Introduction
Albert Einstein
"Relativity" comes in two varieties, Special Relativity and General Relativity. Both are creations of Albert Einstein.
Special Relativity was published in March of 1905. In that paper, Einstein adopted two (2) postulates. In September of that same year he published his second paper on relativity, deducing from relativity that E = mc2 but dropping one of the two postulates. A true revolution in the understanding of relativity came in 1908 when Minkowski pointed out that both postulates should be dropped, and that relativity should be deduced instead from the Pythagorean Theorem. Since that time, no scientist (including Albert Einstein) has ever used the two postulates, ... except to teach young people relativity.
Why this is, is beyond me!
Here, we will ignore the obsolete postulates and obtain relativity in transparent fashion from elementary middle school geometry.
Pythagorean Theorem
First, consider the following famous and very fundamental proof of the Pythagorean Theorem:
Proof of the Pythagorean Theorem
Proof of the Pythagorean Theorem.
OK, did you follow the proof? No? Then stare at the figure some more! It was discovered on a piece of pottery from 1000 BCE, with one Chinese character meaning "Behold!"
Notice that this proof involves NO numbers, and NO algebra. It is a purely geometrical proof. If you like, you may make up the two figures from eight identical copies of a right angle triangle.
Now let us look at ANOTHER proof of the Pythagorean Theorem:
This proof is drastically different, in that it employs algebra. Look at the algebra with fresh eyes: what can those arbitrary symbols have to do with space? Our first proof was carried out simply by looking at space itself, and using as an axiom that if equals are removed from equals, what remains are also equals. That was it! In this new, vastly more elaborate proof, squiggles are introduced that bear no relationship at all to space: but they have been crafted with rules ("algebra") that result in them mimicking space successfully!
Now, which of our two proofs is better? Our second proof has the enormous virtue that it can be generalized beyond the two dimensions of a surface. Consider the Pythagorean Theorem in three dimensional space:
Two applications of the Pythagorean Theorem result in deduction of the Pythagorean Theorem in three dimensions. By the way, notice that this is a perspective drawing! We did not always know how to create such drawings; our ability to do so was a product of the renaissance. And yet you hardly notice it!
And now we come to your vast and almost instantaneous reward: I want you to write down the Pythagorean Theorem in an eleven-dimensional universe.
Aren't you amazed that you can do it?
Special Theory of Relativity
We have now arrived at the Special Theory of Relativity. Just write down the Pythagorean Theorem in FOUR dimensions, and ponder it. Could time be the fourth dimension? Use the symbol t for the fourth dimension. What's wrong with this?
Well, it does not work, because as we've done it, what we've called time is just another space dimension!
We need some distinction between time and space, because we know there IS a distinction: we can visit Rome (space) but not Julius Caesar (time).
So how do we need to modify the Pythagorean Theorem? Well, there's not much available to modify, is there? There's just the numbers, x, y, z, t.
Hmm, numbers! What kind of numbers? Well, real numbers, of course! However, strange as it seems to think of doing it, you CAN use imaginary numbers to express distances! Let me prove you don't get into real trouble by doing so:
Of course, you could also use imaginary numbers to represent time: who's going to stop you?
So now we know we have options! We want to create a DISTINCTION between space and time. How can we possibly do that?
YES, of course: use imaginary numbers for space, and real numbers for time! Or, vice versa!
The result of course, is:
or
depending on which you chose to be imaginary, space or time.
Well, congratulations! You have just discovered Einstein's Theory of Special Relativity!
Suppose we'd gone through this exercise in 1900 (never heard of A. Einstein!). We still would not know what this equation MEANS.
Except that we'd know it was supposed to give the separation of two events in spacetime.
The two events are separated in the x direction by amount x, in the y direction by amount y, and in the t (time) direction by amount t.
Of course we use different UNITS for space and time, but our hypothesis is that time is "just another dimension," except for the signs business, so there must be some conversion factor: call it "c" for "conversion." By our hypothesis, there are c meters in one second. That is, when your watch's second hand advances by one second, you have moved c meters into the future. Of course we don't know the value of c, and we are going to want to devise an experiment to find c.
In the meantime, let's just put the conversion factor into the pythagorean theorem:
or
Now we really have Special Relativity!
We will find what our equations really mean (and we will revisit Einstein's first postulate) when ... Dick Henry does the next part....
Preview Image
"NASA HPC Centers Support Gravitational Wave Breakthrough" - Goddard Space Flight Center’s Gravitational Astrophysics Laboratory achieved a breakthrough recently: computation of the signature gravitational wave pattern that is radiated when two black holes that orbit one another experience orbital decay and merge. The solution of this very difficult three-dimensional, time-variable problem in numerical relativity has been a “holy grail” of the field, according to Joan Centrella, Chief of the Gravitational Astrophysics Laboratory. Centrella's team crunched Einstein’s theory of general relativity equations on the Columbia supercomputer to create a three-dimensional simulation of merging black holes. This was the largest astrophysical calculation ever performed on a NASA supercomputer. The simulation provides the foundation to explore the universe in an entirely new way, through the detection of gravitational waves. (Image credit: Chris Henze, NASA ARC.)
Citation
Henry, Richard, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2009. "Relativity." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published November 26, 2007].
<http://www.cosmosportal.org/articles/view/137291/>
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