Addressing sustainable mobility in urban areas has become a priority in today’s society, given the growing population and increasing vehicular flow in these areas. Intelligent Transportation Systems have emerged as innovative and effective technological solutions to address these challenges. Research in this area has become crucial, as it contributes not only to improve mobility in urban areas, but also to positively impact the quality of life of its inhabitants. To address this, a dynamic clustering methodology for vehicular trajectory data is proposed which can provide an accurate representation of the traffic state. Data was collected for the city of San Francisco, a dynamic clustering algorithm was applied and then an indicator was applied to identify areas with traffic congestion. Several experiments were also conducted with different parameterizations of the forgetting factor of the clustering algorithm. The results showed in terms of precision that the dynamic clustering methodology achieved high match rates compared to the congestion indicator applied to static cells.

The volume of vehicular traffic in large cities has increased in recent years, the devices that collect vehicular GPS data such as cameras, GPS receivers and others generate millions of records at every instant of time generating problems in processing and storage of these data which becomes important for researchers. Intelligent Transportation Systems perform vehicle monitoring and control by collecting GPS trajectories, this large volume of information is necessary to have an optimal storage process. Its processing by means of compression techniques and simplification algorithms allow to reduce the necessary storage space. This paper presents a GPS trajectory simplification algorithm that considers noise reduction, point simplification and analysis of road network information. The results obtained on two data sets from the cities of California and Beijing are satisfactory, achieving a higher compression ratio without affecting data quality

Prior research shows that individuals who have exhibited antisocial behavior are in poorer health than their same-aged peers. A major driver of poor health is aging itself, yet research has not investigated relationships between offending trajectories and biological aging. We tested the hypothesis that individuals following a life-course persistent (LCP) antisocial trajectory show accelerated aging in midlife. Trajectories of antisocial behavior from age 7 to 26 years were studied in the Dunedin Multidisciplinary Health and Development Study, a population-representative birth cohort (N=1037). Signs of aging were assessed at age 45 years using previously validated measures including biomarkers, clinical tests, and self-reports. First, we tested whether the association between antisocial behavior trajectories and midlife signs of faster aging represented a decline from initial childhood health. We then tested whether decline was attributable to tobacco smoking, antipsychotic medication use, debilitating illnesses in adulthood, adverse exposures in childhood (maltreatment, socioeconomic disadvantage) and adulthood (incarceration), and to childhood self-control difficulties. Study members with a history of antisocial behavior had a significantly faster pace of biological aging by midlife, and this was most evident among individuals following the LCP trajectory (β, .22, 95%CI, .14, .28, p.001). This amounted to 4.3 extra years of biological aging between ages 25-45 years for Study members following the LCP trajectory compared to low-antisocial trajectory individuals. LCP offenders also experienced more midlife difficulties with hearing (β, -.14, 95%CI, -.21, -.08, p.001), balance (β, -.13, 95%CI, -.18, -.06, p.001), gait speed (β, -.18, 95%CI, -.24, -.10, p.001), and cognitive functioning (β, -.25, 95%CI, -.31, -.18, p.001). Associations represented a decline from childhood health. Associations persisted after controlling individually for tobacco smoking, antipsychotic medication use, midlife illnesses, maltreatment, socioeconomic status, incarceration, and childhood self-control difficulties. However, the cumulative effect of these lifestyle characteristics together explained why LCP offenders have a faster Pace of Aging than their peers. While older adults typically age-out of crime, LCP offenders will likely age-into the healthcare system earlier than their chronologically same-aged peers. Preventing young people from offending is likely to have substantial benefits for health, and people engaging in a LCP trajectory of antisocial behaviors might be the most in need of health promotion programs. We offer prevention and intervention strategies to reduce the financial burden of offenders on health care systems and improve their wellbeing.

