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The conformal scalar propagator as an explicit function of the Schwarzschild black-hole space-time has been established in the preceding three projects. The present project examines the precise relation which that propagator has to the amplitude the Schwarzschild black hole emit a scalar particle in a particular mode of positive energy. It is thereby established that the effect of infinite time-dilation on the event horizon results in both, the thermal radiation in the background of a static exterior space-time geometry and the detraction from thermality if energy-conservation is properly imposed as a constraint on scalar propagation. The derivation of the latter signifies an equivalent approach to the result of Parikh and Wilczek from the semi-classical perspective of the distant observer. From that perspective for that matter, Hawking radiation emerges in both contexts as the exclusive consequence of infinite time-dilation on the event horizon. These results are consistent with Hawking radiation as the exclusive consequence of the causal structure of the Schwarzschild black-hole space-time both, in a static exterior geometry and in such dynamical exterior geometry as energy-conservation signifies. The extension of the thermal case to other spin-fields and black-hole geometries is discussed., Comment: 22 pages, 3 figures. To appear in International Journal of Modern Physics D

Given a bipartite quantum system in an energy eigenstate, the dynamical description for one component can be derived via entanglement using the other component as a clock. This is the essence of the Page and Wootters mechanism. Moreover, if the clock is subject to a gravitational-like interaction, relative time evolution is then described by a Time-Dilated Schr\"odinger equation, in which the so-called Redshift Operator describes a purely quantum effect, analogue to gravitational time-dilation. Here we adopt a non-perturbative approach and present a finite-dimensional generalisation of this mechanism, expressing the quantum time-dilation effect as an effective interaction involving previously non-interacting system components. We name this a Time-Dilation induced Interaction Transfer (TiDIT) mechanism and discuss an example using two coupled spins as a quantum clock model. Our approach is suitable for implementations in current quantum technology and provides a new tool for exploring gravity at the intersection with quantum physics.

We present a precise measurement of cosmological time dilation using the light curves of 1504 type Ia supernovae from the Dark Energy Survey spanning a redshift range $0.1\lesssim z\lesssim 1.2$. We find that the width of supernova light curves is proportional to $(1+z)$, as expected for time dilation due to the expansion of the Universe. Assuming type Ia supernovae light curves are emitted with a consistent duration $\Delta t_{\rm em}$, and parameterising the observed duration as $\Delta t_{\rm obs}=\Delta t_{\rm em}(1+z)^b$, we fit for the form of time dilation using two methods. Firstly, we find that a power of $b \approx 1$ minimises the flux scatter in stacked subsamples of light curves across different redshifts. Secondly, we fit each target supernova to a stacked light curve (stacking all supernovae with observed bandpasses matching that of the target light curve) and find $b=1.003\pm0.005$ (stat) $\pm\,0.010$ (sys). Thanks to the large number of supernovae and large redshift-range of the sample, this analysis gives the most precise measurement of cosmological time dilation to date, ruling out any non-time-dilating cosmological models at very high significance., Comment: 14 pages, 13 figures. Updated in response to reviewer feedback. Accepted for publication in the Monthly Notices of the Royal Astronomical Society (MNRAS)

Light-pulse atom interferometers based on single-photon transitions are a promising tool for gravitational-wave detection in the mid-frequency band and the search for ultralight dark-matter fields. Here we present a novel measurement scheme that enables their use as freely falling clocks directly measuring relativistic time-dilation effects. The proposal is particularly timely because it can be implemented with no additional requirements in Fermilab's MAGIS-100 experiment or even in the 10-m prototypes that are expected to start operating very soon. This will allow the unprecedented measurement of gravitational time dilation in a local experiment with freely falling atoms, which is out of reach even for the best atomic-fountain clocks based on microwave transitions. The results are supported by a comprehensive treatment of relativistic effects in this kind of interferometers as well as a detailed analysis of the main systematic effects. Furthermore, the theoretical methods developed here constitute a valuable tool for modelling light-pulse atom interferometers based on single-photon transitions in general., Comment: 8 + 18 pages including appendices, 6 figures

The fluctuation relations, which characterize irreversible processes in Nature, are among the most important results in non-equilibrium physics. In short, these relations say that it is exponentially unlikely for us to observe a time-reversed process and, thus, establish the thermodynamic arrow of time pointing from low to high entropy. On the other hand, fundamental physical theories are invariant under time-reversal symmetry. Although in Newtonian and quantum physics the emergence of irreversible processes, as well as fluctuation relations, is relatively well understood, many problems arise when relativity enters the game. In this work, by considering a specific class of spacetimes, we explore the question of how the time-dilation effect enters into the fluctuation relations. We conclude that a positive entropy production emerges as a consequence of both the special relativistic and the gravitational (enclosed in the equivalence principle) time-dilation effects.

