Measuring emission coordinates in a pulsar-based relativistic positioning system

A four-dimensional relativistic positioning system for a general spacetime is constructed by using the so-called emission coordinates. The results apply in a small region around the world line of an accelerated observer carrying a Fermi triad, as described by the Fermi metric. In the case of a Schwarzschild spacetime modeling the gravitational field around the Earth and an observer at rest at a fixed spacetime point, these coordinates realize a relativistic positioning system alternative to the current GPS system. The latter is indeed essentially conceived as Newtonian, so that it necessarily needs taking into account at least the most important relativistic effects through post-Newtonian corrections to work properly. Previous results concerning emission coordinates in flat spacetime are thus extended to this more general situation. Furthermore, the mapping between spacetime coordinates and emission coordinates is completely determined by means of the world function, which in the case of a Fermi metric can be explicitly obtained.

We study decoherence effects in neutrino flavor oscillations in curved spacetime with particular emphasis on the lensing in a Schwarzschild geometry. Assuming Gaussian wave packets for neutrinos, we argue that the decoherence length derived from the exponential suppression of the flavor transition amplitude depends on the proper time of the geodesic connecting the events of the production and detection in general gravitational setting. In the weak gravity limit, the proper time between two events of given proper distance is smaller than that in the flat spacetime. Therefore, in presence of a Schwarzschild object, the neutrino wave packets have to travel relatively more physical distance in space to lapse the same amount of proper time before they decoher. For nonradial propagation applicable to the lensing phenomena, we show that the decoherence, in general, is sensitive to the absolute values of neutrino masses as well as the classical trajectories taken by neutrinos between the source and detector along with the spatial widths of neutrino wave packets. At distances beyond the decoherence length, the probability of neutrino flavor transition due to lensing attains a value which depends only on the leptonic mixing parameters. Hence, the observability of neutrino lensing significantly depends on these parameters and in-turn the lensing can provide useful information about them.

We discuss the Gupta-Bleuler quantization of the free electromagnetic field outside static black holes in the Boulware vacuum. We use a gauge which reduces to the Feynman gauge in Minkowski spacetime. We also discuss its relation with gauges used previously. Then we apply the low-energy sector of this field theory to investigate some low-energy phenomena. First, we discuss the response rate of a static charge outside the Schwarzschild black hole in four dimensions. Afterwards, motivated by string physics, we compute the absorption cross sections of low-energy plane waves for the Schwarzschild and extreme Reissner-Nordstrom black holes in arbitrary dimensions higher than three.

The basic elements of Coll positioning systems (n clocks broadcasting\nelectromagnetic signals in a n-dimensional space-time) are presented in the\ntwo-dimensional case. This simplified approach allows us to explain and to\nanalyze the properties and interest of these relativistic positioning systems.\nThe positioning system defined in flat metric by two geodesic clocks is\nanalyzed. The interest of the Coll systems in gravimetry is pointed out.\n

The precision of optical atomic clocks is approaching a regime where they resolve gravitational time dilation on smaller scales than their own extensions. Hence, an accurate description of quantum clocks has to take their spatial extension into account. In this article, as a first step toward a fully relativistic description of extended quantum clocks, we investigate a quantized version of Einstein's light clock fixed at a constant distance from a large massive object like the Earth. The model consists of a quantum light field in a one-dimensional cavity in Schwarzschild spacetime, where the distance between the mirrors is fixed by a rigid rod. By comparing a vertical and a horizontal clock, we propose an operational way to define the clock time when the clock resolves gravitational time dilation on scales smaller than its extension. In particular, we show that the time measured by the vertical light clock is equivalent to the proper time defined at its center. We also derive fundamental bounds on the precision of these clocks for measurements of proper time and the Schwarzschild radius.

The numerical integration of particle trajectories in curved spacetimes is fundamental for obtaining realistic models of the particle dynamics around massive compact objects such as black holes and neutron stars. Generalized algorithms capable of handling generic metrics are required for studies of both standard (Schwarzschild and Kerr metrics) and nonstandard (e.g., Schwarzschild metric plus nonclassical perturbations or multiple black hole metrics) spacetimes. The most commonly employed explicit numerical schemes (e.g., Runge–Kutta) are incapable of producing highly accurate results at critical points, e.g., in the regions close to the event horizon where gravity causes extreme curvature of the spacetime, at an acceptable computational cost. Here we describe a generalized algorithm for the numerical integration of time-like (massive particles) and null (photons) geodesics in any given 3 + 1 split spacetime. We introduce a new, exactly energy-conserving implicit integration scheme based on the preservation of the underlying Hamiltonian, and we compare its properties with a standard fourth-order Runge–Kutta explicit scheme and an implicit midpoint scheme. We test the numerical performance of the three schemes against analytic solutions of particle and photon orbits in Schwarzschild and Kerr spacetimes. We also prove the versatility of our framework in handling more exotic metrics, such as Morris–Thorne wormholes and quantum-perturbed Schwarzschild black holes. The generalized approach is also discussed in the perspective of future extensions to more complex particle dynamics, e.g., the addition of the Lorentz force acting on charged particles, which allows for test particle diagnostics in GRMHD simulations.

