1
$\begingroup$

One way to define the Randers metric is using the data $(h,W)$ associated to the Zermelo problem. Here $h$ is the Riemannian metric and $W$ is the wind. In order to define the Randers metric we must have $h(W,W)<1$.

Now the question is that what happens if $h(W,W)\geq 1$? I mean we are faced with a new metric? Why in Randers $h(W,W)\geq 1$ is a necessary condition?

$\endgroup$
  • 3
    $\begingroup$ If the wind is too strong, the Finsler function can take negative values. In terms of the Zermelo problem: if the wind is stronger than the maximum speed of your boat, then not every point is reachable by boating. $\endgroup$ – Willie Wong Mar 27 '18 at 20:13
  • $\begingroup$ Probably my comment does not answer your question; maybe you should edit it to clarify what you mean by "what happens"? $\endgroup$ – Willie Wong Mar 27 '18 at 20:16
  • $\begingroup$ @WillieWong Thanks for your comment. Could you explain a bit more why the Finsler function can take negative values? $\endgroup$ – Majid Mar 27 '18 at 23:38
  • 1
    $\begingroup$ If $h(W,W) <1$ then $F(v)=\sqrt{h(v,v)}+h(v,W)$ is positive by Cauchy-Schwarz. On the other hand if $h(W,W)\ge 1$ then you should find $F(-W) \le 0.$ $\endgroup$ – Anthony Carapetis Mar 28 '18 at 0:11
  • $\begingroup$ @AnthonyCarapetis why did you write $F(v)$ in this way? As far as I know it must be like $F(v)=\frac{\sqrt{h^2(W,v)+\lambda h(v,v)}}{\lambda}-\frac{h(y,W)}{\lambda}$, in which $\lambda=1-h(W,W).$ $\endgroup$ – Majid Apr 11 '18 at 13:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.