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Dash, M. & Franzén, M. (2012). Time dilation have opposite signs in hemispheres of recession and approach. PHILICA.COM Article number 312.

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Time dilation have opposite signs in hemispheres of recession and approach

Manmohan Dashunconfirmed user (Willgood Inst. & Independent, not @ Armita! - Will confirm account asap., Amrita University)
Mikael Franzénunconfirmed user (i3tex AB & Willgood Inst. not @ Umea Uni! - Will confirm account asap., Umea University)

Published in physic.philica.com

Abstract
In this paper we give a detailed analysis of the factual observation that time dilation or Doppler shift of frequency are oppositely signed relative to our line of sight in different regions of motion. We show that this depends on how these regions are connected to the actual path of motion and to the location of the observer/detector. For a static globe or circle/sphere of reference, we define two hemispheres; shifts will be red-shifted in the hemisphere of recession and blue-shifted, or violet shifted, in the hemisphere of approach, a fact which is not often mentioned. Indeed, it is a complete sphere where the motion can take place with respect to the line-of-sight, which we have studied in this paper, not just the direction of motion along a certain specific path. The transverse and longitudinal effects are special cases of this general effect. The transverse effect is always red-shifted consistent with the known effect of Relativity and it lies at the intersection of the hemispheres of approach and recession. By saying moving clocks run slower we conveniently forget this important effect. They also run faster, in the hemisphere of approach. We demonstrate this important result from the basic principles of the Special Theory of Relativity.

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Author information:

Manmohan Dash*
Informal association with Willgood Institute, registered in Sweden; author’s mail address: Mahisapat, Dhenkanal, Odisha, India, 759001†

Mikael Franzén‡
Willgood Institute, Luckvägen 5, 517 37 Bollebygd, Sweden
and
i3tex AB, Klippan 1A , 414 51 Gothenburg, Sweden

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*Electronic address: mdash@vt.edu
†previously affiliated with VT, USA and KEK, Japan.
‡Electronic address: mikael.franzen@willgood.org

Article body


 

 

I. INTRODUCTION

We will perform a detailed analysis of how relative to the line-of-sight a general motion of a source and equivalently an observer will affect observation of time dilation and Doppler shift. We use only two main references, so we mention this fact here; [1] and [2]. It is a straightforward but non-trivial analysis. It applies to any situation where one would like to have a binomial expansion of the Lorentz factors γ and β to any desired degree of accuracy by retaining terms up-to one’s requirement. This will be useful in many studies.

Let us now define the context in which we have used terminology like “sphere of relative motion”, “hemispheres of recession” and “spheres of reference”; see Figure.1 and Figure.2 at the end of section IV. Our definition is a natural extension of the concept of reference frames. This is helpful towards identification of the relations of different parts to each other in the problem/situation that we present. Simply put, one defines two spheres/circles, the first one is a sphere of reference whose center is where a detector or origin of reference is located. The detector can also be placed e.g. at the surface of another object whose center is at the center of this reference. Any concentric sphere that encloses or excludes a segment of the trajectory of another object is also a sphere of reference. Now a second sphere of reference can be defined. This second sphere will encompass a segment of the trajectory path so the path lies in either the lower or upper hemisphere with respect to the line of sight. Let us call it a sphere of relative motion as it defines whether an object is receding away from, or approaching the center of reference. The upper hemisphere of this second sphere, where an object is gradually moving away from the reference point, defined at the center of the first sphere, can be called a hemisphere of recession. On the other hand, if the object is in the lower hemisphere of the second sphere, we can call it a hemisphere of approach, as the object will gradually approach the center of reference. One must note that these spheres are not very arbitrary in nature, but rather, dependent on the specific nature of the trajectories, e.g. a satellite orbiting a planet in a circle will naturally always traverse the intersection of the two hemispheres, so it is a transverse motion to the line of sight. In hyperbolic motion, such as that of a flyby satellite moving away from a planet towards some other destination, part of the motion lies in the upper hemisphere or the hemisphere of recession. When dealing, as described above, with earthbound motion such as perigee, {the point of closest approach to earth}, the opposite is true. The segment of the trajectory is naturally proportional to the radial speed vector. The radial - speed - vector is very important for any study of celestial applications. Similarly a meteor falling towards Earth always lies in the hemisphere of approach. One sees that one can cautiously adjust the sphere of reference and the sphere of relative motion so that there is no ambiguity in the situation. Only then can one consistently apply the laws of Relativity, such as those that govern time dilation and Doppler shift of frequencies.