The role of Black Carbon (BC) as a contributor to glacial retreat is of particular interest to the scientific community and decision makers, due to its impact on snow albedo and glacier melt. In this study, a thermal-optical instrument (LAHM) was used to measure effective Black Carbon (eBC) in a series of surface snow samples collected from the Vallunaraju glacier, Cordillera Blanca, between April 2019 and May 2020. The time series obtained indicates a marked seasonal variability of eBC with maximum concentrations during the dry season and dramatic decrease during the wet season. The concentrations detected ranged between a minimum of 3.73 ng/g and 4.23 ng/g during the wet season and a maximum of 214.13 ng/g and 181.60 ng/g during the dry season, in the accumulation and ablation zone. Using SNICAR model, the reduction of albedo was estimated at 6.36% and 6.60% during the dry season and 0.68% and 0.95% during the wet season, which represents an average radiative forcing of 4.52 ± 1.84 W/m2 and 4.69 ± 1.59 W/m2 in the accumulation zone, and 0.49 ± 0.27 W/m2 and 0.68 ± 0.43 W/m2 in the ablation zone. The melting of snow due to the eBC translates into 80.18 ± 37.30 kg/m2 and 83.16 ± 32.75 kg/m2 during the dry season, and 7.91 ± 4.29 kg/m2 and 10.85 ± 6.62 kg/m2 during the wet season, in the accumulation and ablation zones, respectively. Finally, the HYSPLIT trajectory assessment shows that aerosols predominate in the Amazon rainforest, especially when forest fires are most abundant according to VIIRS images.

Vehicular traffic in urban areas continues to be a constant challenge today, due to the continuous population growth and the increase of vehicles in circulation. Vehicular congestion, as a recurrent problem, generates a negative impact on urban mobility and citizens’ quality of life. It is hypothesized that a dynamic clustering method of vehicle trajectory data can provide an accurate and up-to-date representation of real-time traffic behavior. To evaluate this hypothesis, data were collected from three different cities: San Francisco, Rome and Guayaquil. A dynamic clustering algorithm was applied to identify traffic congestion patterns and an indicator was applied to identify and evaluate the congestion conditions of the areas. The results show a high precision and recall in congestion classification compared to a method based on static cells.

As the problem of air pollution becomes more urgent, the precise determination of the characteristics of air pollution and the associated mechanisms of transport of pollutants becomes an increasingly urgent task. The simulation results of the Copernicus Atmospheric Monitoring Service (CAMS) [18] were used as initial data sets. The area of the Crimean Peninsula (within coordinates 44.50 latitude, 33.50 longitude) in 23 - 26 January, 2019 was chosen for the study. That time concentrations of pollutants (aerosol particles with a diameter of 2.5 (PM2.5) and 10 micrometers (PM10)) steadily increased and reached the maximum value. Four kinds of meteorological data sets were used in the work. The first and second data sets were the global NCEP/NCAR and (ERA-5) re-analyses. The third and fourth meteorological data sets were obtained using the Weather tracking and Forecasting model (WRF) as domains on high-resolution grids: 9 km and 2 km. The analysis of meteorological data sets was carried out to analyze the nature of the air masses movement in the studied region. Using the HYSPLIT model simulation was carried out to determine area of the sources of negative impact. The emissions transfer from the supposed sources of negative impact on the Crimea region is analyzed. The sensitivity of simulation the inverse and direct trajectories of air masses transfer was studied based a set of simulation results. On the basis of HYSPLIT model the simulation was carried out separately using meteorological data sets having different spatial and temporal resolutions. In general, the simulation results allowed us to conclude that the HYSPLIT model in combination with high-resolution meteorological input data of the WRF model are good to identify the sources of possible negative effects on atmospheric air located at long distances from the study site.

