Of Rotation, LT and Acceleration

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Lorentz transformation is a rotation in the Minkowski space. In order to see it, let&#8217;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, $\left[ {\begin{array}{*{20}c}{\cos \theta } & { - \sin \theta } \\{\sin \theta } & {\cos \theta } \\\end{array}} \right]$
Vì vậy,, the matrix equation expands to ở đâu $\theta$ is the angle of rotation. This is how a point $X=(x,y)$ in the original frame transforms to in the rotated frame.

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

This expands to 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&#8217;t define the invariant distance any longer. The geometry requires a different metric tensor. Do đó, 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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