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The confusing aspects of Einstein's approach
The feature that upsets most non-physicists is that the clocks of the two parties cannot both have slower times than the other party and both cannot have shorter rulers than the other party. It is common sense that if one party's rulers are shorter then the other party's rulers must be longer. Similarly with their clocks, if one party's clocks run slower then the other party's clocks must run faster. The average person could accept this as an optical illusion of some sort, but at the same time both party's rulers cannot really be shorter. From his point of view, it is science's job to sort this out and inform the world what is "really" happening here. The average person has a strong sense that there is an absolute reference frame for space and time. They feel that space and time are unchangeable standards against which we measure changes in length and changes in temporal speed of physical processes. The physical processes can change, but it is meaningless in the minds of most people to say that space and time themselves change. After all, what use is a ruler made of rubber or a clock that varies in speed?
It is even more confusing that Einstein's changes in time and space metric are associated with the speed of the object or observer in question and not associated with a position. Thus a particular spot in space can have many different reference frames flying through it, each with a different amount of warping of its space and time. It's not like the "space" at a given spot is warped a fixed amount; it's more like it just appears to be warped differently due to the relative speed of different observers.
Einstein's theory does not rule out an absolute space and time. In his theory, we are free to pick whatever reference frame we wish and personally declare it to be our "absolute" reference frame in which we observe phenomena to understand what "really" happens. In fact, most physicists do just that. When faced with a problem that needs solving, they usually pick a reference frame through which to figure out what is "really going on". So probably what is most confusing to a beginner to relativity is Einstein's language. He had his own vision of what was or was not underneath the oddities and he set the way we describe these phenomena to reflect his view.
"A powerful agent is the right word: it lights the reader's way and makes it plain." [Mark Twain]
As I mentioned above, a lot of the confusing aspects of relativity can be avoided by substituting "locally perceived distance" for Einstein's "distance", and "locally perceived time" for Einstein's "time". Also, it helps to view the moving party as having warped instruments and warped human senses which make their measurement flawed and not "real". At some level these substitutions are questions of language, but at another level they relate to how we view the world at a very fundamental level. More on this will be discussed later in this set of postings.
Lorentz transforms as a pragmatic calculational tool
Both Einstein and Lorentz would agree that Lorentz transforms are very practical. In the very rapidly moving space ship, life would appear very normal. To the crew it would appear that all objects were their normal lengths (even if Lorentz felt they were really shortened) since any shortening would be uniformly applied to all objects inside the space ship, including all the crew's bodies, even their eyes. Also all temporal events would be slowed down the same, their clocks and even their bodily processes including their hearts and mental processes. The only thing that would appear odd to the crew of the space ship would be that the Earth would appear to be foreshortened and temporal events on the Earth would appear slowed down. But luckily, the crew could use the Lorentz transforms to convert between their space perceived coordinates and time and those perceived on Earth.
"Perceived" is the operative word. Lorentz transformations relate perceived space and time in one reference frame to perceived space and time in another reference frame.
We can liken Lorentz transforms to the use of time zones around the world. Life goes on about the same in the different zones; the sun rises roughly at 6 AM, is overhead at noon and sets roughly at 6 PM. For most activities a person need not be aware that the time in his zone is offset from that in other zones. But knowing the transforms, i.e. the amount to add or subtract from your time is useful for certain activities, such as traveling and for communicating with folks in other time zones. Similarly, the Lorentz transforms would be useful to people changing high speed reference frames and communicating with people in high speed reference frames, if either of these activities ever becomes a reality in the future. In today's world, Lorentz transforms are useful for understanding and predicting how high speed subatomic particles will behave and for understanding small timing effects we notice in very accurate clocks aboard satellites and space probes. Both are important for our technology.
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