Serological human birth cohort studies have identified maternal infection during pregnancy as a risk factor for development of disorders such as Autism Spectrum Disorder and schizophrenia in offspring. Similarly, in experiments using animal models, maternal immune activation (MIA) has been shown to alter neuroanatomical and behavioral development in offspring. This study employs magnetic resonance imaging (MRI) and magnetic resonance spectroscopy (MRS) in conjunction with behavioral assays to refine our understanding of the impact of MIA on neurobiological development in exposed animals. On gestational day nine, pregnant dams were injected with either polyinosinic:polycytidylic acid (POL) to induce MIA or saline (SAL) as a control. Whole-brain MRI, localized proton MRS, and behavioral tests (open field, three chambered social approach, and prepulse inhibition) were acquired at two timepoints, during adolescence (postnatal day [PND] 35) and adulthood (PND 60). Whole-brain voxel-wise volumetric analyses revealed that MIA offspring exhibited altered volume in the hippocampus and caudate putamen (CPu) between adolescence and early adulthood. MRS data were assessed at each timepoint separately; MIA offspring during early adulthood but not adolescence exhibited trending reductions in γ-aminobutyrate (GABA) (p = 0.06) and myo-inositol (Ins) (p = 0.08) compared to saline controls. However, these metabolite differences did not reach levels of significance, even before multiple comparison corrections. Open field testing revealed that during adolescence, MIA offspring displayed a more anxious phenotype than controls wherein they spent less time in the anxiogenic center zone of the open field arena (p < 0.007), but this difference normalized by adulthood. There were no significant differences in sociability preference, novelty preference, or prepulse inhibition comparing the groups. Results suggest that early gestational exposure to MIA results in subtle neuroanatomical changes in the trajectories of development, trending behavioral changes in adolescent offspring, and slight neurochemical changes in young adult offspring. Maternal infection alone may not be enough; additional genetic or environmental risk factors may be required to elicit the more typical symptoms of neuropsychiatric disorders.

Road segment can be estimated from GPS trajectories by averaging. However, most existing averaging strategies suffer from high complexity or poor accuracy. For example, finding the optimal mean for a set of sequences is known to be NP-hard, while using Medoid compromises the quality. In this paper, we introduce three extremely fast and practical methods for solving the problem. The methods first analyze three descriptors and then use either a simple linear model or a more complex curvy model depending on an angle criterion. The results outperform Medoid and provide equal or better accuracy than the best existing methods while being very fast, and therefore suitable for real-time processing.

There has been a recent revival of interest in the notion of a `trajectory' of a quantum particle. In this paper we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm `trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis {\em et al.}}

In a previous paper\cite{simplified}, a variation of the Collision-time Statistics method was applied to identify the relevant perturbers for line broadening under the action of a constant magnetic field. As discussed, that version was simplified and inadequate for low magnetic field and/or large perturber mass (ions). The purpose of the present work is to augment the previous work, so that such cases, corresponding to the Larmor radius exceeding the maximum impact parameter, can be handed efficiently. The results may also be used to construct analytic, i.e. impact/unified models under the usual assumptions in these models.

This paper presents research on the application of trajectory design, optimization, and control to an orbital transfer from Mars-Phobos Distant Retrograde Orbits to the surface of Phobos. Given a Distant Retrograde Orbits and a landing location on the surface of Phobos, landing trajectories for which total Δv for a direct 2-burn maneuver is minimized are computed. This is accomplished through the use of Particle Swarm Optimization in which the required Δv and time of flight are optimization parameters. The non-uniform gravitational environment of Phobos is considered in the computation. Results show how direct transfers can be achieved with Δv in the order of ∼30 m/s.

We discuss slip bonds, catch bonds and the tug-of-war mechanism using mathematical arguments. We propose simple 2-dimensional surface models to demonstrate how the potential energy surface of a molecule under an external force behaves. Examples are selectines. Catch bonds in particular are explained in more detail. We demonstrate that Newton trajectories are a powerful tool to describe these phenomena. Newton trajectories form the theoretical background of Mechanochemistry.

As paradigm of complex behavior, multifractals describe the underlying geometry of self-similar objects or processes. Building on the connection between entropy and multifractals, we show here that the generalized dimension of geodesic trajectories in General Relativity coincides with the four-dimensionality of classical spacetime.

