Special Theory of Relativity

Space and Time

Inertial reference frames

• Reference frames in which Newton’s first law is valid.– In other words, a reference frame that is

either at rest, or moving at a constant velocity.

• Noninertial reference frames are ones in which the reference frame is rotating or vibrating– Objects will tend to move towards the outside

of the circle when rotating.

Relativity principle

• The basic laws of physics are the same in all inertial reference frames.

• Example: You are on a moving train playing Ping-Pong.

• All the physics of Ping-Pong will remain the same as long as the train is moving at a constant speed.

Coin Example

• Suppose moving down a highway at a constant speed in a car you flip a coin above your head within the car.

• How does the motion look to a person in the car?

• How does the motion of the coin appear to a person observing the car passing?

Coin Example

• For a person in the car, the object falls straight down.

• For an observer on Earth watching the car, the coin follows a curved path like parabolic motion.

• For each inertial frame of

reference, the motion follows

the laws of physics.

Velocity in relativity

• Suppose your friend is on a flatbed truck throwing a baseball to you at 60mph.

• What is the speed of the ball when you catch it if the truck is – At rest?– Moving towards you at 40mph?– Moving away from you at 40 mph?

Acceleration in relativity

• A train is moving East at 45 mph

• A person walking on the train West accelerates from 0 to 5 km/hr in 1 second.

• You are an observer at a station the train is passing.

Acceleration in relativity

• Train reference frame is accel = 5 km/hr/s

• Earth reference frame: a = (45-40)/1 or

a = 5 km/hr/s

• The acceleration of a body is the SAME in all inertial reference frames according to classical mechanics.

Other constants

• In all reference frames, mass is also constant.

• Therefore, if mass and acceleration are constant, then force is constant in all reference frames!

Laws of mechanics

• It can be shown that all laws of mechanics are the same in all inertial reference frames.

• This is implies that no one inertial frame is special in any sense. Or…

• All inertial reference frames are equivalent!

Maxwell messes things up

• In the 1870’s Maxwell

predicted the speed of light to

be 3x108 m/s.

• But, in what reference frame?

Relativistic Speeds

• Suppose a rocket ship travels at a speed of 1.0x108 m/s.

• An observer on a rocket ship observes light to be (3x108 – 1x108) 2x108 m/s.

• Maxwell stated that c should be constant.

• This seemed to imply that there must be some special reference frame where c would have this value.

Another example on a smaller scale

• Suppose your friend is still throwing a ball to you from a flatbed truck.

• No matter what, moving towards you or not, the ball would be moving at 60mph.

• This means the truck is in its own special inertial reference frame according to Maxwell.

Relativity Principle

• So to recap…

• All laws of mechanics are the same in all inertial reference frames…

• Except laws of electricity and magnetism.

• Stupid Maxwell!

Into the Ether

• Scientists in the late 1800s were in search of a reference frame that was absolute.

• …A reference frame where light would have different speeds relative to the ether.

Michelson-Morley

Experiment

• Suppose that the “Ether wind” is moving to the right in the diagram just shown.

• Then, the velocity of the beam going to the right is c+v.

• The velocity of the beam going up will be sqrt(c2-v2)

Interference

• If the beams were traveling at different speeds and arrive at the detector at different phases, there should be interference.

• By changing the distance of the mirrors, Michelson-Morley can calculate v, the speed of the ether wind.

The null hypothesis

• No significant interference pattern was observed!

• Tried at different times of day and year (different orientations with the sun), but no interference patterns.

• No ether velocity was found!

Einstein to the rescue

• What would I see if I rode a light beam?

Riding the light

• If you are riding a light wave the observer would see more light moving away from the rider at 3x108 m/s as well.

• There speed of light will be the same in all reference frames.

Einstein’s conclusion

• Postulate #1: The laws of physics have the same form in all inertial reference frames.

• Postulate #2: Light propagates through empty space with a definite speed c independent of the speed of the observer.

Why so special?

• Special is in comparison to Einstein’s later theory of “general relativity”.

• Special relativity (1905) refers to inertial frames.

• General relativity (1916) deals with noninertial reference frames (accelerating, like rotating).

Violating commonsense

• The 2nd postulate means that the speed of light is the same for any observer.

• If you are moving toward or away from a source of light, the speed of light will be the same as observed by someone at rest.

Gedanken Experiments

• Gedanken is German for “thought”.

• Einstein was famous for following up the mathematics with “thought experiments” to explain his theory of special relativity.

• We will examine some of these now…

Simultaneity

• Simultaneous – two events occur precisely at the same time.

• How can we tell if events are simultaneous?

• If the events are separated by a large distance, it is difficult since we must account for the time light has to travel to determine if the events are simultaneous.