Approaches to relativistic positioning around Earth and error estimations

Joel Franklin's textbook `Advanced Mechanics and General Relativity'comprises two partially overlapping, partially complementaryintroductory paths into general relativity at advanced undergraduatelevel.Path I starts with the Lagrangian and Hamiltonian formulations ofNewtonian point particle motion, emphasising the action principle andthe connection between symmetries and conservation laws. The conceptsare then adapted to point particle motion in Minkowski space,introducing Lorentz transformations as symmetries of the action.There follows a focused development of tensor calculus, paralleltransport and curvature, using examples from Newtonian mechanics andspecial relativity, culminating in the field equations of generalrelativity. The Schwarzschild solution is analysed, including adetailed discussion of the tidal forces on a radially infallingobserver. Basics of gravitational radiation are examined, highlightingthe similarities to and differences from electromagneticradiation. The final topics in Path I are equatorial geodesics in Kerrand the motion of a relativistic string in Minkowski space.Path II starts by introducing scalar field theory on Minkowski spaceas a limit of point masses connected by springs, emphasising theaction principle, conservation laws and the energy–momentumtensor. The action principle for electromagnetism is introduced, andthe coupling of electromagnetism to a complex scalar field isdeveloped in a detailed and pedagogical fashion. A free symmetricsecond-rank tensor field on Minkowski space is introduced, and theaction principle of general relativity is recovered from coupling thesecond-rank tensor to its own energy-momentum tensor.Path II then merges with Path I and, supplanted with judicious earlyselections from Path I, can proceed to the Schwarzschild solution.The choice of material in each path is logical and focused. A notableexample in Path I is that Lorentz transformations in Minkowki spaceare introduced efficiently and with a minimum of fuss, as symmetriesof a geodesic action principle. Another example is a similarlyefficient and hands-on introduction of Killing vectors. A consequenceof this focus is that some perhaps traditional material is omitted. For example, Lorentz contraction appears briefly in the incompatibility discussion of special relativity and Newtoniangravity but is not introduced in a more systematic manner. The style is informal and very readable, with detailed explanations,frequent summaries of what has been achieved and pointers to what isabout to follow. There are plenty of examples and some 150 well-chosenexercises, and the author's website hosts relevant Maple samplescripts for tensor manipulations and variational problems. The textconveys an enthusiasm for explaining the subject, frequentlyreminiscent of the Feynman lectures.The presentation emphasises explicit calculations and examples,largely avoiding technical definitions of abstract mathematicalconcepts. The author negotiates the challenge between readability andtechnical accuracy with admirable skill, striking a balance that willbe much appreciated by the target audience. For example, the notion ofspherical symmetry in curved spacetime is introduced informally as ageneralisation of a spherically symmetric vector field in Minkowskispace, and spherically symmetric vacuum and electrovacuum solutionsare then carefully discussed so that a formal definition of sphericalsymmetry is not required. A rare instance that may border onoversimplification is the brief discussion of curvature scalars versusspacetime singularities.Towards the end of the book, the text mentions with increasing explicitness that inserting a gauge condition or an ansatzin an action before varying may not always give the correctequations of motion. It would be useful to be more explicit about thispoint already earlier in the book. In particular, the text refers tothe reparametrisation-invariant square root action of a relativisticpoint particle as being `in proper time parametrisation', while theactual calculations of course impose the proper time condition only inthe equation of motion after the action has been varied.Two presentational conventions surprised me. First, the speed of lightis throughout kept explicitly as c: might advanced undergraduatesappreciate being trusted with geometric units, reinstating c bydimensional analysis when desired? Second, in Minkowski space fieldtheory, the overall coefficient in the action is chosen so that thetime derivative term is negative, with the consequence that theHamiltonian is negative (as explicitly noted in an exercise) and thedefinition of the energy-momentum tensor must include a minus sign toachieve the usual choice T00 > 0. This convention eliminates someminus signs in the computations with the spin two field: does thiscomputational saving outweigh the adjustment awaiting those whocontinue with the topic at graduate level?Overall, Franklin's book is an excellent addition to the literature,and its readability and explicitness will be appreciated by the targetaudience. Should I be teaching an introductory undergraduate class ingeneral relativity in the near future, I would seriously consider this book for the main class text.