Complex problem-solving (CPS) tasks have become an increasingly popular tool for understand-ing and assessing cognitive ability. These tasks have been repeatedly shown to be predictors of academic and workplace success above and beyond traditional measures of general intelligence and fluid intelligence. To date, there has been little exploration of the underlying mechanisms that drive this additional predictive utility. In this study, we examined the role of a variety of non-cognitive personality and investment traits that could drive performance on CPS tasks. Adult participants (n = 152) were recruited via M-Turk and completed a battery of personality and in-vestment trait measures, a measure of general mental ability, and a 61-trial microworlds-style CPS task. Generalised linear mixed-effects models revealed a wide variety of personality and in-vestment traits influenced task performance above and beyond general mental ability. Specifical-ly, two clusters of traits emerged as important determinants of performance: one cluster that in-fluenced the capacity to deal with the introduction of system randomness (Conscientiousness and Extraversion) and one cluster that influenced the capacity to deal with the introduction of system delays (NFC, Learning Goal Orientation, and Intellect). These findings suggest that CPS tasks do capture more than just general mental ability and may be good predictors of academic and workplace success because they tap into both cognitive ability and the motivation and willingness to engage in cognitive exploration and mental effort.

The “NWCSAF High Resolution Winds (NWC/GEO-HRW)” software is developed by the EUMETSAT’s “Satellite Application Facility on support to Nowcasting and very short range forecasting (NWCSAF)”, inside its stand-alone software package for calculation of meteorological products with geostationary satellite data (NWC/GEO). The whole NWC/GEO software package can be obtained after registration at the NWCSAF Helpdesk, www.nwcsaf.org. It is easy to get, install and use. The code is easy to read and fully documented. And in the NWCSAF Helpdesk, users find support and help for its use. “NWCSAF High Resolution Winds” provides a detailed calculation of Atmospheric Motion Vectors (AMVs) and Trajectories, locally and in near real time, using as input NWP model data and geostationary satellite image data. The latest version of the software, v2018, is able to process MSG, Himawari-8/9, GOES-N and GOES-R satellite series images, so that AMVs and Trajectories can be calculated all throughout the planet Earth with the same algorithm and quality. In the “2014 and 2018 AMV Intercomparison Studies”, “NWCSAF High Resolution Winds” has shown to be one of the two best AMV algorithms for both MSG and Himawari-8/9 satellites. And the “Coordination Group for Meteorological Satellites (CGMS)” has recognized in its “2012 Meeting Report”: 1. “NWCSAF High Resolution Winds” fulfills the requirements to be a portable stand-alone AMV calculation software due to its easy installation and usability. 2. It has been successfully adapted by some CGMS members and serves as an important tool for development. It is modular, well documented, and well suited as stand-alone AMV software. 3. Although alternatives exist as portable stand-alone AMV calculation software, they are not as advanced in terms of documentation and do not have an existing Helpdesk. Considering this, a full description and validation of the “NWCSAF/High Resolution Winds” is shown here for the first time in a peer-reviewed paper. The procedure to obtain the software for operational meteorology and research is also explained.

The performance of laser beams in tracking Lissajous scan trajectories is severely limited by beam jitter. To enhance the performance of fast steering mirror (FSM) control in tracking Lissajous scan trajectories, this paper proposed a fractional order active disturbance rejection controller (FOADRC) and verified its effectiveness in improving system scanning tracking accuracy. A dynamic mathematical model of fast steering mirror was studied and the design of parameters for the control mode of the closed-loop system was determined. A reduced order linear active disturbance rejection controller suitable for FSM systems was designed and the corresponding fractional order proportional differentiation (FOPD) controller were determined according to the mathematical model. The use of the designed controller enabled high-performance tracking of high-frequency Lissajous scanning curves (X-axis 500 Hz, Y-axis 350 Hz) and met the need for high-frequency repetitive scanning. The controller has the characteristics of simple implementation and low computational complexity and is suitable for closed-loop control applications in engineering.

