities, corresponding to spin values close to or distant from the limiting case of extreme black holes, respectively. We derive important limiting angular frequencies for naked singularities. We especially study very weak singularities as resulting from the spin variation of black holes. We also explore the main properties of zero angular momentum observers for different classes of black hole and naked singularity spacetimes.     

A nondispersive de Broglie wave packet

It is assumed that the motion of a particle in spacetime does not depend on the motion relative to it of any observer or of any frame of reference. Thus if the particle has an internal vibration of the type hypothesized by de Broglie, the phase of that vibration at any point in spacetime must appear to be the same to all observers, i.e., the same in all frames of reference. Each observer or reference frame will have its own de Broglie wave for the particle. The phase of the particle's vibration must, by definition, be the same as that of all possible de Broglie waves at the point where the particle is. By superimposing all these possible de Broglie waves, a wave packet is formed centered in space on the particle. The formation of such a packet is discussed with the help of spacetime diagrams; the packet does not spread with time. The relevance of this packet to the wave mechanics of Schrödinger is discussed; it is also pointed out that any vibration can lead to such a packet.     

Separable geodesic action slicing in stationary spacetimes

A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent the world lines of freely falling fiducial observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function, although the time coordinate lines still follow Killing trajectories to retain the explicitly stationary nature of the coordinate grid. This explains some of the properties of the original Painlevé-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the Gödel spacetime. For the static spherically symmetric case the slicing can be chosen to be intrinsically flat with spherically symmetric geodesic observers, leaving all the gravitational field information in the shift vector field.     

Geodesic Motion in Spinning Spaces

In this paper the generalized equations for spinning space are investigated and the constants of motion are derived in terms of the solutions of these equations. We study the geodesic motion of the pseudo-classical spinning particles in the spacetime produced by an idealized cosmic string and the non-extreme stationary axisymmetric black hole spacetime. The bound state orbits in a plane are discussed. We also show, for a conical spacetime and the Kerr spacetime, that the geodesic motion of spinning particles is different.     

Simultaneity on the Rotating Disk

The disk that rotates in an inertial frame in special relativity has long been analysed by assuming a Lorentz contraction of its peripheral elements in that frame, which has produced widely varying views in the literature. We show that this assumption is unnecessary for a disk that corresponds to the simplest form of rotation in special relativity. After constructing such a disk and showing that observers at rest on it do not constitute a true rotating frame, we choose a “master” observer and calculate a set of disk coordinates and spacetime metric pertinent to that observer. We use this formalism to resolve the “circular twin paradox”, then calculate the speed of light sent around the periphery as measured by the master observer, to show that this speed is a function of sent-direction and disk angle traversed. This result is consistent with the Sagnac Effect, but constitutes a finer analysis of that effect, which is normally expressed using an average speed for a full trip of the periphery. We also use the formalism to give a resolution of “Selleri’s paradox”.     

Geodesic Motion in Spinning Spaces

In this paper the generalized equations for spinning space are investigated and the constants of motion are derived in terms of the solutions of these equations. We study the geodesic motion of the pseudo-classical spinning particles in the spacetime produced by an idealized cosmic string and the non-extreme stationary axisymmetric black hole spacetime. The bound state orbits in a plane are discussed. We also show, for a conical spacetime and the Kerr spacetime, that the geodesic motion of spinning particles is different.     

On the Accelerated Observer’s Proper Coordinates and the Rigid Motion Problem in Minkowski Spacetime

Physicists have been interested in accelerated observers for quite some time. Since the advent of special relativity, many authors have tried to understand these observers in the framework of Minkowski spacetime. One of the most important issues related to these observers is the problematic definition of rigid motion. In this paper, I write the metric in terms of the Frenet–Serret curvatures and the proper coordinate system of a general accelerated observer. Then, I use this approach to create a systematic way to construct a rigid motion in Minkowski spacetime. Finally, I exemplify the benefits of this procedure by applying it to two well-known observers, namely, the Rindler and the rotating ones, and also by creating a set of observers that, perhaps, may be interpreted as a rigid cylinder which rotates while accelerating along the axis of rotation.     

