Rapidity and Relativity

[Schneibster]

One of the most difficult things to get your head around when trying to figure out what's going on in relativity theory is the idea that time is a dimension. Here is a good way to think about dimensions that will help you understand what physicists are thinking of when they talk about them. Previously posted elsewhere.

An important precursor concept to this way of looking at reality is the concept of dimension. And there is a way you've been taught to think about dimensions that is wrong; this is not surprising, since it was first conceived by a Greek a few thousand years ago. This way of thinking has been superceded, and the update has not yet made its way down the educational system, despite having been invented a hundred years ago (my goodness, how influential those Greeks were). Here is the way to think about dimensions if you wish to understand relativity:

Let's begin with one dimension. Movement is possible in only one direction. Rotation is impossible; there is no direction in which to rotate.

We're done. We can do only one thing.

Next?

Let's combine a second dimension with this first one we started with. Now, we can move not merely in one direction, and not merely in two, but in an infinite number of directions. By merely adding one dimension to the one we already had, we have not doubled our possibilities, we have increased them infinitely. This will not be the last time that happens. But we've done something else. In two dimensions, we can do something completely new: objects can rotate. So we've found something out: the minimum number of dimensions required to support rotation is two.

There's something else I want you to notice. What is the direction of the axis of rotation? Uh-oh; there isn't one. It sticks out in a direction that doesn't exist; remember, we only have two dimensions. And we've discovered something else again: "axis of rotation" is a meaningless term in the simplest system in which rotation can occur. So how can we define the direction in which rotation can occur? Simple: plane of rotation. And we will never refer to an axis of rotation again, because we now know that it's meaningless. We will always use "plane of rotation" which is always meaningful in all dimensionalities in which rotation can exist.

Now, how many planes of rotation are there in two dimensions? Simple. One. That's it. There's only one direction you can rotate in, and it's defined by those two dimensions. Anything in our two dimensions that's rotating, is rotating in that plane, period.

We're done here. We can do two things: move, and rotate.

Next?

OK, now let's combine a third dimension with those two we already had. What do we gain? Well, first, we now have added two more infinities of directions in which we can move objects. I told you this would not be the last time that happened. We've done it again. What's more, we've MORE than done it; we've added an infinity of infinities of directions we can move objects in. OK, now what about rotations? We combined in one more dimension; how many planes of rotation do we have now?

Interestingly, adding one dimension to one dimension produced one plane of rotation; but adding one more dimension to that two didn't just add one more plane of rotation; it added two more. Now we have three planes of rotation. We'll come back to this later. There is a mathematical law here that we need to understand, that defines how many planes of rotation we get in a given number of dimensions; this is a basic feature of reality. If you want a geek equation, like they've been talking about in Wired for weeks, the one that determines the number of planes of rotation in a manifold is it.

Now, let's take just one more look at our "axis of rotation" concept before we discard it forever. We'll do this so that we can understand precisely why it confused our understanding. Note that the axis of rotation of an object rotating in one of our three available planes of rotation always points in the direction of the dimension that is not part of the rotation. So whenever we try to envision extra dimensions, the first thing we do is try to imagine the extra axes of rotation. We now see that that's wrong. We should be thinking about the extra planes of rotation, not the extra axes. Those axes might point in directions that don't exist in our dimensional framework; so they'll do nothing but confuse us. But the planes will always exist as combinations of two of our dimensions, and in fact we can always count how many planes of rotations there are in a space by counting the combinations of its dimensions in pairs; however many possible combinations there are, that's how many ways things can rotate. There's a hint at our mathematical law.

So let's name our dimensions. We'll call the one we started with "x." The second one we added, we'll call "y." So the single plane of rotation we had in our two-dimensional framework, what should we call that? How does "x-y" grab you? Looks pretty good to me. OK, what shall we call the third dimension? "z," of course. And the two new planes? "x-z" and "y-z" sound pretty good to me.

OK, now we take the step that takes us beyond our normal abilities. Let's add a fourth dimension.

First, how many directions have we added? Well, the first time, we added an infinity; and the second, we added an infinity of infinities. This time, we'll add three infinities of infinities. OK, now how about rotational planes? Well, first, let's name this new dimension. We'll call it "t." (No, don't jump ahead of me here. We'll get to it in just a little bit. Just accept it and don't wonder why for the moment.) So now how many planes do we have? Well, we had x-y, x-z, and y-z; looks to me like we now have added x-t, y-t, and z-t. So we didn't just add one more plane of rotation, we added three. Oh, now we understand why we couldn't envision four dimensional space; we only expected to get one more rotation axis... errrmmm, oops! I thought we weren't going to talk about axes any more!

Now you can see why we don't want those axes of rotation, and why we have such confusion as a result of them. Everything is much clearer when we think about rotation planes. We have no problems understanding them; we can't quite visualize them, being 3D entities ourselves, but at least we can see how things must be in 4D space. This is as close as you'll ever get to being able to visualize 4D. If I've done well, then you can almost see it in your head, and you can see how to manipulate objects in it.

OK, now what does it mean to say space is curved through a fourth dimension? You can almost see it. Just imagine a sheet of rubber, with various weights in various places- and then imagine the sheet thickening, but remaining curved in various places just as it was before it was thickened. And (this is just a little harder, but it's the last piece) imagine that even though the top and bottom of the thickness of the sheet aren't defining the curve in the middle of the thickness, that curve is still there. And that's what it means to curve a three-dimensional manifold, like our normal conception of space, through a fourth dimension.

