# Special Relativity

Einstein's theory of Relativity came in two parts. The first was published in 1905 and dealt only with objects that are moving with respect to each other at constant velocities. Thus each object has an indistinguishable reference frame from the other. The theory is built on 2 basic assumptions.

Postulate #1: The Laws of Nature are the same for everyone

When you are on an airplane, train, bus, or whatnot and you have the window shades down and the ride is smooth you do not notice that you are moving with respect to the Earth. If you do physics experiments to test the laws of nature you will find the same form for these laws as anyone at rest with respect to the Earth would find.

We begin to realize that there is no absolute standard of rest when we then realize that Earth is not stationary either, but rather it is moving about the Sun with considerable speed. The Sun is in motion about the center of the Milky Way Galaxy with consierable speed, and the Milky Way Galxaxy is moving with respect to the center of mass of the local group of galaxies and so on...

If we approximate these motions as ones with constant velocity between them (no accelerations, that would mean a force is acting on the frame) then we should find no differences in the laws of nature from one frame to the next.

Our point of view will be relative and so we seek to understand how we can view events from one frame while in another.

## Relative Speed

Einstein began his journey toward developing special relativity when he asked the question of what the world would like if he could travel on a beam of light.

Before understanding this we must first be sure to understand what the world looks like at slower speeds. So let us do some thought experiments.

Let's just focus on the relative motion of a pair of twins: Bob and Jill. They are both far from Earth's (or any other planet's) gravity for simplicity. They each have a small space craft which has big windows on the side that allow them to see inside each others craft. Let's say that Bob is moving to the right past Jill with some constant velocity, v.

Bob could claim that he is not the one moving, but rather it is Jill who is moving to the left past him with the same constant velocity, v. Who is right? They both are. Each frame is equally valid. Since there are no accelerations on either frame there is no way to distinguish one from the other.

As Jill watches Bob go by she sees him throw a ball toward the from of his craft. Bob throws the ball with velocity, u, with respect to his craft. What is the speed of the ball as seen by Jill.

Common sense tells us that we should simply add the speed of the ball to the speed of the craft. v' = u + v.

Now, Bob pulls out a flash light and sends a beam of light toward the front of his space craft. What is the velocity of the light as observed by Jill? Should it not simply be: c' = c + v?

No!

If the speed of light were able to simply add we would have a paradox in many experiences.

For example, consider a star in a binary system. Sometimes it is moving toward Earth with its orbital speed, v, and others it is moving away. When it comes toward you the speed of the light coming toward you will be faster than the light that you see when the star is moving away. At a great enough distance or great enough speed of the binary star you could see the light of the star moving toward you arrive at the same time as the light moving away from the other side of the orbit. You could see the star appear in multiple positions at once! This is never observed.

Also, consider the situation of two cars, A and B, headed toward an intersection. Both cars are equally far from the intersection and are headed toward it at 100 km/hr. They will collide when they reach the intersection. However, if you are standing straight down the street in the direction that car A is heading you will see the light from car A moving faster than the light from car B. Thus, you would see car A get to the intersection before car B and thus there should be no collision.

What!? That's not possible. The collision happens, it should not depend on where we are located as to whether or not we see the collision to occur.

These two illustrations say that something is wrong with our assumption that we can just add the velocity of light to the motion of whatever emitted it. In fact they say that in order for things to make sense the speed of light should be the same for everyone no matter what speed they have relative to it.

Indeed, electromagnetic theory at the end of the 19th century was saying the exact same thing, only no had yet realised it. The speed of light is a constant for all observers. That is the second assumption of special relativity.

Postulate #2: The speed of light (in a vacuum) is the same for everyone

This has in fact been experimentally verified again and again for the past 90-some-odd years. No matter who emits the light and at what relative velocity, everyone measures it to have the same speed, c = 3 x 108 m/s.

Thus when Bob shines his flashlight he measures the speed to be, c, and when Jill observes the light of Bob's flashlight she also measures the speed of the light to be c, not c + v.

## Universal Speed Limit

The fact that light must be measured the same by all observers in fact requires that no material object can exceed, or even attain, the speed of light. Consider that Jill now goes speeding of in her space craft. She throws the craft into high gear, firing the rockets to incredible speed. Now she turns on her head lights. In her frame the lights go zooming out ahead of her at the speed of light (no faster, no slower). In her own frame it is impossible for her to overtake her headlights since the speed of light must always be constant. Can anyone else observe her to move faster than light? People in different frames will measure her to have different speeds relative to themselves, but they will all agree on two things. (1) The light of her headlights is moving out in front of her and (2) it has the speed, c. Clearly, is she is being outraced by her headlights and her headlights are traveling at the speed of light, she must be traveling slower than the speed of light.

This is not merely some technological challenge. No material object may even reach the speed of light. This is the law of nature. There are no penalties for violating the law, because it cannot be broken.

## Time Dilation

Now consider another thought experiment. Imagine that Bob is in a train moving to the right with some speed, v, relative to the ground on which Jill is standing watching him go by. Bob has a ball that he is casually tossing up and catching as it falls back down. From Bob's perspective the ball goes straight up and straight down and therefore has no motion foward. But Jill sees the ball take a different path. From her perspective the ball has a forward velocity given it by the train so that the ball appears to take a slanted path. It leaves Bob's hand and slants forward and then falls back to Bob's hand, which has moved in the time interval because of the motion of the train. The ball therefore takes a longer path than the one that Bob sees. That's fine because the ball travels faster as seen by Jill than Bob. So there's no paradox. As the train's speed increases the slant grows larger and thus the ball takes a longer path and thus must be traveling faster.

