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In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), when the authors introduce the differentiable structure on page 70, they claim:

It should be realized that the representation of light in special relativity by means of ordinary cones rather than by hypersurfaces of hour-glass shape depends partly on a particular choice of differential structure.

The question is: What do they mean by saying particular choice of differential structure? In what condition may the light cone be deformed and become like hour-glass as shown in this figure?

Thank you in advance for any help. enter image description here

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  • $\begingroup$ With a metric $ds^2 = c^2dt^2 - dx^2$, the light cone $ds^2$ corresponds to $dx= \pm dt$. Now make the diffeomorphism $x = x_0 \ln y$, you have $ds^2 = c^2dt^2 - x_0^2 \frac{dy^2}{y^2}$, and the light cone is $dy = \pm y \frac{cdt}{x_0}$, which gives you a "deformed" light-cone. $\endgroup$ – Trimok Aug 15 '14 at 13:01
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    $\begingroup$ @Trimok thanks for your example,but diffeomorphism does not change the differentiable structure. $\endgroup$ – Dory Aug 15 '14 at 14:58
  • $\begingroup$ Oh I see. Yes, This has to do with the possibility of different differential structures for the same manifold. This is maths then. I found a paper which maybe gives an idea of what all this is about. $\endgroup$ – Trimok Aug 16 '14 at 9:14

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