Recent advance in electric propulsion systems have demonstrated that these engines can be used for for long-duration interplanetary voyages. Constant specific impulse engine described as a thrust-limited engine is an example of this type of engine, processing the ability to operate at a constant level of impulse. The determination of minimum-fuel, planar heliocentric Earth-to-Mars low-thrust trajectories of spacecraft using a constant specific impulse is discussed considering the first-order necessary conditions derived from Lawden’s primer vector theory. The minimum-fuel low-thrust Earth-to-Mars optimization problem is then solved in two-dimensional, heliocentric frame using both indirect and direct methods. In the indirect method, two-point-boundary-value problems are derived to solve boundary value problems for ordinary differential equations. In the direct method, a general-purpose optimal control software called GPOPS-II is adopted to solve these optimal control problems. Numerical examples using two different optimization methods are presented to demonstrate the characteristics of minimum-fuel planar low-thrust trajectories with on-off-on thrust sequences at three chosen flight times and available maximum powers. The results are useful for broad trajectory search in the preliminary phase of mission designs.

The precise mathematical description of gaze patterns is still an unsettled matter of debate. This can, consequently, affect the analysis of this type of data in its practical applications. In this scope, we show evidence that a Lévy-flight description of eye-gaze trajectories is not only appropriate but its scale invariant properties are remarkably useful to assist in diagnosing children with Attention Deficit and Hyperactivity Disorder (ADHD), alongside with the usual cognitive tests. Using this picture, we find that the distribution of the characteristic exponent of Lévy-flights statistically is different in children with ADHD. Furthermore, we observe that these children deviate from a strategy that is considered optimal for searching processes, in contrast to non-ADHD children. We focus on the case where both eye-tracking data and data from a cognitive test are present and show that the study of gaze patterns in children with ADHD can help in identifying this condition. Since eye-tracking data can be gathered during cognitive tests without needing extra time-consuming specific tasks, we argue that it is in prime position to provide assistance in the arduous task of ADHD diagnosing.

In this paper, a new method is proposed to detect traffic regulations at intersections using GPS traces. The knowledge of traffic rules of regulated locations can help various location-based applications in the context of Smart Cities, such as the accurate estimation of travel time and fuel consumption from a starting point to a destination. Traffic regulations as map features, however, are surprisingly still largely absent from maps, although they do affect traffic flow which in turn affects vehicle idling time at intersections, fuel consumption, CO_2 emissions and arrival time. In addition, mapping them using surveying equipment is costly and any update process has severe time constraints. This fact is precisely the motivation for this study. Therefore, its objective is to propose an automatic, fast, scalable and inexpensive way to identify the type of intersection control (e.g. traffic lights, stop signs). A new method based on summarizing the collective behavior of vehicles crossing intersections is proposed. A modification of a well-known clustering algorithm for detecting stopping and decelerating events is presented. These detected events are then used to categorize vehicle crossing of intersections into four possible traffic categories (p1: free flow, p2: deceleration without stopping events, p3: only one stopping event, p4: more than one stopping event). The percentages of crossings of each class per junction arm, together with other speed/stop/deceleration features, extracted from trajectories, are then used as features to classify the junction arms according to their traffic control type dynamic model). The classification results of the dynamic model are compared with those of the static model, where the classification features are extracted from OpenStreetMap. Finally, a hybrid model is also tested, where a combination of dynamic and static features is used, which outperforms the other two models. For each of the three models, two variants of the feature vector are tested: one where only features associated with a single junction arm are used (one-arm model) and another where features also from neighbouring junction arms of the same junction are used to classify an arm (all-arm model). The methodology was tested on three datasets and the results show that all-arm models perform better than single-arm models with an accuracy of 94\% to 97\%.

