Metric and Measurement

by Nikolai V. Shokhirev

Distance and Metric

(see also Science and Common sense).

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Distance

The most basic aspect of measurement is to establish relations of the type "closer-farther", "smaller-bigger", "shorter-longer", etc. This implies that systems (objects) can be arranged in some order. Is this always the case? The answer is "Yes" in the one-dimension case. 

1D-Case

Let us consider for example, a straight road. 

We can unambiguously say that the point "B" is closer to "A" than "C" or "A" is the leftmost point and "C" is the rightmost point (after we define what is "left" and "right"). This unambiguity takes place because each point can be characterized only by one parameter. We can call it "distance" and associate some numeric values with it. For example, one can say that "B" is 20000 strides away from "A", and "C" is 40000 strides from "A". It is also possible to say that it takes two days to reach "B" and two more days to get to "C". 

The fundamental feature of this one-dimensional world is that if one goes from "A" toward "B" and "C" then "B" is always before "C" regardless of the length of the stride or the traveler's speed. We can also formulate that one cannot avoid "B" going from "A" to "C".    

We can use this one-dimensional approach for simple systems that can be well characterized by one parameter.

Multi-dimensional Case 

When the number of parameters is more than one, then one can go from "A" to "C" without having to pass through point "B":

The definition of a distance, size or proximity is not a trivial task in multi-dimensional systems. Such definitions are not necessarily unique. One of the solution is to make a system artificially one-dimensional. For example, in medieval Europe the capacity of vine barrels was determined by its diagonal length:  

A regular approach to the definition of distance is based on mathematics.

Metric

In homogeneous two-dimensional space it is obvious to define a distance between the point 1 and point 2 "as the crow flies":                       

Given X1, X2, Y1 and Y2  the distance D can be calculated as:

        (1)

Such a mathematically defined distance is called the metric. The above definition is the Euclidian metric (not the crow's one). It is also known as the L2  metric. This is not the only possible definition. The L1 metric is the sum of the absolute values of the intervals along the x and y-axes. It is called the Manhattan metric because it is the distance a car would drive in a city laid out in square blocks:  

Euclidian distance (5-block length) and
Manhattan distance (7-block length).   

 In Manhattan even crows fly along blocks. Metrics can be more complex for non-plane surfaces like sphere (see for example Global distances).

Metric Tensor

The above 2D examples illustrate the parameters of the same type (length). It is also assumed that the same length unit is used for the x and y directions. If you measure x in inches and y in feet then the equation (1) transforms to

        (2)

where D is in inches. The factors at x and y are (1, 122). For L1 metric the equation for distance reduces to

         (2a)

In the Minkowski four-dimensional space three coordinates are the ordinary spacial coordinates (x, y, z) and the fourth parameter is time. There is no way to measure all of the four coordinates using the same unit. The "distance" is then defined as:

        (3)

where c is the speed of light.

If we measure D, X, Y and Z in miles, c in miles per hour and T in hours then the factors at x, y, z, and t are (1, 1, 1, -c2), where c = 670000000 mph or 300000 km/sec.

The conversion factors (1, 1, 1, -c2) are, in fact, the diagonal elements of the so-called metric tensor:

(This is not the exact definition but it is enough for our discussion). In some coordinate systems off-diagonal elements of the tensor can have non-zero values as well.

Further we will use this generalized distance definition (usually in the L1 sense) and extend it to the areas of human activity other than science and technology. 


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