Special Relativity

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.

Common sense tells us that we should simply add the speed of the ball to the speed of the craft.

Now, You pull out a flashlight and send a beam of light toward the front of your spacecraft. What is the velocity of the light as observed by Jackie?
Should it not simply be:

Example 1: Consider a star in a binary system. Sometimes it is moving toward Earth with its orbital speed, v, and other time 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 orbital speed you could see the light from the star moving toward you arrive at the same time as the light from the star moving away from you, on the other side of the orbit. You could see the star appear in multiple positions at once! This is never observed.

 Example 2: Consider 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.

Consider Jackie racing by You in her spaceship. She has a green light in the front of her craft and a red light in the rear. At the moment that Jackie and You are lined up (the shortest distance between each other) the two lights flash in Jackie's ship. Since You are equal distances from the lights You will see the light reach You at the same time, and conclude that the flashed were simultaneous. You will see Jackie lit up first by the green and then by the red since she is moving toward the red.

From Jackie's point of view she 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.

Imagine that a fellow Bob is heading toward Jackie at 80% the speed of light from your frame of reference. Jackie is heading toward Bob at 90% the speed of light as seen by You. How fast does Jackie see You and Bob coming at her?

By just adding velocities you would get 170% the speed of light for Bob. That is clearly not right. She must see Bob moving slower than the speed of light but also faster than 90% the speed of light. As it turns out she will see it coming at her at 98.8% the speed of light.

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 forward motion.

But Jackie 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 Jackie 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 imagine a similar situation. This time Jackie 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 Jackie gets in her spacecraft with the light clock and goes speeding off. You observe her passing by and watch 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 You is the same as that seen by Jackie. But You see the light take a longer path than Jackie sees it to take. Therefore You think the light takes a longer time to make one tick. But to Jackie in the spacecraft the light clock is running just like it did before she got in the spacecraft.

You therefore must conclude that Jackie's time is running slower than yours. The faster Jackie's speed relative to You the longer the light's path and thus the longer it will take to complete one tick. So as the speed increases Jackies's clock seems to run slower and slower.

It's not just the light clock, it's time itself that is running slower in Jackie's spaceship.

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

Now, imagine that Jackie goes zipping by Your spaceship. She makes a measurement of your spaceship's length. She does this by timing how long it takes for her to cover the distance between the front end and back end of your ship moving at the relative speed between them. Because You see her time running slower than your own she will seem to pass over your length in less time than you would expect. Thus she measures your ship's length to be shorter than what it is at rest. Likewise You 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 Jackie goes off to Vega in her spacecraft at 99.9% the speed of light. Because Jackie 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 You 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. You are now 48 years older than Jackie.

One could make even longer journies. Imagine that Jackie heads off to the nearest spiral galaxy, Andromeda (M31) at 2.5 million years distance. At sufficient speed 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.

Imagine that Jackie is in her spacecraft 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 Jackie 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 Jackie. Hence your push on Bob's ship happens over a shorter time than the push on Jackie's ship. Therefore your equal pushes produce different accelerations on Jackie 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 Jackie's.

F = ma

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 infinite 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.