Of Rotation, LT and Acceleration – Page 2

This post uses Easy LaTeX Pro to display equations.

Lorentz transformation is a rotation in the Minkowski space. In order to see it, let’s first look at rotation in Euclidean space, which can be written as X’ = R X. In the 2-D case, the matrix of rotation R is,
$\left[ {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array}} \right]$
So, the matrix equation expands to
$x' = x \cos \theta - y \sin \theta \\ y' = x \cos \theta + y \sin \theta$
where $\theta$ is the angle of rotation. This is how a point $X=(x,y)$ in the original frame transforms to $X'=(x',y')$ in the rotated frame.

Similarly, LT in Minkowski space is X’ = L X. Lorentz Transformation matrix (in our 2-D case) is,
$\left[ {\begin{array}{*{20}c} \gamma & { - \beta \gamma } \\ { - \beta \gamma } & \gamma \\ \end{array}} \right]$
where $\beta = v/c$ and $\gamma = 1/\sqrt {1 - \beta ^2 }$

This expands to
$t' = (t - vx/c^2)/\sqrt{1-(v/c)^2} \\ x' = (x - vt)/\sqrt{1-(v/c)^2}$

Note that rotation (and so LT) is a linear transformation, which means that the matrix R (or L) has to be independent of the vector it transforms. What happens when the matrix is a function of x, y or t? The geometry becomes non-flat and the metric tensor we defined doesn’t define the invariant distance any longer. The geometry requires a different metric tensor. Therefore, rotation or LT as we defined it and the associated single component equations is not valid any more. I will illustrate it further using 2-D rotation in the next post and show what they mean when they say that space-time is curved.

Continued…

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