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dimensionality of mass part two

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Posted by rtharbaugh on May 11, 2003 at 08:45:38:

In Reply to: dimensionality of mass part one posted by rtharbaugh on May 11, 2003 at 07:28:42:

Dimensional analysis of mass.

Mass is sometimes said to be contained in a point-like partical. Is this a reasonable usage of the notion of mass and point? Yes, it is reasonable, but only in cases where the non-pointlike nature of mass can be neglected. For example, we measure the center of mass of a galaxy as occuring at a point, and that is reasonable when considering the behavior of a very large number of galaxies. But the idea breaks down when considering the motions of stars within the galactic structure.

Closer to home, consider the center of mass of a boomerang. It is a toy contradiction to imagine that the center of mass of the boomerang occurs in the empty space between the two arms of the boomerang, where there is presumably no (boomerang) matter and so no mass. Similarly, we can speak of the center of mass of a two body system such as the earth and moon occuring in space somewhere between the earth and moon and not within either of them. We see that the idea of center of mass and the idea of mass are two different things. The center of mass of a hollow sphere is another case highlighting this difference.

So it is reasonable and useful to speak of the center of mass as being pointlike, but this tells us nothing about the nature of mass itself. In particular, it tells us nothing about whether a partical, having mass, can be said to occupy a single point. A point has no dimension.

So can a partical, having mass, be said to occupy one dimension? Call it an X partical, meaning that it exists in x but not in y and not in z. I have already shown that such a partical must also have extension in t, and that extension in t is just extension in y, there being no reason to collapse t into x. So it is not reasonable, and may not be useful, to say that a partical exists only in x.

Particals then must exist in at least two dimensions. This can be thought of as (x,y), or (x,t), where t=y, and t=y just means that t is displayed for our viewing pleasure along the spatial dimension we call y.

Does string theory give us insights into the nature of mass? Not if we evade the question of the nature of mass by simply stating that mass is equivalent to the length of the string. Not even if we add that the energy of vibration of the string adds to the mass.

There is usefulness in saying that mass can be point-like, or that a partical can be point-like, but this should not be confused with the idea that the partical or the mass occupies a single point. Center of mass or position of partical may be said to be point-like, but we cannot reason from that that a partical or a mass occupies a single point. Instead, we are still stuck with the notion that mass and matter are multidimensional systems. Mass and matter do not exist within a point (only nothing can exist within a point) but in the relationship between points.

Mass has also been defined as that which resists accelleration. Since it resists some accelleration, it must also be said to accept some accelleration. Accelleration is change of velocity with respect to time, or dxdt^-2. I count three variables there, one in x and two in t. Using the N-1 routine, that counts as at least two dimensions.

Now I believe I have shown in two ways why mass as a property of matter must be said to be at least two dimensional. It cannot be useful to define mass simply as a length of one dimensional string. Instead, mass has to be thought of in string theory as a property of the surface or world sheet or brane which the string occupies in some length of time.

Now it may be useful to imagine what happens to the brane of a string when the direction in time undergoes change. An open string could be translated through time along the axis of its length without sweeping out any world sheet at all. If the time direction is perpendicular to the string, it will sweep out the maximum brane. If the time direction is at some angle to the perpendicular, the area of spacetime swept by the string will be some intermediate value. SO we see how in the many times interpretation (MTI) the mass of a string depends not only on its length and topology but also on the orientation of the sweep through timespace.

Thanks for being here,


RTH

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