Now let's imagine that we have a similar situation. This time Jill has a clock that works by measuring the time difference between a beam of light emitted from the bottom of the clock which reflects off a mirror on the top of the clock and then is detected back on the bottom of the clock: one tick of the light clock. Now Jill gets in her space craft with the light clock and goes speeding off. Bob, observes her passing by and watches the light clock. The light in the clock will now take a path just like that of the ball in the previous thought experiment, a slanted path. The difference in this case is that the speed of the light as seen by Bob is the same as that seen by Jill. But Bob sees the light to take a longer path than Jill sees it to take. Therefore Bob thinks the light takes a longer time to make one tick. But to Jill in the spave craft the light clock is running just like it did before she got in the space craft. Bob therefore thinks that Jill's time is running slower than his. The faster the speed of Jill relative to hime the longer the light's path and thus the longer it will take to complete one tick. So as the speed increases Jill's clock seems to run slower and slower. And it's not just the light clock, it's time itself that is running slower in Jill's space ship.

What does Jill see? To make this easier to understand, imgagine that Bob is in another space craft as we earlier considered, and they have a relative speed between them of, v. Now, it doesn't matter who is moving. Bob thinks he is stationary and Jill is moving away at v, likewise Jill thinks that she is stationary and it is Bob that is moving away from her. Both viewpoints are valid. Bob has a clock just like hers. Therefore she will see his time to run slower than hers. Wow, that's weird!

This effect is called time dilation. From your point of view time runs slower for anyone moving relative to you. As the their speed approaches the speed of light time seems to stop.

This effect has been experimentally verified. Very accurate atomic clocks have been taken aboard jet airplanes that flew in circles for hours. When the clocks aboard the plane were compared to clocks on the ground they had lost a few nanoseconds of time. Exactly the amount predicted by special relativity. The effect is obviously inperceptible at normal speeds. But as speeds approaching the spped of light it is much more noticeable.

In particle accelerators, physicists accelerate particles to near the speed of light and smash them into each other. In the collisions new particles are formed. One such particle is the muon. When at rest this particle live only 2.2 microseconds before decaying into other particles. But in experiments where it is created moving at 99% the speed of light it last for 15.6 microseconds, which is exactly the amount predicted by special relativity.

## Simultaneity

Another fact that results from the constancy of the speed of light is that no two relative observers can agree on the simultaneity of events. Consider Jill racing by Bob in her space ship. She has a green light in the front of her craft and a blue light in the rear. At the moment that Jill and Bob are lined up (the shortest distance between each other) the two lights flash in Jill's ship. Since Bob is equal distances from the lights he will see the light reach him at the same time, and conclude that the flashed were simultaneous. However, Jill is racing toward the green light and intercepts that light first and then the red light catches up to her. Since the lights are also equal distances from her she must conclude that the green light flashed before the red light did. Curiouser and Curiouser.

Note: All observers must observed the same order of events that occur at the same location. For example if you pick up and eat a cookie, no one will see you to eat the cookie first and the pick it up.

## Length Contraction

Now, imagine that Jill goes zipping by Bob's space ship. She makes a measurement of Bob's space ship length. She does this by timing how long it takes for her to cover the distance between the front end and back end of his ship moving at the relative speed between them. Because Bob sees her time running slower than his she will seem to pass over his length in less time than he would expect. Thus she measures his ship's length to be shorter than what it is at rest. Likewise he measures her ship to be shorter than what she thinks it is.

This effect is called length contraction. It occurs only along the direction of motion. From your point of view, lengths of objects moving by you (or distances between objects moving by you) are shorter in their direction of motion than they would be if they were at rest with respect to you. The faster the objects are moving, the shorter the lengths. As they approach the speed of light their lengths contract to zero.

This effect allows us to reach the stars in a reasonable amount time. Consider a trip to the star Vega, which lies at a distance of 25 light-years. At conventional speeds it would take eaons to reach Vega. But imagine that Jill goes off to Vega in her space craft at 99.9% the speed of light. Because Jill can view herself as stationary and Vega is rushing at her at 99.9% the speed of light the distance will contract to just over 1 light-year. So she will make it there in just a little longer than a year. She can then return and the total round trip time for her is only about 2 years. However, on Earth Bob saw her take off at near the speed of light and so if it takes light 25 years to make it to Vega and another 25 to make it back, 50 years will have passed on Earth. Bob is now 48 years older than Jill.

You could make even longer journies. Imagine that Jill heads off to the nearest spiral galaxy, Andromeda (M31) at 2.5 million years distance. At sufficient sppeed she could make it there in 25 years. The round trip is then only 50 years. If she left when she was 20 she'd return when she was 70. However, on Earth 5 million years would have passed.

## Mass Increase

Another important effect of special relativity is that of mass increase. Imagine now that Jill is in her space craft and you are floating out in space right next to the craft ready to give it a push. Bob's ship comes doodiddlin' along at a relative speed, v. Now both Jill and Bob are identical in mass and so are their ships (at rest anyway). You give them both an identical push at the moment that Bob passes you. That means you applied an identical force. But, Bob's clock is running slow as seen by you and Jill. Hence your push on Bob's ship happens over a shorter time than the push on Jill's ship. Therefore your equal pushes produce different accelerations on Jill and Bob's ships. Bob's ship accelerates less because it felt the force for a shorter time. The only way this can be is if you measure Bob's mass to be greater than Jill's.

From your point of view, objects moving past you have a greater mass than they have at rest. The faster an object is moving the greater the increase in mass. As you increase the speed toward the speed of light the mass becomes infinte. Thus it would take infinte force or energy to accelerate any mass to the speed of light. This is why it was stated earlier that no material object may move at the speed of light.

This fact also is the origin of Einstein's equation: E = mc2