We derive the predicted time dilation of delocalized atomic clocks in an optical lattice setup in the presence of a gravitational field to leading order in quantum relativistic corrections. We investigate exotic quantum states of motion whose relativistic time dilation is outside of the realm of classical general relativity, finding a regime where $^{24}\mathrm{Mg}$ optical lattice clocks currently in development would comfortably be able to detect this quantum effect (if the technical challenge of generating such states can be met and the expected accuracy of such clocks can be attained). We provide a detailed experimental protocol and analyse the effects of noise on our predictions. We also show that the magnitude of our predicted quantum relativistic time dilation effect remains just out of detectable reach for the current generation of $^{87}\mathrm{Sr}$ optical lattice clocks. Our calculations agree with the predicted time dilation of classical general relativity when restricting to Gaussian states.

A fundamental prediction of relativistic cosmologies is that, due to the expansion of space, observations of the distant cosmos should be time dilated and appear to run slower than events in the local universe. Whilst observations of cosmological supernovae unambiguously display the expected redshift-dependent time dilation, this has not been the case for other distant sources. Here we present the identification of cosmic time dilation in a sample of 190 quasars monitored for over two decades in multiple wavebands by assessing various hypotheses through Bayesian analysis. This detection counters previous claims that observed quasar variability lacked the expected redshift-dependent time dilation. Hence, as well as demonstrating the claim that the lack of the redshift dependence of quasar variability represents a significant challenge to the standard cosmological model, this analysis further indicates that the properties of quasars are consistent with them being truly cosmologically distant sources., Comment: 18 pages, 3 figures. Author version of a manuscript under review at Nature Astronomy

One of the concepts of Relativity theory that challenges conventional intuition the most is time dilation and length contraction. Usual approaches for describing relativistic effects in quantum systems merely postulate the consequences of these effects as physical constraints. Here, we propose to rebuild Special Relativity from quantum mechanical considerations. This is done by dropping one of its fundamental postulates: the universality of the speed of light. Lorentz transformations are obtained by a quantum mechanism. We use the fact that quantum states depend on the Galilean reference frame where they are defined. In other words, quantum states outside an observer's Galilean reference frame are distorted. Then, we show in a theorem the existence of time-dependent observables that are sensible to this distortion in such a way that their expectation value is a Lorentz-covariant function of time. We then postulate this mechanism as the source of the phenomena of Special Relativity. As a corollary of the main theorem, we show the existence of a Lorentz covariant momentum and mass operators which yield the relativistic momentum and mass. In this theory, the fundamental limit of the speed of light imposes a transparency condition for faster-than-light particles: they are allowed but they are not observable. The transparency effect could explain dark matter in a more general theory following this quantum formalism., Comment: 8 pages, 5 figures, submitted to Int. J. Mod. Phys. D

We present a novel test of the cosmological principle: the idea that, on sufficiently large scales, the universe should appear homogeneous and isotropic to observers comoving with the Hubble flow. This is a fundamental assumption in modern cosmology, underpinning the use of the Friedmann-Lema\^itre-Robertson-Walker metric as part of the concordance $\Lambda$CDM paradigm. However, the observed dipole imprinted on the Cosmic Microwave Background (CMB) is interpreted as our departure from the Hubble flow, and such a proper motion will induce a directionally-dependent time dilation over the sky. We illustrate the feasibility of detection of this 'time dilation dipole' and sketch the practical steps involved in its extraction from a catalogue of sources with intrinsic time-scales. In essence, whilst the scale of this dilation is small, being of order of 0.1%, it will in principle be detectable in large scale surveys of variable cosmological sources, such as quasars and supernovae. The degree of alignment of the time dilation dipole with the kinematic dipole derived from the CMB will provide a new assessment of the cosmological principle, and address the tension in dipole measures from other observations., Comment: 9 pages, 4 figures, accepted for publication in MNRAS

We derive the time dilation formula for charged quantum clocks in electromagnetic fields. As a concrete example of non-inertial motion, we consider a cyclotron motion in a uniform magnetic field. Applying the time dilation formula to coherent state of the charged quantum clock, we evaluate the time dilation quantum-mechanically., Comment: 11 pages, 2 figures, to appear in PRD

