Gamow's Cyclist: A New Look at Relativistic Measurements for a Binocular Observer
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Subject Areas:
Special Relativity
Keywords:
Terrell Rotations; Visual Appearance
Author for correspondence:
P. D Stevensone-mail: [email protected]
Gamow’s Cyclist: A New Lookat Relativistic Measurementsfor a Binocular Observer
E. A. Cryer-Jenkins and P. D. Stevenson Department of Physics, University of Surrey,Guildford, GU2 7XH, United Kingdom
The visualisation of objects moving at relativisticspeeds has been a popular topic of study sinceSpecial Relativity’s inception. While the standardexposition of the theory describes certain shape-changing effects, such as the Lorentz-contraction, itmakes no mention of how an extended object wouldappear in a snapshot or how apparent distortionscould be used for measurement. Previous work on thesubject has derived the apparent form of an object,often making mention of George Gamow’s relativisticcyclist thought experiment. Here, a rigorous re-analysis of the cyclist, this time in 3-dimensions,is undertaken for a binocular observer, accountingfor both the distortion in apparent position and therelativistic colour and intensity shifts undergone bya fast moving object. A methodology for analysingbinocular relativistic data is then introduced, allowingthe fitting of experimental readings of an object’sapparent position to determine the distance to theobject and its velocity. This method is then appliedto the simulation of Gamow’s cyclist, producing self-consistent results.
1. Introduction
In 1938, George Gamow envisioned a beautiful thoughtexperiment in his book
Mr Tompkins’ Adventures inWonderland [1]: in it, the titular hero is transported toa strange world in which the speed of light is onlyslightly faster than that of a bicycle and he sees a passingcyclist to be Lorentz contracted, in apparent agreementwith Einstein’s Theory of Special Relativity. This view,that a moving extended object would appear “simply"contracted, was widely held in the first half of the 20thcentury with Einstein and Lorentz both promulgatingthe visibility, and the ability to be photographed, ofrelativistic length contraction [2]. c (cid:13) The Author(s) Published by the Royal Society. All rights reserved. a r X i v : . [ phy s i c s . c l a ss - ph ] J un r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Despite an early attempt to correct this misconception by Anton Lampa in 1924 [3], it wouldnot be until the late 1950s when two papers, one by Roger Penrose [4] and another by James Terrell[2] (after whom this family of visual aberrations is named), that the actual visual appearance ofrelativistic objects became widely acknowledged.Images are generated by photons arriving at an observer simultaneously, not by thosesimultaneously emitted by an object. As light has both a constant and finite speed, what one reallysees is an extended object made up of a patchwork of itself at different times. Terrell’s conclusionwas a simple one: a quickly moving object appears rotated with an increasing proportion ofits rear face becoming visible at greater speeds; an elegant solution that quickly found its wayinto the works of others and into the educational literature [5–7]. In the years following, re-analyses of Terrell’s work (most significantly by Mathews and Lakshmanan [8]) painted a pictureof a more complex set of distortions which were better understood as a non-linear shear andextension/contraction parallel to the direction of movement. Mathews and Lakshmanan’s re-analysis was swiftly joined by a vast array of literature expounding the various phenomenologicalconsequences of this treatment [9–18], some pieces citing Gamow’s cyclist as a metric with whichto consider the extent of these deformations [9,19]. These results describe what Mr Tompkinswould photograph in these relativistic situations (assuming the photographic device able to detectphotons of any wavelength) but not, as Mr Tompkins is a human and not a camera, what hewould actually see .In this work, we have re-analysed relativistic visualisation from the perspective of a binocularobserver, introducing an analytical time delay between the “eyes" of such an observer andapplying it to the analytical transformations of an object’s actual location to its apparent one aswell as the relativistic Doppler and intensity shifts of a body isotropically emitting radiation inits rest frame. This treatment is then applied to a 3-dimensional simulation of Gamow’s cyclist,comprised of spheres and cylinders, thus giving a more complete picture of what Mr Tompkinsmight see.After reviewing the analytical results of the visual relativitic effects, we produce a sphericalpolar method, using binocular observers, of fitting experimental position data from movingobjects to determine their actual velocity and distance to the observer, allowing corrections to bemade to multi-aperture relativistic photographs and other quantities, such as emission intensitiesand spectra, to be determined. Finally, it is suggested that these methods could be integratedwithin a relativistic probe, such as one proposed in the work of Christian and Loeb [20], ora synthesised, ground-based aperture that would allow the probing of exoplanets and otherastronomical bodies.
