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The Relativistic Rotation

The classical rotation is not self-consistent in the framework of the special theory of relativity. the Relativistic rotation is obtained, which takes the relativistic effect into account. It is demonstrated that the angular frequency of classical rotation is only valid in local approximation. The properties of the relativistic rotation and the relativistic transverse Doppler shift are discussed in this work.


I. INTRODUCTION
The special and general theory of relativity which was developed in the beginning of the twentieth century, especially through Einstein's work [1,2,3], has its roots far back in the past. But the rotational reference systems utilized in the present researches of physics are all classical [4,5], although there exists the special and general theory of relativity. In the classical rotation, the velocity v of one point relative to the center of circle is proportional to its distance r from the center of circle and its angular frequency ω 0 . However, this relationship is not suitable within the special relativistic theory.
The velocity formula of classical rotation is given as v = rω 0 . It could be understood from the following three aspects. First, the foundation of experiments, i.e., the experience of daily life. Second, Galilean transformations [4]. Third, the general conclusion derived from the first and the second points. Following the inference of Galilean transformations, the velocity formula of classical rotation is generally come into existence under any rotational conditions, such as the condition of long-distance and high-velocity. It means that the formula v = rω 0 is self-consistent in classical.
However, the application of Galilean transformations in high-velocity condition is proved early to be not reliable [4,5]. It is not correct for the object with high velocity and must be replaced with the principle that could embodies the relativistic properties. Therefore, the extending velocity formula v = rω 0 of classical rotation to arbitrary is suspicious, and it is need to reevaluate. We should start from the special relativistic theory to reconsider the rotation with high velocity. the proper relativistic metric of rotational reference system should be confirmed. The four-dimension metric of the uniform rotational system is classical in the general theory of relativity now [4,5]. It means the proper relativistic present of the rotational system couldn't be given in the general theory of relativity. And the uniform rigid rotation is instantaneous and local inertial, so we could research it in the frame of the special theory of relativity.
To perform the relativistic analysis of uniform rotation, the Galilean transformations and its deduced conclusion, i.e., the formula v = rω 0 is extended to be correct in infinite distances, should be discarded. Nevertheless, we hold the opinion as usual experience that the classical formula of rotation is applicable within short distances. In every observational point, what the observer measures is just the rotational velocities in very close distances, while for the rotational velocities far away from the observers, the calculation using the transformation of velocity [4] in the special theory of relativity is needed.
We analysis the classical rotation firstly, and point out its non-self-consistent disadvantage. A rotational formula of angular velocity with relativistic character is obtained, which approach to classical velocity in the limit of non-relativity. It shows that the classical rotation is only local and slow-velocity approximation of the relativistic one.

II. THE RELATIVISTIC ROTATION
In classical rotation the velocity V of one point relative to another point is equal to the product of its angular frequency ω 0 and its distance from center point R.
We show in Fig.1 We can see from formula (2) that the velocity obtained from classical limit can not coincide with the one derived from the relativistic one due to the use of the transformation formula of the velocity in the special theory of relativity. It implies that the classical rotation is not self-consistent in the framework of the special theory of relativity.
No one will suspect the accuracy of classical rotation formula in daily life. In fact, the correction term in the denominator of Eq.2 is nearly equal to zero under the ordinary condition. Nevertheless, when the classical rotation formula is extended to infinite, i.e.
Rω 0 ≈ c, the Galilean transformations is still applied but not the special relativity. It is shown that this kind of formula can not fulfill the special theory of relativity. This motivates us to improve the classical rotation.
because the general theory of relativity only gives the principle of equivalence, has nothing to do with the rotational system [4,5], the relativistic rotational system can not be obtained through it. Therefore, we start from the special theory of relativity to study the rotational system, and to derive the kinematical equation of the relevant rotational system. From the kinematical point of view, the investigation of the rotation based on the special theory of relativity is suitable.
We can see from formula (2) that the velocity (Eq.1) of classical rotation is not correct for the whole description of rotation under the framework of special relativity. In order to obtain the suitable expression of the rotational velocity, we discard the assumption of formula (1) for the whole-description and only consider it as the local approximation. The formula (1) is only suitable in the limit of Rω 0 ≪ c. Furthermore, we think that there is not intrinsic discrepancy between the inertial and noninertial system if the gravitation is not taken into account, and the instantaneous localization of rotational system is approximately regarded as the inertial one. Moreover, the two adjacent systems of instantaneous localization fulfill the principle of special relativity.
Taking point A as the reference point, we would determine the velocity of point B relative to point A in the framework of special relativity using the former conditions. The velocity of each point relative to point A is The velocity of inertial system B relative to the one of A is the value of V n in the limit of n approaching infinite. According to formula (3), we obtain we could rewrite formula (4) as The U n could be expressed in terms of continued fraction The recursive properties of the continued fraction is as follow [6,7] U n,1 = p 1 q 1 = 1 1 , U n,2 = p 2 q 2 = 2−η 2 , · · · , U n,i = p i q i , · · · , U n = U n,n = pn where the series of p i and q i could be solved in terms of the characteristic equation of recursive series [8] x 2 − 2x + η = 0 (8) The two solutions of formula (8) are Then, the general solutions of p i and q i are It is obtained with the original condition of formula (7).
Inserting formula (11) into formula (10) and considering the formula (7), we can get Inserting η = 1 − (∆Rω 0 /c) 2 and ∆R = R/n into the former formula (12), the relativistic velocity in the limit of n → ∞ is which is the expecting formula of relativistic rotation velocity. We calculate numerically the results of formula (3), and the consistent results are reached with formula (13). Comparing with the classical rotation formula (1), We can see that the classical rotation is the approximation of formula (13) in the limit of c → ∞. For the case far from classical rotation, an additional correction term −R 3 ω 3 0 /c 2 is obtained from formula (13) in comparison with formula (1), i.e., Considering the three points (A, B, and C) in the same line, we could calculate easily the their velocity using the relativistic transfer formula of velocity and formula (13) These show that the velocity relation in formula (13) is self-consistent in the framework of special relativity, and now the relativistic space-time interval are different with the classical one [4]. It could be expressed in cylindrical coordinates It is also observed from the comparison between formula (1) and (13)

III. TRANSVERSE DOPPLER EFFECT IN RELATIVISTIC ROTATION
The expression of transverse Doppler effect can be obtained from special relativity in classical rotation [5]. For instance, there is a light-source with frequency ν 0 at point B in Fig.1. Following the Doppler formula in special relativity and formula (1), the frequency of the light at point A in classical rotation is given as However, in relativistic rotation, through the Doppler formula in special relativity and formula (13), the observed frequency of the light at point A is as follow For further understanding, we plot in Fig.2 It is obviously showed in formula (20) that the discrepancy is hardly observed in ordinary conditions. However, we may observe this discrepancy in the large-scale rotations such as the rotation of galaxy.

IV. SUMMARY
The classical rotation which considers the rotation formula (1) as the whole-conditions is not self-consistent in the special relativistic description. We propose the velocity expression Rω 0 ∼ c. With the flourish development of astrophysics, the self-consistent relativistic rotation would make one understand the phenomena of celestial rotations more suitably than the classical one. Moreover, the relativistic rotation may provide a feasible way for the basic and theoretical description of rotation.