Spinning Test Particles as Locally Nonrotating Observers in Kerr Spacetime

We derive a class of exact solutions of Mathisson-Papapetrou equations of motion for spinning test particles. The world lines of the particles are those of the so-called locally non-rotating observers in Kerr spacetime.     

Robust prediction of complex spatiotemporal states through machine learning with sparse sensing

Complex spatiotemporal states arise frequently in material as well as biological systems consisting of multiple interacting units. A specific, but rather ubiquitous and interesting example is that of “chimeras”, existing in the edge between order and chaos. We use Machine Learning methods involving “observers” to predict the evolution of a system of coupled lasers, comprising turbulent chimera states and of a less chaotic biological one, of modular neuronal networks containing states that are synchronized across the networks. We demonstrated the necessity of using “observers” to improve the performance of Feed-Forward Networks in such complex systems. The robustness of the forecasting capabilities of the “Observer Feed-Forward Networks” versus the distribution of the observers, including equidistant and random, and the motion of them, including stationary and moving was also investigated. We conclude that the method has broader applicability in dynamical system context when partial dynamical information about the system is available.     

Linking covariant and canonical general relativity via local observers

Hamiltonian gravity, relying on arbitrary choices of ‘space,’ can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between ‘spatial’ and ‘temporal’ variables. The key is viewing dynamical fields from the perspective of a field of observers—a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the ‘space of observers’ is fundamental, and spacetime geometry itself may be observer-dependent.     

Gravitational Collapse in Quantum Einstein Gravity

The existence of spacetime singularities is one of the biggest problems of nowadays physics. According to Penrose, each physical singularity should be covered by a “cosmic censor” which prevents any external observer from perceiving their existence. However, classical models describing the gravitational collapse usually results in strong curvature singularities, which can also remain “naked” for a finite amount of advanced time. This proceedings studies the modifications induced by asymptotically safe gravity on the gravitational collapse of generic Vaidya spacetimes. It will be shown that, for any possible choice of the mass function, quantum gravity makes the internal singularity gravitationally weak, thus allowing a continuous extension of the spacetime beyond the singularity.     

Length measurement in accelerated systems

We investigate the limitations of length measurements by accelerated observers in Minkowski spacetime brought about via the hypothesis of locality, namely, the assumption that an accelerated observer at each instant is equivalent to an otherwise identical momentarily comoving inertial observer. We find that consistency can be achieved only in a rather limited neighborhood around the observer with linear dimensions that are negligibly small compared to the characteristic acceleration length of the observer.     

Scalar field quantization in stationary,non-static spacetimes

A quantization procedure is given for the scalar field on stationary, axisymmetric background spacetimes with orthogonal 2-surfaces. The procedure is based on observers orthogonal to surfaces of constant Killing time, and thus agrees with the usual procedure for static spacetimes. For stationary but nonstatic spacetimes the procedure differs from the usual one but nonetheless leads to a natural quantization scheme. Applying the procedure to flat space in rotating coordinates gives the standard, inertial Minkowski vacuum. For the Kerr spacetime, the procedure yields a particle definition which is well-defined everywhere outside the horizon. The above observers are just nonrotating ZAMO's, and the vacuum state smoothly interpolates between the “in” and “out” Boulware vacua.     

Covariant energy–momentum and an uncertainty principle for general relativity

We introduce a naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. The extension links seamlessly to the action integral for the gravitational field. The demand that the general expression for arbitrary systems reduces to the Tolman integral in the case of stationary bounded distributions, leads to the matter-localized Ricci integral for energy–momentum in support of the energy localization hypothesis. The role of the observer is addressed and as an extension of the special relativistic case, the field of observers comoving with the matter is seen to compute the intrinsic global energy of a system. The new localized energy supports the Bonnor claim that the Szekeres collapsing dust solutions are energy-conserving. It is suggested that in the extreme of strong gravity, the Heisenberg Uncertainty Principle be generalized in terms of spacetime energy–momentum.     

