Inertial Research Articles

Back Reaction of the Electromagnetic Radiation and the Local Inertial Frame

We discuss on the problem of the electromagnetic radiation from the accelerated charged particle and the back reaction of that. The free-falling charged particle radiates the electromagnetic wave. The charged particle on the surface of the earth does not radiate the electromagnetic wave. The existence or the non-existence of the electromagnetic radiation from the charged particle and the back reaction of that is independent of the observer, which is consistent with the energy conservation. A paradox comes from combining this phenomenon with the equivalence principle in the theory of the general relativity. We consider the Hawking effect in the context of this paradox. We give our resolution on this paradox.

Kinematics of the barn-pole paradox

Several ways are shown to solve this famous paradox of special relativity, including two that involve calculations exclusively in one of the inertial frames.

Assessing the Preservation of Lumbar Lordotic Curvature in Everyday Sitting Conditions Assessed with an Inertial Measurement System.

Background/Objectives: Lumbar lordotic curvature (LLC), closely associated with low back pain (LBP) when decreased, is infrequently assessed in clinical settings due to the spatiotemporal limitations of radiographic methods. To overcome these constraints, this study used an inertial measurement system to compare the magnitude and maintenance of LLC across various sitting conditions, categorized into three aspects: verbal instructions, chair type, and desk task types. Methods: Twenty-nine healthy participants were instructed to sit for 3 min with two wireless sensors placed on the 12th thoracic vertebra and the 2nd sacral vertebra. The lumbar lordotic angle (LLA) was measured using relative angles for the mediolateral axis and comparisons were made within each sitting category. Results: The maintenance of LLA (LLAdev) was significantly smaller when participants were instructed to sit upright (-3.7 ± 3.9°) compared to that of their habitual sitting posture (-1.2 ± 2.4°) (p = 0.001), while the magnitude of LLA (LLAavg) was significantly larger with an upright sitting posture (p = 0.001). LLAdev was significantly larger when using an office chair (-0.4 ± 1.1°) than when using a stool (-3.2 ± 7.1°) (p = 0.033), and LLAavg was also significantly larger with the office chair (p < 0.001). Among the desk tasks, LLAavg was largest during keyboard tasks (p < 0.001), followed by mouse and writing tasks; LLAdev showed a similar trend without statistical significance (keyboard, -1.2 ± 3.0°; mouse, -1.8 ± 2.2°; writing, -2.9 ± 3.1°) (p = 0.067). Conclusions: Our findings suggest that strategies including the use of an office chair and preference for computer work may help preserve LLC, whereas in the case of cueing, repetition may be necessary.

Changes in trunk postural control with different types of mounts with relevance for hippotherapy

BackgroundHippotherapy uses horseback riding movements for therapeutic purposes. In addition to the horse's movement, the choice of equipment and types of positions are also useful in the intervention. Trunk postural control demands can be manipulated through varying mounting materials and patient positioning on the horse. The quantification of trunk dynamic parameters under the interaction of the different types of riding material and the position variations act upon trunk postural control provides insight into how children with neurotypical development respond to a similar hippotherapy session. This study aimed to evaluate the postural control of the trunk of children with neurotypical development when the riding materials and positioning of the feet and hands support were modified during horse riding. MethodsFifteen children with neurotypical development, aged 6–12 years old, of both sexes, participated in a hippotherapy session. Postural control was measured by the mean amplitudes of oscillation of the lumbar (L5) and thoracic (T5) segments in the mediolateral, anteroposterior and rotational planes with a portable inertial measurement system, in situations using the saddle or blanket, hands resting on the straps or thigh and feet free or supported on the stirrup. ResultsDifferences were observed in lumbar and thoracic oscillation in the mediolateral direction. Values were significantly higher with the use of the blanket when the hands were on the handle and feet in the stirrups, compared to the use of the saddle. When using the blanket and feet positioned in the stirrups, statistical differences with higher values occurred with the hands on the handle compared to the thigh positioning. ConclusionThe use of the blanket equipment associated with the support of the hands on the handles and feet on the stirrups produces greater mediolateral oscillation in the subjects. DescriptorsBalance. Stability. Children. Hippotherapy. Kinematics.

