مفارقة إرنفست

مفارقة إرنفست Ehrenfest paradox، هي مفارقة فيزيائية تتعلق بالنقاش عن جسم أسطواني صلب يدور بسرعة كبيرة حول محوره. ولأن الشعاع الهندسي للاسطوانة يكون عموديّاً بالنسبة إلى الحركة الدورانية يبقى ذلك البعد ثابتاً نظريّاً، فلا يصيبه تقلص الأبعاد الناتج من التحرك بسرعة قريبة من الضوء، وفق معادلات لورنز عن النسبية الخاصة..

في المقابل، إذا جرت مراقبة النقطة عند آخر الشعاع (على سطح الأسطوانة) فإنها تتحرك طوليّاً بسرعة كبيرة، ما يعني أن محيط الأسطوانة لا بد من أن يتقلّص وفق معادلات لورنز ذاتها. وبذا، يكون الشعاع ثابتاً ومتقلصّاً وفق المعادلات ذاتها، ما يتنافى مع الحس السليم.

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 جوهر التناقض

Imagine a disk of radius R rotating with constant angular velocity $\omega$.

Ehrenfest paradox - Circumference of a rotating disk should contract but not the radius, as radius is in perpendicular to the direction of motion.

The reference frame is fixed to the stationary center of the disk. Then the magnitude of the relative velocity of any point in the circumference of the disk is $\omega R$. So the circumference will undergo Lorentz contraction by a factor of ${\sqrt {1-(\omega R)^{2}/c^{2}}}$.

However, since the radius is perpendicular to the direction of motion, it will not undergo any contraction. So ${\frac {\mathrm {circumference} }{\mathrm {diameter} }}={\frac {2\pi R{\sqrt {1-(\omega R)^{2}/c^{2}}}}{2R}}=\pi {\sqrt {1-(\omega R)^{2}/c^{2}}}$. This is paradoxical, since in accordance with Euclidean geometry, it should be exactly equal to $\pi$.

 زعم إرنفست

Ehrenfest considered an ideal Born-rigid cylinder that is made to rotate. Assuming that the cylinder does not expand or contract, its radius stays the same. But measuring rods laid out along the circumference $2\pi R$ should be Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated Born rigid disk should shatter.

Thus Ehrenfest argued by reductio ad absurdum that Born rigidity is not generally compatible with special relativity. According to special relativity an object cannot be spun up from a non-rotating state while maintaining Born rigidity, but once it has achieved a constant nonzero angular velocity it does maintain Born rigidity without violating special relativity, and then (as Einstein later showed) a disk-riding observer will measure a circumference:

$C^{\prime }={\frac {2\pi R}{\sqrt {1-v^{2}/c^{2}}}}$.

 تاريخ موجز

This figure shows the world line of a Langevin observer (red helical curve). The figure also depicts the light cones at several events with the frame field of the Langevin observer passing through that event.

 انظر أيضاً

• أحداثيات بورن
• Length contraction
• Relativistic disk
• Bell's spaceship paradox
• مفارقة السلم
• مفارقة فيزيائية
• Supplee's paradox
• مفارقة التوأم

 المصادر

أوراق ذات قيمة تاريخية:

بعض المراجع الكلاسيكية "الحديثة":

• Reichenbach, Hans (1969). Axiomatization of the Theory of Relativity. Berkeley: University of California Press. LCCN 68021540.
• Cantoni, V. (1968). "What is wrong with Relativistic Kinematics?". Il Nuovo Cimento. 57 B: 220–223.
• Grøn, Ø. (1975). "Relativistic description of a rotating disk". Amer. J. Phys. 43 (10): 869–876. Bibcode:1975AmJPh..43..869G. doi:10.1119/1.9969.
• Landau, L. D. & Lifschitz, E. F. (1980). The Classical Theory of Fields (4th ed.). London: Butterworth-Heinemann. ISBN 0-7506-2768-9.{{cite book}}: CS1 maint: multiple names: authors list (link) See Section 84 and the problem at the end of Section 89.

بعض الأعمال التجريبية ونقاشات لاحقة:

• Davies, P. A. & Jennison, R. C. (1975). "Experiments involving mirror transponders in rotating frames". J. Phys. A: Math. Gen. 8, No.9 (9): 1390–7. Bibcode:1975JPhA....8.1390D. doi:10.1088/0305-4470/8/9/007.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Ashworth, D. G. & Jennison, R. C. (1976). "Surveying in rotating systems". J. Phys. A: Math. Gen. 9 (1): 35–43. Bibcode:1976JPhA....9...35A. doi:10.1088/0305-4470/9/1/008.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Davies, P. A. (1976). "Measurements in rotating systems". J. Phys. A: Math. Gen. 9, No.6 (6): 951–9. Bibcode:1976JPhA....9..951D. doi:10.1088/0305-4470/9/6/014.
• Boone, P. F. (1977). "Relativity of rotation". J. Phys. A: Math. Gen. 10 (5): 727–44. Bibcode:1977JPhA...10..727B. doi:10.1088/0305-4470/10/5/007.
• Ashworth, D. G. & Davies, P. A. (1979). "Transformations between inertial and rotating frames of reference". J. Phys. A: Math. Gen. 12, No.9 (9): 1425–40. Bibcode:1979JPhA...12.1425A. doi:10.1088/0305-4470/12/9/011.{{cite journal}}: CS1 maint: multiple names: authors list (link)

مراجع حديثة مختارة:

• Rizzi, G.; & Ruggiero, M.L. (2002). "Space geometry of rotating platforms: an operational approach". Found. Phys. 32 (10): 1525–1556. doi:10.1023/A:1020427318877.{{cite journal}}: CS1 maint: multiple names: authors list (link) They give a precise definition of the "space of the disk" (non-Euclidean), and solve the paradox without extraneous dynamic considerations. See also the eprint version.

• Rizzi, G. ; & Ruggiero, M. L. (2004). Relativity in Rotating Frames. Dordrecht: Kluwer. ISBN 1-4020-1805-3.{{cite book}}: CS1 maint: multiple names: authors list (link) This book contains a comprehensive historical survey by Øyvind Grøn, on which the "brief history" in this article is based, and some other papers on the Ehrenfest paradox and related controversies. Hundreds of additional references may be found in this book, particularly the paper by Grøn.
• Pauri, Massimo; & Vallisneri, Michele (2000). "Märzke-Wheeler coordinates for accelerated observers in special relativity". Found. Phys. Lett. 13 (5): 401–425. doi:10.1023/A:1007861914639.{{cite journal}}: CS1 maint: multiple names: authors list (link) Studies a coordinate chart constructed using radar distance "in the large" from a single Langevin observer. See also the eprint version.
• Nikolic, Hrvoje (2000). "Relativistic contraction and related effects in noninertial frames". Phys. Rev. A. 61 (3): 032109. arXiv:gr-qc/9904078. Bibcode:2000PhRvA..61c2109N. doi:10.1103/PhysRevA.61.032109. Studies general non-inertial motion of a point particle and treats rotating disk as a collection of such non-inertial particles. See also the eprint version.

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 وصلات خارجية

• The Rigid Rotating Disk in Relativity, by Michael Weiss (1995), from the sci.physics FAQ.
• Einstein's Carousel (section 3.4.4), by B. Crowell