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Oblique-Length Contraction Factor in the Special Theory of Relativity Volume 1 PROGRESS IN PHYSICS January, 2013 Oblique-Length Contraction Factor in the Special Theory of Relativity Florentin Smarandache University of New Mexico 705 Gurley Ave. Gallup, NM 87301, USA. E-mail: [email protected] In this paper one generalizes the Lorentz Contraction Factor for the case when the lengths are moving at an oblique angle with respect to the motion direction. One shows that the angles of the moving relativistic objects are distorted. 1 Introduction 3 Time-Dilation Factor According to the Special Theory of Relativity, the Lorentz Time-Dilation Factor D(v) is the inverse of Lorentz Factor: Contraction Factor is referred to the lengths moving along 1 the motion direction. The lengths which are perpendicular on D(v) = [1, + ] f or v [0, c] (3) the direction motion do not contract at all [1]. v2 ∈ ∞ ∈ 1 2 In this paper one investigates the lengths that are oblique r − c to the motion direction and one finds their Oblique-Length Δt = Δt0 D(v) (4) Contraction Factor [3], which is a generalization of the ∙ Lorentz Contraction Factor (for θ = 0) and of the perpen- where Δt = non-proper time and, Δt0 = proper time. D(0) = 1, dicular lengths (for θ = π/2). We also calculate the distorted meaning no time dilation [as in Absolute Theory of Relativity angles of lengths of the moving object. (ATR)]; D(c) = limv c D(v) = + , which means according to the Special Theory→ of Relativity∞ (STR) that if the rocket 2 Length-Contraction Factor moves at speed ‘c’ then the observer on earth measures the elapsed non-proper time as infinite, which is unrealistic. v = c Length-Contraction Factor C(v) is just Lorentz Factor: is the equation of the vertical asymptote to the curve of D(v). v2 4 Oblique-Length Contraction Factor C(v) = 1 2 [0, 1] f or v [0, 1] (1) r − c ∈ ∈ The Special Theory of Relativity asserts that all lengths in the direction of motion are contracted, while the lengths at right L = L0 C(v) (2) angles to the motion are unaffected. But it didn’t say anything ∙ about lengths at oblique angle to the motion (i.e. neither per- where L = non-proper length (length contracted), L0 = proper pendicular to, nor along the motion direction), how would length. C(0) = 1, meaning no space contraction [as in Abso- they behave? This is a generalization of Galilean Relativity, lute Theory of Relativity (ATR)]. i.e. we consider the oblique lengths. The length contraction C(c) = 0, which means according to the Special Theory factor in the motion direction is: of Relativity (STR) that if the rocket moves at speed ‘c’ then the rocket length and laying down astronaut shrink to zero! v2 C(v) = 1 . (5) This is unrealistic. 2 r − c Suppose we have a rectangular object with width W and length L that travels at a constant speed v with respect to an observer on Earth. Fig. 1: The graph of the Time-Dilation Factor Fig. 2: A rectangular object moving along the x-axis 60 Florentin Smarandache. Oblique-Length Contraction Factor in The Special Theory of Relativity January, 2013 PROGRESS IN PHYSICS Volume 1 Fig. 3: Contracted lengths of the rectangular object moving along the x-axis Fig. 4: The graph of the Oblique-Length Contraction Factor OC(v, θ) Then its lengths contract and its new dimensions will be periodic of period π, since: L0 and W0: where L0 = L C(v) and W0 = W. The initial diagonal of the rectangle ABCD∙ is: OC(v, π + θ) = C(v)2 cos2(π + θ) + sin2(π + θ) 2 2 δ = AC = BD = √L + W = qC(v)2[ cos θ]2 + [ sin θ]2 | | | | (6) − − (11) = √L2 + L2 tan2 θ = L √1 + tan2 θ = pC(v)2 cos2 θ + sin2 θ while the contracted diagonal of the