OK, that's a good place to break. Next piece, the shape of the time dimension (and the space dimensions, too).

[Schneibster]

Here's the next piece, I'm retrieving at this point so sorry if there's some context errors. Starting from here is a set of three or four posts I made that discuss relativity, both Galilean and Einsteinian, and dimensionality, and that lead toward the rapidity concept.

First of all, every observer has to be somewhere, moving at some speed. Unless that observer is undergoing acceleration, there is no way for them to measure their speed. It is therefore mathematically convenient for every observer to assume they are motionless. And in fact, they cannot prove they are not, unless they are undergoing acceleration. There is no experiment they can perform to decide how fast they are going, other than to pick some arbitrary point other than the one they occupy and declare it "motionless" and measure their motion relative to it. That arbitrary point then becomes the "observer" and they become the "observed." This is indistinguishable, physically and mathematically speaking, from declaring their point to be motionless and the other point moving. The math will work out correctly either way.

Now, there's a word that's been kicked around on this thread, "transform." What does it mean? A transform is a set of equations that allows you to transform the mathematical description of objects moving from the point of view of one observer to the point of view of another. In its simplest form, it's merely a coordinate change; if the two observers agree they are not moving relative to one another, then you need merely shift the position of "zero." To do this, you define three axes to give you references, with their origin at the location of the first observer, and then you note that the second observer is at a distance measured along each of these axes from the first. From then on, every time the second observer claims to see an object at some location, all the first observer needs to do to substantiate their claim is to transform the second observer's coordinates into their own, then look there and see if the object is there. If the object is motionless according to the second observer, then (remembering that we already said the observers agree they are motionless to one another) s/he will.

This is the Galilean transform. Mathematically,
x' = x + d1
y' = y + d2
z' = z + d3
t = t
Where,
(x, y, z, t) is the position of the object according to the second observer
(x', y', z', t) is the position of the object according to the first observer
and
(d1, d2, d3, 0) is the distance between the observers.
Note that I have included time. Without really mentioning it, I've used time the same way I have used x, y, or z- as a dimension. This is an important point: despite the fact that we see time as different, in actuality it is just the same as any of the three space dimensions. The reason we see it differently is not because it is intrinsically different from space, but because the geometric arrangement of time with respect to the three space dimensions is different from the way those dimensions are arranged with respect to one another. I'll address this again later; you'll want to see some proof. And I'll provide it, or at least I'll prove it to the extent that it can be proven, scientifically speaking. But for now, just assume that I already have proven it, and accept that it's true. It will make things easier. Just make a mental note that I've said something I haven't yet proven, and watch to see that I ultimately prove it, in the end.

Now, the Galilean transform works fine for things that aren't moving; but does it work for things that are? Actually, because we see the speed of light as finite and maximal, it can't. If it did, the speed of light could not be finite, if it were maximal. Thinking about it, you'll realize that you can't see something moving at greater than the speed of light. Einstein's great genius was to realize that this actually would imply that something moving faster than the speed of light would disappear, as far as its forces or other effects on any other object were concerned; and this would imply that any object accelerated to faster than the speed of light would just disappear, and this would violate the law of the conservation of mass. He therefore was able to say that unless we were willing to tolerate violations of mass conservation, we could not be willing to tolerate the idea that something could move faster than light. And such was his genius that he therefore made the idea that nothing could move faster than light an assumption, that is, a postulate, rather than something that needed to be proven. And the above chain of reasoning is how it is justified: moving something faster than light violates mass conservation.

I have spoken before of the interconnectedness of scientific thought. One finds this all through physics; it is present, though less common, in other areas (not because physics is somehow "better;" merely because we know more about it than any other area of science, having studied it more. One might even argue that physics is the simplest of all the sciences, that we know more about it because other areas are more complex. But that is another thread). In this case, Einstein makes a set of postulates, and proceeds to derive a description of moving objects from it, and then proposes that we measure the real behavior of actual moving objects and see if it is right; and if it is, then we must accept the postulates. And we have, and it is, and we do. That's what the Theory of Special Relativity is.

So what are these postulates, anyway? Well, we've discussed the first of them above: The speed of light is finite and maximal. We've also discussed another, without naming it: Every unaccelerated observer's coordinate system yields physics (the math describing motion) that can be transformed into every other's. This second postulate is also stated as the Principle of Relativity, which is viewed as so important to the theory that it gave it its name. What it states, more clearly, is that physics looks the same wherever you are, and no matter how fast you are going. It actually extends a prior principle, with the same name, which makes the first statement, but not the second; that is, wherever you are, but not however fast you're going. This is called "Galilean relativity." Einstein adds another dimension to things; time. Because, as everyone knows, speed is distance over time.

