Euclidean Relativity’s Take On Space And Time
March 17, 2010 4 Comments
You know how space time are intermingled? Euclidean relativity is another theory which explain our current perceptions regarding space and time. It is just a transformation of minkowskian invariant….below is space time metric in Minkowskian invariant
where x, y, and z are space dimensions
and τ =proper time dimension
The above equation can be written as below in euclidean invariant
where all terms have usual meaning.
The temporal velocity component is
This clearly defines as the coordinate for the fourth Euclidean dimension, and it says that the velocity components in all four dimensions involve derivatives with respect to , which then can no longer represent the fourth dimension. It can only be an extra, fifth dimension, (provided we index the other four , , , and respectively, with ). This fifth dimension is sometimes treated as a parameter in Euclidean approaches similar to special relativity, but here it will be treated as a genuine extra Euclidean dimension. A general expression for speed in the time dimension (henceforth refereed to as time-speed) is now:
while the scalar value of time speed
So how Euclidean relativity interpret space time ?
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 space-time. 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 4-dimensional space-time is his 4-dimensional space, while he lives in a 5-dimensional space-time or “Hyperspaceland”. For the man who shrunk, our 3-dimensional space is his 3-dimensional space-time. He lives in “Flatland”. What is it then that makes the time dimension appear to us as something different than a spatial dimension?
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 is just a division of displacement (change of position) in two different dimensions.
A division of displacement in two spatial dimensions, let’s say from dimension 1 (breadth) and dimension 3 (height), is for us merely a “dimensionless” and rather meaningless number. However, for the old man whose size shrunk this division represents a spatial speed in his 2-dimensional world because dimension 3 is his time dimension. This implies that the position of his whole 2-dimensional environment must be changing in that third dimension, otherwise there would be nothing to divide the displacement in his spatial dimension nr. 1 by.
This displacement in the 3rd 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 3rd 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. But how can we express this speed in the time dimension in terms of a division of displacements 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”.
For the man who grew after eating the very nutritious food, the answer is obvious. For him, the original time dimension turned into a normal spatial dimension and another, 5th dimension became his new time dimension. Our speed in time becomes a spatial speed in the 4-dimensional space of the tall man. If he now wants to express that speed in his 4-dimensional space he divides the spatial displacement by the time duration from his 5th 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 5th dimension divided by a displacement in yet another higher 6th dimension. And so on, and so on. It’s a recursive, or fractal-like system.
What I think?
here is an better illustration of diagram
[I will upload better photo of above diagram later]