pernero wrote:If you ask any astrophysicist or semi-intellectual person what the fourth dimension is, they'll say that it's spacetime.
It sounds more like you've been asking your high school buddies.
Start from the start, and it's not so hard.
Start with a point on a number line. You can describe to someone where it is easily by telling them which number the point is on.
What about a point on a plane. Well, draw arrows at 90 degree angles to each other. Label one "x" and the other "y". Then draw out a grid to the point in question. Then give the (x, y) coordinate of the grid. That's two dimensions.
What about a point in three dimensions? You can't do it on a chalkboard or on paper any more. But you can use three rulers in the corner of a room at 90 degree angles. Label the rulers x, y, and z. Give the (x, y, z) coordinate.
What about four dimensions? Well, sadly, we've run out of physical dimensions, so we can't do it in our room. Space only has three dimensions, so we're out of luck in creating an actual point in four dimensions. But mathematically, we just continue what we were doing and ignore the limitations of real life:
1 dimension - number line - one real number
2 dimesions - a chalk board - two (ordered) real numbers
3 dimensions - a room - three (ordered) real numbers
Take a guess. How do we describe a point in 4 dimensions? You guessed it, four real numbers. And similarly, we can talk about 5 or 6 or 7 or 29. In fact, as mathematicians, when a theorem applies to any finite-dimensional real space, we just call it "Rn
". Lots of things that are true in R2
are true in Rn
for any n. Lengths are measured using the "root-means-squared" function we derive from the Pythagorean theorem. So if we have a point (x, y, z, w), the distance from the origin is the value sqrt(x^2 + y^2 + z^2 + w^2). Of course, we could use any letter for w, but w and t are the two most common ones.
The key aspect of something being a dimesion is that we need an additional coordinate to describe it. It is "independent" of the rest. If you have a cube, certainly it's height has no bearing on its width or its length. This "independence" is why we have to use rulers at 90 degree angles. Right angles are very, very strongly connected to this notion of independence.
So where else can we find this independence in real life? Time comes to mind. Say we put a box in the room and take it out shortly thereafter. HOW LONG we keep the box in the room has no bearing on its height or its width or its length. It satisfies this independence, so we can just go ahead and call it a dimension. We can describe the box by four parameters: its (x, y, z, t) where x y and z are its physical dimensions and t represents (in seconds) how long the box stayed in the room. It's a "length of time." So there, it's a dimension.
Now time, even in classical physics, isn't like the other dimensions. The spacial dimensions are directions we can move stuff back and forth. You go to school, you go back home, and the next day, you're at school yet again. But time doesn't allow you to really "move back". It's a much less obvious dimension. Not only that, but it seems that your spacial position is a function of time. That is, at any given point in time, you are at exactly one location. This means, geometrically, you are a CURVE in 4-D space. This isn't so weird, though, because this is the essence of a graph. In calculus or physics, you might graph a ball's height as a function of time. What you're really drawing is a graph of spacetime, using ink to represent the ball.
Just to rid your idea of the specialty of time as a unique forth dimension, let's take another example of a 4D space. As you know, to describe a point on a sheet of paper, you need two numbers, the (x, y) coordinate. Well, how do you describe the location of TWO points, point A and point B? You need four coordinates, (x_A, y_A) and (x_B, y_B). But the positions of either point doesn't depend on another, so you could just as easily write (x_A, y_A, x_B, y_B). And what's in a name? We can just rename that as (x, y, z, w) and voila. A "point" in four dimensions. Now it's *really* two points in two dimensions. But even more *really* than that is it doesn't matter. Words like "real" and "exists" are pretty stupid words. What matters is what you can describe, and you can describe a pair of points just as easily as two planar points as you could as if it were a point in 4 dimensional space.
Let's talk about relativity for a bit. There are three steps in relativity. Galileian Relativity, discovered in the 1600s was that there is no distinction between "motion" and "rest". A kid in an airplane can't tell if he's on the ground of soaring through the air (at a constant speed). The air may be more turbulent or something, but it could just be an Earthquake. Galileo gave us this theory that one man's rest is another man's motino.
Einstein's Special Relativity invented 300 years later made an important correction based on experiments in light and electromagnetism. Light waves can move through space at a constant speed. But very strangely, this constant speed did NOT depend on the observer. Take this example. A man watches a plane fly overhead. He sees the plane moving at 600mph. Another man is on a train and sees the same plane. He measures the speed at 500mph. Do the same experiment with light instead of a plane. The man on the ground sees a light wave moving at 670 million miles per hour. The man on the train will also see the light wave moving at the EXACT SAME SPEED. That is creepy if you think about it. The man on the ground will swear to god that the man on the train is wrong. He sees the light moving at 670 million mph. If the man on the train, moving 100mph sees the light moving at 670 million mph, then the light must actually be moving 670 million PLUS 100 mph. There's a paradox to be resolved.
Einstein's resolution was Everything You Know is Wrong. It is taken as an axiom that light moves at a constant speed relative to the person measuring it. From that axiom, we can prove that time slows down for moving observers, length contracts, and no two events separated in space can happen "at the same time". But conversely, if I see you moving, you see me moving. To me, YOU are the one who is slowing down in time. To you, *I* am slowing down in time. It's called relativity because you have to pick a biased observer and your measurements are "relative" to that observer in accordance to the theory. Who is *really* slowing down? You can't answer that question without picking a frame of reference -- a bias observer.
So, that was our physics interlude. What does this mean for time's "dimension status"? Well, it turns out it affects how you translate from one frame to another. To any fixed observer, time works just like it does in classical physics. But when you want to change your POV (point of view), you have to use a different set of operations. In Galileo's theory, you used Galileian transformations, which basically subtracted your new POV's velocity from everything. In Special Relativity, it's not a simple subtraction. There's an extra factor based on the new POV's velocity in terms of the percentage of the speed of light. This new way to change POV is called a Lorentz Transformation, if you want the full details.
Now special relativity has limitations. It only works when velocities are fixed. When all objects are just cruising about space with no forces acting on anything. The problem is that that doesn't really happen in life. It took Einstein over a decade to work out the rest of his theory, General Relativity.
General Relativity goes and makes things complicated. A lot of Einstein's time working on the theory was learning the math behind it. It's not simple Euclidean geometry and calculus. It requires a branch of mathematics called Riemann Geometry. Riemann Geometry started off as a way to work out how to do calculus on the surfaces of 3D objects. The Earth is a good example. We can talk about "distance" between New York and Los Angelos, but the truth is this distance isn't just the distance of a straight line between the two cities. Such a line would travel through the Earth's crust. Not a path we can take on a plane. Instead, we restrict our motion to the surface of the sphere. The surface, on a very small scale, looks almost like a plane, even though it's part of a three dimensional object. This concept is generalized into what's called a manifold. Riemann Geometry gives us the tools to do natural geometric things in this "richer" space. What is the shortest distance between two points? What is the angle of a triangle embedded on the surface? How "curved" is this point on the surface? It's not a subject for the casually curious, sadly. It requires a pretty good understanding of calculus, analysis, and topology.
So bottom line, even if that didn't all make sense. "The forth dimension", along with "division by zero", "infinity", and "time travel", is just one of those phrases which sounds a LOT more exotic than it really is. There's interesting stuff to space and time, but it's MUCH more interesting than trying to count higher than three. If you ever hear anyone talk about "four dimensional space", they guaranteedly don't know as much about the subject as you think they know.