2. Mathematics of Relativistic Distortions
In this section, we review - in accordance with Einstein’s Postulates of Special Relativity [21] - theequations that translate various physical quantities from their actualities to their apparencies,already present in literature [8]. We make use of the trigonometrical relations that hold inEuclidean, flat space and this result is thus valid only in the case that strong gravitational fieldsare not present. The terms β = v/c and γ = (1 − β ) − / represent their standard quantities inSpecial Relativity. (a) Apparent Positions We follow Mathews and Lakshmanan [8] in using objective to refer to properties of the (moving)object in a frame in which it is moving that are due to “purely relativistic effects” that couldbe determined by some contact method without the intervention of light signals. The apparent quantities then take into account the communication of properties of the object by light travellingto the observer which is perceived at a single point in spacetime. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... For a point P , travelling parallel to the x -axis in the positive x -direction with coordinates P ( vt (cid:48) , y, z ) , the apparent position, x (cid:48) , is given by x (cid:48) = γ ( x + βcT ) − γc (cid:113) γ ( x + βcT ) + y + z (2.1)for an observer on the x -axis where t (cid:48) is the time at which a photon is emitted and T is the time atwhich the same photon is received at the observer. x , y and z are the t (cid:48) = 0 objective coordinatesof the point in the observer’s coordinate frame and v is the point’s velocity. For an observer at theorigin, we therefore have that x = 0 , y = y and z = z .The other apparent coordinates, y (cid:48) and z (cid:48) , retain their objective values, y and z respectively.Here, we will only consider points with x -coordinate given by x ( t (cid:48) ) = vt (cid:48) so that x (0) = 0 . (b) Relativistic Radiative Effects Along with visual distortions, we also consider the colour change of the object in question,both in the interest of phenomenologically examining what Mr Tompkins might see and also forscientific measurement, as photographs of relativistic objects must be corrected for Doppler andluminosity shifts as a result of the motion of a radiating source [22–24]. Maintaining the previousconfiguration of the point, P , the relativistic Doppler shift between the received wavelength, λ (cid:48) r ,at the observer and source wavelength, λ S , emitted by P is given by λ (cid:48) r = γ (cid:18) − γβ ( x + βcT ) (cid:113) γ ( x + βcT ) + y + z (cid:19) λ s (2.2)Similarly, we find that the intensity at the observer, I (cid:48) , for an isotropically (in its own restframe) emitting source with rest intensity, I , is given by I (cid:48) = γ (cid:18) − γβ ( x + βcT ) (cid:113) γ ( x + βcT ) + y + z (cid:19) I (2.3)
3. Measurements from Binocular Distortion
In this section, we introduce formalism with respect to binocular observations; herein, we definetwo types of observers:
Class 1
An observer with a single aperture such as a camera
Class 2
An observer with two apertures, capable of depth perception generated by visual parallaxsuch as a humanAll apertures in these definitions are considered to have an infinitesimal exposure time andperfect focus.Works by Boas [9] and Nowojewski [19] have gone some way towards realising the actual form ofGamow’s cyclist but, as with the other considerations made before, they deal with the aberrationspresented to a Class 1 observer. As such, Mr Tompkins would not see the bicycle as describedby Gamow, Nowojewski, Boas or others [25,26] due to him being a Class 2 observer. We alsochoose to account for colouration and intensity shifts [24] to provide a more complete picture ofour relativistic cyclist, including the difference between distortions presented to each apertureproviding a method of determining the distance from and speed of the bicycle. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... (a) Time Delay between Observers A Class 2 observer’s two apertures will perceive the same object differently at different points inits trajectory as the distance between them is not negligible for relativistically moving bodies. Assuch, it is useful to introduce a time difference between apertures so that one equation can be usedwith identical values apart from a time difference term, ∆T , between the primary and secondary apertures. Here, we consider a Class 2 observer with apertures equally spaced a distance d eitherside of the origin, the “left" aperture at x = − d being the primary. The object, as before, is made ofpoints with coordinates ( vt (cid:48) , y , z ) , travelling with velocity v ; we see this construction presentedin Figure 1. Figure 1.
