Transverse Doppler Effect Research Articles

An imaging algorithm for missile-borne SAR with downward movement based on variable decoupling

The ordinary SAR imaging algorithms are inapplicable to missile-bo-rne SAR due to the high speed dive and high-squint of missile. Aiming at this problem, this paper firstly sets up the model of echo and analyses of two-dimensional spectrum. By using azimuth nonlinear chirp scaling (NLCS) based on variable decoupling, an imaging algorithm for missile-borne SAR is proposed.It can effectively compensate the scene with longitudinal and transverse Doppler shift and improve the focusing quality while it can also simplify the geometric image correction operation. Simulation results are provided to confirm the effectiveness of the proposed algorithm.

Thim’s experiment and exact rotational space-time transformations

Thim measured the transverse Doppler shift using a system consisting of a stationary antenna and pickup, in addition to a number of intermediate antennas mounted on the rim of a rotating disk. No such shift was detected, although the experiment should have had enough sensitivity to measure it, as predicted by the Lorentz transformations. However, using the Lorentz transformations to analyze the results of experiments involving circular motion, while commonly done, is inappropriate because such an analysis involves non-inertial frames, which are outside the range of validity of special relativity. In this paper, we re-analyze Thim's experiment using exact rotational space-time transformations, finding that his null result is consistent with theoretical predictions.

Experimental detection of transverse particle movement with structured light

One procedure widely used to detect the velocity of a moving object is by using the Doppler effect. This is the perceived change in frequency of a wave caused by the relative motion between the emitter and the detector, or between the detector and a reflecting target. The relative movement, in turn, generates a time-varying phase which translates into the detected frequency shift. The classical longitudinal Doppler effect is sensitive only to the velocity of the target along the line-of-sight between the emitter and the detector (longitudinal velocity), since any transverse velocity generates no frequency shift. This makes the transverse velocity undetectable in the classical scheme. Although there exists a relativistic transverse Doppler effect, it gives values that are too small for the typical velocities involved in most laser remote sensing applications. Here we experimentally demonstrate a novel way to detect transverse velocities. The key concept is the use of structured light beams. These beams are unique in the sense that their phases can be engineered such that each point in its transverse plane has an associated phase value. When a particle moves across the beam, the reflected light will carry information about the particle's movement through the variation of the phase of the light that reaches the detector, producing a frequency shift associated with the movement of the particle in the transverse plane.

Measuring gravitational redshifts in galaxy clusters

Wojtak et al. have stacked 7800 clusters from the Sloan Digital Sky Survey (SDSS) in redshift space. They find a small net blueshift for the cluster galaxies relative to the brightest cluster galaxies, which agrees quite well with the gravitational redshift predicted from general relativity. Zhao et al. have pointed out that, in addition to the gravitational redshift, one would expect to see transverse Doppler (TD) redshifts, so 〈δz〉 = −〈Φ〉 + 〈β2〉/2 with β the 3D source velocity in units of c, and that these two effects are generally of the same order. Here, we show that there are other corrections that are also of the same order of magnitude. The fact that we observe galaxies on our past light cone results in a bias such that more of the galaxies observed are moving away from us in the frame of the cluster than are moving towards us. This causes the observed average redshift to be |$\langle \delta z \rangle = -\langle \Phi \rangle + \langle \beta ^2 \rangle / 2 + \langle \beta _x^2 \rangle$|⁠, with βx the line-of-sight velocity. That is if we average over galaxies with equal weight. If the galaxies in each cluster are weighted by their fluence, or equivalently if we do not resolve the moving sources, and make an average of the mean redshift giving equal weight per photon, the observed redshift is 〈δz〉 = −〈Φ〉 − 〈β2〉/2, so the kinematical effect is then opposite to the usual transverse Doppler effect. In the Wojtak, Hansen & Hjorth experiment, the weighting is a step-function because of the flux limit for inclusion in the spectroscopic sample and the result is different again, and depends on the details of the luminosity function and the spectral energy distributions of the galaxies. Including these effects substantially modifies the blueshift profile. We show that in-fall and out-flow have very small effect over the relevant range of impact parameters but out-flow becomes significant and needs to be taken into account for measurements on larger scales.

