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 3x10 8 m/s. But, in what reference frame?

Relativistic Speeds Suppose a rocket ship travels at a speed of 1.0x10 8 m/s. An observer on a rocket ship observes light to be (3x10 8 – 1x10 8 ) 2x10 8 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(c 2 -v 2 )

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 3x10 8 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 2 nd 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 1 st 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

2 nd 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 2 nd 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(D 2 +L 2 ), 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”