Construction used when considering the time difference between two apertures, evenly spaced around theorigin, O , a distance d apart. After arriving at the x -axis, photons experience no additional path difference before reachingan aperture. To find this time difference, we consider the path difference from Point A to each aperture.Defining v as positive, Point A moves from left to right and a “newer" realisation of the object willinitially be presented to the left aperture until it passes the point of equidistance at t (cid:48) = 0 , afterwhich it is the right aperture realising a “newer" image. We will therefore define this ∆T as thetime difference added on to the T value of the left aperture as it is positive for t (cid:48) < , motivating itsassignment as the primary aperture. We can see this path difference is c∆T = (cid:115) y + z + (cid:18) vt (cid:48) − d (cid:19) − (cid:115) y + z + (cid:18) vt (cid:48) + d (cid:19) (3.1)which provides the expected result; namely that the time difference tends to a constant valuewhen the object is far to either side of the observer and is equal to 0 for t (cid:48) = 0 , i.e. when the objectis equidistant from both apertures.We then use the trigonometric relation between emission and observation times c ( T − t (cid:48) ) = (cid:113) ( x + vt (cid:48) ) + y + z (3.2)for t (cid:48) = 0 and, as we are considering the left aperture as the primary aperture, x = d , to obtainan expression for the time of apparent equidistance, T t (cid:48) =0 as T t (cid:48) =0 = 1 c (cid:115)(cid:18) d (cid:19) + y + z . (3.3) r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... We will see that this is a useful expression in the fitting of visual data as it provides the observer’stime at which the object is apparently equidistant from both apertures, allowing the determinationof object distance from the fitting of experimental position data. (b) Parameter Fitting
Suppose we can identify a point on the surface of an object approaching the observer withvelocity v in the increasing x -direction and coordinates ( vt (cid:48) , y , z ) , the apparent azimuthal anglesubtended is given by standard trigonometry φ P = arctan (cid:18) y x (cid:48) ( T ) (cid:19) (3.4)for the primary aperture, which, we remind, is assigned as being on the left (noting that φ isdefined relative to the positive x -axis). The secondary aperture, on the right, will receive, at thesame observation time, a “later" image of the object for t (cid:48) < , resulting in an apparent azimuthalangle given by φ S = arctan (cid:18) y x (cid:48) ( T + ∆T ) (cid:19) (3.5)As we have seen previously, x (cid:48) is given, in terms of objective coordinates, by x (cid:48) = γ ( x + βcT ) − γc (cid:113) γ ( x + βcT ) + y + z (3.6)We also require an expression for ∆T in terms of T . We again make use of our expression for t (cid:48) interms of objective coordinates and observer time T , given by t (cid:48) = γ c ( T c + x β ) − γc (cid:113) γ ( x + βcT ) + y + z (3.7)which we can substitute into our equation for ∆T to achieve a non-linear equation for x (cid:48) ( T + ∆T ) , the apparent position according to the secondary aperture, in terms only of Tx (cid:48) | ( T + ∆T ) = γ ( x + βc ( T + ∆T )) − γc (cid:113) γ ( x + βc ( T + ∆T )) + y + z (3.8)where ∆T ≡ ∆T ( x, y , z , T ) The power of expression (3.8) is that it allows the determination of relativistic quantities asthe differences in polar and azimuthal angles between apertures in Class 2 observers. Thesedifferences have characteristic shapes that can be fitted to determine the velocity of an object andits distance from the observer. We now produce typical plots of φ P and φ S as well as the differencebetween them which can be employed for fitting as it provides a second known quantity, d . Figure2 illustrates the azimuthal position of the object for both apertures, indicating a clear cross overas the object appears to “overtake" itself travelling from the left to the right.All figures are functions of the observer’s time, T , and all angles are in radians unless otherwisestated.The difference between these apparent azimuthal angles, defined by ∆φ = φ P − φ S = arctan (cid:18) y x (cid:48) ( T ) (cid:19) − arctan (cid:18) y x (cid:48) ( T + ∆T ) (cid:19) (3.9)has a characteristic shape, illustrated in Figure 3, which is asymmetric due to the difference inapparent velocity for an approaching or receding object.Figure 3 illustrates a maximum negative angular difference of -0.34 rad (for β = 0 . ), equivalentto 19.5 degrees and maximum positive difference of 0.06 rad, or 3.4 degrees. As such, these r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 2.
