Suppose I take 2d Minkowski space [tex]ds^2=-dt^2+dx^2[/tex] and put a test particle in there. I would expect that since we have a flat space with no matter inside that it should just "sit still" so to speak i.e. not move anywhere.(adsbygoogle = window.adsbygoogle || []).push({});

However, there will be an integral of motion (since we have a timelike Killing vector) given by [tex]E=\dot{t}[/tex] where dot denotes differentiation with respect to proper time.

Then I can use the fact that [tex]ds^2=-1[/tex] for timelike geodesics and rearrange to get [tex] -1=-E^2+\dot{x}^2 \Rightarrow \dot{x} = \sqrt{E^2-1} \Rightarrow x(\tau)=\sqrt{E^2-1} \tau[/tex]

In other words the particle will move to infinity along a timelike geodesic. Why is it moving?

Thanks.

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# Particle in Minkowski

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