New objective Liutex vector based on an optimization procedure

The objectivity of a vortex identification method, which is critical in capturing vortices in rotating machineries, requires that the visualized vortical structures are independent of the movement of the observer. However, the common methodology to objectivize an Eulerian vortex identification method with relative/net vorticity or relative/net velocity gradient suffers from the issues of requirement to select a spatial volume to do vorticity average and inconsistency with the non-objective counterpart in an inertial reference frame. In this regard, a new objective Liutex vector method to visualize vortices is developed in the current study, which is based on the observed intermittency of Liutex represented vortices in turbulence. The intermittency indicates that a large volume fraction of the considered domain is void of Liutex, and the rotating motion of the observer is very likely to decrease this volume fraction. Therefore, an optimization procedure is proposed to approximate the rotating angular velocity of the observer by maximizing the zero Liutex volume percentage. With the known angular velocity to offset the rotational effects of the observer’s reference frame, objective version of Eulerian vortex identification methods can be introduced. As an example, the new objective Liutex method has been applied to turbulent channel flow at Reτ=180 in arbitrarily rotating coordinate systems, and the results show that the new method is both objective, i.e. independent of observer’s motion including translation and rotation and consistent with the original Liutex method in an inertial reference frame.

Schwinger correlation of Dirac fields in accelerated frames

We study the Schwinger correlation of Dirac fields in the noninertial frames under the influences of both constant and pulsed electric fields. We use both the entanglement negativity and quantum mutual information between particle and antiparticle as the indicator of the Schwinger correlation observed by the accelerated observers. We find that the Schwinger correlation in the inertial frames is the largest. With the increase of acceleration of the observers, the Schwinger correlation becomes smaller and smaller, but does not vanish in the limit of infinite acceleration. For the given acceleration, the Schwinger correlation is a nonmonotonic function of the electric field intensity, and there is an optimal value of electric field intensity for which the Schwinger correlation is the largest. In the case of pulsed electric fields, the Schwinger correlation is also the nonmonotonic function of pulsed width, which suggests the existence of optimal pulsed width for observing Schwinger correlation.

Effect of video angle on detection of induced front limb lameness in horses

BackgroundLameness examinations are commonly performed in equine medicine. Advancements in digital technology have increased the use of video recordings for lameness assessment, however, standardization of ideal video angle is not available yielding videos of poor diagnostic quality. The objective of this study was to evaluate the effect of video angle on the subjective assessment of front limb lameness. A randomized, blinded, crossover study was performed. Six horses with and without mechanically induced forelimb solar pain were recorded using 9 video angles including horses trotting directly away and towards the video camera, horses trotting away and towards a video camera placed to the left and right side of midline, and horses trotting in a circle with the video camera placed on the inside and outside of the circle. Videos were randomized and assessed by three expert equine veterinarians using a 0–5 point scoring system. Objective lameness parameters were collected using a body-mounted inertial sensor system (Lameness Locator®, Equinosis LLC). Interobserver agreement for subjective lameness scores and ease of grading scores were determined.ResultsInduction of lameness was successful in all horses. There was excellent agreement between objective lameness parameters and subjective lameness scores (AUC of the ROC = 0.87). For horses in the “lame” trials, interobserver agreement was moderate for video angle 2 when degree of lameness was considered and perfect for video angle 2 and 9 when lameness was considered as a binary outcome. All other angles had no to fair agreement. For horses in the “sound” trials, interobserver agreement was perfect for video angle 5. All other video angles had slight to moderate agreement.ConclusionsWhen video assessment of forelimb lameness is required, a video of the horse trotting directly towards the video camera at a minimum is recommended. Other video angles may provide supportive information regarding lameness characteristics.

Maximal steered coherence in the background of Schwarzschild space-time

In the past two decades, the exploration of quantumness within Schwarzschild spacetime has garnered significant interest, particularly regarding the Hawking radiation’s impact on quantum correlations and quantum coherence. Building on this foundation, we investigate how Hawking radiation influences maximal steered coherence (MSC)-a crucial measure for gauging the ability to generate coherence through steering. We find that as the Hawking temperature increases, the physically accessible MSC degrade while the unaccessible MSC increase. This observation is attributed to a redistribution of the initial quantum correlations, previously acknowledged by inertial observers, across all bipartite modes. In particular, we find that in limit case that the Hawking temperature tends to infinity, the accessible MSC equals to 1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\sqrt{2}$$\\end{document} of its initial value, and the unaccessible MSC between mode A and B¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{B}$$\\end{document} also equals to the same value. Our findings illuminate the intricate dynamics of quantum information in the vicinity of black holes, suggesting that Hawking radiation plays a pivotal role in reshaping the landscape of quantum coherence and entanglement in curved spacetime. This study not only advances our theoretical understanding of black hole thermodynamics but also opens new avenues for investigating the interface between quantum mechanics and general relativity.