In case of the maximally rotating Black Holes (BH) through Kerr-Neumann frames, or as described in Boyer-Lindquist coordinates metrics, the rotation axis of the BHs inputs a frame dragging effect i.e., relativistically a Lens-Thirring Precession that accelerates the photon trajectories oscillates with a shaped induced rotations through the ring singularity, between alternate universes, as a means of an induced geodesics that takes a sharp turning points back and forth provided, in the prograde photon sphere, due to magnetorotational instability, the path tracing of a photons circulates as a smooth fiber bundles over the event horizon curves, that when gets interpolate between mixed trajectories behaves as a geodesics and thus forms a smooth Jacobi-fields through Jacobi-lines by mutual intersection of geodesics over the photon sphere which when somehow gets leaked inside the event horizon, then gets sucked in with not sufficient escape velocity for retardation and trapped in the compact singularity, oscillating back and forth through alternate universes.

Abstract: In recent years, low frequency (LF) electroencephalogram (EEG) signals have been decoded to obtain kinematic trajectories with the aim of achieving closed loop natural control. However, decoded trajectories suffer variability and low correlation against the measured movement. This paper suggests that if EEG features of motor intents are directly associated with fixed points along a given range of motion, then the natural kinematic parameters can be quantified. A novel classification method is proposed to evaluate this hypothesis along with equations for obtaining the movement parameters (velocity, acceleration). Furthermore, this implies that the force of an end-effector and work done in moving the end-effector may be computed, leading to natural control of EEG-based robotic arms and neuroprostheses. Finally, expectations are that combining the proposed method with existing classification methods will increase the accuracy and scalability of brain-computer interface (BCI) applications.

Artificial intelligence and Quantum science supports path-breaking technologies for main technological change. The investigation goal of this study is to examine their scientific and technological development to explain drivers of these disruptive technologies. Analyses of data, based on publications and patents in quantum science and artificial intelligence, with a technometric model show that that quantum science and technology has a growth rate of 1.07, artificial Intelligence ha a rate of growth of 1.37, whereas the interaction of quantum science and artificial intelligence generates a convergent technological trajectory that has an accelerated rate of growth of 1.58, higher than trends of these technologies taken individually. These findings here clarify one of the fundamental determinants of the evolution of path-breaking technologies that can support decision making of policymakers, technology analysts and R&amp;D managers to direct R&amp;D investments towards fruitful inter-relationships between radical technologies that foster scientific and technological change in knowledge economies.

This cross-country four-year longitudinal study investigated the development of processing speed throughout primary school education. The analyses were conducted on data collected from 441 schoolchildren in Grades 1, 2, 3 and 4 (aged 6.42 to 11.85 years) in the Kyrgyzstan and Russia. Mixed effects growth modeling was applied to estimate average and individual growth trajectories for processing speed at two cross-country samples. Latent class growth modeling was conducted to identify different types of growth trajectories for processing speed and to compare the distribution of the types within the analyzed samples. According to the results, processing speed significantly increases across primary school years. The trajectory is characterized by nonlinear changes with more intensive growth between Grades 1 and 2, which later slows down until Grade 4. There were no significant cross-country differences in the initial score of processing speed or developmental changes in processing speed across primary school years. The development of processing speed is described by a model with three quadratic growth types but minimally differs. It is concluded that in both samples, the development of processing speed may be characterized by homogeneity, with the most intensive growth from Grade 1 to Grade 2 and subsequent linear improvement until Grade 4.

In this work, we propose a transition wave equation from quantum to classical regime in the framework of the von Neumann formalism for ensembles and then obtain an equivalent scaled equation. This leads us to develop a scaled statistical theory following the well-known Wigner-Moyal approach of quantum mechanics. This scaled nonequilibrium statistical mechanics has in it all the ingredients of the classical and quantum theory described in terms of a continuous parameter displaying all the dynamical regimes in-between the two extreme cases. Finally, a simple application of our scaled formalism consisting of reflection from a mirror by computing various quantities including probability density plots, scaled trajectories and arrival times are analyzed.