We also note as a reminder of convenience that the transverse effects are always 2nd order in speed of recession and the longitudinal effects are 1st and 3rd order in terms of the Lorentz factor, β. Higher order terms appear in the same pattern. Transverse effects contain β2 and β4 and longitudinal effects contain alternative odd terms given by factors from binomial expansion as presented in our analysis. The general case is accommodated by cosines of a suitable angle, which appear only in the orders of odd power terms. The analysis is valid for 0 abs(β) 1, where abs stands for the modulus of a number or the regular absolute value function. The relation between fractional frequency shift fν and fractional time dilation or contraction ft related to fν are computed here and these are equal in their modulus. This is the case when the fν’s are small. When they are not small, additional review is necessary for the relation between the fs in ν, t, λ and v. We note that this also follows a binomial analysis like the one presented in this paper, because the fνs are fractions.

 

II. BINOMIAL THEOREM

The form and properties of binomial expansion is given in [1] for:-1 < x < 1

By using the above, we derive the Lorentz factors γ and its products with β or its functions, and any powers in the following steps, γ = (1 - β2)¯1?2; expanding in the power of β2

 

 

 

 Where   and

,

 therefore;

  and .

The following are expression for ;

to order β2 , , .

 

III. DOPPLER SHIFT AND ABERRATION IN SPECIAL RELATIVITY

 A. Description of the general form of Doppler shift; recession, approach and transverse motion.

The time dilation and the Doppler shift are described here. You will notice that the usual form of time dilation given in textbooks and literature is nothing but a transverse Doppler-shift. Here we show how the Doppler transverse is always red-shifted and how the hemispheres of approach and recession are always oppositely shifted where the hemisphere of recession is entirely red-shifted. The special cases of longitudinal and transverse effects follow ipso-facto. A rigorous binomial and a basic logarithmic treatment is given.

The general form of special relativistic Lorentz transformation in terms of frequency is given in [2],

the 2nd part of above equation is obtained by letting,  β → -βthe reciprocity of the Lorentz Transformation of inertial frames. We can see that both νand ν are proper frequencies and that each with respect to the other is an observed frequency. Each of the alternative cases can induce a relative change in the signs of geometric and coordinate system connections, but not the Physics. Let us rewrite the above, still in its general form;

 

Now, let us apply the result of all the care we took in deriving the binomial expansion of the Lorentz factors;


, or

;

 

The above equations were obtained by using the binomial approximations to the 2nd order in β and we have now an equation to the 3rd order in β. This might change slightly if we first accommodate up-to 3rd order and then obtain our result by chopping after 3rd order. This may give rise to new terms within the 3rd order. But for most purposes β itself is very small, usually 10-5, e.g. for satellites falling in cosmos at speeds ~ 50 times faster than jet airplanes in flight. That makes the β2 terms 10-10 and β3 terms 10-15. We would then make an error in that order by not accommodating enough terms before the expansion. 

 

The above equation gives the most general Doppler shift terms in the chosen order and it accommodates the transverse and longitudinal effects with angular modifications giving the general shift. fν is called fractional shift in frequency. This is a very important variable and most calculations can be made very conveniently using such fractional shifts; this also relates to the fractional shifts in speed or wavelength and in time dilation/contraction. We have done another derivation, {see NOTE - 3 in section IV}, that relates the fractional shift in frequency to that in time. The result is that fν is equal to ft in magnitude but differs by a minus sign. It is valid only when fν is in itself a small fraction such that binomial expansion can be used for such a fraction. In this idealized limit fν = -ft , which means a redshift is a time dilation and a blue-shift is a time contraction, speed fractions will have the same signs as time fractions: ,   , or , .

Thus cosθ will vary between 0 to 900and we’ll accommodate the other variations of θ into sign of β. So we have 3 cases;

  • (i)   Doppler hemisphere of recession, θ = 0, β > 0, longitudinal Doppler effect (recession)
  • (ii)  Doppler hemisphere of approach, θ = 0, β < 0, longitudinal Doppler effect (approach)
  • (iii) Doppler plane of intersection of hemispheres of approach and recession, θ = 900, β > 0, the transverse Doppler effect.