A modern interpretation of Newton's theorem of revolving orbits

Newton's Theorem of Revolving Orbits derives the force that is necessary to explain a particular precession that leaves the shape of an orbit unchanged. Newton showed that for an orbiting body that is already subject to any central force, the inclusion of an additional inverse-cube central force will change that body's angular speed without affecting its radial motion; this leads to the orbit's precession. After reviewing the relevant concepts of Kepler orbits, we present a full description and explanation of Newton's theorem using modern physical and mathematical approaches aimed at a general audience. We specifically highlight the use of the transformation between a rotating reference frame and an inertial reference frame, making this a suitable topic for an upper-level undergraduate mechanics course.

High-throughput enrichment of portal venous circulating tumor cells for highly sensitive diagnosis of CA19-9-negative pancreatic cancer patients using inertial microfluidics

The carbohydrate antigen 19-9 (CA19-9) is commonly used as a representative biomarker for pancreatic cancer (PC); however, it lacks sensitivity and specificity for early-stage PC diagnosis. Furthermore, some patients with PC are negative for CA19-9 (<37 U/mL), which introduces additional limitations to their accurate diagnosis and treatment. Hence, improved methods to accurately detect PC stages in CA19-9-negative patients are warranted. In this study, tumor-proximal liquid biopsy and inertial microfluidics were coupled to enable high-throughput enrichment of portal venous circulating tumor cells (CTCs) and support the effective diagnosis of patients with early-stage PC. The proposed inertial microfluidic system was shown to provide size-based enrichment of CTCs using inertial focusing and Dean flow effects in slanted spiral channels. Notably, portal venous blood samples were found to have twice the yield of CTCs (21.4 cells per 5 mL) compared with peripheral blood (10.9 CTCs per 5 mL). A combination of peripheral and portal CTC data along with CA19-9 results showed to greatly improve the average accuracy of CA19-9-negative PC patients from 47.1% with regular CA19-9 tests up to 87.1%. Hence, portal venous CTC-based microfluidic biopsy can be used with high sensitivity and specificity for the diagnosis of early-stage PC, particularly in CA19-9-negative patients.

Relativistic analysis of the Michelson-Gale experimental result.

The result of the Michelson-Gale experiment, which shows fringe shifts by the interference between two light beams traversing a rectangular loop in opposite directions, has been nonrelativistically analyzed based on the Galilean transformation. We relativistically analyze it via the transformation under the constant light speed (TCL) and via the framework of Mansouri and Sexl (MS). The TCL provides a coordinate transformation between the isotropic frame and a rotating frame, in which the two-way speed of light is a constant c irrespective of direction on the surface that has the same radius of rotation. When using TCL, we assume that the Solar System is isotropic so that the one-way speed of light is c in it. On the contrary, considering its movement, the analysis is carried out without the assumption of isotropy based on the MS framework. The analysis results via the TCL and via the MS framework correspond to each other and are in agreement with the result of the experiment. It is shown that the difference between the travel times of the counter-propagating light beams, which results in the fringe shift, takes place due to the two factors, the anisotropy of the one-way speed of light in inertial frames and the different rotation radii at different latitudes on the Earth surface.

