Special Relativity &

General Relativity

Textbook: 11.1, 11.2, 11.3Homework: pg. 568 # 5, 6;

pg. 579 # 1, 2, 6, 8, 9, 12

Special Relativity

Einstein’s Postulates:The relativity principal: All of the laws of

physics are valid in all inertial reference frames.

The Constancy of the Speed of Light: Light travels through empty space with a speed of c = 3.00 x 108 m/s, relative to all inertial reference frames.

Simultaneity

Two events that are simultaneous in one frame of reference are not necessarily simultaneous in another frame of reference

The order in which events occur depends on your frame of reference

Time Dilation

A clock moving past an observer appears to run slow

ts = proper time (time measured when at rest with respect to clock)

tm = dilated time

tm ts

1 v 2

c 2

Length Contractions

Since observers must agree on relative velocities, length must change based on the frame of reference.

Lm = Length between points A and B where A and B are moving with velocity v

Ls = Length between points A and B where A and B are stationary

Lm 1 v

c

2

Ls

Paradoxes The Curious Case of the Muon

- Muon’s are elementary particles that are produced when our upper atmosphere is bombarded by cosmic rays

- The Muon’s lifetime is very short (2.2 x 10^-6 s)- According to Newtonian mechanics even traveling at high speeds Muon’s should

not get to the surface of the Earth- The Muon’s clock is running slow so it seems to last longer- In the Muon’s frame of reference the distance is contracted

The Twin Paradox- The twin that leaves ages at a slower rate than the twin that stays on Earth- The symmetry that we have seen is ruined because the twin in the shuttle has to

accelerate- High acceleration causes your clock to run slower - MUCH slower

The Ladder in the Barn- How can one observer say that the ladder fits into the barn and the other says it

doesn’t?- For one observer both ends are simultaneously inside the barn- The other observer does not see both ends inside the barn simultaneously

Relativistic Momentum

m = rest mass [kg]

2

21

mvp

vc

Relativistic Energy

Conservation of mass-energy

2

2

21total

mcE

vc

2restE mc

22

2

21k total rest

mcE E E mc

vc

Pg 572 1. Are airline pilots’ watches running slow in comparison with

clocks on the ground? Why or why not?

2. A beam of unknown elementary particles travels at a speed of 2.0 x 108 m/s. Their average lifetime in the beam is measured to be 1.6 x 10-8 s. Calculate their average lifetime when at rest. [1.2 x 10-8 s]

3. A Vulcan spacecraft has a speed of 0.600c with respect to Earth. The Vulcans determine 32.0 h to be the time interval between two events on Earth. What value would they determine for this time interval if their ship had a speed of 0.940c with respect to Earth? [75.0 h]

4. The K+ meson, a subatomic particle, has an average rest lifetime of 1.0 x 10-8 s. If the particle travels through the laboratory at 2.6 3 108 m/s, by how much has its lifetime, relative to the laboratory, increased? [2x]

Pg 576 5. A spaceship passes you at the speed of 0.90c. You

measure its length to be 50.0 m. What is its length when at rest? [115 m]

6. You are a space traveller, moving at 0.60c with respect to Earth, on your way to a star that is stationary relative to Earth. You measure the length of your trajectory to be 8.0 light-years (ly). Your friend makes the same journey at 0.80c with respect to Earth. What does your friend measure the length of the trajectory to be? [6.0 ly]

7. A spacecraft travels along a space station platform at 0.65c relative to the platform. An astronaut on the spacecraft determines the platform to be 3.00 x 102 m long. What is the length of the platform as measured by an observer on the platform? [3.95 x 102 m]

Pg 576 8. A star is measured to be 40.0 ly from Earth, in the

inertial frame in which both star and Earth are at rest. (a) What would you determine this distance to be if you travelled

to the star in a spaceship moving at 1.00 x 108 m/s relative to Earth? [37.7 ly]

(b) How long would you determine the journey to take? [113 a]

9. The proper length of one spaceship is twice the proper length of another. You, an observer in an inertial frame on Earth, find the two spaceships, travelling at constant speed in the same direction, to have the same length. The slower spaceship is moving with a speed of 0.40c relative to Earth. Determine the speed of the faster spaceship relative to Earth. [0.89c]