The Larson Research Center

There was a discussion last month, on John Baez’ blog, entitled dimensional analysis, which Baez characterizes as a one of several “knarly issues” he wants to discuss.   “They’re ‘gnarly’,” he says, “not because they’re technical, but because they involve slippery concepts.  Their clarification may require not so much hard calculations as patient, careful thought.” He begins by observing:

It’s common in physics to assign quantities “dimensions” built by multiplying powers of mass (M), length (L) and time (T).  For example, force has dimensions MLT ⁻².  Keeping track of these dimensions can be a powerful tool for avoiding mistakes and even solving problems.

This raises some questions:

    * What’s so special about mass, length and time?  Do we have to use three dimensions?  No - we often use fewer, and sometimes it’s good to use more.  But is there something inherent in physics that makes this choice useful?
    * What’s the special role of dimensionless quantities - those with dimensions M⁰L⁰T⁰?  In what sense is a dimensionless quantity like the fine structure constant more fundamental than a dimensionful one like the speed of light?

 I thought I had these pretty much figured out, until Vera Kehrli pointed out two things that surprised me:

    * Dimensionless constants often depend on our choice of units.
    * Dimensionful constants often don’t depend on our choice of units

For example, the speed of light is

c=299,792,458m/s

Here a meter, m, has dimension L.  A second, s, has dimension T.  The speed of light, c, has dimensions
LT ⁻¹.  To make the dimensions match, it follows that the number 299,792,458 must be dimensionless.

Now suppose someone comes and changes our units.  Say they redefine the meter to be twice as long as it had been.  Then m doubles and the number 299,792,458 gets halved, keeping c the same.  So we see:

    * The dimensionful constant c does not depend on our choice of units.  If we double m, we halve C, but c stays the same. Of course this number is what it is, regardless of our units.  But if we say

      c=Cm/s

then the dimensionless quantity C depends on the definition of m and s. 

    * The dimensionful constant c does not depend on our choice of units.  If we double m, we halve C, but c stays the same. 

All perfectly trivial - yet physicists like to run around saying the fine structure constant is more fundamental than the speed of light because it’s dimensionless and therefore doesn’t depend on our choice of units!  They mean something sensible by this, but what they mean is not what they’re saying.

It’s good to compare two examples:

The fine structure constant:

α=e²/4πε₀ℏc≃1/137.036

is a dimensionless quantity built from quantities that seem very fundamental - the electron charge −e, the permittivity of the vacuum ϵ 0, Planck’s constant ℏ and the speed of light c.  (Ultimately, Benjamin Franklin is responsible for the conventions that make the electron charge be called −e instead of e.  But that’s another story.)

The speed of light in meters per second:

C=c/(m/s)=299,792,458

is also dimensionless, but it’s built from quantities that seem less fundamental.  c seems fundamental, but m and s seem less so.  After all, the definition of a second is “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of a caesium-133 atom at rest”.   Like the speed of light, the value of this quantity is a fact about physics - but a more complicated fact.   The length of the standard meter rod in Paris is an even more complicated fact, which has the disadvantage of being tied to a specific artifact!   With this definition of m, the dimensionless quantity C tells us something funny about the universe: something about how the speed of light, the frequency of a specific kind of light emitted by caesium, and the length of the meter rod in Paris are related.  It’s a bit like how α tells us some relationship between the electron charge, the permittivity of the vacuum, Planck’s constant and the speed of light - but it seems less “fundamental”, whatever that means.

But the definition of a meter no longer involves a rod in Paris - that’s obsolete; I mentioned it just to illustrate a point.  The current definition says a meter is “1/299,792,458 times the distance light in a vacuum travels in one second”.  And this makes a different point. Again the value of this quantity is a fact about physics - we could radio an alien civilization the definition of a meter, and if they knew enough physics, including the definition of a second they could build a rod the right length. But with this definition of m, the dimensionless quantity C=c/(m/s) seems to tell us nothing about our universe! 

(Actually it tells us some funny blend of information about the speed of light and the definition of m and s.)

One might argue that C is less fundamental than α because we could get any value of C by changing our definitions of m and s. But that can’t be the whole point, since we could also get any value of α by changing our definitions of e,ε₀,ℏ and c.  So, there must be some other reason why α seems important and C seems completely silly.  What’s going on, exactly?

Most of the ensuing discussion devolved into a discussion of the meaning of units, and the confusion that results in assigning units to coordinates per se, when it’s the change of coordinates, or the difference between them, that is the only meaningful concept of multi-dimensional magnitudes.  The trouble is, it seems necessary to assign units to the coordinates in order to work in a meaningful way with the vectorial concepts, but the dimensions of the units assigned to the coordinates get confused with the dimensions of the physical magnitudes involved.  In one post, Baez described the confusion in a detailed example. Here’s part of that comment:

I always get confused about this when I try to solve a GR problem with the help of dimensional analysis.  I wind up spending so much time analysing the basic issues that I get bogged down before I reap the rewards of this work!  So then I give up and act like a mathematician where everything is dimensionless… and then the next time this situation comes up, I’ve forgotten what I learned before.

So in fact, I need to start from scratch again here.  Let me relive my blunderings in public - it could be educational.

When I see a coordinate like x i my instant gut feeling is to assign this units of length. But then I think “diffeomorphism invariance” and imagine a general coordinate transformation

y i=f i(x 1,…,x n)

so I think “oh-oh, it’s forbidden to apply an arbitrary smooth function to a dimensionful quantity!”    So then I start wanting the coordinates to be dimensionless. 