Conversion Factors for Physical Property Measurements in different Rest Frames Using a Revised Version of Relativity Theory Based on the Uniform Scaling Method

The history of the development of the Uniform Scaling Method is reviewed. It has its beginning with the Law of Causality and Newton’s First Law of Kinetics (Law of Inertia). On this basis it can be concluded that the rates of clocks do not change spontaneously, i.e. without the application of some external force. Since the ratio Q of the rates of any two such (inertial) clocks must therefore itself be a constant, it follows that whenever they are used to measure a specific time difference between two events, their respective results must always be in the same ratio: Δt’=Δt/Q. This relationship is referred to as Newtonian Simultaneity since it guarantees that if one of the clocks finds that the events occurred simultaneously (Δt’=0), the other must do so as well (Δt=0). A useful way to interpret Q is as a conversion factor between the different units of time employed in the two rest frames. If we assume that the value of the speed of light is the same in both rest frames, it follows that they must agree on the unit of speed, i.e. the conversion factor in this case is unity (Q0 ). It therefore follows that the conversion factor for the unit of distance must be exactly the same as that for time, namely also Q. This claim of the Uniform Scaling Method is thus seen to be in direct contradiction to the length contraction tenet of Einstein’s theory of relativity introduced in 1905. Experiments with accelerated electrons carried out by Bucherer in 1909 indicate that their inertial mass increase in direct proportion to γ (v) = (1-v2 c-2)-0.5 (v is the speed of the electrons and c = 299792458 ms-1 is the speed of light in free space), the same ratio as observed for the periods of atomic lines by Ives and Stilwell in 1938. Consistency therefore requires that the conversion factor for inertial mass is the same (Q) as for time. Since inertial mass, time and distance are the three fundamental quantities on which all other physical properties are defined (mks system), it follows that the conversion factor for any other property must be an integral multiple of Q. An analogous set of definitions can also be justified for the corresponding conversion factors in the same two rest frames caused by gravitational effects. In this case the conversion factors are always integral multiples of the quantity S, which therefore plays as analogous role as Q in kinetic scaling. Since experiments with circumnavigating atomic clocks carried out in 1971 by Hafele and Keating found that the two types of effects are completely independent of one another, it is therefore useful to define a product Z of the corresponding Q and S factors for each physical property, including those that are used to define the results of electromagnetic measurements. These total conversion factors are stored in Tables 1 and 2 of the manuscript. They are shown to guarantee that if a physical law is valid in one of the rest frames, it will also hold in the other. This relationship allows for an Addendum to Galileo’s Relativity Principle: The laws of physics are the same in all inertial rest frames, but the units in which their results are expressed can differ from one rest frame to another.

Optimal Design and Performance Evaluation of HMDV Inertial Suspension Based on Generalized Skyhook Control

The hub motor driven vehicle (HMDV), due to the increase in unsprung mass and the influence of uneven electromagnetic radial forces from road excitations, results in the deterioration of body acceleration, significantly impacting ride comfort. This seriously impacts ride comfort. However, the inerter proves effective in reducing low-frequency vibrations in body acceleration, and skyhook control significantly enhances ride comfort. To fully exploit the benefits of the inerter, comprehensively elevate skyhook control performance, and effectively counteract the worsening of body acceleration caused by the hub motor, this study integrates skyhook control with an inertial suspension system. This integration markedly improves the suspension’s low-frequency isolation performance, thereby substantially enhancing vehicle ride comfort. Firstly, establish a quarter-vehicle dynamics suspension model based on the Switched Reluctance Motor (SRM) driven wheel motor system. Subsequently, confirm the constraints of two different generalized skyhook inertial suspension structures separately and optimize their parameters to determine the most suitable parameters for each structure. Finally, analyze the impact of these two suspension types on body acceleration to derive the optimal structure. Simulation results demonstrate that, compared to traditional suspensions, the optimized generalized skyhook inertial suspensions reduce the Root Mean Square (RMS) values of body acceleration and suspension working space by 18.8% and 22%, respectively. This reduction is particularly significant in the 0[Formula: see text]2[Formula: see text]Hz frequency range, effectively enhancing ride comfort. It also adequately validates the effective utilization of the advantages of inertial suspensions and skyhook control.

Dissipativity Analysis of Memristive Inertial Competitive Neural Networks with Mixed Delays

Without altering the inertial system into the two first-order differential systems, this paper primarily works over the global exponential dissipativity (GED) of memristive inertial competitive neural networks (MICNNs) with mixed delays. For this purpose, a novel differential inequality is primarily established around the discussed system. Then, by applying the founded inequality and constructing some novel Lyapunov functionals, the GED criteria in the algebraic form and the linear matrix inequality (LMI) form are given, respectively. Furthermore, the estimation of the global exponential attractive set (GEAS) is furnished. Finally, a specific illustrative example is analyzed to check the correctness and feasibility of the obtained findings.

