By R.F.J. van Linden 1. Spatial dimensions and the time dimension Imagine a man standing on a staircase with steps of about one third of his own height. He will be able to see the surface of the step he is standing on as well as the next two that are in front of him. The fourth step is harder to see, for his view on it is partly blocked (he may see it from the bottom or the front side). The fifth is even harder to see, or perhaps even invisible. Suppose he eats something very nutritious and his body suddenly grows a full step in height. He is now able to see four steps but his view on the fifth step is partly blocked. That fifth step now looks exactly like the fourth step did before he grew. The looks of the other four steps are not really different from the three steps he used to see before. He grows older and, like older people do, he shrinks in size. He shrinks a full two steps and now he can clearly see two steps but the third is partly blocked. That third step now looks identical to the original fourth. We
can translate this staircase into our spacetime. The visible steps are
our spatial dimensions and the partly visible higher step is our time dimension.
The time step/dimension is actually just another spatial step/dimension but that
only becomes clear when you move up and down in the dimensions. What
you call a spatial dimension from your own
"dimensional viewpoint" may be a time dimension from another observer's
dimensional viewpoint. For the man who
grew tall, our 4dimensional spacetime is his 4dimensional space,
while he lives in a 5dimensional spacetime or "Hyperspaceland". For
the man who shrunk, our 3dimensional space is his 3dimensional spacetime.
He lives in "Flatland".
We measure speed by dividing a covered distance by a time duration. Traveling one meter per two seconds, our speed is 1/2 m/s. If we call the time duration (the seconds) a "length" in the dimension time we recognize that speed actually is a division of covered distance in a spatial dimension by covered distance in the time dimension. A division of covered distance in two spatial dimensions, let's say dimension nr. 1 (breadth) and dimension nr. 3 (height), for us results merely in a "dimensionless" and rather meaningless number. However, for the old man whose size shrunk and lives in Flatland this division represents a spatial speed in his 2dimensional world because dimension 3 is his time dimension. This implies that the position of his whole 2dimensional environment must be changing in that third dimension, otherwise there would be nothing to divide the displacement in his spatial dimension nr. 1 by. In other words, he will only be able to measure it as a speed if his Flatland moves as a whole in the third dimension. This displacement in the 3^{rd} dimension looks to us like a spatial speed but the man who shrunk cannot see this as a spatial speed because he is not able to see this 3^{rd} dimension as a spatial dimension. For him it intuitively feels like a progress in his time dimension. Our intuitively felt "motion in time" actually is similar. Our 3D environment apparently moves as a whole in our time dimension too. And that enables is to measure speeds in our 3D spatial world. So even if we do not measure any speed of objects in our spatial environment, e.g. if everything around us is standing motionless, that spatial environment still has a speed in the time dimension. But how can we express this speed in the time dimension in terms of a division of covered distances in two different dimensions? Seconds per second doesn't make sense because that actually uses the same dimension twice and will merely result in "1". And we don't have a spatial distance covered that could be used to measure that speed somehow. For the man who grew tall after eating the very nutritious food, the answer is obvious. For him, the original time dimension turned into a normal spatial dimension and another, 5^{th} dimension became his new time dimension. Our speed in time becomes a spatial speed in the 4dimensional space of the tall man. If he wants to express that speed in his 4dimensional space he divides the spatial distance covered by the time duration from his 5^{th} dimension. So our inability to "see" our speed in time as a real motion is due to our inability to measure displacements in the fifth dimension that is used to calculate it. Our speed in time turns into a regular spatial speed as soon as one is able to perceive an extra dimension, like the tall man does. But how then do we calculate the speed in time (that will henceforth also be referred to as "timespeed") for the tall man? Obviously it must be a displacement in his 5^{th} dimension divided by a displacement in yet another higher 6^{th} dimension. And so on, and so on. It's a recursive, or fractallike system.