Static observers in curved spaces and non-inertial frames in Minkowski spacetime

Static observers in curved spacetimes may interpret their proper acceleration as the opposite of a local gravitational field (in the Newtonian sense). Based on this interpretation and motivated by the equivalence principle, we are led to investigate congruences of timelike curves in Minkowski spacetime whose acceleration field coincides with the acceleration field of static observers of curved spaces. The congruences give rise to non-inertial frames that are examined. Specifically, we find, based on the locality principle, the embedding of simultaneity hypersurfaces adapted to the non-inertial frame in an explicit form for arbitrary acceleration fields. We also determine, from the Einstein equations, a covariant field equation that regulates the behavior of the proper acceleration of static observers in curved spacetimes. It corresponds to an exact relativistic version of the Newtonian gravitational field equation. In the specific case in which the level surfaces of the norm of the acceleration field of the static observers are maximally symmetric two-dimensional spaces, the energy?Cmomentum tensor of the source is analyzed.     

Collisional super-Penrose process and Wald inequalities

We consider collision of two massive particles in the equatorial plane of an axially symmetric stationary spacetime that produces two massless particles afterwards. It is implied that the horizon is absent but there is a naked singularity or another potential barrier that makes possible the head-on collision. The relationship between the energy in the center of mass frame \(E_{c.m.}\) and the Killing energy E measured at infinity is analyzed. It follows immediately from the Wald inequalities that unbounded E is possible for unbounded \(E_{c.m.}\) only. This can be realized if the spacetime is close to the threshold of the horizon formation. Different types of spacetimes (black holes, naked singularities, wormholes) correspond to different possible relations between \(E_{c.m.}\) and E. We develop a general approach that enables us to describe the collision process in the frames of the stationary observer and zero angular momentum observer. The escape cone and escape fraction are derived. A simple explanation of the existence of the bright spot is given. For the particular case of the Kerr metric, our results agree with the previous ones found in Patil et al. (Phys Rev D 93:104015, 2016).     

On causality violation on a Kerr-de Sitter spacetime

The causal properties of the family of Kerr-de Sitter spacetimes are analyzed and compared to those of the Kerr family. First, an inextendible Kerr-de Sitter spacetime is obtained by joining together Carter’s blocks, i.e. suitable four dimensional spacetime regions contained within Killing horizons or within a Killing horizon and an asymptotic de Sitter region. Based on this property, and leaving aside topological identifications, we show that the causal properties of a Kerr-de Sitter spacetime are determined by the causal properties of the individual Carter’s blocks viewed as spacetimes in their own right. We show that any Carter’s block is stably causal except for the blocks that contain the ring singularity. The latter are vicious sets, i.e. any two events within such block can be connected by a future (respectively past) directed timelike curve. This behavior is identical to the causal behavior of the Boyer–Lindquist blocks that contain the Kerr ring singularity. These blocks are also vicious as demonstrated long ago by Carter. On the other hand, while for the case of a naked Kerr singularity the entire spacetime is vicious and thus closed timelike curves pass through any event including events in the asymptotic region, for the case of a Kerr-de Sitter spacetime the cosmological horizons protect the asymptotic de Sitter region from a-causal influences. In that regard, a positive cosmological constant appears to improve the causal behavior of the underlying spacetime.     

On the Finiteness of Light Rays Between a Source and an Observer on Conformally Stationary Space-Times

We use a convexity condition to prove the finiteness of the number of light rays joining a pointlike source with a pointlike observer in a stationary relativistic spacetime. The result is extended to the case of conformally stationary metrics under suitable boundedness assumptions on the conformal factor. We discuss our hypotheses in some explicit examples of standard static and stationary Lorentzian manifolds.