Thursday, March 09, 2006

The Fourth Dimension

Before the twentieth century many people believed that every person experienced only three dimensions. This included most physicists who described the world within mathematics as mainly three dimensional and that time was absolute. In the beginning of the twentieth century came the reformulation of the basic principle of physics. This was the theory of special relativity.
The great mind of Albert Einstein has introduced to the world that time is no different than space, they are interchangeable. For example, what if you had an appointment in the heart of Manhattan? You are given instructions to meet someone as follows. Go to the corner of Fourth Street and Ninetieth Street. Then proceed in the building up to the tenth floor. Here you were given the three spatial directions to meet someone. However, you might be standing there for eternity because you never had the indication of when you were suppose to meet someone. Therefore, time is just as important as all three spatial dimensions.
In class discussion we talk about the arrow not moving at an instant. Now that we concluded that space and time are one in the same we have concluded that just as there no possible way to move an infinite amount of positions in a finite amount. Similarly, with time there is no way to capture an instant of a moment in a time equal to zero. This would violate the laws of special relativity. We can perceive this idea in our mind of a slice of the reality loaf and assume that it is an instant of time, but this is impossible. To capture a picture at an instant you would have to open and close you eyelids in an instant which would be physically impossible.
With the argument of Zeno I have concluded that it impossible to view an instant of a motion. It is easy to presume that at an instant something is not moving when flash the image in our mind, but with our human senses this impossible. If you looked at object moving and took a snap shot it would appear to be not moving in the photograph. The truth is that the object was moving during the time when the shutter opened and then closed. Furthermore, it is impossible to slice the reality loaf in an infinitely thin slice.

(Note: This is Chad's post.)


Blogger linford86 said...

While I am fully aware of Einstein's Special Relativity - I am, after all, a physics major - I do not buy your argument that it is impossible to find a slice of the "reality loaf" based merely on the idea that it is physically impossible to view such a slice. Note that much of Einstein's special relativity is derived from the following equation:

d=sqrt(x^2+y^2+z^2-(ct)^2), which is just the Pythagorean Theorem applied to 4rth dimensional space.

Notice that the equation I just gave corresponds to a particular distance in space-time, which does exist, regardless of whether or not any physical device could ever capture it (note that regardless of whether or not a camera could capture an infinitesmal moment of time is in no way related to a supposed inability for infinitesmal moments of time to exist.) Also note that if distances exist in space-time, it must be that these distances are constructed from some unit, whether that unit is infinitesmal or finite determines entirely whether or not space-time is discrete. These smallest units of space-time - whatever form they take on - can be thought of as slices through the "reality loaf". Putting it another way, one could take a 3 dimensional slice through Minkowskian space-time in the same way that you can take a 2 dimensional slice through 3 dimensional space. In general, an n-1 slice can be taken through n dimensional space. Note that all of this is in no way related to whether or not we could ever actually observe such things, and is entirely dependent upon whether or not space-time has a geometric character (Einstein would argue, by the way, that space-time does, indeed, have a geometric character - that's the very basis of special and general relativity.) I would argue that although it might be physically impossible to capture a slice through space-time, that does not relate to whether or not it is a geometrical or logical possibility. Thus, I fail to see how adding relativity to the picture corrects any paradox that may exist here; to me, it seems that all relavity would do is to help us better understand the issue, not rectify it.

2:29 PM  
Blogger linford86 said...

If space-time truly does have a geometric character, as special relativity would have us believe, and if space-time is correctly described by the equations given by special relativity, then it must be the case that one can take an infinitely thin slice of the reality loaf.

If space-time is geometric in character, then it would be possible to have a 2 unit long line in space-time.

Space-time has a geometric character.

Therefore, it is possible to have a 2 unit long line in space-time.

The points on a 2 unit long line are infinitely dense.

If the points on a 2 unit long line are infinitely dense, then between any two numbers on the line is another number.

If that is the case, then you cannot find a number that is closest to the 1 unit mark on the line, but not equal to the one unit mark on the line (because you can always find a closer number than any number you happen to choose.)

This would mean that the point corresponding to the one unit mark must be infinitely thin; that is, it must have no extension whatsoever. Of course, for each point on the line, you can identify a plane; a plane is a slice; thus, you can take a slice through space-time.

Of course, this assumes that space-time has the same characteristics as the set of real numbers. That may not be the case; but if it isn't, then irrational distances are not allowed to exist; if rational distances are not allowed to exist then it can be shown through a relatively simple mathematical proof that Euclidean geometry no longer correctly describes the space (more importantly, the Pythagorean theorem no longer works.) This is related to another post I already put on the blog. I don't have time at the moment to post the entire proof, but I think it should be relatively easy to see from what I've put up here in the past.

2:42 PM  

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