Special Theory of Relativity Space and Time
Jan 01, 2016
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”