Vector Arithmetic

Before discussing how a flowing aether can be represented by a tetrad field, it is useful to discuss notation and to explore quantities that invariant under transformations within a Minkowski space-time.

Notation

Scalars are represented using lowercase letters. A bold lowercase letter represents a spatial three-vector, which can be broken into the x, y and z components using subscripts. Four-vectors are represented using uppercase letters. Matrices are represented using uppercase bold letters.

Four-vectors are represented by a column vector, but using ASCII it is useful to have a horizontal format. These notations are made clear by examining the ways in which the position four-vector can be expressed.
X = (ct; x) = (ct; x_x, x_y, x_z) = \left[\begin{array}{c} ct\\ x_x \\ x_y \\ x_z \end{array}\right]

For a diagonal matrix it is useful to express only the diagonal elements in a horizontal format.
D = diag(d_tt, d_xx, d_yy, d_zz)

An antisymmetric matrix can be expressed by a pair of vectors.
M = (j; k) = \left[\begin{array}{cccc} 0&j_x&j_y&j_z\\ -j_x&0&k_z&-k_y \\ -j_y&-k_z&0&k_x \\ -j_z&k_y&-k_x&0 \end{array}\right]

Invariant Quantities

If we use arbitrary quantities to express our physical laws, then it is not usually clear how these physical laws may changed when we move from one reference frame to the next. This requires us to show that a general expression is unchanged by a transformation between reference frames.

Alternatively, it is possible to determine quantities that are constant regardless of which particular reference frame is selected. Typically these represent scalar values or geometrical objects embedded within the space-time.

If we can define our equations in terms of these invariant quantities, then the invariant nature of our physical laws is made clear.

Points

One of the most fundamental invariant objects are markers of specific events in space and time. While our index for a point will change with each reference frame, the actual points remain the same. The coordinates of a point can be expressed as a column vector, but using ASCII it is useful to have a horizontal format. P represent the space-time coordinates in one coordinate system, then the coordinates in another reference frame can be found by applying the normal transformation.

P’ = L P + D

Displacement Vectors

Another fundamental concept is a displacement, defined as the difference between two points. Here we let D=(cΔt;Δx). Transforming the points and subtracting shows that the translation terms cancels. This is because displacements are vectors quantities, independent of their location in space.

D’ = L D

Dot Product

In Euclidean space, is is common to define a dot product, whose uses are broad. One use is to define a norm in terms of the dot product of a vector with itself.
a⋅b = a_x b_x + a_y b_y + a_z b_z
a⋅a = a_x^2 + a_y^2 + a_z^2.

These quantities are not preserved by general transformations, as the time component needs to come into play. Due to the hyperbolic relationship between space and time, however, the sign of the time component is found to be opposite that of space.

Is Space or Time More Imaginary?

The dot product for a one-dimensional space appears to resemble the real part of a complex multiplication. This begs the question whether space or time is more like a complex number.

One significant difference between imaginary values and real values is that real values have a definite sense of order defining sign, while with imaginary values the two signs are essentially a convention. Our experience of time has a definite direction, while left and right can be seen to simply be a convention by considering two people standing face to face.

This suggests that it is time that should square to a positive value. While all authors agree that this is a matter of sign convention, some authors prefer to keep consistency with the Euclidean dot product.

Minkowski Dot Product

The required dot product can now be expressed using a diagonal matrix called the metric, η = diag(1;-1,-1,-1).
AB = A^T η B = A_t B_t – A_x B_x – A_y B_y – A_z B_z
AA = A_t^2 – A_x^2 – A_y^2 – A_z ^2

Vector Norms

The norm of a vector A is defined as the dot product with itself.
N = AA

The sign of N separates vectors into three distinct categories.

Time-like Vectors

When N is positive the time-like component is the primary difference and it is possible to find a frame where the two points are stationary. A points to the past or future depending on the sign of A_t. Here it is useful to define a proper time relative to the frame where they are coincident.
c^2Δτ^2 = N

Space-like Vectors

When N is negative the space-like component is the primary difference and it is possible to find a reference frame where the two points are simultaneous but separated spatially. Here it is useful to define a proper distance relative to the frame where they are simultaneous.
Δσ^2 = -N

Null Vectors

Finally, When N=0 the points are on a light cone, requiring the invariant speed for a constant velocity between them. Although the points are distinct, there is no invariant measure of the magnitude of such vectors.

Hidden tensors

The matrix and vector notation used so far have hidden the underlying tensor nature of the quantities being discussed. Tensor notation is useful because the invariance follows from a tensor representation, but can be viewed as esoteric, if not intimidating, to those unfamiliar with tensors.

In tensor notation the dot product looks like this:
A\cdot B = A^iB^j\eta_{ij}

For the cross product, the tensoral definition will be given, followed by the vectoral relation it defines. This is intended to fulfill the following purposes:

  1. Provide a feel for how much more familiar the presentation here is to those familiar with vectors, but not tensors.
  2. Provide a framework where those familiar with tensors can derive the invariant equations.
  3. Provide precise definitions of the generalizations being used for the more pedantic readers.

While explaining the arithmetic of tensors is beyond the scope of this discussion, new definitions of new tensors should be defined as a courtesy to the reader.

The tensor \epsilon^{ijkl}=-\epsilon_{ijkl} is called the permutation tensor, and is fundamental to the concept of a cross product and its generalizations. For this discussion \epsilon^{0123}=1.

Axial Vectors

It is useful to define a generalization of the cross product. First, we need to examine the nature of vectors under reflections.

Directional and Axial Vectors

The intuitive idea of a vector is separated into two classes when reflections are considered. Directional vectors that are inverted on a reflection while the orientation of axial vectors is preserved.

So far we have been dealing with directional vectors that represent an arrow pointing in a particular direction. An axial vector represents an axis, such as an angular velocity represented by a vector along the axis of rotation and having a magnitude of its speed.

The cross product of two vectors of the same class is an axial vectors, while the cross product of vectors of different classes is a directional vector.

Axial Vectors as Matrices

Directional vectors can be represented by the familiar column vectors that have already been implied. Axial vectors are better represented by anti-symmetric matrices.

In 3D Euclidean space this results in three independent parameters that can be also be expressed as a vector. In 4D Minkowski space-time this results in six independent parameters that can be represented as a spatial three-vector and a velocity-like three-vector.

Cross Products

Let A=(a;a) and B=(b;b) be directional vectors, while M=(m’;m) and N=(n’;n) are axial vectors.

A × B = A^iB^j-A^jB^i = (ab – ba; a × b)
M × N = M^{ij}N^{kl}\eta_{jk}(m’ × n + m × n’; m × nm’ × n’)
A × M = A^iM^{jk}\eta_{ij} = M × A = (a⋅m’; am’ +  a × m)

Axial Dot Products

The dot product formula above does not apply to axial vectors.
N = \frac{1}{2}M^{ij}N^{kl}\eta_{ik}\eta_{jl} m⋅n – m’⋅n’

N =  M can again be positive, negative and 0, defining circular, hyperbolic and null axial vectors respectively.

The helical quantities are defined as the dot product of an axial vector and a directional vector, which  is not an invariant scalar in Minkowski space-time. It is possible to define an invariant vector that includes such helical quantities, with the time-like component the classical value.
A M = \frac{1}{2}A^iM^{jk}\epsilon_{ijkl}\eta^{lm} = ( a⋅m; ama × m‘)

  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: