Lecture 20 Relativistic Effects

1. Lecture 20Relativistic Effects Chapter 26.6  26.10 Outline • Relativity of Time • Time Dilation • Length Contraction • Relativistic Momentum and Addition of Velocities

2. Basis for Special Relativity • Einstein’s relativity: • The laws of physics are the same in any inertial frame of reference. • The speed of light (c = 300,00 km/s) is the same for all observers, irrespective of their relative speeds. • Consequences: • Relativity of Time • Time Dilation • Length Contraction

3. Relativity of Time

4. Explanation of Time Relativity

5. Simultaneity in Relativity • Conclusions from the box car experiment: • Two simultaneous events in one reference frame can be non-simultaneous in another reference frame, moving with respect to the first one. • Simultaneity is not absolute, but depends on the observer’s state of motion. • There is no preferred inertial reference frame.

6. ct/2 Time Dilation d Experiment vt/2 tp= 2d/c  time interval for light to reach the mirror at the stationary reference frame (proper time) t  this interval in the moving reference frame Speed of light c is equal in both frames ct 2 v t 2  =  + d2 2 2 2d t =  c(1 v2/c2) t = tp   = 1/(1 v2/c2)

7. Conclusions from Time Dilation t = tp  is always grater than 1 The time interval between two events measured by an observer moving with respect to the clock is longer than that between the same two events measured by an observer at rest with the clock. A clock moving past an observer runs slower than an identical clock at rest with the observer. Proper time is the time interval between two events as measured by an observer who sees them at same position.

8. Twin Paradox Twin paradox is a consequence of time dilation. The age of the traveling twin is indeed smaller than that of the non-traveling one, because the former experiences acceleration and lives in a non-inertial reference frame. Only the non-traveling twin can apply Einstein’s relativity and determine the time dilation.

9. Length Contraction The proper length Lp is the length of an object measured at rest with it. In a moving reference frame, the length L is shorter. L = Lp/ = Lp (1 v2/c2) Length contraction occurs only in the direction of motion. Example: A right triangle with the sides of 10 m each at the right angle. How would the angles change if the triangle moves at 0.9 c along one of the short sides? Solution: l = 10 (1  0.92) = 1.9 m tan ' = 1.9/10 = 0.19  ' = 10.8  = 45

10. Length Contraction '  10 10 10 1.9 v = 0.9 c

11. Other Effects of Relativity Momentum is the product of mass and velocity p(rest) = mv p(relativistic) = mv Relativistic momentum obeys the law of momentum conservation even at velocities, close to c. Relativistic addition of velocities (works for parallel or antiparallel paths) vx  speed of object x vy speed of object y vxy relative speed of objects x and y vx + vy vxy =  1 + (vxvy)/c2

12. Relativistic Addition of Velocities

13. Equivalence of Mass and Energy Kinetic energy of a non-relativistic object KE=mv2/2 For a relativistic object: KE = mc2 – mc2 The term independent of velocity is the rest energy ER = mc2 Total energy, defined as E = mc2, is the sum of the kinetic and rest energies. Any object has energy by virtue of its mass alone. Mass can be transferred into energy and back.

14. Summary • There are two postulates of special relativity: • The laws of physics are the same in any inertial frame of reference. • The speed of light is the same for all observers, irrespective of their relative speeds. • The consequences (relativity of time, time dilation, length contraction, velocity addition) seem unusual for our experience, but can be verified by experiments or observations of distant objects.