Einstein developed two postulates which make up the special theory of relativity. These two postulates detail what is witnessed by an observer in an "inertial frame of reference." (A frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner.)
EINSTEIN'S FIRST POSTULATE
This first postulate is called the "Principle of relativity." It states that the laws of physics are the same in every inertial frame of reference.
Example; Suppose you are driving a car and two children in the back seat are tossing a ball back and forth. By observing the motion of the ball, you will not be able to tell the rate of motion of the car. This is because in every inertial frame, Newton's laws of motion are the same. You will only be able to determine the rate of motion of the ball in the context of the moving car.
Example 2. Suppose you have a positive end, and a negative end of two different magnets. Whether you move magnet A towards magnet B, or magnet B to magnet A, the attractive force will be equal.The laws of magnetism remain consistent in every inertial frame of reference.
EINSTEIN'S SECOND POSTULATE
This postulate states that the speed of light in a vacuum is the same in all inertial frames of reference and is independent of the motion of the source.
Example 3. Imagine two people observing the speed of light in a vacuum. The first observer is not moving with respect to the source of light. The second observer is moving away from the light source. Both observers are in inertial frames of reference. Einstein's second postulate states that the two observers must measure the same speed of light even though one observer is moving with respect to the other observer.
Example 4. A meteor is traveling through space at 500 meters per second. An astronaut standing on the meteor throws a rock in the direction the meteor is moving at 5 meters per second relative to the meteor. relative to an observer on earth, the rock is clearly moving at a velocity of 505 meters per second. Let's say the astronaut now shines a laser in the direction of motion of the meteor. The observed speed of light by the observer on earth is simply THE SPEED OF LIGHT (c), NOT c + 500 meters per second. The speed of light cannot exceed it's vacuum state despite being fired from an already moving object.
Also implied by Einstein's second postulate is the following conclusion. It is impossible for an inertial observer to travel at c, the speed of light in a vacuum. Suppose that the meteor in Example 4 is moving at c. If the astronaut shines the laser, the observer on earth witnesses the meteor and the light in the same position moving at the same velocity. But the second postulate states that the laser beam moves at c relative to the meteor, so therefore they can not be at the same point in space. The only result to avoid this conflict is that it is impossible for an inertial observer, such as the astronaut, to be moving at c.
THE GALILEAN COORDINATE TRANSFORMATION
Let's say your are the observer on earth in example 4. The point at which the astronaut throws the rock is the ORIGIN or O. the meteor moves at a certain velocity, let's call it u. The origin for the astronaut at the time of release, t=0, is the same for both him and yourself. The astronauts origin moves at the speed u times the amount of time that has passed, or u times t. The total amount of space covered by the rock is equal to the amount of space covered by the rock relative to the astronauts origin, plus (u times t).