Testing gravity theories via transverse Doppler and gravitational redshifts in galaxy clusters

There is growing interest in testing alternative gravity theories using the subtle gravitational redshifts in clusters of galaxies. However, current models all neglect a transverse Doppler redshift of similar magnitude, and some models are not self-consistent. An equilibrium model would fix the gravitational and transverse Doppler velocity shifts to be about 6sigma^2/c and 3sigma^2/2c in order to fit the observed velocity dispersion sigma self-consistently. This result comes from the Virial Theorem for a spherical isotropic cluster, and is insensitive to the theory of gravity. A gravitational redshift signal also does not directly distinguish between the Einsteinian and f(R) gravity theories, because each theory requires different dark halo mass function to keep the clusters in equilibrium. When this constraint is imposed, the gravitational redshift has no sensitivity to theory. Indeed our N-body simulations show that the halo mass function differs in f(R), and that the transverse Doppler effect is stronger than analytically predicted due to non-equilibrium.

Gravitational redshifts from large-scale structure

The recent measurement of the gravitational redshifts of galaxies in galaxy clusters by Wojtak et al. has opened a new observational window on dark matter and modified gravity. By stacking clusters this determination effectively used the line of sight distortion of the cross-correlation function of massive galaxies and lower mass galaxies to estimate the gravitational redshift profile of clusters out to 4 Mpc/h. Here we use a halo model of clustering to predict the distortion due to gravitational redshifts of the cross-correlation function on scales from 1 - 100 Mpc/h. We compare our predictions to simulations and use the simulations to make mock catalogues relevant to current and future galaxy redshift surveys. Without formulating an optimal estimator, we find that the full BOSS survey should be able to detect gravitational redshifts from large-scale structure at the ~4 sigma level. Upcoming redshift surveys will greatly increase the number of galaxies useable in such studies and the BigBOSS and Euclid experiments should be capable of measurements with precision at the few percent level. As has been recently pointed out by McDonald, Kaiser and Zhao et al, other interesting effects including relativistic beaming and transverse Doppler shift can add additional asymmetric distortions to the correlation function. While these contributions are subdominant to the gravitational redshift on large scales, they represent additional opportunities to probe gravitational physics and indicate that many qualitatively new measurements should soon be possible using large redshift surveys.

The broad emission-line region: the confluence of the outer accretion disc with the inner edge of the dusty torus

(Abridged) We investigate the observational characteristics of BLR geometries in which the BLR clouds bridge the gap, both in distance and scale height, between the outer accretion disc and the hot dust, forming an effective surface of a "bowl". The gas dynamics are dominated by gravity, and we include the effects of transverse Doppler shift, gravitational redshift and scale-height dependent macro-turbulence. Our simple model reproduces many of the phenomena observed in broad emission-line variability studies, including (i) the absence of response in the core of the optical recombination lines on short timescales, (ii) the enhanced red-wing response on short timescales, (iii) differences between the measured delays for the HILs and LILs, and (iv) identifies turbulence as a means of producing Lorentzian profiles (esp. for LILs) in low inclination systems, and for suppressing significant continuum--emission-line delays between the line wings and line core (esp. in LILs). A key motivation of this work was to reveal the physical underpinnings of the reported measurements of SMBH masses and their uncertainties. We find that SMBH masses derived from measurements of the fwhm of the mean and rms profiles show the closest correspondence between the emission lines in a single object, even though the emission line fwhm is a more biased mass indicator with respect to inclination. The predicted large discrepancies in the SMBH mass estimates between emission lines at low inclination, as derived using the line dispersion, we suggest may be used as a means of identifying near face-on systems. Our general results do not depend on specific choices in the simplifying assumptions, but are in fact generic properties of BLR geometries with axial symmetry that span a substantial range in radially-increasing scale height supported by turbulence, which then merge into the inner dusty TOR.

Testing Einstein's time dilation under acceleration using Mössbauer spectroscopy

The Einstein time dilation formula was tested in several experiments. Many trials have been conducted to measure the transverse second-order Doppler shift by Mössbauer spectroscopy using a rotating absorber, to test the validity of this formula. Such experiments are also able to test if the time dilation depends only on the velocity of the absorber, as assumed by Einstein's clock hypothesis, or whether the present centripetal acceleration contributes to the time dilation. We show here that because the experiment requires γ-ray emission and detection slits of finite size, the absorption line is broadened, by geometric longitudinal first-order Doppler shifts immensely. Moreover, the absorption line is non-Lorentzian. We obtain an explicit expression for the absorption line for any angular velocity of the absorber. The analysis of the experimental results in all previous experiments which did not observe the full absorption line itself were wrong and the conclusions doubtful. The only proper experiment was done by Kündig (1963 Phys. Rev.129 2371), who observed the broadening, but associated it with random vibrations of the absorber. We establish necessary conditions for the successful measurement of a transverse second-order Doppler shift by Mössbauer spectroscopy. We indicate how the results of such an experiment can be used to verify the existence of a Doppler shift due to acceleration and to test the validity of Einstein's clock hypothesis.