Simultaneity

• If two events appear to occur at the same time, then the one farther from the observer must have occurred earlier.

Thought experiment #1

• 1st part: assume that an observer is halfway between two events, A and B.

• If the observer, halfway between, sees the light from both events at the same time, we can conclude they occurred simultaneously.

• illustration 2

The real question

• If two events are simultaneous to an observer in one reference frame, are they also simultaneous to another observer moving with respect to the first?

Thinking…

• Suppose two observers are fixed in position, but are moving relative to each other (like staying still on a moving train).

• Observer 1 can say that observer 2 is moving to the right with speed v

• Observer 2 can say that observer 1 is moving to the left with speed v.

Still thinking…

• Suppose now two simultaneous events occur that are observed and measured by both observers.

• For observer 1, the events appear simultaneous.

• For observer 1 looking at observer 2, they will appear to be not simultaneous because they are moving.

Done thinking…

• Two events which are simultaneous to one observer are not necessarily simultaneous to a second observer.

• Simultaneity is therefore not absolute, it is relative.

• illustration 3

2nd thought experiment

• Since simultaneity is different for different reference frames, that means that time is also relative.

• This brings a new thought experiment…

Time dilation

• Suppose there is an observer on Earth and an observer on a space ship traveling past Earth.

• On the space ship, a light source is shined onto a mirror and then reflected back to a receiver connected to a clock.

Time dilation

• The observer on the space ship will see the time traveling as t = 2D/c, where D is the distance from the source to the mirror.

• The person on Earth will observe the light traveling over a distance in a 2nd dimension (not just back and forth from the mirror)

Time dilation

• The observer on earth will see the light traveling at a distance of 2*sqrt(D2+L2), where L is the distance traveled by the space ship.

• Mathematically, we can show that the time traveled between two events is greater for the observer on Earth than for the observer on the space ship.

• illustration 4

Evidence for time dilation

• Time dilation only works at relativistic speeds.

• An experiment in the 1970’s showed that muons will have a longer lifetime when traveling at high speeds than when at rest.

Space Travel

• Suppose we want to reach a star 100 light years away.

• Even if we can travel at the speed of light, it would take 100 years to reach the star.

• But time dilation shows that the time involved would be less for the astronaut.

Time dilation example

• Mathematically, we can show that travelling at 0.999c, the astronaut would only feel like 4.5 years have passed.

• But, is it just the clocks that would slow down for the astronaut??

Twin Paradox

• The astronaut would experience 4.5 years of normal sleeping ,eating, reading, and so on.

• People on Earth would experience 100 years.

Twin Paradox

• If one twin stays on Earth, and another goes on a relativistic speed trip, the one on the trip would age less than the one at home.

Now for the paradox

• What about the view point for the traveling twin?

• Earth is moving away at a high speed, so time will pass more slowly on Earth.

• So won’t the twin on Earth therefore age less in the reference frame of the traveling twin?

To help solve this paradox…

• Let’s have the ship leaving earth send a signal of light every 6 minutes going away from Earth for 1 hour.

• Let’s say that the speed of the ship is such that Earth will receive the signals every 12 minutes.

• During the hour trip, the ship gives out 10 flashes.

• How many flashes will Earth receive?

• How long until Earth receives the last signal?

• Suppose the ship left at 12 PM.

• It would be 1PM on the ship, but 2PM on Earth when the last signal is sent.

Return trip

• Now the ship miraculously turns around without decelerating or accelerating and returns to Earth at the same speed…

• The return trip, the ship still sends 10 signals, one every 6 minutes.

• Earth sees them every 3 minutes.

• Earth will see the last one after 30 minutes

Spaceship time vs. Earth time

• On the space ship, it will be 2 PM.

• On Earth it will be 2:30 PM

• There is still a time dilation!

• The twins will be different ages.

What if Earth sends the signal?

• Using the same analysis with the Earth person sending the signal to the Space ship (this is the paradox part)

• What time is it for the spaceship twin? What time is it for the Earth twin?

• exploration 3

Implications for space travel

• Suppose we want to go to Prycon which is 11.4 light years away.

• If we travel at 99% of the speed of light, it would take 23 Earth years to travel there and back (just double it)

• But the astronaut would only age 3 years.

• The mission control would welcome back an astronaut 23 years later, but the astronaut would only be 3 years older!

Is this practical?

• No, it would take billions of times the energy used to get spaceships just into Earth’s orbit.

But what if we could…

• We could time travel forward into the future!

• If we travel really fast, we could see some elapse in time for the traveler, but a lot of time elapsed here on Earth.

Length contraction

• Similar to time dilation…

• The length of an object is measured to be shorter when it is moving relative to the observer that is at rest.

• In other words, moving objects are shorter than stationary….

• Think “warp speed”