The other two postulates are also implicit above, but even harder to see. The first is: Space and time form a four-dimensional continuum. Remember I said I'd prove that? Well, I will; but Einstein didn't prove it, he took it as a postulate. The second is implicit too: There exist frames of reference defined in the spacetime continuum in which unaccelerated objects move at constant velocity. Now, you'll recognize this one from somewhere else, if you think about it; it's Newton's First Law of Motion, stated somewhat differently, and with an extra proviso; that of non-acceleration. You might note that some forms state "in a straight line at constant velocity;" that first part is actually unnecessary, because we can easily show that any object not moving in a straight line is also not moving at a constant velocity, since velocity is not merely speed but also direction. And in addition to all of that, if you really think about it, Newton's First Law assumes this, rather than explicitly stating it; and that is part of the genius of Einstein.

From these four postulates, the Theory of Special Relativity springs. All of the complexities of modern spacetime physics come from this; and it, along with a few more postulates and one really incredible revelation, Einstein's crowning achievement of genius, forms the Theory of General Relativity. The predictions of the Special and General Theories have been confirmed time and again; and there is yet another test coming up, in a couple of months. In April of this year, 2007, the final reduction of the data from the Gravity Probe B mission will be done, and we will know then whether yet another prediction of these theories is correct: something called "frame dragging." Further out, the LIGO experiments will be complete somewhere around 2010 or 2011, and will confirm still another prediction: "gravity waves." But never forget that the most important predictions have all come true; the extra revolution of the major axis of Mercury's orbit; the bending of light around the Sun during an eclipse (and around a galaxy when the light of a galaxy behind it passes through its gravity field); and the frequency change in electromagnetic radiation when it loses potential energy in a gravity field.

Now it's time for a digression. I'll make that in the next post.

[Schneibster]

The conversation now must turn to electricity. All through the eighteenth and nineteenth centuries, scientists were playing with electricity. Pretty soon, they knew that it had intimate connections to chemistry and magnetism. And late in the nineteenth century, a man named James Clerk Maxwell turned his attention to it, and started looking for a framework in which he could derive all of the Laws that those before him had empirically discovered. His genius, as great as that of Einstein, Fermi, Dirac, Feynman, or Heisenberg, told him that these laws had to have some common underpinning, some single principle or principles from which they sprang. And he found it. Many argue that this was the birth of modern physics; some place it later, but none ignore Maxwell's incredible accomplishment, or fail to acknowledge that most of what came later sprang from the insights it gave us.

Maxwell considered how magnetism and electricity had to be related to one another, and how they had to behave with respect to space in order for the empirically discovered Laws to work the way they did. He used calculus to define a set of eight integral equations. Later, Oliver Heaviside and Willard Gibbs saw how to combine them into a set of four, from which the eight could be derived, and from them all the equations that electrical and electronics engineers use to this day: one that says how the electric field strength in space is related to the charge, one that says how the magnetic field behaves in space, one that defines how the magnetic field relates to the movement of charge, and one that defines how the magnetic field relates to the electric field. This was a stunning achievement, but the best was yet to come. When Maxwell and others began to consider the implications of these four equations, they realized that they implied that the electric and magnetic fields were actually two aspects of the same thing, and that all of it was related to the electric charge. They called this thing, electromagnetism. As a field, they realized that it must combine these two fields as aspects of this single underlying phenomenon, the effect of charge.

As scientists studied them more, further revelations awaited them. Soon someone noticed that the equations implied the existence of two basic parameters of space and matter: the electric permittivity, and the magnetic permeability. These defined how electric and magnetic fields behaved in various environments, varying with the substance, and achieving certain basic values in empty space. And like the two fields, they also were related to one another. After some analysis, it became clear that their relationship was based on the speed of propagation of the two fields, and that the two fields combined to form a wave.

But that was not all. Other scientists had been measuring the speed of light; and as methods got more and more sophisticated, those measurements became more and more accurate. When scientists studying Maxwell's equations used empirical measurements to determine what the speed of the waves these equations defined, imagine their surprise to find that it was so close to the speed of light that it couldn't be a coincidence. This was astonishing; suddenly, they were seeing light in a whole new way. It was a combination of the electric and magnetic fields generated by electric charges.

Soon, scientists were generating waves of their own; invisible waves, but not undetectable. Heinrich Hertz showed that these waves could propagate through space, just like light, and be detected by appropriate apparatus at a distance without any obvious connection between the point where they were generated and the point they were detected. These waves were called "radio." And elsewhere, William Roentgen made other waves that not only were invisible (but still not undetectable) but apparently could move through solid matter. He called these "X-rays," and so we still call them today.

All of these, and more besides, are aspects of the same underlying thing: they are the fields and the radiation of the electromagnetic force, one of the four fundamental forces in our universe. And, by the way, the easiest one to discover, and still today the one we know the most about.

Now, the story shifts again. When Newton studied light, he discovered many things; and he had an idea that light was made from tiny packets, which he called "corpuscles." We call these today, "particles," and we have a special name for the particles of light; but all of that comes much later. For a time, scientists accepted his ideas; but at the beginning of the nineteenth century, a scientist named Thomas Young performed an experiment that, for all of the nineteenth century and a bit of the twentieth, convinced scientists that light had to be made of waves, not particles. This experiment is called the dual or double slit experiment. What it shows is that light beams don't just pass through one another without any effect; instead, they interfere. Interference is what happens when waves interact with one another; in some places they add together, where their phase is the same, in others, they cancel out, where their phase is opposite. Now, two particles adding together is understandable; but how can two particles add up to no particles? It simply doesn't make any sense. Scientists therefore abandoned Newton's ideas of "corpuscles," and when Maxwell's fields came along, they just wrote them right into their story. Maxwell's waves were light waves, and that was that.

