PHY 1371Dr. Jie Zou1 Chapter 39 Relativity (Cont.)

PHY 1371Dr. Jie Zou2 Outline Consequences of the special theory of relativity Simultaneity and the relativity of time Time dilation Twins paradox Length contraction The Lorentz transformation equations

PHY 1371Dr. Jie Zou3 Time dilation Time dilation: The time interval  t measured by an observer moving with respect to a clock is longer than the time interval  t p measured by an observer at rest with respect to the clock.  t p : The proper time interval. The proper time interval is the time interval between two events measured by an observer who sees the events occur at the same point in space. Time dilation is not observed in our everyday lives since the factor  deviates significantly from 1 only for very high speeds.

PHY 1371Dr. Jie Zou4 The factor 

PHY 1371Dr. Jie Zou5 Example 39.1 The period of a pendulum is measured to be 3.00 s in the reference frame of the pendulum. What is the period when measured by an observer moving at a speed of c relative to the pendulum. Note: A moving clock is measured to run more slowly than a clock in your reference frame by a factor . All physical processes, including chemical and biological ones, are measured to slow down when those processes occur in a frame moving with respect to the observer.

PHY 1371Dr. Jie Zou6 The twin paradox-an intriguing consequence of time dilation When the twins depart, they are 20 yr old. Speedo sets out to Planet X, 20 ly from the Earth at a speed of 0.95c relative to the inertial frame of Goslo and then returns back immediately to the Earth at the same speed 0.95 c. Twin paradox: Which twin is the traveler and which is really younger as a result of this experiment? Resolve the apparent paradox: Only Goslo, who is in a single inertial frame, can apply the simple time-dilation formula to Speedo’s trip.

PHY 1371Dr. Jie Zou7 Length contraction The proper length L p of an object: the length measured by someone at rest relative to the object. Length contraction: The length of an object measured by someone in a reference frame that is moving with respect to the object is always less than the proper length. If an object has a proper length L p when it is measured by an observer at rest with respect to the object, then when it moves with speed v in a direction parallel to its length, its length L is measured to be shorter according to

PHY 1371Dr. Jie Zou8 Example 39.3 A spacecraft is measured to be m long and 20.0 m in diameter while at rest relative to an observer. If this spacecraft now flies by the observer with a speed of 0.99 c, what length and diameter does the observer measure? Note that length contraction takes place only along the direction of motion.

PHY 1371Dr. Jie Zou9 The Lorentz transformation equations The equations that are valid for all speeds and enable us to transform coordinates from S to S’ are the Lorentz transformation equations: S  S’: x’ =  (x-vt), y’ = y, z’ = z, t’ =  [t-(v/c 2 )x] S’  S: x =  (x’+vt’), y = y’, z = z’, t =  [t’ + (v/c 2 )x’] Lorentz transformation equations for describing pairs of events (difference form): S  S’:  x’ =  (  x - v  t),  t’ =  [  t – (v/c 2 )  x] S’  S:  x =  (  x’ + v  t’),  t =  [  t’ + (v/c 2 )  x’] Here,  x’ = x 2 ’ – x 1 ’ and  t’ = t 2 ’ – t 1 ’ are the differences measured by observer O’ and  x = x 2 – x 1 and  t = t 2 – t 1 are the differences measured by observer O. y and z coordinates are not affected by motion along the x direction.

PHY 1371Dr. Jie Zou10 Example 39.6 Simultaneity and time dilation revisited: Use the Lorentz transformation equations in difference form to show that (A) simultaneity is not an absolute concept and that (B) a moving clock is measured to run more slowly than a clock that is at rest with respect to an observer.

PHY 1371Dr. Jie Zou11 Homework Ch. 39, P. 1278, Problems: #5, 6, 7, 22, 23, 24.