The Relativity of Phase Space

In physics the fundamental knowledge we want to determine of a particle or of a body consists of four data – three coordinates of a point in space and an instant in time, collectively known as an event. Because it changes between inertial frames in accordance with the Lorentz Transformation, we call that four-element set {x,y,z,ct} a four-vector. If two observers occupy inertial frames separated from each other by a velocity V oriented entirely in the frames’ mutual x-direction (for algebraic simplicity), then they can derive Minkowski’s theorem by relating the measurements that they make of one event relative to a common origin of their frames, using the Lorentz Transformation to convert the four-vector {x,y,z,ct} into {x’,y’,z’,ct’}. Expressed in differential form, that transformation gives them

(Eq’ns 1)

From those equations they can derive the four-dimensional analogue of the Pythagorean theorem,

(Eq’n 2)

That equation means that the four-dimensional distance between any two events remains invariant under a Lorentz Transformation; the distance has the same value for all observers.

We can rederive that result in a more general way through the recognition of the fact that vectors belong to a more general class of mathematical objects called matrices. We write a vector as a matrix by drawing it either as a row of elements (as above) or as a column of elements, each being the transpose of the other. Under the rules of matrix algebra (see appendix below) if we consistently represent vectors as row matrices (or as column matrices), then we must represent the product of two vectors as the product of one vector multiplied by the transpose of the other,

(Eq’n 3)

If we take two four-vectors {u} and {v}, subject them to a Lorentz Transformation to get {u’}=L{u} and {v’}=L{v}, and multiply the transformed four-vectors together, we get

(Eq’n 4)

In going from the second line to the third line in that equation the product of the Lorentz Transformation matrix with its transform yields the identity matrix. Thus we prove and verify the proposition that the product of any two four-vectors stands invariant relative to the Lorentz Transformation.

Next we want to identify another four-vector. Ultimately in physics we want to convert the coordinate variables in the four-vector {X}=(x,y,z,ct) in such a way that time becomes the sole independent variable and the spatial coordinates become dependent variables, x=x(t), y=y(t), and z=z(t). That conversion works through equations of motion, which involve the two concepts of linear momentum and energy.

If we have a particle of rest mass m_{0} moving at
velocity **v** (which gives it relativistic mass m), then the particle
carries linear momentum **p**=m**v**. That particle also possesses total
relativistic energy

(Eq’n 5)

which leads to

(Eq’n 6)

We thus get an invariant function that has the same algebraic form as does Equation 2. That fact leads us to describe the particle’s four-momentum as

(Eq’n 7)

so that we have the analogue of Equation 2 as

(Eq’n 8)

Now we can locate our particle in an eight-dimensional relativistic phase space, describing it with two four-vectors. If we multiply those two four-vectors together, we get the action associated with the particle,

(Eq’n 9)

That product stands invariant under the Lorentz Transformation, which means that between any two events all observers will measure the same amount of action.

That fact enables us to work out the Lorentz
Transformation as it applies to the four-momentum. To prepare to work out that
derivation more easily than we could otherwise do we note that dy’=dy and dz’=dz.
Further, we know that dp_{y}’=dp_{y} and dp_{z}’=dp_{z}
because if we were to accelerate from one frame to the other, no force would act
in those lateral directions, so the momentum in those directions cannot change
and, thus, cannot have different values in the different frames. Thus we get
Equation 9 as

(Eq’n 10)

Comparing like terms gives us the desired transformation,

(Eq’n 11)

Appendix: Matrix Algebra

Mathematicians define a matrix as a square or rectangular
array of elements that obeys a small set of certain rules. In terms of its
component elements we represent a matrix as A=[a_{ij}], in which i
designates the row in which the element sits and j designates the column in
which it sits. We now have two basic processes that we apply to matrices:

1) addition: we can only legitimately add matrices that
have the same number of rows and columns. We have A+B=C by adding their elements
[a_{ij}]+[b_{ij}]=[c_{ij}] for all values of i and j.
That process obeys both the commutative and associative laws of ordinary
addition.

2) multiplication: aside from the trivial multiplication
of a matrix by a scalar (áA=[áa_{ij}]),
we have the multiplication of one matrix by another. In terms of their component
elements we have AB=C as a_{ik}b_{kj}=c_{ij}, in which
we use the Einstein convention that we sum over all values of the repeated
index. Of course, the A matrix must have as many columns as the B matrix has
rows. If we have a pair of 2x2 matrices, we have their product as

(Eq’n A-1)

Note that we have a relation between the transpose of a
matrix (the matrix A^{T}=[a_{ji}] obtained by transposing the
indices on the elements of A=[a_{ij}]) and commutation under
multiplication. If we have C=AB, then we have C^{T}=[c_{ji}]=b_{jk}a_{ki}=B^{T}A^{T}.

Matrix multiplication gives us a handy way to represent a transformation of coordinates. We organize the coordinates into a vector and then represent the transformation as a square matrix. Multiplying the vector by the matrix yields a new vector, the transform of the original vector. As an example consider the Pythagorean transformation of the vector description of a straight line segment in a plane.

Impose an x-y coordinate grid on a flat plane and then superimpose upon it an x’-y’ coordinate grid that’s turned an angle θ in the counterclockwise direction relative to it. If we draw a straight line from the grids’ common origin to an arbitrarily chosen point, then we can represent that line by the components projected onto the x- and y-axes of our grid; that is, by the vector (x,y). We can also project that line onto the axes of the primed grid to get the vector (x’,y’). But instead of obtaining (x’,y’) through direct measurement, we can calculate it by transforming (x,y). In that case we have multiplied our vector by the transformation matrix Θ,

(Eq’n A-2)

Carrying out that multiplication gives us the standard two-dimensional transformation equations,

(Eq’n A-3)

If we solve those equations for x and y, we find that we can reverse the transformation of Equation A-2 by multiplying the vector (x’,y’) by the transpose of the matrix Θ,

(Eq’n A-4)

If we combine Equations A-2 and A-4, we get

(Eq’n A-5)

which necessitates that

(Eq’n A-6)

the identity matrix. That fact remains true to mathematics for any transformation of coordinates plotted on a grid of mutually orthogonal axes, axes that all cross each other at a right angle. Among such transformations we find the Lorentz Transformation.

That transformation goes beyond simple three-dimensional geometry and stands before us as

(Eq’n A-7)

in which β=v/c, the fraction of the speed of light between the two inertial frames under consideration and we have the Lorentz factor as

(Eq’n A-8)

In order to make that transformation work properly we must explicitly represent the time coordinate as an imaginary number in the four-vector location of an event, {X}=(x,y,z,ict), even if we don’t always write it out. This transformation gives us the means to work out the basic relativistic rules of phase space.

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