Graph depicting the azimuthal angles, φ , subtended by a point with y = 0 . , z = 0 . travelling at β = 0 . (black representing the primary aperture and red the secondary) for d = 0 . . changes are on the order of degrees and are much larger than the smallest resolution of angularmeasurement devices used in astronomy [27]. It is also intuitive from the previously defined timedifference that, when ∆T = 0 , the intersection with the x -axis in Figure 3 is given by Equation(3.3); as the object is equidistant from both apertures at t (cid:48) = 0 , it is at coordinates ( d , y , z ) withrespect to the left aperture. Using the analytical data from Figure 3, we see (again taking c = 1 forsimplicity) T t (cid:48) =0 = (cid:113) (0 . + (0 . + (0 . = 0 . which is visibly the x intersection in Figure 3, providing the distance to the object.Similar fitting can be carried out for the apparent polar angle (although this is more difficultfor smaller/more distant objects), allowing detector sensitivity to be investigated and providingan estimate of the error in object distance and velocity. Here, we define the apparent polar angleagain using standard trigonometry θ P = arctan z (cid:113) x (cid:48) ( T ) + y (3.10)for the primary aperture and θ S = arctan z (cid:113) x (cid:48) ( T + ∆T ) + y (3.11)for the secondary.It should be noted that θ is defined as the angle subtended by the line joining object andobserver and the x-y plane and it thus tends to 0 for t (cid:48) → ±∞ ; a closer object also subtends alarger θ . r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 3.
Graph depicting the difference in azimuthal angles, ∆φ , subtended by a point with y = 0 . , z = 0 . for anobserver with d = 0 . and β = These quantities have been plotted in Figure 4, illustrating the different times at which anobject would appear larger for each aperture.
Figure 4.
Graph depicting the polar angles, θ subtended by a point with y = 0 . , z = 0 . travelling at β = 0 . (blackrepresenting the primary aperture and red the secondary) for d = 0 . . r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... The difference between these apparent polar angles, ∆θ = θ P − θ S = arctan (cid:18) z (cid:113) x (cid:48) ( T ) + y (cid:19) − arctan (cid:18) z (cid:113) x (cid:48) ( T + ∆T ) + y (cid:19) , (3.12)also has a characteristic shape, illustrated in Figure 5, which is asymmetric for the same reasonsconsidered previously. Figure 5.
Graph depicting the difference in polar angles, ∆θ , subtended by a point with y = 0 . , z = 0 . for anobserver with d = 0 . and β = It also has two intersections with the x-axis which may seem counter-intuitive; however, thistoo arises from the asymmetry of the construction. The right aperture initially sees an “older"instance of the object that is further away and thus smaller. As the object passes the left aperture,the right begins to see a larger instance and the angular difference becomes negative. Finally,after the object has just passed the point of equidistance between apertures, the right aperturenow receives a “newer" instance of the object that is further away and thus smaller, reversing theangular difference to positive once more.The fitting of the intersection at larger T provides a β -independent measure of the position,identical to the azimuthal intersection, while the intersection at smaller T is β -dependent; thesepoints tend to the same time coordinate, T t (cid:48) =0 , given by Equation (3.3), for larger values of β (asin Figure 5) and greater distances as the more extreme instances of the object overtaking itself aresmoothed out.Figure 5 illustrates a maximum negative angular difference of -0.015 rad (for β = 0 . ), equivalentto 0.86 degrees and maximum positive difference of 0.19 rad, or 10.89 degrees. As noted before,these angles, while for a relatively close object, are sufficiently significant to be readily measuredand are orders of magnitude larger than the resolution of current astronomical angle measuringdevices. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... From fitting Figures 3 and 5 with the aforementioned equations, we can obtain values andcombined uncertainties for the distance of the object from the observer and the speed with whichthe object is moving; these quantities can then be utilised to correct relativistic photographs oremission spectra of moving objects. We will now consider the phenomenological example ofGamow’s cyclist, considering both their appearance to a human observer and applying Equations(3.9) and (3.12) to a point on the bicycle as it moves past, testing the validity of the method.
4. Gamow’s Cyclist
In this section, we consider the famous and oft-mentioned thought experiment that is Gamow’srelativistic cyclist, re-imagining it in 3-dimensions for a Class 2 observer and applying themethodology set out in the previous section to determine its velocity and distance from theobserver. (a) 2-Dimensional Appearance
First, we consider a simple bicycle composed of straight lines and circles, depicted in Figure 6,in order to provide a clear metric with which to measure the distortion of well-known shapes.The transformation of extended objects can be done by considering them as comprising of acontinuum of points, each transformed according to Equation (2.1).
