First is to realize the shadow effect is really from the basic particles that make up your and my mass and the basic particles of the Moon and the Earth. The incoming basic particles mostly miss everything when going through the Earth. The basic solid core of the basic particles is like one to 90,000 times the effective diameter of the sphere of influence the basic particle has. So most particles go right through the Earth. Newton's calculus states the sum of the shadows of the basic particles of the Moon and Earth act as if they were all at the center of the Moon and Earth. If Newton's Calculus is correct, then the shadows and mass of the basic particles can be considered to act as if they were all at the centers.
Your powerful rocket achieves a velocity exceeding the speed of light within a few moments. Since that isn't possible, the question is absurd. The answer is everywhere! This is like the problem of localizability in theoretical physics. If the photon moves at the idealized velocity of c, no time passes for it and by quantum mechanics' uncertainty principle, it is in no particular position in the spacetime continuum; thus it is in contact wih every spacetime point of the 4 manifold. Every photon in the universe is in contact with everywhere in the universe. In order to avoid this enigma physicists like Dirac invented even stranger devices like delta functions which are infinite at a point but zero everywhere else. Some avoidance. Of course, we can introduce exotic geometries like wormholes to mimic the intent of your enigma 6, but these are hypothetical constructs that are fraught with their own absurdities. Also, we can lean on Special Relativity as you mention and talk about from whose perspective is the velocity being assessed. Relative to the rocket ship frame there is no theoretical limiting velocity. I.E. if you accelerate for a few years at the rate of gravity, you will achieve velocities far exceeding that of light. For external observers though you just shrink and disappear only to reappear at a distance slightly less than what it would take a photon to travel. The enigma here is can you beat a photon in a race. To this day this is a controversial issue and remains unclear. We have no knowledge of photon structure, what it means to move in the continuum, or even what is the continuum. Similarily we don't know what the constituents of the magnetic field are. At every level of physics even quark level where for practicality purposes the photon is asserted to be composed of a quark-antiquark pair in a specially bound state, entities like photon or magnetic field are assumed as infinitely divisible givens. You don't ask what they are, you only ask what they do. Some ontology. So, your enigma is wrongly posed, the idea is based on a wrong perception (semi-classical approximation), and what is really going may not be determinate at all. Now there is an enigma!
I have to argue your solution to the doubler enigma. The box would never be half full because in order to be so it would have to have an even number of balls to be divided into an integer. Since you can't have half a ball put into the box it will always contain an odd number of balls since the number of balls will be equil to the sum of 2^n as n goes from 0 to 360, for the # of minutes passed. The fact that one ball is added first makes the sum at every minute an odd number. As a result the box will be just a little more than half full at 5:59. Chris
First off, I agree that, of course, time cannot move at the speed of light. They are fundamentally different 'things' and time is used to measure velocity. Thus, it just doesn't work that way. However, you are forgetting Einstein's Relativity. It has been proven that the faster one moves, the more time slows down. If you reach the speed of light, there would be a complete conversion of matter into energy because mass increases as speed increases and energy use increases as mass increases. This can be represented by an asymptotic curve as one approaches the speed of light. Theoretically, that speed cannott be reached but if it were, it would require infinite energy, which would be the conversion of the infinite mass of the moving object. Anyway... If you are able, somehow, to move faster than light, a simple extrapolation from the equations shows that you would move backward in time. This is because velocity of an object affects the space-time continuum(as Einstein coined) in such a way as to curve time. On the other hand, if one were to reach an infinite speed, one would never stop, and I rather think that this would be detrimental to the universe as a whole, in that infinite speed would mean that you would be everywhere at once in the entire universe. Infinite speed, due to the warping of time and space by velocity, implies a singularity. Put the two together and you have the entire universe becoming a singularity, and this is, to say the least, a rather bad thing. :) That's all for now. Later! -Matt Miller
I also think in this manner. We only realize a line from one point to the next, but if we shift our position, the line shortens and we could consider it a shorter distance. It is only from the position of objects which are stationary outside of us that we can mark the actual distance of the line, using geometry. If space were curved, we wouldn't notice it. When we look at a star, we only realize it to be a line's distance away from us when actually the star could be a reflection on a gigantic mirror that's just outside our solar system. one could go so far as to infer that all of reality is just a mirrored reflection and this would fit in nicely with your theory. One could start travelling to a star and, for all we know, it could take only half a light year to reach, or longer than a calculated time/distance, because apace is curved in some bizarre way. The only way I think anyone will be able to say where things lie for sure is to experience them. In our physical world, this is near impossible because each individual lies directly outside everything else, but there is a way: astral projection is one way that I know of to slip into objects and experience them directly. I don't know. There has to be a better way. Mystery and Science are married, you know. If you want to, you can e-mail me
I have a couple of ideas on multi-dimensional shapes you may find
interesting.
