- #1

Sciencemaster

- 105

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- TL;DR Summary
- Can one derive a coordinate transformation for the de Sitter space from its invariant interval akin to the Lorentz Transform in Minkowski Space?

Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate transforms be physically significant?

I understand that the concept of reference frames and their coordinate systems works differently in GR than in the flat spacetime of SR, but given that the de Sitter Metric is maximally symmetric, I was wondering if this would be possible and perhaps even physically meaningful.

For instance, an axiom used in deriving the Lorentz Transformations is that $x=\pm ct$. In de Sitter space, the invariant quantity would instead be $r=\pm \left(1-\frac{r^2}{\alpha^2}\right)ct$. Could this be used to derive a coordinate transform and would it have any physical significance?

Also, if this transform can be derived, would it be linear? I would imagine it is due to the symmetry in the metric, and it would make deriving the transformations much easier, but I'm not absolutely sure since $g_{rr}$ and $g_{tt}$ depend on $r$.

I understand that the concept of reference frames and their coordinate systems works differently in GR than in the flat spacetime of SR, but given that the de Sitter Metric is maximally symmetric, I was wondering if this would be possible and perhaps even physically meaningful.

For instance, an axiom used in deriving the Lorentz Transformations is that $x=\pm ct$. In de Sitter space, the invariant quantity would instead be $r=\pm \left(1-\frac{r^2}{\alpha^2}\right)ct$. Could this be used to derive a coordinate transform and would it have any physical significance?

Also, if this transform can be derived, would it be linear? I would imagine it is due to the symmetry in the metric, and it would make deriving the transformations much easier, but I'm not absolutely sure since $g_{rr}$ and $g_{tt}$ depend on $r$.