When one considers waves, a couple of questions become interesting. The first is, "what's waving?" and the second is, "how do these waves move with respect to moving objects?" So scientists concocted elaborate theories of something called "the aether," to use the English spelling. They hypothesized that it filled all of space; and that when Maxwell's waves waved, what waved was the aether. But more than that; you see, when one considers Newton's Laws, another question emerges, which subsumes the second question about these waves of light, and that question is, "Newton's Laws talk about motion; with respect to what is this motion defined?" Newton had an answer, but even he knew that there was more to figure out, and he couldn't say precisely what that more was. He invented the first of the gedankenexperiments, thought puzzles that Einstein would later make famous. This gedankenexperiment is called "Newton's Bucket," and it led to one of the greatest misunderstandings in all of physics; it was this misunderstanding that Einstein's Theory of Special Relativity cleared up, and in the minds of many, the reason it has the word "relativity" in its name.

Newton proposed that one consider a bucket suspended from a rope, filled with water. If one twists the bucket, and thereby the rope, and then releases the bucket, what happens? Well, it spins. But what happens then? The water in the bucket humps up at the outside, and shrinks down in the middle. But Newton asked a very important question: "spinning with respect to what?" At first, it seemed clear that the answer was, "with respect to Earth." But the question was still there, and when scientists started to think about things moving through emptiness, with no gravity, this question became much more difficult. Of course, the bucket wouldn't do; but it was easy enough to see that one could convert the system from a bucket of water to two rocks connected by a rope, and the distortion of the water's surface to the tension on that rope. And without gravity, this posed a very serious conundrum for scientists.

When they found the aether from Maxwell's equations' waves, they realized that this had to be their sine qua non, the ultimate answer to the question, "with respect to what," the what that they were looking for. So from this one idea, they got double duty: something to wave, and something to define absolute motion with respect to.

Not only that, but the longer they studied Maxwell's equations, the more they realized that there was something essential missing. These equations, you see, defined motion in terms of the wave; but there was no other reference point for it. And this introduced yet another problem: two observers, moving at different velocities, could see the same wave moving at the same velocity after transformation (you DO remember transformation from the previous post, right? ). But this COULD NOT BE. How could the same wave seem to be moving at the same speed for observers moving at different speeds or even in different directions?

The aether, of course, answered the other questions, but ultimately, it could not answer this one.

But there was a worse problem, and eventually, that problem would lead to our modern understanding of physics, and the death of the hypothesis of the aether.

The fly was something called the "Michelson-Morley Experiment." Albert Michelson and Edward Morley performed this experiment first in 1887, twenty-six years after Maxwell had published the first paper containing his equations, and twenty-three after the seminal paper in which Maxwell finally linked all of them together and proposed the unification of electricity and magnetism into a single overarching phenomenon. For a while, there was not much notice; but soon, everyone saw that their experiment threw the whole aether idea into the circular file.

In this experiment, Michelson and Morley hypothesized first that since the Earth was moving around the Sun, it must at some point be moving relative to the aether. And in fact, most scientists thought that the Sun was moving too, so they figured it must be moving with respect to the aether all the time. This would mean in turn that there must be an "aether wind," as the movement of the Earth passed it through the motionless aether, and they proposed to measure it.

They failed utterly. Comprehensively. They tried again and again, with ever more precise equipment, under ever more controlled conditions, and no matter what they did, there was no aether wind.

The obvious conclusion was, there was no aether. Furor ensued; mass confusion in the physics community. It was only made worse by George FitzGerald, an Irish physicist who worked with Heaviside and Gibbs, and Hertz, on Maxwell's equations. FitzGerald proposed in 1889 that the reason that Michelson and Morley's experiment failed was because movement through the aether compressed all objects, and even space itself, along the direction of movement, just enough that the movement could not be measured. This rather famously led to comparison's with Lewis Carroll's tale of the White Knight, who said,
"But I was thinking of a plan
To dye one's whiskers green,
And always use so large a fan
That it could not be seen."
Ridicule ensued, until no less figure than Hendrik Antoon Lorentz, who did not know of FitzGerald's ideas, proposed a new set of transforms in 1892 to replace those of Galileo, confirmed by Newton in his Laws of Motion. And there was FitzGerald's "fan," and it had slipped, and the green whiskers were plain for all to see. Of course, no one realized it for what it was until Einstein came along; but Lorentz's idea had better grounding than FitzGerald's, since it managed to explain not only the Michelson-Morley experiment, but also the problem of Maxwell's equations' lack of a frame of reference. However, credit was shared when everyone realized that both had discovered the idea independently, so to this day the idea is called the Lorentz-FitzGerald contraction.

Now, Lorentz's idea was superior to FitzGerald's, in that it proposed that all objects contracted in the direction of motion. So it didn't merely account for the problem of the aether wind, it also accounted for the problem with Maxwell's equations. But neither Lorentz, nor anyone else, took the final step of understanding, until Albert Einstein saw that there must be another answer, deeper, subsuming both Maxwell's equations, and Lorentz's. And ultimately, answering the old question of Newton's Bucket, as well.

