(Dated: September 3, 2015)

Transcription

1 LORENTZ-FITZGERALD LENGTH CONTRACTION DUE TO DOPPLER FACTOR Louai Hassan Elzein Basheir 1 Physics College, Khartoum University, Sudan. P.O. Box: Zip Code: (Dated: September 3, 2015) This paper is prepared to show the mathematical proof of the Lorentz-Fitzgerald length contraction to be an effect of the Doppler factor instead of the Lorentz factor. 1

2 THEORETICAL BACKGROUND Derivation of Lorentz-Fitzgerald Length Contraction 1 At the beginning, we are going to mention the derivation of the Lorentz-Fitzgerald length contraction. Therefore consider the situation schematically illustrated in Figure 1, where observers in S have placed a mirror M perpendicular to their X-axis, thus, observers in S reason that (noticing that we switch the symbols that have been used in the reference, i.e. S, x, y, z, t becomes S, x, y, z, t and vice versa.) c = 2 x t where t is the time interval required for the light pulse to travel from A to M and back to A. Thus, the distance they measure between A and M is given by x = ½c t, (1) where t should be interpreted as a length measurement. Observers in system S are operationally perform the necessary measurements for the length as the following; consider the motion of the light pulse from A to M as viewed by observers in the S -frame and depicted in Figure 2. x = (c v) t A B. (2) Thus, the time interval t A B is given by t A B = x c v. Now, consider the motion of the light pulse from M back to the Y-axis, as viewed by observers in system S. In this case, the distance x is immediately obtained from Figure 3 as x = (c + v) t B C. (3) Solving the above equation for the time interval t B C gives t B C = x c + v. which is not the same as t A B as far as observers in system S are concerned. Let observers in S define the time interval t as that time required for the pulse of light in S to travel from A to M and back to A, as viewed in their reference frame. Clearly, then t = t A B + t B C = x c v + x c + v Page 2 of 6

3 which, when solved for x, yields ( ) x = ½c t 1 v2 c 2 Substitution of the time dilation formula gives x = ½cγ t (1 β 2 ) (4) Now, from Eqs.(1) and (4) we have the famous Lorentz-Fitzgerald length contraction equation in the form x = x (5) γ In view of this result, the length measurement obtained by S -observers on a moving object is always less than the corresponding proper length measured by S-observers on the object at rest (i.e., x < x), when v is not small and the length of the object is parallel to the axis of relative motion. ANALYSIS Lorentz-Fitzgerald Length Contraction Return to Doppler Effect We are going to use the same derivation of Lorentz-Fitzgerald length contraction, since we have from equation 2 x = (c v) t A B But from the time dilation formula one can write the time interval t A B as t A B = γ t A B = γ(½ t) where γ t A B is the time interval required for the light pulse to travel from A to M as measured in S-frame, hence one obtains x = (c v) γ(½ t) = γ(1 β) ½c t, Finally, substituting Eq.(1) yields x = γ(1 β) x (6) Similarly, equation 3 gives x = γ(1 + β) x (7) where γ(1 β) and γ(1 + β) are the Doppler factors. From equations (6) and (7), one discovers x A B x B C (8) Page 3 of 6

4 CONCLUSION We have found the relativistic effect on the spatial component of spacetime to be asymmetric along the direction of the motion, that is, the approaching frame of reference and receding frame of reference are not equal relativistically; observers whom observe approaching frames of references measure contraction in spatial length while observers whom observe receding frames of references measure expansion in spatial length, and as a result of Doppler factor the spatial coordinate transfers from one inertial frame of reference to another like wavelength. Page 4 of 6

5 REFERENCES 1 Marshall l. Burns, Modern Physics for Science and Engineering, First Edition (electronic copy) p.42 to 48. Figure 1. observers in S. A horizontal pulse of light being reflected from a vertical mirror in S, as viewed by Page 5 of 6

6 Figure 2. A horizontal pulse of light propagating towards a vertical mirror in S, as viewed by observers in S. Figure 3. observers in S. A horizontal pulse of light being reflected from a vertical mirror in S, as viewed by Page 6 of 6