Figure 6.
Illustration of a simple bicycle composed of straight lines and circles.
Here, we consider this bicycle for a single observer as it travels past, deformed using Mathewsand Lakshmanan’s transformations in 2 dimensions, not accounting for any kind of intensityor colouration distortion. The visual deformations undergone by the relativistic bicycle arepresented in Figure 7 with the observer represented as a black dot; we see the exact deformationspredicted by Mathews and Lakshmanan in 2 dimensions, namely a combination of non-uniformshear and extension/contraction parallel to the direction of movement. As the bicycle approachesthe observer (with an apparent velocity greater than c [19,28,29]), it appears grossly extended withthe circular wheels deformed into elongated ellipsoids. As it passes the observer, the wheels andspokes appear concave and the entire frame is contracted (receding now with apparent velocity ≤ c [30]). It can also be noted that all line sections parallel to the direction of movement remainparallel, such as the cross bar of the bicycle, and the height of the bicycle remains unchanged.This construction goes a long way in illustrating the increased elongation and contraction anobject experiences before and after passing an observer with ratio (cid:113) β − β before reaching theobserver and (cid:113) − β β after passing them. However, there are other considerations to be taken intoaccount for what Mr Tompkins would really see; not only are there colouration shifts, there is also r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 7.
Figure illustrating deformation of a 2D bicycle with β = 0 . with the observer represented as a black dot. the matter of binocular vision becoming confused with the overlap of contradictory images andMr Tompkins’ perception of 3 dimensions.Employing the time difference term, ∆T , from Equation (3.1) we can visualise Figure 7 but fora Class 2 observer with the left aperture again chosen as the primary, providing a good indicationof what Mr Tompkins’ eyes would actually see, the result displayed in Figure 8.As visible in Figure 8, Mr Tompkins’s sensory apparatus, which relies on visual parallax todetermine depth [31–33], would see apparent fluctuations in distance to the cyclist as the imageproduced in the secondary aperture (which is denoted by the red bicycle) overtakes that of theleft. This disparity between images that so confuses a human is, however, what can be exploitedas the basis of a relativistic measuring tool. (b) 3-Dimensional Appearance We now construct a simple 3-dimensional bicycle and rider, comprised entirely of tubes andspheres, and displayed in Figure 9, which - for simplicity - is entirely comprised of red pixelswith wavelength 700nm.This 3D model was generated from an array of spatial coordinates for a series of pixels, alongwith emission time t (cid:48) , rest wavelength and intensity of each pixel. These are then translatedinto apparent coordinates, observation time T , relativistic Doppler wavelength and intensityat the observer through the equations derived previously. The colours and intensities are thentransformed and the results animated at 15 frames per second. The crux of this methodology is toallow the transformation of intensities and emission wavelengths back into their rest quantitiesthat enable the analysis of relativistic objects. From Section 3, we have seen that the fitting ofbinocular profiles, obtained by two (or potentially more to improve the fitting) observers, can be r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 8.
Figure illustrating deformation of a 2D bicycle with β = 0 . for a Class 2 observer with aperture spacing d = 0 . .We see the features pointed out in the previous section with different parts of the bicycle catching up with itself at differenttimes. The image received at the primary aperture is in black and at the secondary is in red. used to determine both the distance to an object and also its velocity.In the computer simulation, we track the first point on the wheel (with t (cid:48) = 0 coordinates x = 0 , y = 3 and z = 2 and velocity β = 0 . in this simulation), imagining that the observer is tryingtheir best to keep their view on the front end of the bicycle; as such, the entire elongated bicycledoes not fit within the visual range of the observer. The distance between apertures is fixed at d = 0 . . The leftmost image in each frame is the appearance presented to the left aperture; likewisefor the rightmost image corresponding to the right aperture. The time at which the forward-mostpixel in the front wheel is observed as being equidistant between both apertures is T t (cid:48) =0 .For T < T t (cid:48) =0 , we are presented with a frontal view of the bicycle; while the headlight effectrelativistically beams the majority of the isotropically emitted radiation forwards, it is Dopplershifted out of the visual spectrum (and into the 200nm ultraviolet range) and is thus invisible toa human observer (represented as black in the simulation):A human observer without equipment to detect light outside the visible range would thereforeinfer the appearance of the cyclist as it blocks the background they would otherwise see. Theywould, however, begin to notice the extreme elongation of the cyclist as it approaches, asvisible from the side in Figure 7. Figure 10 also illustrates that, for large negative times, thereis little difference in the distortions presented to both apertures, analytically illustrated by theconvergence to 0 for T → ±∞ in Figures 3 and 5. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 9.