We consider two non-relativistic quantum clocks interacting with a Newtonian gravitational field produced by a spherical mass. In the framework of Page and Wootters approach, we derive a time dilation for the time states of the clocks. The delay is in agreement up to first order with the gravitational time dilation obtained from the Schwarzschild metric. This result can be extended by considering the relativistic gravitational potential: in this case we obtain the agreement with the exact Schwarzschild solution., Comment: 8 pages, 2 figures

We study gravitational back-reaction within relational time formulations of quantum mechanics by considering two versions of time: a time coordinate, modelled as a global quantum degree of freedom, and the proper time of a given physical system, modelled via an internal degree of freedom serving as a local quantum "clock". We show that interactions between coordinate time and mass-energy in a global Wheeler-DeWitt-like constraint lead to gravitational time dilation. In the presence of a massive object this agrees with time dilation in a Schwarzchild metric at leading order in $G$. Furthermore, if two particles couple independently to the time coordinate we show that Newtonian gravitational interaction between those particles emerges in the low energy limit. We also observe features of renormalization of high energy divergences., Comment: Essay written for the Gravity Research Foundation's 2023 Awards for Essays on Gravitation

We present a unified treatment of the gravitational and cosmological redshift, the Doppler effect due to the moving observer or light source, and the time dilation in the gravitational field in the framework of general relativity. The primary purpose of this paper is to extend the description of Narlikar (1994) on the unified approach towards the redshifts and the Doppler effect in a more generalized form, with the help of the four facts extracted from the comprehensive review article by Ellis (1971). We apply it to the cases of moving observer or light source in the gravitational field and obtain the Doppler effect term, in addition to the standard gravitational or cosmological redshift. The secondary purpose is to explicitly show that the time dilation of a moving clock in the gravitational field can also be understood within the same framework of the unified treatment. We examine the time dilation of the moving clock on geodesic in the gravitational field. We also derive the time dilation of the moving clock on elliptical orbit, based on the same unified treatment. The tertiary purpose is to show that we can understand special-relativistic effects without using the Lorentz transformation. We derive the special-relativistic formulae such as the Doppler effect and aberration of light, the kinetic time dilation, and the Lorentz contraction in the general-relativistic framework.

Time dilation is a difference in measured time between two clocks that either move with different velocities or experience different gravitational potentials. Both of these effects stem from the theory of relativity and are usually associated with classically defined trajectories, characterized by position, momentum, and acceleration. However, when spatial degrees of freedom are treated in a quantum way and a clock is allowed to be in a coherent superposition of either two momenta or two heights, additional quantum corrections to classical time dilation appear, called kinematic and gravitational quantum time dilations, respectively. We show that similarly to its classical counterpart, kinematic quantum time dilation is universal for any clock mechanism, while gravitational quantum time dilation is not. We also show that although both of these effects reduce to incoherent averaging of different classical time dilation contributions, there exists an additional quantum time dilation effect that has no classical analog and can be extracted from higher-order corrections to the system's Hamiltonian.

In this work, we obtain a relation for average quantum time dilation between two clocks A and B in the near-horizon region of a black hole supported by the Rindler metric and conformal tortoise coordinate. It is indicated that this relation is identified with time dilation in classical and flat background limits., Comment: 20 pages, 0 figure. The paper is dedicated to the fond memory of one of the authors, Farhad Darabi who passed away on 13 Sept 2022

A proper time observable for a quantum clock is introduced and it is found that the proper time read by one clock conditioned on another clock reading a different proper time obeys classical time dilation in accordance with special relativistic kinematical time dilation. Here, we extend this proposal to a weak gravitational field in order to investigate whether the weak equivalence principle holds for quantum matter. We find that for general quantum states the quantum time dilation in a weak gravitational field obeys a similar gravitational time dilation found in classical relativity. However, unlike the special relativistic case, the time dilation involves the external time (a background coordinate time at the observer on the Earth) as well as the proper times of two clocks. We also investigate a quantum time dilation effect induced by a clock in a superposition of wave packets localized in momentum space or in position space and propose the setup to observe the gravitational effect in the quantum interference effect in the time dilation., Comment: 12 pages, 2 figures, version published in PRD