rectangle A0B0C0D0 is: = OCq (v, θ). More exactly about the OC(v, θ) range: δ0 = A0C0 = B0D0 | | | | OC(v, θ) [C(v), 1] (12) = (L )2 + (W )2 = L2 C(v)2 + W2 (7) ∈ 0 0 ∙ 2 2 2 2 2 2 but since C(v) [0, 1] , one has: = pL C(v) + L tanpθ = L C(v) + tan θ. ∈ p p OC(v, θ) [0, 1]. (13) Therefore the lengths at oblique angle to the motion are ∈ contracted with the oblique factor The Oblique-Length Contractor 2 2 δ0 L C(v) + tan θ OC(v, θ) = = OC(v, θ) = C(v)2 cos2 θ + sin2 θ (14) δ pL √1 + tan2 θ (8) q 2 2 C(v) + tan θ 2 2 2 is a generalization of Lorentz Contractor C(v), because: when = 2 = C(v) cos θ + sin θ r 1 + tan θ θ = 0 or the length is moving along the motion direction, then q OC(v, 0) = C(v). Similarly which is different from C(v). OC(v, π) = OC(v, 2π) = C(v). (15) δ0 = δ OC(v, θ) (9) ∙ π Also, if θ = 2 , or the length is perpendicular on the mo- where 0 OC(v, θ) 1. tion direction, then OC(v, π/2) = 1, i.e. no contraction oc- ≤ ≤ For unchanged constant speed v, the greater is θ in 0, π 3π 2 curs. Similarly OC(v, 2 ) = 1. the larger gets the oblique-length contradiction factor, and re- ciprocally. By oblique length contraction, the angle 5 Angle Distortion π π Except for the right angles (π/2, 3π/2) and for the 0, π, and θ 0, , π (10) 2π, all other angles are distorted by the Lorentz transform. ∈ 2 ∪ 2 Let’s consider an object of triangular form moving in the is not conserved. direction of its bottom base (on the x-axis), with speed v, as In Fig. 4 the horizontal axis represents the angle θ, while in Fig. 5: π π the vertical axis represents the values of the Oblique-Length θ 0, , π (16) Contraction Factor OC(v, θ) for a fixed speed v. Hence C(v) ∈ 2 ∪ 2 is thus a constant in this graph. The graph, for v fixed, is is not conserved. Florentin Smarandache. Oblique-Length Contraction Factor in The Special Theory of Relativity 61 Volume 1 PROGRESS IN PHYSICS January, 2013 α2 C(v)2 + β2 OC(v, ] A + B)2 γ2 OC(v, ] B)2 ]C0 = arccos ∙ ∙ − ∙ . 2α β OC(v) OC(v, ] A + B) ∙ ∙ ∙ As we can see, the angles ]A0, ]B0, and ]C0 are, in gen- eral, different from the original angles A, B, and C respec- tively. The distortion of an angle is, in general, different from the distortion of another angle. Submitted on November 12, 2012 / Accepted on November 15, 2012 References Fig. 5: 1. Einstein A. On the Electrodynamics of Moving Bodies. Annalen der Physik, 1905, v. 17, 891–921. 2. Smarandache F. Absolute Theory of Relativity and Parameterized Spe- cial Theory of Relativity and Noninertial Multirelativity. Somipress, Fes, 1982. 3. Smarandache F. New Relativistic Paradoxes and Open Questions. Somipress, Fes, 1983. Fig. 6: The side BC = α is contracted with the contraction fac- tor C(v) since| BC|is moving along the motion direction, there- fore B0C0 = α C(v). But the oblique sides AB and CA are contracted| | respectively∙ with the oblique-contraction factors OC(v, ]B) and OC(v, ]π C), where ]B means angle B: − A0B0 = γ OC(v, ]B) (17) ∙ and C0A0 = β OC(v, ]π C) = β OC(v, ]A + B) (18) ∙ − ∙ since ]A + ]B + ]C = π. (19) Triangle ABC is shrunk and distorted to A0B0C0 as in Fig. 6. Hence one gets: α0 = α C(v) ∙ β0 = β OC(v, ] A + B) (20) ∙ γ0 = γ OC(v, ] B) ∙ In the resulting triangle A0B0C0, since one knows all its side lengths, one applies the Law of Cosine in order to find each angle ]A0, ]B0, and ]C0. Therefore: α2 C(v)2 + β2 OC(v, ] A + B)2 + γ2 OC(v, ] B)2 ]A0 = arccos − ∙ ∙ ∙ 2β γ OC(v, ] B) OC(v, ] A + B) ∙ ∙ ∙ α2 C(v)2 β2 OC(v, ] A + B)2 + γ2 OC(v, ] B)2 ]B0 = arccos ∙ − ∙ ∙ 2α γ OC(v) OC(v, ] B) ∙ ∙ ∙ 62 Florentin Smarandache. Oblique-Length Contraction Factor in The Special Theory of Relativity.
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