Ergonomic Analysis of Dental Work in Different Oral Quadrants: A Motion Capture Preliminary Study among Endodontists.

Dentists, including endodontists, frequently experience musculoskeletal disorders due to unfavourable working postures. Several measures are known to reduce the ergonomic risk; however, there are still gaps in the research, particularly in relation to dental work in the different oral regions (Quadrants 1-4). In this study (of a pilot character), a total of 15 dentists (8 male and 7 female) specialising in endodontics were measured while performing root canal treatments on a phantom head. These measurements took place in a laboratory setting using an inertial motion capture system. A slightly modified Rapid Upper Limb Assessment (RULA) coding system was employed for the analysis of kinematic data. The significance level was set at p = 0.05. The ergonomic risk for the entire body was higher in the fourth quadrant than in the first quadrant for 80% of the endodontists and higher than in the second quadrant for 87%. For 87% of the endodontists, the ergonomic risk for the right side of the body was significantly higher in the fourth quadrant compared to the first and second quadrant. The right arm was stressed more in the lower jaw than in the upper jaw, and the neck also showed a greater ergonomic risk in the fourth quadrant compared to the first quadrant. In summary, both the total RULA score and scores for the right- and lefthand sides of the body ranged between 5 and 6 out of a possible 7 points. Considering this considerable burden, heightened attention, especially to the fourth quadrant with a significantly higher ergonomic risk compared to Quadrants 1 and 2, may be warranted.

Potential Prodromal Digital Postural Sway Markers for Fragile X-Associated Tremor/Ataxia Syndrome (FXTAS) Detected via Dual-Tasking and Sensory Manipulation.

FXTAS is a neurodegenerative disorder occurring in some Fragile X Messenger Ribonucleoprotein 1 (FMR1) gene premutation carriers (PMCs) and is characterized by cerebellar ataxia, tremor, and cognitive deficits that negatively impact balance and gait and increase fall risk. Dual-tasking (DT) cognitive-motor paradigms and challenging balance conditions may have the capacity to reveal markers of FXTAS onset. Our objectives were to determine the impact of dual-tasking and sensory and stance manipulation on balance in FXTAS and potentially detect subtle postural sway deficits in FMR1 PMCs who are asymptomatic for signs of FXTAS on clinical exam. Participants with FXTAS, PMCs without FXTAS, and controls underwent balance testing using an inertial sensor system. Stance, vision, surface stability, and cognitive demand were manipulated in 30 s trials. FXTAS participants had significantly greater total sway area, jerk, and RMS sway than controls under almost all balance conditions but were most impaired in those requiring vestibular control. PMCs without FXTAS had significantly greater RMS sway compared with controls in the feet apart, firm, single task conditions both with eyes open and closed (EC) and the feet together, firm, EC, DT condition. Postural sway deficits in the RMS postural sway variability domain in asymptomatic PMCs might represent prodromal signs of FXTAS. This information may be useful in providing sensitive biomarkers of FXTAS onset and as quantitative balance measures in future interventional trials and longitudinal natural history studies.

Study on the method of transfer alignment based on the distributed inertial network of guided submunition

Aiming at the problem of poor transfer alignment accuracy between missiles with a high length-to-diameter ratio as a carrier missile and the internally carried guided submunition, this study proposes a transfer alignment method based on the distributed inertial network of the guided submunition. Initially, considering the influence of the deflection deformation of the carrier missile body and the dynamic nonlinear deformation of the lever arm on the transfer alignment accuracy, the corresponding error model is established and compensated for the sub-inertial guide velocity to improve the accuracy of the observed quantities. Additionally, in response to the high failure rate of low-cost inertial guidance systems equipped with guided submunition, we used an inertial network data fusion architecture based on the isolation of device failure monitoring, which effectively isolates the faulty data through the fusion of inertial measurements at the sensor level and optimal fusion of filtering to further improve the reliability of transmission alignment. The effectiveness of the method proposed in this paper is verified through a large number of simulation experiments. Compared with the traditional transfer alignment method between a single master inertial guide and a sub inertial guide, the method in the paper has obvious improvement in the transfer alignment accuracy. In addition, this method is not affected by the failure of inertial devices in the distributed inertial network, and can continuously maintain a high transfer alignment accuracy. For this kind of missile-borne distributed inertial network with a sub-node sets layer, it is found after comparison that: the transmission alignment error is basically the same for different slave nodes in the same sub-node sets; between different sub-node sets, the closer to the master inertial guide the transmission alignment error is smaller. This study provides a new solution idea and certain reference value for solving the problem of poor transfer alignment accuracy between missiles with a high length-to-diameter ratio as a carrier missile and the internally carried guided submunition.

