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Exploring the Twin Paradox and Temporal Dynamics in Special Relativity

Delve into the intriguing world of special relativity as we examine the Twin Paradox and its implications on time perception. Discover how Betty and Ann's journeys through spacetime result in differing ages due to relativistic effects. This overview will cover the Lorentz transformation, velocity transformation, and the invariant interval, providing a comprehensive understanding of how travel influences time. Explore thought-provoking scenarios that challenge our perception of simultaneity and the nature of time itself.

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Exploring the Twin Paradox and Temporal Dynamics in Special Relativity

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  1. Special Relativity4

  2. Topics • Paradoxes • Twin • Temporal • The Lorentz Transformation • Velocity Transformation • The Interval • Summary

  3. Betty’s Now in 2022 Ann’s Now in 2030 2030 E-mail sent by Betty in 2016, Betty’s time, received by Ann in 2028, Ann’s time. 2028 Betty’s Now in 2016 Betty’s Now in 2010 2020 Ann’s Now in 2020 E-mail sent by Ann in 2012, Ann’s time, received by Betty in 2016, Betty’s time. 2013.6 2012 2010 Ann’s Now in 2010 8 light years Twin Paradox Betty 8 years younger! Puzzle: Why does Betty become younger?

  4. Send super-luminal signal to star A in 2010, arriving in 2016 according to Betty. But for Ann, signal sent in 2010 arrives in 2007! Betty’s Now in 2010 Betty’s Now in 2016 8 light years 2008 Ann and Betty’s Now in 2007.2 2007.2 A Super-luminal signal sent from A arrives in 2008, preventing signal sent in 2010! Temporal Paradox 2020 Ann’s Now in 2020 Ann’s Worldline 2013.6 8 light years Ann’s Now in 2010 2010

  5. t t' P x' C Q x O A B The Lorentz Transformation Define t = tBC + tCP x = xOA + xAB What is the map between the two different frames of reference? x' = xOQ t' = tQP tCP = tQP / g tBC = v x / c2 xOA = xOQ / g xAB = v t

  6. t t' P x' C Q x O A B Lorentz Transformation Lorentz transformation (t,x,y,z) → (t’,x’,y’,z’) t = g (t – v x/c2) x’ = g (x – v t) y’ = y z’ = z tCP = tQP / g tBC = v x / c2 xOA = xOQ / g xAB = v t

  7. t t' P x' Q x O B Velocity Transformation Suppose point P is moving at velocity u = (ux, uy, uz) in the S frame. To find its velocity in the S’ frame we start with: S’ frame S frame

  8. t t' P x' Q x O B Velocity Transformation Example: Consider u’x S’ frame S frame

  9. t t' P S’ frame x' x O B Q S frame Velocity Transformation Similarly, we arrive at Review Example 1-4

  10. P A O B The Interval The distance between O and P is independent of the components used to calculate it: OB2 + BP2 = OP2 = OA2 + AP2 OP2 is said to be invariant. The formula for computing it is called the metric. In 1908, Herman Minkowski introduced a similar idea for spacetime Hermann Minkowski (1864 -1909)

  11. t t' P x' Q x O B The Interval Try OP2 = (ct)2 + x2 + y2 + z2 What is the spacetime distance, i.e., the interval, between event O and event P? This does not work! Minkowski showed that the correct formula is OP2= (ct)2– (x2 + y2 + z2) = (ct’)2– (x’2 + y’2 + z’2) Extra Credit: Show that OP2 is invariant

  12. The Interval In general, the interval Ds2 between two events is either timelike Ds2 = c2Dt2 – Ds2 or spacelike Ds2 = Ds2 – c2Dt2 or null Ds2 = c2Dt2 - Ds2 = 0 depending on whether the temporal difference cDt or spatial difference Ds dominates

  13. The Interval Since the interval Ds2 is an invariant we can immediately deduce the following: when the spatial separation between two events Ds = 0 their temporal separation is invariant: Dt = Ds / c Proper Time when the temporal separation between two events Dt = 0 their spatial separation is invariant: Ds= Ds Proper Distance

  14. Summary • Super-luminal travel would lead to temporal paradoxes • The temporal separation of events depends on the path taken between them and is greatest for an inertial, i.e., non-accelerating, path yields the longest temporal separation • The distance between events in spacetime is computed from the interval Ds2 = c2Dt2 – Ds2

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