And now the digression ends; in my next post, we'll take up Einstein's story again, and see how his Theory of Special Relativity led to a final resolution of all of these conundrums; then generated yet another one; and finally resolved that one too, in the Theory of General Relativity in which Special Relativity was subsumed, the geometry of space explained, and gravity finally fully understood hundreds of years after Newton's Law of Universal Gravitation. And finally how this theory became one of the two most important theories in modern physics.

[Schneibster]

OK, so now we have seen how transforms work. We've seen how Einstein took Lorentz's transform, and Maxwell's equations, and subsumed them into a startling revelation about the nature of space and time; but let's look a little deeper.

What are Lorentz's transform equations, anyway? How are they different than Galileo's and Newton's?

They propose that objects shrink along the direction of travel. But in Einstein's hands, they propose a lot more than that. First, let's have a quick look at them. For convenience's sake, we'll examine them as algebra, and we'll stick to an object moving only in x, however x is defined; to go beyond that requires some pretty tricky math, and is probably beyond the necessary scope of this discussion. I'll go a little farther later; we'll talk about not only trigonometry, but hyberbolic trigonometry; but for now, let's stick to the easy stuff.

x' = \gamma (x - v t)\
y' = y \
z' = z \
t' = \gamma \left( t - \frac{v x}{c^{2}} \right) \

where \gamma = { 1 \over \sqrt{1 - v^2/c^2} }, is called the Lorentz factor. This last is the equation FitzGerald proposed for compression along the direction of motion through the aether.

If you look here, you'll notice something interesting: the definition of distance becomes intertwined with the definition of time, as soon as you start to move. It is this that causes all of the diverse effects of relativity. The change in mass, the change in the experience of time, the change in energy, the change in length: all are results of this.

I've made a mistake; I hit "submit" when I meant to hit "preview." I'll continue in the next post.

[Schneibster]

So now we've seen the Galilean transform, and the Lorentz transform, and we know why the Lorentz transform exists; but we still have to show that it really does apply. Only then can we say what we should see, and why we should see it, and go measure it and see if it really is that way.

Let's step back a bit. Remember the First Postulate of Special Relativity? Here it is: Space and time form a four-dimensional continuum. But what does that really mean?

First, we have to discuss how rotations look in various dimensionalities. Let's start with one dimension. Obviously, no rotation is possible in one dimension. The lowest dimensionality in which the concept "rotation" has any meaning is two. So how many axes can we rotate an object around in two dimensions? The answer is, "one," and if you think about it, you'll suddenly realize that that axis points in a direction that does not exist in those two dimensions. In other words, in order to have rotation in two dimensions you already have to have a third.

Now, that's pretty weird. But it teaches a lesson: rotation is always weird. You'd better never forget that; it's the most important thing to know about rotations. Not only that, but I've already bought into a concept that physics actually doesn't use: that of an axis of rotation. This is an imaginary concept, according to physics; and nothing could make this clearer than the above example of rotations in two dimensions. The axis cannot exist in two dimensions. It points in an undefined direction. The concept that physicists use is much less known, though in the end, far more intuitive: physicists talk not about axes of rotation, but about planes of rotation. Note most carefully that this formulation allows the definition of rotation in two dimensions to exist entirely within those two dimensions. This makes this a much better way to think of things.

OK, let's test our idea: we'll apply it in three dimensions. What are the planes?

Simple enough: x-y, x-z, and y-z. No problem there.

Now, back to our postulate. If we consider the addition of a fourth dimension, time, what are our planes of rotation?

Obviously, just as x-y subsumes the previous two-dimensional definition, this one must subsume the x-y, y-z, x-z definition; and it does. But how many new planes does it add? Here's where it gets interesting: the intuitive answer, thinking of the rejected concept of "axes," implies only one; but the correct answer is, "three." They are x-t, y-t, and z-t.

Suddenly, we realize that our identification of the three "axes of rotation" as "x, y, and z" is merely a fluke; in other dimensionalities, the number of these figments of our imagination is NOT tied to the number of dimensions, and this is true not only of more, but less dimensions than three. A basic idea that many have no doubt stumbled over is suddenly revealed to be completely without merit. Our new conception of planes of rotation shows us that two dimensions has one plane, three has three planes, and four has six. We can go further: in each case, we see that the number of planes is equal to the number of combinations of two dimensions; another way to look at this is, the number of planes of rotation in the current dimensionality is equal to the sum of planes and dimensions in the next lower dimensionality. Thus, in five dimensions, there would be not seven, but ten planes of rotation; the six planes plus the four dimensions of four-space. Or, another way, if we call the fifth dimension a, we would have x-y, x-z, x-t, x-a, y-z, y-t, y-a, z-t, z-a, and t-a for the possible planes.

Let's step back to our Postulate of the four-dimensional spacetime continuum again. We now see that any of the three dimensions can be rotated into time, or all three of them together.

Wait a minute. What did I just say there? Let me say it again: objects can be rotated not only in space, but in spacetime.

Now, what the heck does that mean?

The answer to that question is the key to understanding the implications of the Theory of Relativity.

