I'm old, gray and idiosyncratic: so expect a lot of personal side comments.
The purpose of this blog is to document the development of a visualization for the Kerr Metric using very basic tools; mostly Matrix tools from whatever Computer Algebra systems that seem reliable. To start I will be using maxima/wxmaxima. The attempt is to mathematically investigate the strange and wonderful (?) world of the Kerr metric and the English interpretations of it.
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The purpose of this blog is to document the development of a visualization for the Kerr Metric using very basic tools; mostly Matrix tools from whatever Computer Algebra systems that seem reliable. To start I will be using maxima/wxmaxima. The attempt is to mathematically investigate the strange and wonderful (?) world of the Kerr metric and the English interpretations of it.
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Mathjax tests: $ \sqrt[q]{r} $
\[
\left[ \begin{array}{cc} dt' & dr'\end{array}\right]=\left[\begin{array}{cc} dt & dr\end{array}\right]\left[T\right]=\left[\begin{array}{cc} dt & dr\end{array}\right]\left[\begin{array}{cc} \frac{\partial t'}{\partial t} & \frac{\partial r'}{\partial t}\\ \frac{\partial t'}{\partial r} & \frac{\partial r'}{\partial r} \end{array}\right]=\left[\begin{array} {cc} dt & dr\end{array}\right]\left[\begin{array}{cc} A & B\\ C & D \end{array} \right]
\]
\[
\left[ \begin{array}{cc} g{}_{t't'} & g{}_{t'r'}\\ g{}_{r't'} & g{}_{r'r'} \end{array}
\right]
\]
If the above didn't come out as nice mathematical expressions please get or enable mathjax.
--------
\left[ \begin{array}{cc} dt' & dr'\end{array}\right]=\left[\begin{array}{cc} dt & dr\end{array}\right]\left[T\right]=\left[\begin{array}{cc} dt & dr\end{array}\right]\left[\begin{array}{cc} \frac{\partial t'}{\partial t} & \frac{\partial r'}{\partial t}\\ \frac{\partial t'}{\partial r} & \frac{\partial r'}{\partial r} \end{array}\right]=\left[\begin{array} {cc} dt & dr\end{array}\right]\left[\begin{array}{cc} A & B\\ C & D \end{array} \right]
\]
\[
\left[ \begin{array}{cc} g{}_{t't'} & g{}_{t'r'}\\ g{}_{r't'} & g{}_{r'r'} \end{array}
\right]
\]
If the above didn't come out as nice mathematical expressions please get or enable mathjax.
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It is motivated by some statements that seem not too obvious, or doubtful, made in various places. Or if not doubtful, at least provide context. For instance statements about: causality and the first event horizon.
The technique I have chosen is a propagating light wavefront/null geodesic. Looking at the wavefront allows one to see all accessible timelike points in the space; in the sense that one choice would give where you might possibly get to via timelike world lines and the other choice gives all possible points that you might have come from via timelike world lines. The choice of using null geodesics/wavefronts allows casualty visualization via an affine parameter control knob. Also, the wavefront is independent of the velocity of the source emitting it.
The start will be practicing the wavefront generation on the Schwarzschild metric. My intent is a log the difficulties involved and describe solutions.