Then this desire gets heightened when I remember that the metric is what takes tangent vectors and spits out lengths (actually squares of lengths).  So, I want to pack units of length 2 into my metric tensor somehow. 

But then I think: the metric tensor is an element of S 2T *M, the symmetric square of the cotangent bundle.  If this has units of length 2, I must want cotangent vectors to have units of length.

And then I think: no, it’s not the metric tensor g that has units of length 2, it’s when we apply this tensor to a pair of tangent vectors, say v and w, that we get something with dimensions of length 2, namely g(v,w).

So where do I put the units of length?  Do I put two of them in g, or one in v and one in w?  If I do the latter, I’m saying tangent vectors have units of length.  But a minute ago I was wanting cotangent vectors to have units of length!
How can I be so confused? I’m supposed to know something about physics, but apparently I don’t even know if tangent vectors or cotangent vectors have units of length!

Although applying the concepts of units, dimensions, and directions, used in scalar science, would seem to be useful in vectorial science, it is not possible for LST physicists to see it clearly, unless they grasp the idea that all physical entites consist of motion, combinations of motion, or relations between motion, and that motion exists in three dimensions, with two, reciprocal, aspects, space and time.

If I had the chance to converse with these guys, I would start by asking an unusual question: “Is it possible to measure the dimensions of length or time independently?” I’ve asked this question many times, and have never received a positive answer to date, because to measure length always requires motion; that is, we have to move a measuring rod into place, counting the units of measure as we do so, or after we do so; or else we have to send a sound wave, or a light wave, of known velocity, and measure the time of travel along the distance to derive the length, or devise some other way, but, the point is, any method of measuring length or time that we can conceive involves measuring motion.

Since this is the case, then, logically, when we measure distance, or time, we are actually measuring the past motion that previously separated given locations in space and time, in a sense. If this is true, then is it not also true that we cannot regard length, or time, as independent physical entities? That is to say, if we are to be logically consistent in our reasoning, space and time should have no physical meaning outside the meaning they have as the reciprocal aspects of motion.

Hence, when we select a system of units of length and time, selecting the value of one fixes the value of the other in that system of units, if we specify the relative value, or the velocity, that they must have for our purposes.

For instance, when we consider the velocity c, and choose a unit of measure based on an observed physical constant, say on the Rydberg frequency for hydrogen, regarding this constant as a natural unit of motion, measured in units of cycles per second, we can then determine the corresponding natural unit of time, in terms of seconds, a derived unit in a system of units we select.

Since velocity, expressed as a frequency, is an oscillation, assuming that the value of the Rydberg frequency, expressed in cycles per second, is a natural unit of oscillation, then its motion includes two natural units of length. Therefore, we need to double the value of the Rydberg frequency to determine the natural unit of time in this constant, expressed in seconds, which is simply the reciprocal of the doubled frequency.

Now, of course, equipped with a natural unit of time, expressed in units of seconds, we can calculate a natural unit of length, based on meters, the corresponding unit of space in our selected system of units, by multiplying c times the reciprocal of the natural unit of time, expressed in our system’s units of time, seconds.

Does this not give us a beginning to finding a way out of the dilemma being discussed, at least in terms of the meaning of units, whether dimensionless units, or dimensionful units? By assuming motion as our fundamental unit of measure, we gain a new perspective on the issue articulated by Baez:

A system of units gives a coordinate system on some space of quantities we’re trying to measure. So, if we understand coordinate systems thoroughly, we should understand systems of units.

We can easily understand coordinate systems in terms of 1D units of motion, because this is the domain of the reals, the basis of Euclidean geometry. If we take Hestenes’s suggestion, and stick to the reals in Cl₃, we can define vectors in terms of Cl₃’s four, independent, linear spaces, and describe “space” in terms of units of points, lines, areas, and volumes, a coordinate system of relative space locations, conforming to Euclidean geometry.

However, if the locations of “space” and “time” within this system of multi-dimensional units have no independent physical meaning, as indicated by our inability to measure them independently, but only have meaning when considered together, in terms of motion, and we can determine a natural unit of motion as described above, doesn’t that imply that the units to use in this system of units, which we want to exploit in exploring invariance principles, should be a system of units of motion, rather than a system of units of space and time (spacetime)?

This thought just bends the mind when you think about it, because the dimensions of motion are space and time. Yet, the velocity is a pure number too, the reciprocal relation between two real numbers. The fact that we give dimensions to these two reciprocal numbers, does not mean that velocity also has these dimensions. For instance, the velocity of an expanding gas or liquid, does not have the dimensions of length and time, but the dimensions of volume and time. The velocity of an expanding planar wave, does not have dimensions of length and time, but the dimensions of area and time.

If this is so, then why can’t the space aspect of c-speed have dimension zero, as well as dimension 1, 2, or 3? That is to say, why should we think of the dimensions of space, in the equation of motion, as length, area, or volume? Isn’t it just as logical to regard motion as the relation of two changing scalar values, as it is to require that space have dimensions of length, area, or volume?

Indeed, if we were to describe the outward expanding motion of the universe as a whole, not from a particular point of reference, wouldn’t the dimensions of both the space and time aspects of the equation have to be scalar, i.e. dimensionless numbers?  The answer is obvious, but what might not be so obvious is that, if we do choose a particular point of reference, then we can see that the expanding motion is motion in all three dimensions, depending on the dimensions of measurement.

(to be continued)