From a practical point of view, this means that Newton's laws of motion are valid in all inertial systems, which means those at rest or those moving with constant speed relative to one considered at rest. This is the law of inertia: a body at rest continues at rest and a body in motion continues in motion in a straight line unless influenced by an external force. A Galilean coordinate system is one where the law of inertia is valid. The laws of mechanics of Galileo and Newton are valid in a Galilean coordinate system. If K is a Galilean coordinate system, then every other system K' is a Galilean coordinate system if it lies at rest or moves according to the law of inertia relative to K. Relative to K', the mechanical laws of Galileo and Newton are as valid as they are relative to K.
In other words, if a mass m is at rest or is moving with constant acceleration (the constant acceleration could be equal to zero in which case the velocity would remain constant) in a straight line relative to a Galilean coordinate system K, then it will also be at rest or moving with constant acceleration in a straight line relative to a second coordinate system K' provided the law of inertia is valid in system K' (in other words, provided it is a Galilean coordinate system).
Therefore, if we want to observe an effect in a moving system at constant speed, we can apply the Newton laws directly. If the moving system speeds up (or we speed up relative to it, like looking at the stars from the earth) then we will have to introduce imaginary forces to compensate this effect.
These fictitious forces are called centrifugal force and coriolis force.
Newton's Laws of Motion are mechanically accurate for speeds that are slow compared with the velocity of light. For speeds that approach the speed of light, it is necessary to apply the discoveries of Einstein's Special Theory of Relativity.
In order to describe what happens mechanically in the universe, physicists use mass, length and time. In the physics of Galileo and Newton, these quantities remain the same throughout the universe.
With Einstein's Special Theory of Relativity, these quantities can change.
Special relativity
Special relativity (or the special theory of relativity) is a theory in physics that was developed and explained by Albert Einstein in 1905. It applies to all physical phenomena, so long as gravitation is not significant. Special relativity applies to Minkowski space, or "flat spacetime" (phenomena which are not influenced by gravitation).
Einstein knew that some weaknesses had been discovered in older physics. For example, older physics thought light moved in luminiferous aether. Various tiny effects were expected if this theory were true. Gradually it seemed these predictions were not going to work out.
Eventually, Einstein (1905) drew the conclusion that the concepts of space and time needed a fundamental revision. The result was special relativity theory, which brought together a new principle "the constancy of the speed of light" and the previously established "principle of relativity".
Galileo had already established the principle of relativity, which said that physical events must look the same to all observers, and no observer has the "right" way to look at the things studied by physics. For example, the Earth is moving very fast around the Sun, but we do not notice it because we are moving with the Earth at the same speed; therefore, from our point of view, the Earth is at rest. However, Galileo's math could not explain some things, such as the speed of light. According to him, the measured speed of light should be different for different speeds of the observer in comparison with its source. However, the Michelson-Morley experiment showed that this is not true, at least not for all cases. Einstein's theory of special relativity explained this among other things.
Suppose that you are moving toward something that is moving toward you. If you measure its speed, it will seem to be moving faster than if you were not moving. Now suppose you are moving away from something that is moving toward you. If you measure its speed again, it will seem to be moving more slowly. This is the idea of "relative speed"—the speed of the object relative to you.
Before Albert Einstein, scientists were trying to measure the "relative speed" of light. They were doing this by measuring the speed of star light reaching the Earth. They expected that if the Earth was moving toward a star, the light from that star should seem faster than if the Earth was moving away from that star. However, they noticed that no matter who performed the experiments, where the experiments were performed, or what star light was used, the measured speed of light in a vacuum was always the same.
Einstein said this happens because there is something unexpected about length and duration, or how long something lasts. He thought that as Earth moves through space, all measurable durations change very slightly. Any clock used to measure a duration will be wrong by exactly the right amount so that the speed of light remains the same. Imagining a "light clock" allows us to better understand this remarkable fact for the case of a single light wave.
Also, Einstein said that as Earth moves through space, all measurable lengths change (ever so slightly). Any device measuring length will give a length off by exactly the right amount so that the speed of light remains the same.
The most difficult thing to understand is that events that appear to be simultaneous in one frame may not be simultaneous in another. This has many effects that are not easy to perceive or understand. Since the length of an object is the distance from head to tail at one simultaneous moment, it follows that if two observers disagree about what events are simultaneous then this will affect (sometimes dramatically) their measurements of the length of objects. Furthermore, if a line of clocks appear synchronized to a stationary observer and appear to be out of sync to that same observer after accelerating to a certain velocity then it follows that during the acceleration the clocks ran at different speeds. Some may even run backwards. This line of reasoning leads to general relativity.
Other scientists before Einstein had written about light seeming to go the same speed no matter how it was observed. What made Einstein's theory so revolutionary is that it considers the measurement of the speed of light to be constant by definition, in other words it is a law of nature. This has the remarkable implications that speed-related measurements, length and duration, change in order to accommodate this.
The mathematical bases of special relativity are the Lorentz transformations, which mathematically describe the views of space and time for two observers who are moving relative to each other but are not experiencing acceleration.
Let there be an observer K who describes when events occur with a time coordinate "t", and who describes where events occur with spatial coordinates "x", "y", and "z".
This is mathematically defining the first observer whose "point of view" will be our first reference.
Let us specify that the time of an event is given: by the time that it is observed "t" (say today, at 12 o'clock) minus the time that it took for the observation to reach the observer.
Now with these definitions in place, let there be another observer K' who is
where "x"' axis is coincident with the "x" axis,
and with the "y' "and "z' "axes - "always being parallel" to the "y" and "z" axes.
This means that when K' gives a location like (3,1,2), the x (which is 3 in this example) is the same place that K, the first observer would be talking about, but the 1 on the y axis or the 2 on the z axis are only "parallel" to some location on the K' observer's coordinate system,
and
The Lorentz Transformations then are <br>