In this work we study the effects of non-minimal coupling between electromagnetism and gravity, which are motivated by quantum effects such as vacuum polarization. We investigate the modification of both asymptotically dipole and uniform magnetic fields around a Schwarzschild black hole that come as the result of non-minimal coupling. While in both cases the magnetic field gets enhanced or suppressed with respect to the case of minimal coupling, depending on the sign of non-minimal coupling parameter, in the case of a background uniform magnetic field the direction of the magnetic field also alters in the vicinity of the black hole horizon. We have discussed the possible astrophysical and cosmological sources for which the vacuum polarization may be at play, while also discussing the observational effects, in particular the possibility of synchrotron radiation from the vicinity of a black hole. We conclude that such observations could be used to constrain the value of the non-minimal coupling parameter.

The next-to-leading term of the renormalized stress-energy tensor of the quantized massive field with an arbitrary curvature coupling in the spacetime of the Schwarzschild black hole is constructed. It is achieved by functional differentiation of the DeWitt-Schwinger effective action involving coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely coefficients ${a}_{3}$ and ${a}_{4}$. It is shown, by comparison with the existing numerical results, that inclusion of the second-order term leads to substantial improvement of the approximation of the exact stress-energy tensor even in the closest vicinity of the event horizon. The back reaction of the quantized field upon the Schwarzschild black hole is briefly discussed.

A semiclassical investigation of the electromagnetic radiation emitted by a charged particle in a radially freely falling motion in Schwarzschild spacetime is carried out. We use quantum field theory at tree level to obtain the one-particle-emission amplitudes. We analyze and compare the energy spectrum and total energy released, which are calculated from these amplitudes, for particles with varying initial positions and for particles originating from infinity with varying kinetic energy. We also compare the results with those due to a falling charged ``string'' extended in the radial direction.

The effect of Hawking radiation on tripartite measurement uncertainty in a Schwarzschild black hole background is analyzed in this study. Two scenarios are examined. In the first, quantum memory particles approach a Schwarzschild black hole and are positioned near the event horizon, while the particle being measured remains in the asymptotically flat region. In the second scenario, the measured particle moves toward the black hole, and the quantum memories stay in the asymptotically flat region. This study considers two initial quantum states, namely GHZ and W states. Our findings reveal that in both cases, measurement uncertainty increases steadily with rising Hawking temperature. When comparing the GHZ and W states, the GHZ state initially exhibits lower measurement uncertainty at low Hawking temperatures than the W state, indicating greater resilience to Hawking radiation. Additionally, when the quantum memories remain in the asymptotically flat region while the measured particle falls toward the black hole, the uncertainties for GHZ and W states do not align at high temperatures. The GHZ state consistently demonstrates lower measurement uncertainty, showcasing its superior robustness against Hawking radiation.

Pseudo-Newtonian potentials and the radiation emitted by a source swirling around a stellar object

An important feature of Schwarzschild spacetime is the presence of orbiting null geodesics and caustics. Their presence implies strong gravitational lensing effects for matter and radiation, i.e., for excitations of quantum fields. Here, we raise the question whether the lensing manifests itself also in the vacuum of quantum fields, namely by lensing the distribution of vacuum entanglement. To explore this possibility, we use the method of entanglement harvesting, where initially unentangled localized quantum systems are temporarily coupled to the field at different locations. We find that for the Boulware, Hartle-Hawking and Unruh vacua in $3+1$-dimensional Schwarzschild spacetime, the harvesting of vacuum entanglement is indeed greatly amplified near caustics. In particular, we establish that preexisting vacuum entanglement can also be harvested for lightlike separations.

We propose a noncommutative extension of the Minkowski spacetime by introducing a well-defined proper time from the kappa-deformed Minkowski spacetime related to the standard basis. The extended Minkowski spacetime is commutative, i.e. it is based on the standard Heisenberg commutation relations, but some information of noncommutativity is encoded through the proper time to it. Within this framework, by simply considering the Lorentz invariance we can construct field theory models that comprise noncommutative effects naturally. In particular, we find a kind of temporal fuzziness related to noncommutativity in the noncommutative extension of the Minkowski spacetime. As a primary application, we investigate three types of formulations of chiral bosons, deduce the lagrangian theories of noncommutative chiral bosons and quantize them consistently in accordance with Dirac's method, and further analyze the self-duality of the lagrangian theories in terms of the parent action approach.