Let's see the mathematics of rotations first. Perhaps this will lead us to a better understanding. (Of COURSE it will. )

We've talked about the transform of two orthogonal frames of reference into one another; the Galilean transform. We just add the difference between the origins. But we've ignored a subtlety: how do we define that transform if I decide x points thisaway, but you want your x to point thataway? The answer lies in trigonometry. Of course it does; trigonometry is the mathematics of the description of rotations in space.

Thus, we see that instead of our simple x' = x + d1, we really have to allow for x to point in a different direction, not merely be from a different origin; and that means that x is actually composed partly of x', partly of y', and partly of z'. We'll consider a simple rotation, in the single plane of x-y. Note that this means that your x points somewhere midway between my x and my y. The equations look like this:

x' = ((x + d0) \cos \theta) + ((y + d1) \sin \theta)\
y' = ((y + d1) \cos \theta) - ((x + d0) \sin \theta)\
z' = z \
t' = t \

So now we've not only moved the origins of our coordinate systems to one another, we've also allowed for them to be rotated. You can see (if you've dealt much with math) that I can formulate a system of two or three sets of these four equations, of similar form to the above, to describe rotations that involve two or even three planes of rotation. You can also see that I can reverse the equations to transform your coordinates into mine. Finally, from the Principle of Relativity, you can see that it doesn't matter which of these coordinate systems you define it in, we both see the same thing the same way.

Now, in all of this, it remains true that
t' = t \
So we've not yet started talking about rotations in the three planes that include time; x-t, y-t, and z-t. That's where this gets complicated.

You see, the three spatial dimensions, x, y, and z, however we decide in our personal frame of reference to define them, are not only orthogonal, they also have another property, one that is implicit in the type of trigonometry we've chosen to use. This is circular trigonometry. And the fact that this type of trig really works to describe these transforms, and that when we use it, we really do see precisely what we predict we will using it, tells us about this implicit property of space: each dimension of space is circularly symmetric with respect to the other two. In other words, if I rotate something in space, it stays the same size. There is a special formula for figuring something's size out in our two coordinate systems so it remains the same; this is called the Pythagorean formula. You will be familiar with a derivation of this formula to calculate the missing hypotenuse of a right triangle. But it is far more important than that. What it says is, if we define the length of an object in terms of our three coordinates, x, y, and z, it will remain invariant under rotations. If we say that length is "d," then the definition of d is:

d = (x^2 + y^2 + z^2) \

and this definition is the same no matter how we define x, y, and z, and the size of d does not change no matter whose x, y, and z we choose. It doesn't matter if we use my x, y, and z, or yours; you realize, of course, that we would have to use the Galilean transform, including the rotational formulae, to translate my coordinates into yours; but d would still come out the same. And this is because of the circular symmetry of the three spatial dimensions with respect to one another.

Now I'm going to stop and define some terms. Physicists call the operation of adding d1, d2, and d3 to x, y, and z to get x', y', and z' a translation. They call the operation of using the trigonometric formulae to figure how to change my x, y, and z into yours when you think x is a different direction than I do a rotation. And finally, the combination of these, rotation and translation, is called transformation. So when we speak of a transform, we're talking about combining the formulae for a translation and a rotation to transform some observations I made using my coordinate system into your observations of the same phenomena in yours.

But we now have to consider the relation of the time dimension to the three space dimensions; and here matters are different. We know they must be. If we rotate something in the three extra planes of rotation, we can't directly see that rotation. We can't see time. We experience the passage of time; but we can't see into the future, or into the past. We can only see right now.

If we rotate something in the three invisible planes, some of its length must become time. But we can't see in time. So that means that if we rotate something in those three planes, it will be foreshortened. It will be compressed along the direction it is moving. And it doesn't matter what transform we use; this must be true no matter what the transform for a time-related rotation is.

But if we rotate something in time, something has to change. What is that something?

It is velocity. The proof is in the fact that the very definition of velocity includes time. v = d/t, velocity is distance per time; miles per hour, furlongs per fortnight, meters per second- velocity. Of course, "velocity" also includes direction. Now the importance of the "d" in the Pythagorean formula becomes clear. When we realize that d remains invariant over translations and rotations, we have realized that we will observe an object to have moved the same distance in both our frames of reference. But what if we are NOT motionless relative to one another? What if we are MOVING relative to one another? What will we observe THEN? Obviously, some of that distance will have rotated into time; and since we cannot see time, it will have effectively disappeared. But if the distance has changed, then we have seen the same object move a different distance; and if we define the starting and ending times the same way, then it will HAVE A DIFFERENT VELOCITY. And there is only one way this can be true: a velocity must be a rotation in the three invisible planes of rotation, x-t, y-t, and z-t. What can I add to a velocity to make it different? Only another velocity: my own. And that means that when we see something move, we are seeing it rotate some of its space into time.

We're nearly there. I'll finish up this idea in my next post, if I can; and then I'll branch back again, and we'll understand how quantum mechanics fits into this picture.

[Schneibster]

OK, let's see if I can bring this home in one post, or if I'll need more.

We discussed the following points:

The four postulates of Special Relativity.
The Galilean transform, first as merely translation, then as rotation, and finally in its full form.
Maxwell's equations, the identity of light with electromagnetism.
The aether, the postulate of absolute space, and its two flaws.
Lorentz's formula for calculating the change in an object's length with velocity, necessary to eliminate the two flaws in the aether hypothesis.
Rotations in space, and the implications of the invariance of distance over rotation for the geometry of space.
The existence of rotations in time, and their identity with differences in velocity.


