Length contraction - Special Relativity
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Deduction of the Contraction of Length in Special Relativity using Lorentz Transformation Equation and an Animation to Exhibit the Effect
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Length contraction - Special Relativity
- 2. Measurements Change with Frames Now? S O v 𝑥1 𝑥2
- 3. Measurements Change with Frames Now? S O v 𝑥1 𝑥2
- 4. Measurements Change with Frames Now? S O S’ O’ v 𝑥1 𝑥2 𝑥′1 𝑥′2 𝑥2 & 𝑥1 𝑥2′ & 𝑥1′
- 5. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾
- 6. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1
- 7. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 8. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≪ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 9. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ( )𝛾 𝑣 ≪ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 10. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡) ( )𝛾 𝑣 ≪ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 11. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡) ⇒ 𝑙′ = 𝑥2 − 𝑥1 ( )𝛾 𝑣 ≪ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 12. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡) ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 𝑣 ≪ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 13. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡) ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 ⇒ 𝑙′ = 𝑙 𝑣 ≪ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 14. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡) ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 ⇒ 𝑙′ = 𝑙 𝑣 ≪ 𝑐 𝑣 ≈ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1
- 15. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡) ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 ⇒ 𝑙′ = 𝑙 𝑣 ≪ 𝑐 𝑣 ≈ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1
- 16. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 ⇒ 𝑙′ = 𝑙 𝑣 ≪ 𝑐 𝑣 ≈ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡1) − (𝑥1 − 𝑣𝑡2)
- 17. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 ⇒ 𝑙′ = 𝑙 𝑣 ≪ 𝑐 𝑣 ≈ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡1) − (𝑥1 − 𝑣𝑡2) Length will be measured at an instant. Not at two different times.
- 18. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = 𝑥2 − 𝑥1 = 𝑙 ( )𝛾 ⇒ 𝑙′ = 𝑙 𝑣 ≪ 𝑐 𝑣 ≈ 𝑐 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆 = 𝑙 = 𝑥2 − 𝑥1 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑆𝑐𝑎𝑙𝑒 𝑖𝑛 𝑆′ = 𝑙′ = 𝑥′2 − 𝑥′1 ⇒ 𝑙′ = 𝑥′ 2 − 𝑥′ 1 ⇒ 𝑙′ = (𝑥2 − 𝑣𝑡1) − (𝑥1 − 𝑣𝑡2) Length will be measured at an instant. Not at two different times. ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥2 − 𝑣𝑡 𝛾
- 19. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾
- 20. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐
- 21. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾
- 22. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾
- 23. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 + 𝑣𝑡 1 − 𝑣2 𝑐2
- 24. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 + 𝑣𝑡 1 − 𝑣2 𝑐2 = 𝑥2 − 𝑥1 1 − 𝑣2 𝑐2
- 25. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 + 𝑣𝑡 1 − 𝑣2 𝑐2 = 𝑥2 − 𝑥1 1 − 𝑣2 𝑐2 = 𝑙 1 − 𝑣2 𝑐2
- 26. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 + 𝑣𝑡 1 − 𝑣2 𝑐2 = 𝑥2 − 𝑥1 1 − 𝑣2 𝑐2 = 𝑙 1 − 𝑣2 𝑐2
- 27. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 + 𝑣𝑡 1 − 𝑣2 𝑐2 = 𝑥2 − 𝑥1 1 − 𝑣2 𝑐2 = 𝑙 1 − 𝑣2 𝑐2 𝒍′ = 𝒍𝜸
- 28. Invariance of Length? ⇒ 𝑥′ = 𝑥 − 𝑣𝑡 ( )𝛾 𝑣 ≈ 𝑐 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 𝛾 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 − 𝑣𝑡 𝛾 ⇒ 𝑙′ = 𝑥2 − 𝑣𝑡 − 𝑥1 + 𝑣𝑡 1 − 𝑣2 𝑐2 = 𝑥2 − 𝑥1 1 − 𝑣2 𝑐2 = 𝑙 1 − 𝑣2 𝑐2 𝒍′ = 𝒍𝜸 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖𝑛 𝑆′ ≠ 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖𝑛 𝑆
- 30. Length Contraction S O S’ v 𝑥′1 𝑥′2 𝑥1 𝑥2 𝑥2 & 𝑥1 𝑥2′ & 𝑥1′ 0.866𝑚 1𝑚