Construction of Gamow’s cyclist for which the 3D simulation taking into account colour and intensity was carriedout.
Figure 10.
Figure depicting a bicycle approaching a Class 2 observer with β = 0 . for frames 10 to 27. The left image ineach frame is for the left aperture, likewise for the right. Around T t (cid:48) =0 in Figure 11, colours are shifted into the visual spectrum and enough radiationreaches the observer to be detected. In the case of an non-human observer, this informationcan be used to deduce first the form of the object, then its distance and velocity. It is also thepoint for which an observer notices the most deformation, as noted by Nowojewski [19]. Fora human, a very thin strip of visibility would run vertically across the cyclist, revealing thedistortions described by Mathews and Lakshmanan’s transformations; namely, the front of thebicycle appears squashed and the left side of the cyclist’s body appears rather horrifically twistedas it lags behind the right side. We also observe differences in the distortions presented to bothapertures which, as analysed in the previous section, provide information about the object’smotion and position.For slightly slower objects (such as for β = 0 . ), the visual deformations are less markedbut more of the bicycle is visible as the Doppler and intensity shifts remain within the visiblespectrum. The same distortions, now with a thicker strip of visibility, are presented in Figure 12.It is in the region of the point of closest approach for which the most striking distortions andthe greatest differences between images presented to each aperture occur; it is from this disparitythat the characteristic shapes of the plots produced in Section 3 arise, rendering it key for thedetermination of an object’s distance and velocity.For T > T t (cid:48) =0 , the light is again Doppler shifted out of the visual spectrum (into the 1000nmshort infrared) and the intensity arriving at the observer is minute as the majority is “beamed"in front of the cyclist. We again represent the cyclist as a black solid since it would appearas an outline in front background radiation arriving from other sources, noticing the extremecontraction of the cyclist as the rear wheel appears to catch up with the front suddenly for timesslightly greater than T t (cid:48) =0 . r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 11.
Figure depicting a bicycle abreast of a Class 2 observer with β = 0 . for frames 30 to 45. Figure 12.
Figure depicting a bicycle abreast of a Class 2 observer with β = 0 . for frames 28 to 45. (c) Fitting of Gamow’s Cyclist We now apply the methodology outlined in Section 3 to the case of Gamow’s cyclist. Byconsidering the forward-most pixel on the front wheel of the bicycle depicted in Figure 9, wecan produce similar azimuthal and polar plots to those in Section 3, illustrating the viability ofthis method as a relativistic measuring tool. The simulated parameters are show in Table 1.We present the success of fitting Equations (3.9) and (3.12) to the simulation data using a non-linear bootstrapping regression method [ ? ] in Table 14. The fitted quantities for the azimuthaldifference, ∆φ , are given in Table 2The complimentary fitting for the polar angle is presented in Figure 15. We observe largeruncertainties in the fitted values as the two intersections with the x-axis approach the same value;as the smaller- T intersection is β -dependent and the one at larger- T is not, the close proximity ofthe two increases the fitted uncertainties. The fitted values for the polar difference, ∆θ , are shownin Table 3. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 13.
Figure depicting a bicycle receding from a Class 2 observer with β = 0 . for frames 45 to 55. Quantity Value β y z d Table 1.
Table of values utilised in creating the simulation. y and z are the coordinates of the forward-most pixel on thefront wheel, β is its velocity and d is the aperture spacing. Quantity Value Uncertainty β y z Table 2.
Table of values and uncertainties obtained from fitting of the azimuthal angle difference for a Class 2 observerwith d = 0 . . Quantity Value Uncertainty β y z Table 3.