Now we are ready to show what the Lorentz transform implies. At first, of course, physicists (including Lorentz) applied it naïvely to distances only. But when Einstein looked at Lorentz's transform, he realized what no one else had: it implied that there was an exact formula for transforming space into time, and vice versa, and that this formula implied much, much more than simple distance transformations. For, you see, time is very important; it is important in the discussion of all the forces. Newton's Second Law of Motion tells us that.

f = ma

It's just that simple. The existence of acceleration in the right side tells us that time is doubly important in the definition of force; thus, as time varies, force varies inversely as its square. And if time can rotate into space, and space into time, and if velocity is the definition of this rotation, then we can see that force must vary with velocity.

We have expressions for the force of gravity, and the force of the electric and magnetic fields. Elsewhere, I have shown that the magnetic field is merely the relativistic correction for the force of the electric field. So there are two ways to interpret this: either the force varies with velocity, or the charge does. But what is mass? It is the charge of an object with respect to gravity. Just as the electric charge of an object is proportional to its electric force, the mass of an object is proportional to its gravitational force. Look at the formulae:

F{g} = {gmm'}/{d^2}
F{e} = {kqq'}/{d^2}

We have the forces of gravity and electromagnetism; we have the gravitational and electric constants; we have the masses and charges; and we have the distance squared. Simple, easy, obvious. It is for this reason that physicists talk about invariant mass; one might as well talk about charge varying with velocity as about mass doing so. Since we believe that both charge and mass are conserved quantities, we talk instead about magnetism and observed mass, and distinguish the second from rest mass, which we state is invariant. In fact, physicists rarely talk about observed mass; it is unimportant. This is because it is not invariant; it depends on who is observing it.

So we see the implications of the Lorentz transform; time varies with velocity. This means that from the point of view of one observer, another observer that has some nonzero velocity experiences time at a different rate.

So I've spoken of these rotations; but I haven't said what the Lorentz transform implies about their trigonometry. And that's something very unusual.

You see, the fact that we can't use the Galilean transform for spacetime rotations means that unlike the relationship of the spatial dimensions to one another, the symmetric relation of time to the spatial dimensions must not be circular. Now, there are only four possibilities here: circular, elliptical, parabolic, and hyperbolic. And the Lorentz transform's peculiarities narrow the choice down to a single one: the geometry of the relation of time to space is hyperbolic, and we must use hyperbolic trig to define the effects of these rotations.

Let's have a look at the equations of the Lorentz transform in trigonometric form, rather than the conventional algebra. We'll define our rotation as one in the x-t plane:

x' = t(\sinh s) + x(\cosh s)
y' = y
z' = z
t' = x(\sinh s) + t(\cosh s)

So what's "s?" It's the hyperbolic angle; physicists call it the "rapidity." It's a number that varies between 0 and 1 as velocity varies from unmoving to the speed of light. It measures the magnitude of the difference between the speeds of the two observers as a hyperbolic angle. This difference is called a "boost," often referred to as a "Lorentz boost." It signifies not an acceleration from one speed to another, but the difference in speed between two frames of reference. It is the algebraic equivalent, for the algebraic form of the Lorentz transform, of the hyperbolic trig rapidity s; and it is expressed as a velocity in the algebraic form of the transform.

Hyperbolic trig has some seriously weird implications, among them that there is some rotation whose angle is "infinity." This is totally meaningless in circular trig; there is no direction you can point something that you would define as an angle of "infinity." But hyperbolic trig says that there is some such rotational angle, and that you can never reach that angle by any ordinary rotation.

What does this mean? It means that there is some velocity (angle) you can never reach by any acceleration (rotation). And that velocity is the speed of light. Let's say that again: the fact that the speed of light is maximal is inherent in the geometry of spacetime. And we know it is so because Maxwell's equations tell us there is no frame in which light is not moving at the speed of light; there is no reference. All observers see identical laws of physics; and the laws of physics include Maxwell's equations, and Maxwell's equations say that the speed of light is the same for all possible observers. And this agreement between these two sets of equations is absolute; if Maxwell's equations were different, we would not be able to build generators, or semiconductors; we could not use radios, or take X-rays.

There's just nowhere to hide. If Maxwell's equations are right, and we know they are because electronics works, and if the Michelson-Morley experiment happens, and we know it does, every physics student does it sometime during their career, and if the Lorentz transform correctly converts different observers' observations into one another, and we know it does, because we see the orbit of Mercury, among other things, then relativity must be right. There's no escaping it. We might find a fine correction to it later on; just as relativity itself provided fine corrections to Newton's Laws of Motion, and Law of Universal Gravitation. But that it is essentially true, is unquestionable given our observations. There is no choice. You can't avoid it.

And what of the aether?

It's no longer necessary, save as the "something that waves." And even that is unnecessary; quantum mechanics arrives to give an answer that, while far more complex, and far more difficult to understand, ultimately explains things that the theory of the aether never could. A relic, it has fallen by the wayside; and like many another simplistic theory, phlogiston for example, is mostly forgotten except by historians of science.

What about absolute motion?

There is no absolute motion. Motion is always relative, always at less than the speed of light, and always a rotation in the three invisible planes of rotation.