The maximal acceleration, extended relativistic dynamics and Doppler type shift for an accelerated source

AbstractBased on the generalized principle of relativity and the ensuing symmetry, it has been shown that if the Clock Hypothesis is not true, there is a universal maximal acceleration am. We present here an extension of relativistic dynamics for which all admissible solutions will have have a speed bounded by the speed of light c and the acceleration bounded by am. An additional Doppler type shift for an accelerated source is predicted. The formulas for such shift are the same as for the usual Doppler shift with v/c replaced by a/am. The W. Kündig experiment of measurement of the transverse Doppler shift in an accelerated system was also exposed to a longtitudal shift due to the acceleration. This experiment, as reanalyzed by Kholmetskii et al., shows that the Clock Hypothesis is not valid. Based on the results of this experiment, we predict that the value of the maximal acceleration am is of the order 1019m/s2. Moreover, our analysis provides a way to measure experimentally the maximal acceleration with existing technology.

Lorentz violation in high-energy ions

Recent experiments with high-speed ions have investigated potential deviations from the time-dilation predicted by special relativity (SR). The main contribution of this article is to show that the SR predictions are matched by the experimental results only when the transverse Doppler effect in the observed emissions from the ions are neglected in the analysis. However, the Doppler effect in the emission cannot be neglected because it is similar to the time dilation effect. Thus, the article highlights the need to consider Doppler emission effects when validating SR time dilation using high-speed ion experiments.

Comment on “The Case of Absence of Transverse Doppler Effect”

In the above paper, an extraordinary claim of experimental absence of the transverse Doppler effect is made. As stated in the abstract, An experiment... showing that a 33-GHz microwave signal received by rotating antennas is not exhibiting the frequency shift (transverse Doppler effect) predicted by the relativistic Doppler formula. In this comment, we will reanalyze the experiment and show that the correct application of special relativity predicts exactly a zero effect, in perfect coincidence with the observed result.

Response: The case of absence of transverse Doppler effect

Measurements that were done have yielded a gamma factor due to the quantum-mechanical absorption of photons in the detectors, thereby transferring the mass of photons into the detectors. It is said that if reflectors were used instead of detectors, then no gamma factor would have been obtained. Hence, it has not been demonstrated in the comment that the equations of the relativistic Doppler effect had been misapplied. In fact, it seems that those equations may have been improperly applied in the comment.

Observational Appearance of Luminous Relativistic Movers

Abstract We examine the observational appearance of optically thick luminous gaseous objects that move or rotate at a relativistic speed. In relativistic objects traveling at about the speed of light, Doppler and aberration effects influence their observational appearance. In gaseous objects with a hotter interior, furthermore, there exists a usual limb-darkening effect, which has a direction-dependency, and is therefore affected by the relativistic aberration. In addition, the emitted radiation suffers from a relativistic Doppler boost. Hence, the apparent photosphere of relativistic gaseous objects depends on their velocity and direction as well as the temperature distribution. We have derived the optical depth of the apparent photosphere of such relativistic movers under the plane-parallel approximation, and calculated several typical cases. When the velocity is almost parallel to the line of sight, the relativistic effect on the optical depth becomes important; otherwise, the limb-darkening effect dominates. In the case of a rotating sphere, except for some small fraction in the equatorial direction, the optical depth of the apparent photosphere becomes small due to the relativistic aberration. In the case of a geometrically thin disk, the optical depth becomes large in the left approaching part, whereas it becomes small in the right receding part. In the case of a geometrically thick disk, the projection effect becomes important, and the optical depth is large at the far side, while it is small at the near side. We have also examined light curves of luminous orbiters around a black hole and luminosities and spectra of accretion disks under special relativity, as applications. The light curve of the relativistic orbiter does deviate from a sinusoidal pattern due to the transverse Doppler effect. The pole-on luminosity of accretion disks is enhanced by the limb-dakening effect and reduced by the relativistic one, while the luminosity observed from the inclined direction is reduced by the limb-darkening effect.