Using sunRunner3D to interpret the global structure of the heliosphere from in-situ measurements

Understanding the large-scale three-dimensional structure of the inner heliosphere, while important in its own right, is crucial for space weather applications, such as forecasting the time of arrival and propagation of coronal mass ejections (CMEs). This study uses sunRunner3D (3D), a 3-D magnetohydrodynamic (MHD) model, to simulate solar wind (SW) streams and generate background states. SR3D employs the boundary conditions generated by CORona-HELiosphere (CORHEL) and the PLUTO code to compute the plasma properties of the SW with the MHD approximation up to 1.1 AU in the inner heliosphere. We demonstrate that SR3D reproduces global features of Corotating Interaction Regions (CIRs) observed by Earth-based spacecraft (OMNI) and the Solar TErrestial RElations Observatory (STEREO)-A for a set of Carrington rotations (CRs) that cover a period that lays in the late declining phase of solar cycle 24. Additionally, we demonstrate that the model solutions are valid in the corotating and inertial frames of references. Moreover, a comparison between SR3D simulations and in-situ measurements shows reasonable agreement with the observations, and our results are comparable to those achieved by Predictive Science Inc.'s Magnetohydrodynamic Algorithm outside a Sphere (MAS) code. We have also undertaken a comparative analysis with the Space Weather Adaptive Simulation Framework for Solar Wind (SWASTi-SW), a PLUTO physics-based model, to evaluate the precision of various initial boundary conditions. Finally, we discuss the disparities in the solutions derived from inertial and rotating frames.

Comparability between wearable inertial sensors and an electronic walkway for spatiotemporal and relative phase data in young children aged 6–11 years

BackgroundApproaches to gait analysis are evolving rapidly and now include a wide range of options: from e-patches to video platforms to wearable inertial measurement unit systems. Newer options for gait analysis are generally more inclusive for the assessment of children, more cost effective and easier to administer. However, there is limited data on the comparability of newer systems with more established traditional approaches in young children. Research questionTo determine comparability between the Physilog®5 wearable inertial sensor and GAITRite® electronic walkway for spatiotemporal (stride length, time and velocity, cadence) and relative phase (double support time, stance, swing, loading, foot flat and push off) data in young children. MethodsA total 34 typically developing participants (41% female) aged 6–11 years old median age 8.99 years old (interquartile range 2.83) were assessed walking at self-selected speed over the GAITRite® electronic walkway while concurrently wearing shoe-attached Physilog®5 IMU sensors. Level of agreement was analysed by Lin’s concordance correlation coefficient (CCC), Bland-Altman plots and 95% limit of agreement. Systematic bias was assessed using 95% confidence interval of the mean difference. ResultsExcellent to almost perfect agreement was observed between systems for spatiotemporal metrics: cadence (CCC=0.996), stride length (CCC=0.993), stride time (CCC=0.996), stride velocity (CCC=0.988). The relative phase metrics adjusted for stride velocity showed improved comparability when compared to the unadjusted metrics: swing adjusted (adj) (CCC=0.635); stance adj (CCC: 0.879); loading adj: (CCC=0.626). SignificanceSpatiotemporal metrics are highly compatible across GAITRite® electronic walkway and Physilog®5 IMU systems in young children. Relative phase metrics were somewhat compatible between systems when adjusted for stride velocity.

Minkowskian Approach to the Pseudorange Navigation Equations

Our starting point is the covariant coordinate transformation equation of a relativistic positioning system in Minkowski space–time that maps the receiver’s emission coordinates (proper times broadcast by the emitters) to its coordinates in some inertial reference frame. Bancroft’s analytical (closed-form) solution to the basic pseudorange navigation equations with four emitters is recovered, and the subjacent elements are geometrically interpreted. The case of four static beacons is analysed as a clarifying situation.