So, is there absolute space?

No. Instead, Einstein proposes absolute spacetime. And he proves it, too; what he says is, there is no absolute motion, but there is absolute acceleration. Anyone in a non-inertial frame can perform simple experiments that will show it, without needing to look at objects outside their immediate frame, objects that share their motion. But he says far more than that; in the Theory of General Relativity, Einstein says that gravity and acceleration are indistinguishable. And reasoning from Special Relativity, he shows that acceleration must involve a distortion of spacetime; and if gravity and acceleration are the same, then gravity must also be a distortion of spacetime, but one that happens without motion. This incredible result changed the way we look at our universe, and the ripples from this realization are still making their way through physics.

Are there a set of "Einstein's equations," that describe gravity, just as Maxwell's equations describe electromagnetism?

Yes, there are. They are a set of ten field equations that are reducible in a notation mostly invented by Einstein to a single tensor equation that specifies how spacetime is warped by acceleration (or equivalently, by gravity). These are the field equations of gravity, which define its action just as Maxwell's equations define the action of the electromagnetic force.

Now, how does all this apply to the initial post on this thread? It answers the questions posed there. Let's see that IP again:
Originally Posted by RandomElement View Post
One is travelling in a spaceship between galaxies which are moving apart at great speeds. These galaxies may be rotating around galactic clusters which are also in movement. Space, or at least the objects within it are not stationary and there are numerous eddies and whorls.

From inside the spaceship, one can only measure acceleration and not speed per se. The ship is accelerating at 1 G. How does it "know" to increase to near infinite mass when it approaches the speed of light as there is no fixed reference point only relative ones? How is speed then measured? If speed is measured from the launch point, which is itself moving, then would an alien ship going the exact same speed and direction, but launched from a different galaxy moving in an opposite direction be said to be going a different speed?
First, it doesn't matter what the galaxies are doing, except to how we see them. Second, the only eddies and whorls are due to either acceleration or gravity (and in fact, there is no real difference between the two). Third, from the point of view of an observer on the spaceship, its mass has not increased. Fourth, it does not therefore need to "know" anything. Fifth, the observed mass increase is not really an increase in mass, but a consequence of rotation in the invisible planes of rotation that involve time; specifically, it is the consequence of the change in time, nothing else. Sixth, speed is measured by an arbitrary observer, and observers in different states of motion will measure it differently; yet, all their observations can be transformed into one another using Lorentz's formula. And finally, a difference in speed is a difference in velocity; but a difference in velocity is not necessarily a difference in speed. Not only that, but to different observers, what one sees as a difference in velocity with no difference in speed, another may see as a difference in both. How fast you are going relative to something determines how you see that something. That is the essential truth of relativity.

One last point. I said that I'd prove that time is essentially no different from space. I think I have; please observe the equations of the hyperbolic form of the Lorentz transform above, then examine the Galilean transform in circular trig notation. I don't think I need to point out their near identity. That these equations work in the real world to predict what we observe is the final proof of it all; we see now that there is no difference between space and time, only a difference in the way time is configured with respect to the spatial dimensions from the way the spatial dimensions are configured with respect to one another. And we can even define what this looks like, sort of; it's a curvature, in the form of a hyperbola, of each of the three spatial dimensions with respect to the time dimension, with the hyperbola's axis being the direction of time. This is inexact; the real form of hyperbolic geometry is subtler than this. But it will do for a simple visualization.

[PsychoSerenity]

I've read the first one and the axis vs plane thing seems to make a lot of sense - but I'll have to come back to the others later.

[apophenia]

Meh. tl;dr. You need to choose one per thread. My only comments, having only plowed through 1 1/2 tl;dr's is, a) not all Alephs are created equal, b) when you move into higher dimensions, it's probably profitable to move from planes to hyper-planes (I know -- I've given up making Hawking-Hartle arguments with theists, it just doesn't work), and I may be mistaken -- correct me if I am -- but doesn't the supposition that accelerating bodies need to be treated differently than non-accelerating bodies assume a Euclidean plane or hyper-plane, which, as an isolated body or set of observations it cannot do?

[Schneibster]

Meh. tl;dr. You need to choose one per thread.

I will very probably rewrite them. I will take your advice under consideration; thank you.

My only comments, having only plowed through 1 1/2 tl;dr's is, a) not all Alephs are created equal,

Maybe I'm slow today. Could you be a bit more specific?

b) when you move into higher dimensions, it's probably profitable to move from planes to hyper-planes (I know -- I've given up making Hawking-Hartle arguments with theists, it just doesn't work),

No matter how hyper you make them they can always be broken down into planes. And the results with the planes will be the same as the results with the, errr, hyper-planes. Or whatever. Cali just did that part.

and I may be mistaken -- correct me if I am -- but doesn't the supposition that accelerating bodies need to be treated differently than non-accelerating bodies assume a Euclidean plane or hyper-plane, which, as an isolated body or set of observations it cannot do?

Actually, what we have is a 3 dimensional space with a 1 dimensional time. Space dimensions are different from time dimensions; space dimensions are circularly symmetric with respect to one another. This is different from time dimensions, which are hyperbolically symmetric with respect to space dimensions.

That's in one of the parts you didn't read.