Tuesday 14 November 2017

general relativity - Are the space and time axes of Schwarzschild metric uncurved?


Schwarzschild metric is commonly considered as an expression of curved spacetime:


$$ \mathrm ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2~\mathrm dt^2 + \frac{1}{1 - \frac{2GM}{c^2 r} }~\mathrm dr^2 + r^2 \left(\mathrm d\Theta^2 + \sin^2 \Theta ~\mathrm d\Phi^2\right)$$


However, looking at this equation, I find that the metric on the left side is curved (distorted) but not the coordinate axes dt and dr on the right (which are given in the equation).


dt and dr are the displacement coordinates. The Schwarzschild equation does not calculate them, but they are given. For calculating the warped metric on the left side, they are subject to distortion (multiplication/ division by a factor). But in no way, the space and time coordinate axes are warped. It is only the distance between events which deviates from the distance of Minkowski metric.


It seems to be the same situation as for Minkowski diagrams, where the metric ds is different from the Euclidean metric, but the coordinate axes for space and time (horizontal and vertical) are identic with those already Newton could have used.




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