Einstein's theory of relativity predicts a limit in the speed that a thing can have. This limit is equal to the speed of light c and equals about 300.000 km/s. The faster something moves, the slower time ticks away in that thing. If the speed of a rocket is for example 4/5^{th} of c (about 240.000 km/s) then one second on a clock in the rocket takes as much as 1.67 seconds according to the watch of somebody who stands still on the ground. So according to the man on the ground the clock in the rocket runs slow. He observes that the timespeed in the rocket is slower than his own. Notice here that we quietly switched from a comparison of time durations (1 second versus 1.67 seconds) to a comparison of time speeds. It's straightforward to compare the time durations on the clocks at the man and in the rocket and express these in seconds per second. But in this case the comparison of the timespeeds can also be expressed in seconds per second because in that comparison the common and immeasurable fifth dimension that is used to calculate both speeds cancels out in the division. To see that, divide e.g. 1/x by 2/x. The result is 1/2 and the "x" has vanished. We have seen that whenever an object's spatial speed goes up, its timespeed goes down and vice versa. When the spatial speed is zero, the timespeed must therefore be at its maximum. On the other hand when spatial speed equals its maximum, which is the speed of light c, the timespeed actually has reduced to zero. For a clock in a rocket traveling at that speed it will take forever to tick away a second according to the man on the ground. Thus:
A familiar formula can be used to express this relation between spatial speed and timespeed. It is Pythagoras’ rule for rectangular triangles: A^{2}+B^{2}=C^{2}. If we say that A is the spatial speed and B the timespeed then it appears that the sum of spatial speed and timespeed is always c (the speed of light) if we add them with Pythagoras’ rule like in a triangle: (spatial speed)^{2}+(timespeed)^{2}=c^{2} This allows us to enhance our rules:
Whenever something moves (fast) in space its
length contracts in the
direction of its motion. Furthermore the time coordinate is no longer the same
in all points of the object but has decreasing values in points along the direction of its motion
like in this picture:
We
say that the points are nonsimultaneous which basically means that
adjacent points in the object are in the past or future, relative to each
other ! Another very nice interactive visualization of this effect was made by Adam
Trepczynski on
http://www.adamtoons.de/physics/relativity.swf (you will need the
Adobe Flash Player
browser addon to play it).
Light rays travel at speed c and consist of millions and billions of tiny photons. These photons exist everywhere and travel in all possible directions in our 3D space. The thing that all these photons have in common is that they all travel at exactly that same speed c, no matter what direction they take and no matter who looks at them, even for an observer who travels at high speeds himself! An obvious question now is: "What is the timespeed of the photons?". Well, rule 3 that we derived in section 3 says that if they all travel at speed c in our space then their timespeed must always be zero. Time stands still for all photons. Always and according to every observer. But if there is no timespeed in any photon at all we might just as well say that the whole time dimension does not exist for photons. Similarly, if everything in the universe would never ever move a single inch in space and it all would have started at a single point in space then everything would still be sitting at that same single point and you might as well say that space does not exist. It’s a bit more complex in reality. That’s why this story is called a simplified version. In reality some properties of the photon that are related to its wavenature do actually exist in the time dimension but to explain that here would make it all too confusing. So let's stick with the photon's particlenature that has no clock. That photon lacks dimension number four so it is purely 3dimensional and it always travels at speed c. Going back again to the speed of the rocket we see there that its combined speed, i.e. the Pythagorean sum of its spatial and timespeed is also c. But we know that the timespeed was just a spatial speed for the tall man who ate the very nutritious food! That means that the rocket has a spatial speed c in four dimensions according to that tall man. Even stronger: any rocket, no matter what speed it has according to us, will have a spatial speed c in four dimensions according to the tall man (remember that this speed is rotated in four dimensions). If we now realize that this rocket actually consists of millions and billions of tiny mass particles then it is easy to see that any of these millions and billions of tiny mass particles travels at speed c in four dimensions according to the tall man. And they travel in all kinds of directions too because molecules, atoms, electrons, protons and so on travel crisscross through each other.
It’s a small step on the staircase to see the parallel between the photons and the tiny mass particles. The first apparently live in a 3dimensional world while the latter live in a 4dimensional world. Both travel at speed c in their dimensional environment. The 3dimensional speed of the photons is measured by us using our time in the fourth dimension. The 4dimensional speed of the mass particles is measured by the tall man using his time in the fifth dimension while we can only measure its spatial component using our fourth dimension. We now take the place of the old man who shrunk. He sees only two spatial dimensions and one time
dimension. But these together are the three dimensions that the
photons live in. How does that translate in his world?
If the second photon is in the same plane and travels parallel with
the first photon, the old man will think that the second photon stands still. On
the other hand, if the second photon travels at an angle rectangular to the
first photon, the old man on the back of the first photon will think that the
second photon travels at speed c away from him in his plane. The trick and only explanation is that the time dimension is fully contracted in our space along the direction of our own motion in it, like a sponge that is completely pressed flat in one direction. The same happens with the second photon that travels rectangular to the old man on the back of the first photon. Although it should "vanish" in the past of the old man (which for us is a regular spatial dimension), the old man keeps seeing it because for him this dimension is completely contracted into his flat space. The bottom line is that the old man is able to see photons move at all speeds from zero to c in all kinds of directions in his flat world. So another parallel shows up with the way we see mass particles move in our space.
To summarize what has been told up till now we can say that everything moves at speed c, but you need to be "tall" enough to see that in all worlds. Your speed in time is equal to c, because from your own point of view you yourself always stand still (your spatial speed with respect to yourself is zero). Furthermore, some particles, like the photons, obviously have one less dimension than others, like mass particles. Once you reach this point, a lot more of these symmetries can be found between the 3dimensional and 4dimensional worlds. I will mention a few here to give you an idea but the justification of it requires a more thorough approach that does not fit in the context of this simplified story.
Updated October 14th, 2011
