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a bit of dimensional analysis

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Posted by rtharbaugh on September 15, 2003 at 03:32:44:

Abit of dimensional analysis

Given the Planck quantum of change as h-bar = ML^2T^-1 where M is mass, L^2 is the square of length and T^-1 is the inverse time, and setting h-bar = 1, as in the geometry of the Planck scale, then 1=ML^2T^-1. Perhaps it would be interesting to rearrange this formula to give values for mass, length, and time at the Planck scale. Then we would have

T=ML^2
M=TL^-2
L=(TM^-1)^.5

Let's see if we can translate these formulae into words.

Time is mass multiplied by area. I'm not sure what mass multiplied by area means, but I seem to recall that Maxwell spoke of a mass area. Can we speak of a certain area having a certain thickness of mass?

Well, then Mass is Time divided by area. If we want to know what mass is, we must think of a certain time spread out over a certain area.

What is length? We must take the square root of time divided by mass. Well this seems easy enough on the face of it. We are used to finding lengths of the hypoteneuse of an angle by taking square roots, due to Pythagoras. So can we think of time divided by mass as a sort of length of the side of a triangle? How do you divide, say, 24 hours by a kilogram? Can we speak of minutes per gram, and what does that mean? A gram lasts some number of minutes. Sounds rather like the burning of a fuel, or the rate of a chemical reaction of some kind. Or perhaps the half-life of a particle? Can we speak of length as the time required for the mass of a particle to dissapate into energy?

Now let's see what happens if time and space are the same thing. Then T=L, so we might say we need some other symbol to give us an idea of the spacetime unity. I suggest H, using the convention of capitalization for dimensional analysis, so H for the dimensional analysis symbol of Planck's spacetime quantum. Then we can replace every instance of an L or a T by an H, at least for a first approximation. Later we may wish to consider some variation on T-L=0, such that L=-T or somesuch.

But for now let us look at the formula with H for T and L.

H=MH^2.
M=H^-1
H=(HM^-1)^.5.

Well that looks like a bunch of nonsense. Maybe we can throw this whole approach out by showing that these formulae produce some statement such as M = -M or something. Anyway lets regroup the symbols and see what happens.

H=MH^2 Move all H to the left, then
H^-1=M

M=H^-1. Well at least we are consistant so far. Then,

H=(HM^-1)^.5
H^2=HM^-1
H=M^-1

Ok I guess I see the pattern here. What does it mean? Spacetime is the inverse of Mass? Mass is the inverse of spacetime?

Let me conjecture. When spacetime infalls onto the area of a brane, the brane takes on mass. When mass falls into the event horizon of a singularity, the singularity takes on more spacetime, thus increasing surface area. G, this almost makes sense.

Now the last time I played this game I got hopelessly confused. I could almost wish the xdim boards had followed the ones that vanished, just to save myself the embarrasment of having to revisit my earlier attempt. Oh well. This is a new media, and we have to take the good with the bad. Once you hit the post button, there is no going back, unless you are fortunate enough to be in the good favor of the moderator.

Well, H.M. Georgi said in his artical Effective Quantum Field Theories, in the book The New Physics, edited by Paul Davies, published by Cambridge University Press 1989, page 448,

"...when we set c equal to one, a second can be either a time of one second or the distance light travels in one second. When we set h-bar equal to one, one centimetre can be either a distance of one centimetre or the inverse of the momentum required to produce an angular momentum of h-bar at a distance of one centimetre from the axis. Thus we can measure energy and momentum in units of mass, time and distance in units of inverse mass, force in units of mass squared, etc."

So maybe I am not so far off after all. Time and distance in units of inverse mass. Hmm.

Does this tell us anything about the nature of mass? Or of timespace? What happens if we convert mass to energy via Einstein? Time and space are inverse energy? Wow. Maybe we are getting something out of this after all.

H=(MC^2)^-1
H=(ML^2T^-2)^-1
H=T^2M^-1L^-2

Is this right? The square of time divided by mass area? But since L and T are both H, then

H=H^2M^-1H^-2, or, once again, H=M^-1. Well at least it all fits together.

What is the square of time? What is mass area?

At the beginning of this excursion, I said that time is mass area, and now I have the square of time divided by mass area, so that reduces back to just time. So again I have the tautology that T is the same as H.

It doesn't look like this approach is going to contradict itself anyway. But it is very late and I have spent my night, as usual, inhaling the vapours of strong cleaning agents acting on organic compounds and a variety of metals, so who knows what strange charge states I have got floating around in my think tank.

Wait a minute. Force in units of mass squared. Time and distance in units of inverse mass. Inverse force?

Sorry, I cant think about it any more. Negative energy, hmmm.

Any comments?

Thanks

Richard T. Harbaugh




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