According to relativity, the reading of an ideal clock is interpreted as the elapsed proper time along its classical trajectory through spacetime. In contrast, quantum theory allows the association of many simultaneous trajectories with a single quantum clock, each weighted appropriately. Here, we investigate how the superposition principle affects the gravitational time dilation observed by a simple clock - a decaying two-level atom. Placing such an atom in a superposition of positions enables us to analyze a quantum contribution to a classical time dilation manifest in spontaneous emission. In particular, we show that the emission rate of an atom prepared in a coherent superposition of separated wave packets in a gravitational field is different from the emission rate of an atom in a classical mixture of these packets, which gives rise to a quantum gravitational time dilation effect. We demonstrate that this nonclassical effect also manifests in a fractional frequency shift of the internal energy of the atom that is within the resolution of current atomic clocks. In addition, we show the effect of spatial coherence on the atom's emission spectrum., Comment: 10 + 7 pages

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 towards 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., Comment: updated references

Atomic clock interferometers are a valuable tool to test the interface between quantum theory and gravity, in particular via the measurement of gravitational time dilation in the quantum regime. Here, we investigate whether gravitational time dilation may be also used as a resource in quantum information theory. In particular, we show that for a freely falling interferometer and for a Mach-Zehnder interferometer, the gravitational time dilation may enhance the precision in estimating the gravitational acceleration for long interferometric times. To this aim, the interferometric measurements should be performed on both the path and the clock degrees of freedom., Comment: 6 pages main text, 3 pages appendices, 5 figures

We derive a Hamiltonian for the external and internal dynamics of an electromagnetically bound, charged two-particle system in external electromagnetic and gravitational fields, including leading-order relativistic corrections. We apply this Hamiltonian to describe the relativistic coupling of the external and internal dynamics of cold ions in Paul traps, including the effects of micromotion, excess micromotion, and trap imperfections. This provides a systematic and fully quantum-mechanical treatment of relativistic frequency shifts in atomic clocks based on single trapped ions. Our approach reproduces well-known formulas for the second-order Doppler shift for thermal states, which were previously derived on the basis of semiclassical arguments. We complement and clarify recent discussions in the literature on the role of time dilation and mass defect in ion clocks.

Contacts' temporal ordering and dynamics are crucial for understanding the transmission of infectious diseases. We introduce an interaction-driven model of an airborne disease over contact networks. We demonstrate our interaction-driven contagion model, instantiated for COVID-19, over history-maintaining random temporal networks and real-world contacts. We use it to evaluate temporal, spatiotemporal, and spatial social distancing policies. We find that a spatial distancing policy is mainly beneficial at the early stages of a disease. We then continue to evaluate temporal social distancing, that is, timeline dilation that maintains the activity potential. We expand our model to consider the exposure to viral load, which we correlate with meetings' duration. Using real-life contact data, we demonstrate the beneficial effect of timeline dilation on overall infection rates. Our results demonstrate that given the same transmission level, there is a decrease in the disease's infection rate and overall prevalence under timeline dilation conditions. We further show that slow-spreading pathogens (i.e., require more prolonged exposure to infect) spread roughly at the same rate as fast-spreading ones in highly active communities. This is surprising since slower pathogens follow paths that include longer meetings, while faster pathogens can potentially follow paths that include shorter meetings, which are more common. Our results demonstrate that the temporal dynamics of a community have a more significant effect on the spread of the disease than the characteristics of the spreading processes., Comment: Link to the open-source temporal random network library RandomDynamicGraph (RDG): https://github.com/ScanLab-ossi/DynamicRandomGraphs. Please cite this paper when using the code

We carry out a search for signatures of cosmological time dilation in the light curves of Gamma Ray Bursts (GRBs), detected by the Neil Gehrels Swift Observatory. For this purpose, we calculate two different durations ($T_{50}$ and $T_{90}$) for a sample of 247 GRBs in the fixed rest frame energy interval of 140-350 keV, similar to Zhang et al. We then carry out a power law-based regression analysis between the durations and redshifts. This search is done using both the unbinned as well as the binned data, where both the weighted mean and the geometric mean was used. For each analysis, we also calculate the intrinsic scatter to determine the tightness of the relation. We find that weighted mean-based binned data for long GRBs and the geometric mean-based binned data is consistent with the cosmological time dilation signature, whereas the analyses using unbinned durations show a very large scatter. We also make our analysis codes and the procedure for obtaining the light curves and estimation of $T_{50}$/$T_{90}$ publicly available., Comment: 27 pages, 18 figures. Accepted for publication in JCAP