The Enclosure Element

Before we get to tensors and their uses we need to look at some more familiar geometric concepts. Those concepts have at their centers certain bounded sets of points: if the boundary consists of two points, then we have the bounded set as a one-dimensional object that we call a line, which has the property of length; if the boundary consists of a closed line, we have the enclosed set as a two-dimensional object that we call a surface, which has the property of area; if the boundary consists of a closed surface, we have the enclosed set as a three-dimensional object that we call a geometric solid, which has the property of volume; and so on. Length, area, volume, and their higher-dimensional analogues measure the size of the enclosed set. We want to express that measure in terms of the coordinates that we associate with the set.

We begin by writing the unit vector as

(Eq’n 1)

In that equation **q**_{i} represents a coordinate basis for our
grid; which means, while the unit vectors all have the same unit of measurement
(such as length), the coordinate unit vectors may have different units of
measurement (such as length and angle in a polar-coordinate grid). The
coefficient c_{i} represents a constant that translates the units of the
coordinate basis into the units of the unit vector. We use the unit vectors to
define the unit element of enclosure,

(Eq’n 2)

The multiplications in that equation, represented by the upper case Greek letter pi, consist of vector products and/or their higher-dimensional analogues. To see how those products play out let’s consider a simple example.

Note that in order to make the products in Equation 2 valid the relevant vectors must exist at the same point. If we represent the magnitudes and directions of those vectors with straight lines, the those lines represent adjoining sides of a geometric prism. Assume that the vectors that define our prism all meet each other at right angles and that they lie on the axes of our coordinate grid: those assumptions put one vertex of our prism at the origin of the grid. Now our calculation of an enclosure consists of a simple multiplication.

To calculate the two-enclosure (area) of a rectangle in
the x-y plane we have the vectors **A** and **B** such that

(Eq’ns 3)

We thus calculate the magnitude of the two-enclosure as

(Eq’n 4)

In this case we have our description of the enclosure as the determinant of a diagonal matrix. Imagine that we now rotate the rectangle about its vertex at the origin of our grid by an angle è in the counterclockwise direction. We thus get two new vectors that define the rectangle,

(Eq’ns 5)

To multiply those two vectors together we have the determinant

(Eq’n 6)

Thus, as we expect, our description of the area enclosed by a rectangle
remains invariant under rotations. In addition, if we add the same arbitrary
number to each element of a given column of the matrix, we still get the same
result, ^{2}E=a_{x}b_{y}, thereby proving and verifying
the fact that the area enclosed within the rectangle remains invariant under
translations, again as we expect.

Now we want to generalize that analysis to n-dimensional enclosures. Again we assume that all of the vectors defining our n-dimensional prism must meet each other at right angles and that they lie on the axes of our coordinate grid, so again we calculate the measure of the n-enclosure by multiplying together the nonzero components (one each) of the vectors. Each vector has n components, so we can arrange the full set of vectors into a diagonal nxn matrix. We then have the measure of the enclosure as the determinant of the matrix,

(Eq’n 7)

in which V_{ik} represents the k-th component of the i-th vector.

One of the fundamental properties that defines an array of
numbers as a vector tells us that the magnitude of the vector remains unchanged
when we rotate the coordinate frame in which we represent it. Because we
calculate the measure of an enclosure by multiplying together the vectors that
define it (the vectors that emanate from one of the enclosure’s vertices, the
key vertex), that property also applies to a right prism. We know that the
proposition thus stated stands true to mathematics because if we put the key
vertex of the enclosure at the origin of our coordinate system and rotate the
coordinate frame in the right way, the matrix V_{ik} will become purely
diagonal, with each vector having a single nonzero component, equal to the
vector’s magnitude, so that the measure of the enclosure simply equals the
product of multiplying the defining vectors’ magnitudes together. Because the
magnitudes of the vectors remain invariant under the rotation of the coordinate
frame, their product, the measure of the enclosure, also remains invariant.

With a properly diagonalized matrix representing our vectors, we can rewrite Equation 7 as

(Eq’n 8)

Comparing that result with Equation 2 tells us that, for unit-length vectors,
V_{ii}=c_{i}q_{i}, so we can rewrite Equation 2 in its
differential form as

(Eq’n 9)

In the last step in that equation I exploited the fact that, for matrices A
and B, det |AB|=det|A|det|B| and that we have V_{ii} as the product of
two diagonal matrices.

In order to calculate a value for det|c_{i}| we
must calculate the length of a differential line segment in our coordinate
system. By using differential lengths we ensure that, even if we have a curved
line segment, a straight line segment (specifically, a vector) will give us a
close-enough approximation to it. We obtain a description of the length of a
vector by calculating the square root of the inner product of the vector with
itself. Because we want to use the differential analogues of the unit vectors,
we can generalize that statement to

(Eq’n 10)

In that equation I have replaced the implicitly covariant coordinates of Equation 9 with their contravariant counterparts: because we have based the calculation on a right prism, we have no effective difference between covariant and contravariant in this case.

Next we rewrite Equation 10 as

(Eq’n 11)

in which we have simply commuted the second and third factors. Normally we
couldn’t make such a move, because in matrix algebra multiplication is
non-commutative. But in the limited case in which we have diagonal matrices the
multiplication __does__ become commutative; which means, when we multiply
diagonal matrices together the order in which we multiply them makes no
difference in their product. We represent all four factors on the right side of
Equation 11 with matrices and we can make them all diagonal by a suitable
rotation of the coordinate frame, so the commutation of the second and third
factors is mathematically legitimate.

We also know that

(Eq’n 12)

in which gik represents the metric tensor, a matrix that describes the geometric structure of space and time. So now we know, from comparing Equations 11 and 12, that

(Eq’n 13)

Because the coefficient matrices represent the same vector, we can use either one in Equation 9. But before we make that substitution we must correct one small error that has subtly crept into our calculation.

The metric tensor contains a single negative component, the temporal component, so its determinant comes out as a negative number. But we need the differential measure of the enclosure element to have a real-number value. To make that happen we must multiply the determinate of the metric tensor in Equation 13 by minus one. Then when we make the substitution into Equation 9 we get

(Eq’n 14)

in which g=det|g_{ik}|.

Typically, we use Equation 14 in integrations over various spatio-temporal enclosures in General Relativity and, to a lesser extent, in Special Relativity.

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