Table of values and uncertainties obtained from fitting of the polar angle difference for a Class 2 observer with d = 0 . . Uncertainties in the relativistic parameters fitted were calculated using a bootstrappingmethod, shown to be a better indicator of uncertainty in non-linear fitted parameters thancomparable Monte Carlo methods or linearisation methods [34].Averaging these quantities for azimuthal and polar fitting with weight determined by theuncertainty and propagating error diferentially, we obtain β = 0 . ± . , y = 3 . ± . and z = 2 . ± . , the exact values of the cyclist created for the simulation detailed in Figure 9.The errors in these quantities of course do not reflect any experimental uncertainty as they wereobtained from a deterministic simulation; they do however represent the errors associated withthe fitting of data of this form with Equations (3.9) and (3.12) using a non-linear regressionmethod [35]. It is thus suggested that this methodology has potential as a tool for the accuratemeasurement of relativistic objects, determining both their objective velocity and position, usingClass 2 observers. These quantities can then be used to transform relativistic photographs andspectral measurements back to their rest forms, a more useful format for physical analysis. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 14.
Figure illustrating the simulated azimuthal difference, ∆φ , and the fitted function. The simulation curve is inblack and the fitting of Equation (3.9) is in red. With regard to applications, it is posited by the authors that an interferometer using this fittingtechnique could be either implemented as part of ground based arrays such as those used inaperture synthesis [36] or as a relativistic probe capable of probing astronomical bodies whiletravelling at great speed [20]. In the case of the former, inverse Fourier transforms are utilised toresolve the image for multiple observers; as such, an extension to this work would be to generalisethe solutions for Class N observers and implement differential geometry for a smoothly varyingtime difference, ∆T , across all observers in the aperture that could be used in differential calculus.In the case of the latter, further work must be carried out to generalise the solutions for GeneralRelativistic effects and implement solutions to the geodesic equations for photon trajectories toallow for inclusion within Christian and Loeb’s suggested probe which accounts for gravitationaleffects.
5. Concluding Remarks
In summary, the visual appearance of Gamow’s cyclist has been comprehensively re-analysedin 3-dimensions, accounting for radiative shifts as well as apparent distortions in its shape(which comprise a non-linear shear and elongation/contraction depending on position), and for abinocular observer. The actual appearance of the cyclist for a human observer is then speculated,taking into account the difference in distortions presented to both eyes as well as the limits ofthe visual spectrum. A method of determining the position and velocity of moving objects is alsoillustrated, making use of the azimuthal and polar disparities between apertures, providing anestimate of the uncertainty by comparing the fitting of both curves. This methodology is thenapplied to the simulation of Gamow’s relativistic cyclist, correctly reproducing the simulatedparameters with the uncertainties obtained from non-linear fitting. Finally, it is suggested that r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 15.
Figure illustrating the simulated polar difference, ∆θ , and the fitted function. The simulation curve is in blackand the fitting of Equation (3.12) is in red. these methods could be integrated within a relativistic probe [20] to observe stationary objectsas they move past, or a synthetic aperture which would allow the probing of distant, relativisticobjects.Ethics. There are no ethical issues associated with this research
Data Accessibility.
This article has no additional data.
Authors’ Contributions.
EC-J performed the calcualtions associated with this project. Both authorsdiscussed the ideas involved, were both involved in writing the paper, and agreed the final submitted form.
Competing Interests.
The authors declare no competing interests.
Funding.
PDS acknowledges support from UK STFC grant ST/P005314/1
Acknowledgements.
The authors thank Dr Maxime Delorme for help in constructing the cyclist model inPython
A. Full expression for Right-Aperture Apparent Position
A shortened expression for the apparent position of a point for the right aperture was presentedin Equation (3.8) in the interests of preserving space where the non-linearities in T have beensuppressed within the ∆T term. Here, we give the entire, lengthy equation only in terms of T . r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Note that c = 1 has been used in this equation for (attempted) brevity: x (cid:48) = γ x + β T + (cid:118)(cid:117)(cid:117)(cid:116) y + z + (cid:32) β (cid:18) γ ( T + x β ) − γ (cid:113) γ ( βT + x ) + y + z (cid:19) + d (cid:33) − (cid:118)(cid:117)(cid:117)(cid:116) y + z + (cid:32) β (cid:18) γ ( T + x β ) − γ (cid:113) γ ( x + βT ) + y + z (cid:19) − d (cid:33) − βγ γ x + β T + (cid:118)(cid:117)(cid:117)(cid:116) y + z + (cid:32) β (cid:18) γ ( T + x β ) − γ (cid:113) γ ( x + T β ) + y + z (cid:19) + d (cid:33) − (cid:118)(cid:117)(cid:117)(cid:116) y + z + (cid:32) β (cid:18) γ ( T + x β ) − γ (cid:113) γ ( x + βT ) + y + z (cid:19) − d (cid:33) + y + z References
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