Chapter 28 Relativity

Special Relativity Einstein’s Special Theory of Relativity, published in 1905, was based on two postulates: I. The laws of physics are the same for all frames of reference moving at a constant velocity with respect to each other. II. The free space velocity of light c is constant for all observers, independent of their state of motion. (c = 3 x 108 m/s)

Rest and Motion What do we mean when we say that an object is at rest . . . or in motion? Is anything at rest? We sometimes say that man, computer, phone, and desk are at rest. We forget that the Earth is also in motion. What we really mean is that all are moving with the same velocity. We can only detect motion in reference to something else.

No Preferred Frame of Reference What is the velocity of the bicyclist? Earth 25 m/s 10 m/s East West We cannot say without a frame of reference. Assume bike moves at 25 m/s,W relative to Earth and that platform moves 10 m/s, E relative to Earth. What is the velocity of the bike relative to platform? Assume that the platform is the reference, then look at relative motion of Earth and bike.

Reference for Motion (Cont.) To find the velocity of the bike relative to platform, we must imagine that we are sitting on the platform at rest (0 m/s) relative to it. We would see the Earth moving westward at 10 m/s and the bike moving west at 35 m/s. Earth 25 m/s 10 m/s East West Earth as Reference 0 m/s 10 m/s 35 m/s 0 m/s East West Platform as Reference

Frame of Reference Earth as Reference 25 m/s 10 m/s East West Earth as Reference Consider the velocities for three different frames of reference. 10 m/s 35 m/s 0 m/s East West Platform as Reference 0 m/s 35 m/s East West Bicycle as Reference 25 m/s

Constant Velocity of Light Platform v = 30 m/s to right relative to ground. 10 m/s c Velocities observed inside car 40 m/s 20 m/s c Velocities observed from outside car The light from two flashlights and the two balls travel in opposite directions. The observed velocities of the ball differ, but the speed of light is independent of direction.

Velocity of Light (Cont.) Platform moves 30 m/s to right relative to boy. 10 m/s c 30 m/s Each observer sees c = 3 x 108 m/s Outside observer sees very different velocities for balls. The velocity of light is unaffected by relative motion and is exactly equal to: c = 2.99792458 x 108 m/s

Time Measurements Since our measurement of time involves judg-ments about simul-taneous events, we can see that time may also be affected by relative motion of observers. In fact, Einstein's theory shows that observers in relative motion will judge times differently - furthermore, each is correct.

Relative Time Consider cart moving with velocity v under a mirrored ceiling. A light pulse travels to ceiling and back in time Dto for rider and in time Dt for watcher. Light path for rider d Dto Light path for watcher d x R Dt R

Relative Time (Cont.) Light path for rider d Dto R Dt R x Substitution of:

Time Dilation Equation Einstein’s Time dilation Equation: Dt = Relative time (Time measured in frame moving relative to actual event). Dto= Proper time (Time measured in the same frame as the event itself). v = Relative velocity of two frames. c = Free space velocity of light (c = 3 x 108 m/s).

Proper Time The key to applying the time dilation equation is to distinguish clearly between proper time Dto and relative time Dt. Look at our example: Proper Time d Dto Event Frame Relative Time Dt Relative Frame Dt > Dto

Example 1: Ship A passes ship B with a relative velocity of 0 Example 1: Ship A passes ship B with a relative velocity of 0.8c (eighty percent of the velocity of light). A woman aboard Ship B takes 4 s to walk the length of her ship. What time is recorded by the man in Ship A? Proper time Dto = 4 s v = 0.8c A B Find relative time Dt Dt = 6.67 s

The spacecraft is moving past the earth at a constant speed of Time Dilation The spacecraft is moving past the earth at a constant speed of 0.92 times the speed of light. the astronaut measures the time interval between ticks of the spacecraft clock to be 1.0 s. What is the time interval that an earth observer measures?

Traveling twin ages more! The Twin Paradox Two twins are on Earth. One leaves and travels for 10 years at 0.9c. Traveling twin ages more! When traveler returns, his twin is 23 years older due to time dilation! Paradox: Since motion is relative, isn’t it just as true that the man who traveled should also be 23 years older? After all, the traveler was also seing the man on earth moving as well.

The Twin Paradox Explained The traveling twin’s motion was not uniform. Acceleration and forces were needed to go to and return from space. Traveling twin ages more! The twin on earth ages more and not the one who traveled. This is NOT science fiction. Atomic clocks placed aboard aircraft sent around Earth and back have verified the time dilation.

Moving objects are foreshortened due to relativity. Length Contraction Since time is affected by relative motion, length will also be different: 0.9c Lo L Lo is proper length v L is relative length Moving objects are foreshortened due to relativity.

Length recorded by observer: Example 2: A meter stick moves at 0.9c relative to an observer. What is the relative length as seen by the observer? 0.9c 1 m Lo L = ? Length recorded by observer: L = 43.6 cm If the ground observer held a meter stick, the same contraction would be seen from the ship.

Foreshortening of Objects Note that it is the length in the direction of relative motion that contracts and not the dimensions perpendicular to the motion. Assume each holds a meter stick, in example. 0.9c Wo W<Wo 1 m=1 m If meter stick is 2 cm wide, each will say the other is only 0.87 cm wide, but they will agree on the length.