Derivation of the energy–momentum and Klein–Gordon equations from El Naschie’s complex time

In a previous note, we have provided a formal derivation of the transverse Doppler shift of special relativity from the generalization of El Naschie’s complex time. Here, we show that the relativistic energy–momentum equation, and hence the Klein–Gordon equation, are also natural consequences of the complex time generalization.

Detection of light speed anisotropy via a high-speed Ives–Stillwell experiment

The Mansouri–Sexl theory is a well-known test of the theory of relativity. In one of their three papers (R. Mansouri and S.U. Sexl. Gen. Rel. Grav. 8, 809 (1977).), Mansouri and Sexl deal with the transverse Doppler effect under a set of restrictive conditions. In this paper, we deal with the same effect under the most general conditions and will argue for a novel way of detecting high-order effects in a vacuum via an increased laboratory-speed re-enactment of the Ives–Stilwell experiment. We will demonstrate the presence of a high-order error term and we will set the experimental conditions for detecting such a term.PACS No.: 03.30+p

Kündig's experiment on the transverse Doppler shift re-analyzed

In this paper, we re-analyze the ingenious experiment by Kündig (measurement of the transverse Doppler shift by means of the Mössbauer effect) and show that a correct processing of experimental data gives a relative energy shift ΔE/E of the absorption line different from the value of classically assumed relativistic time dilation for a rotating resonant absorber. Namely, instead of the relative energy shift reported by Kündig (v being the linear velocity of absorber and c being the light velocity in vacuum), we derive from his results . We are inclined to think that the revealed deviation of ΔE/E from relativistic prediction cannot be explained by any instrumental error and thus represents a physical effect. In particular, we assume that the energy shift of the absorption resonant line is induced not only by the standard time dilation effect, but also by some additional effect missed at the moment, and related perhaps to the fact that resonant nuclei in the rotating absorber represent a macroscopic quantum system and cannot be considered as freely moving particles.

Opening angles, Lorentz factors and confinement of X-ray binary jets

We present a collation of the available data on the opening angles of jets in X-ray binaries, which in most cases are small (10 ◦ ). Under the assumption of no confinement, we calculate the Lorentz factors required to produce such small opening angles via the transverse relativistic Doppler effect. The derived Lorentz factors, which are in most cases lower limits, are found to be large, with a mean >10, comparable to those estimated for active galactic nuclei (AGN) and much higher than the commonly assumed values for X-ray binaries of 2‐5. Jet power constraints do not, in most cases, rule out such high Lorentz factors. The upper limits on the opening angles show no evidence for smaller Lorentz factors in the steady jets of Cygnus X-1 and GRS 1915+105. In those sources in which deceleration has been observed (notably XTE J1550−564 and Cygnus X-3), some confinement of the jets must be occurring, and we briefly discuss possible confinement mechanisms. It is however possible that all the jets could be confined, in which case the requirement for high bulk Lorentz factors can be relaxed.

Probing Post-Newtonian Physics near the Galactic Black Hole with Stellar Redshift Measurements

Stars very close to the massive black hole (MBH) in the center of the Galaxy allow us to probe post-Newtonian (PN) physics in a yet unexplored regime of celestial mechanics. Recent advances in infrared spectroscopy enable us to take precise measurements of stellar redshift curves and thereby detect (?2) PN effects (gravitational redshift in the MBH's potential and the transverse Doppler shift). We use simulations to show that these effects can be decisively detected with existing instruments after ~10 years of observations. We find that neglecting these effects leads to statistically significant systematic errors in the derived MBH mass and distance.

Transverse Doppler Effect-Based Frequency Shifter

A block diagram of a quasioptical (QO) frequency shifter (FS) is proposed, in which a transverse Doppler effect has been originally employed. This block diagram provides for a continuous frequency shift much as polarization frequency shifters do. The physical processes that occur in the FS under consideration have been analyzed. It is found that the mirror moving transversely to the incident radiation along with the known force of pressure is acted upon by the lateral force directed towards the mirror motion. The FS block diagram ensures a 4-fold return of the QO beam from the mirror and eliminates the influence of parasitic mirror vibrations on the transmitted radiation phase. The application of the proposed FS may be found to be beneficial in homodyne measuring systems with frequencies ranging from THz and higher up to the optical ones.

Implications of the conjugate complex time on the origin of the relativistic transverse Doppler effect

Implications of the conjugate complex time on the origin of the relativistic transverse Doppler effect