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Proper Time

Proper Time. LL 2 Section 3. An inertial frame with some clocks moving in arbitrary manner. v (t). Y. X. Our frame, our clock. At each different moment, their motion can be considered uniform.

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Proper Time

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  1. Proper Time LL 2 Section 3

  2. An inertial frame with some clocks moving in arbitrary manner. v(t) Y X Our frame, our clock.

  3. At each different moment, their motion can be considered uniform. • At each moment, we can rigidly link a coordinate system with the clocks to make a set of inertial reference systems Y’ Y v, dr t’ K’ X’ K X t

  4. During dt on our clock in K, moving clock travels a distance dr = Sqrt[dx2+dy2+dz2] What time interval dt’ is indicated for this period on the moving clock? • dt’ = dt • dt’ > dt • dt’ < dt 1 2 3

  5. What time interval dt’ is indicated for this period on the moving clock? • dt’ = dt • dt’ > dt • dt’ < dt

  6. Moving object’s proper time is always less than the corresponding time in the rest system. • Moving clocks go more slowly • Time interval observed in rest system is dilated relative to proper time. • These conclusions do not assume uniform rectilinear motion of moving clocks. • They do assume rest frame is inertial.

  7. For uniform rectilinear motion of clock in K’ relative to K, both frames are inertial? • From point of view of observer in K, K’ clock lags. • From point of view of observer in K’, K clock lags. • Contradiction? NO Y’ Y V t’ K’ X’ K X t

  8. To compare clock rates in two reference systems, we need several clocks in one frame, but only one clock in the other frame. • Process is not symmetric • Clock that appears to lag is always the one being compared to the many. • Because time is not absolute, the many clocks are not synchronized. Y Y’ V t’ K’ X’ K X t

  9. Consider a clock that describes a closed path • Moving clock appears to lag relative to clock in K • Converse reasoning is now impossible, because clock in closed trajectory is not in uniform rectilinear motion, so its reference system is not inertial. • Argument that leads to result that clocks at rest lags from point of view of moving clock is not valid, since laws of physics are different in two frames. Y’ Y t’ K’ X’ t K X

  10. Clock at rest follows a straight world line parallel to the t axis. • If clock follows closed trajectory, the world line is a curve through the two points. • Clock at rest always indicates a greater time interval than the clock that leaves and returns to a starting point. t b a x

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