Note that for 2nd case β < 0 is just a convention, actually 0 < β < 1, and there is a sign reversal for approach in our convention. So in all 3 cases the sign of fν and ft, the numerical sign as opposed to our algebraic sign notation, will depend on the fractions as below. We need to work out which of the two inequalities is valid, and in the following we will show that only one of them is valid in all of the cases;

Let us do some more mathematical exercise; let β be given as β = ln(α), {A short note, NOTE - 2 is to be found in section: IV}. This means

under the conditions;
 

Let us now describe approach, recession and transverse motion in terms of properties of the Doppler shift in terms of β;

Approach, case (ii): ln(α) = β < 0 and Recession/Transverse, case (i),(iii): ln(α) = β > 0. So there are 2 conditions and the recognition that (lnα)2 > 0;

 

 

 Hence the < part holds for case (ii) and the > part holds for case (i),(iii).

For case (ii); or  (since  β < 0)      as each time we multiply  a -ve sign we must reverse the inequality.

 

For case (i), (iii) therefore, since in our convention,  β > 0, then;

  
           
 
Thus, THE SAME INEQUALITY IS VALID IN ALL CASES even though the mathematical and physical conditions are different. We note here that although we derived , it is also true that,  which is what we need for our purpose. We show this in our Notes, {NOTE - 4 in section: IV}.
 

B. Conclusions on nature of frequency shifts or time dilation/contraction in different regions of relative motion

Now let us return to fν and ft, or equivalently, and ;

 

 

Case (i), (iii) can be described as:  (i)   {θ = 00,β > 0},  and  {θ = 900,β > 0} (iii). We deal with (iii),(i) before moving onto (ii).

 

  • case (iii):
  • case (i):

  • case (ii)
 

IV. NOTES

  1. In section-III we used a convention where a primed frame is moving with respect to the unprimed at speed v = β in units of speed of light, 0 β 1. This is the convention used by [2]. In this reference ν is proper frequency and ν is observed frequency due to the motion/speed β. Here one has to be careful in preventing oneself from using Lorentz reciprocity inconsistently which results in accruing -ve signs. The Physics in the end has to be consistent. One rule of caution is therefore to check the signs of Δν and ν and the relative primeness of ν and νwhich introduce additional -ve signs at unspecified points in the calculation if used incorrectly. All this could be a source of confusion under Lorentz reciprocation.
     
  2. One can prove the above results without using ln() by just employing the properties of numbers, viz the square of any real number is always positive. But having ln() may be useful if one intends to study a functional form of β. It is widely customary in Relativity to employ logarithmic functions and we think it is very useful for generalization purposes.
     

  3.  

  4.  
  

 
                     
 Figure.1: The timedilation/contraction effects as frequency red-shift/blue-shift, shown for above configuration.
 
 

                     

Figure.2: This diagram explains our terminology, sphere of reference and sphere of relative motion. LOS = Line Of Sight. See NOTE - 10 above.

 

V. SUMMARY/CONCLUSION

Our definition of special notions of reference of observation and reference of the actual motion path enables an investigation of important relativistic effects. We have deduced from an application of the binomial theorem and properties of both logarithmic and real numbers that Doppler shift and time dilation of Special Relativity are oppositely signed in two hemispheres, which naturally define where the motion trajectory lies with respect to the observer and the line of sight. An application on aberration of special relativistic recession and approach is also inherent here. The analysis draws on very basic, yet non-trivial, and not-so-often cited facts of Relativity. This paper serves as an explicit reminder of the mathematical apparatus of Relativity. Thus, the concept and methodology presented in this paper should be applicable to a wide variety of investigations, especially where one seeks accuracy to very high powers of β through binomial expansion. We purposely kept this paper limited to a generalized treatment suitable for future reference.

 

Acknowledgments

We are thankful to the free world for the resources which enabled us to discuss our ideas and communicate the research. In special we would like to mention free software from various sources such as source-forge, LyX, X-fig, and Google. A special mention also goes out to Facebook and WordPress. These sites have been a constant and supportive source for our communication without which discussion and brainstorming prior to professional sharing had not been possible. This research was in part supported by i3tex and the Willgood Institute. Both authors would also like to thank their families for remarkable support. Furthermore, we are really grateful for the valuable feedback provided by Professor Giovanni Modanese. 

 

References

[1]    G.Arfken, H.Weber, "Binomial Theorem", 6th Indian edition, chapter 5.6, (2009).

[2]    R.Resnick, "Introduction to Special Relativity", (1968).

[3]    The OPERA Collaboration, “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam”, arXiv:1109.4897v1 [hep-ex], (2011).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Dash, M. & Franzén, M. (2012). Time dilation have opposite signs in hemispheres of recession and approach. PHILICA.COM Article number 312.


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