The first is on the hypersphere. Now on your web page you state
that we can't know what something in the fourth dimension looks like
since, well, we live in 3 dimensions (R3). I have to disagree, and the
reason why is because living in R3 gives us access to see things in R0, R1,
R2, and R3; (for my point) being a point, line, circle, and a sphere
respectively. With this we can see that four things happen as we go from R2
to R3.
1) A shape in R3 is made up of an infinite number of
"corresponding" shapes in R2.
2) As the infinite number of R2 shapes extend out to the limit
(here radius) of the R3 shape along the "new" axis they get smaller and
smaller until they reach a point.
3) The infinite number of R2 shapes intersect each other in order
to create the shape in R3. The intersection is actually an element of R1.
Here it is a line which has been bent around to make a circle. At this
point making a drawing would help.
4) As a shape in R3 passes through R2 an observer in R2 will see
it as a shape translated to a corresponding shape in R2 starting as a
point, growing to a maximum, and shrinking back to a point.
Since these rules are consistent for jumping through the
dimensions we do know, I find it reasonable to believe that they are
congruent for jumping into dimensions we do not know.
Therefore we can postulate that a hypersphere is made up of an
infinite number of spheres who's intersection is a circle. Imagine a
sphere of radius r with an infinite number of spheres (noted set 1) around
its center who's centers are an infinitely small (here out noted dx) distance
away from the first sphere's center. These spheres will have a radius of r
minus dx. Each of these spheres in set 1 will have a sphere who's center is dx
away from their center and who's radius is r minus 2dx. These spheres belong
to set 2. This continues until we get to set infinity which is made up of
spheres with a radius r minus infinity*dx, a bunch of points. The
hyperradius of the hypersphere is dependent on the radius of the sphere
which is in the center of it. And, what do you know, we have again a
congruency since the radius of a circle yields the radius of a sphere.
The first argument that I can think of to this is that it would
appear that we have objects in the same space at the same time, but
keep in mind that this "invasion of space" is actually acomadated by the extra
dimension.
The hypervolume of this thing would be the infinite sum of diminishing
spheres from both ends of the new dimension times dr; that is, int
(4*pi*r^3)/3 from -r to r dr = (2*pi*r^4)/3
Consider the argument I have for the quintichypersphere. It is much more
difficult to understand this one but here goes...
In keeping with the rules of congruency before mentioned we know
that the quintichypersphere (qhs) is made of on infinite number of
hyperspheres and that their intersection with each other is a sphere. If
you have a picture of what a hypersphere looks like in your head already
then we can go on. The center of each hypersphere will be dx from the
center of the hypersphere preceding it until you get to the first
hypersphere. Let's call the first set of diminishing hyperspheres from
the center one QSet 1. This is analogous to the idea mentioned before. Since
the hypersphere radiates out diminishing spheres to a point, we find that in
the qhs the center hypersphere's diminishing spheres are circumscribed by
the diminishing spheres of QSet 1, and so on. Again we have congruency with
the assumptions made earlier.
The volume of the qhs will be the infinite sum of the
hypervolumes from both the positive and negative ends of the fifth
dimension times dr. As you can see there is a pattern here. The volume
of a multi-dimensional "sphere" where the dimension is greater than 3
is equil to 2^n-3 int (4*pi*r^3)/3 (dr)^n-3 from 0 to r, where n is the
dimension we are refering to. Note that the wedge product is not used for
dr, since if it were the nth dimensional hypervolumes would all be 0.
In order to find the nth-dimensional hypervolume of a shape in
dimension n one must find the volume of the "corresponding" shape in
dimension n-1.;
That's about it. Write back if you can. I'd like to hear some
criticism on this.
Chris Jurgenson.
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