How do sensor research and technologies grow over time? This paper applies the network analysis with a new computational approach to map the structure and evolution of sensor research and technologies over a 30-year time frame (1990-2020).The goal of this study is to analyze the evolution of sensor research for forecasting emerging scientific and technological trajectories. Results show that the scientific interaction within ecosystem (represented with networks) of sensor generates a co-evolution of scientific fields supporting the accelerated growth of different technological tra-jectories, such as: wireless sensors, fiber optic and optical sensors, gas sensors and biosensors. These results suggest main theoretical implications that explain the evolution of sensor research with critical aspects of innovation management to support R&D investments towards new technological trajectories having a high potential of growth.

The fundamental question in the field of sensor research is new directions of scientific fields, which play a vital role in the progress of science and technology. This study confronts this question here by developing a bibliometric analysis, which endeavors to explain the evolution of sensor research and new technologies that are critical to science and society. The database of Scopus concerning scientific documents and patents is used for statistical and computational analyses in these topics. Results suggest that emerging technological trajectories in sensors are wireless sensor networks, wearable sensors and biosensors. Main characteristics of these growing research fields and technologies in sensors are described for fruitful implications of research and innovation policy directed to science advances and technological change in society.

Using the principle of equivalence, it has recently been shown that the electrostatic field lines of a charge, stationary in the gravitational field, bend exactly like the trajectories of photons emitted isotropically from a source at the charge location and that the fraction of electric flux crossing a surface `below' or `above' the charge is exactly similar to the fraction of photon trajectories intersecting these surfaces, with more flux in the downward direction than upward. As one goes much deeper in the gravitational field, all electric field lines increasingly point in the vertically downward direction as is also the case for a stream of photons. Since photon trajectories as well as electric field lines, at any location in the gravitational field, are affected by the local space-time curvature an inference can be drawn that this parallel between the photon trajectories and the electric field lines is a general result. We could then apply these results in the external gravitational field of a black hole, where the trajectories of photons in the gravitational field are already well-known and the behaviour of electric field lines of a stationary charge could be inferred therefrom. Accordingly, we show that the electric field through an external spherical surface surrounding the black hole steadily reduces as the charge location approaches the event horizon (Schwarzschild radius), and like photons from a source inside the Schwarzschild radius cannot escape outside, the electric field lines of a charge within the black hole too remain trapped inside the event horizon. From this one arrives at a conclusion that, contrary to the conventional wisdom, the electric charge contained inside a static black hole cannot be detected or inferred by an external observer. A black hole, said to have no hair with the only external identifying characteristics being mass, electric charge, and angular momentum, is therefore all the more `hairless', as even its charge cannot be ascertained. The derivation of the Reissner-Nordström metric, supposedly describing the gravitational field of a static charged black hole, presumes an external stress-energy tensor of the electrostatic field, as per Gauss law, even for the charges contained within the black hole. However, the absence of electric flux external to a static charged black hole implies that such a charged black hole is not described correctly by the Reissner-Nordström metric and the consequential peculiarities of the space-time geometry, leading in specific cases to the idea of a naked intrinsic singularity and a need for the ``cosmic censorship'' hypothesis, also do not arise here.

It is well known that the photon trajectories follow a curved path in a gravitational field. We explore here the gravitational bending of electric field lines. It seems that the electric field lines of a charge, supported in a gravitational field, follow exactly the trajectories of photons emitted isotropically from a source situated at the charge location. From a detailed examination of the electrostatic field of a charge accelerated uniformly in the instantaneous rest frame, exploiting the strong principle of equivalence, one can determine the bending of the electric field lines of a charge in a gravitational field. The fraction of electric field lines crossing a surface, stationary above or below the charge in the gravitational field, is shown to be exactly similar to the fraction of photon trajectories, emitted from a source placed at the charge location, intersecting that surface. On the other hand, for a freely falling charge in the gravitational field there is no such bending of electric field lines. The field lines continue to extend in radial straight lines from the instantaneous 'present' position of the charge, as do the trajectories of photons streaming away from the instantaneous position of a freely falling source in the gravitational field. The electric field configuration of a freely falling charge in the gravitational field is shown to be exactly the same as that of a charge moving uniformly in an inertial frame with velocity equal to the instantaneous ``present'' velocity of the freely falling charge.