Undergrad Coord Transform in de Sitter Space: Phys Significance &Linearity?

[Sciencemaster]

TL;DR

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 $r$.

 

[PeterDonis]

I understand that the concept of reference frames and their coordinate systems works differently in GR than in the flat spacetime of SR

The concept of "reference frame" that I suspect you are using, namely a global inertial frame in which a family of observers at rest in the frame (i.e., with constant spatial coordinates) are (a) at rest relative to each other, and (b) all in free fall, does not exist in any curved spacetime. That includes de Sitter spacetime.

given that the de Sitter Metric is maximally symmetric, I was wondering if this would be possible

Not the way I suspect you are thinking. See above.

 

[PeterDonis]

Could one derive a set of coordinate transformations that transforms events between different reference frames coordinate charts in the de Sitter metric using the invariant line element

With the correction I made above (see the strikethrough and the bolded text), you can do it in any curved spacetime. In fact, that is the definition of a valid coordinate transformation: that it leaves the line element invariant. But no coordinate chart on a curved spacetime can correspond to a global inertial reference frame.

 

[Orodruin]

Yes, but maybe not in the way you are imagining. The particular coordinate form of the line element is not as important as the geometry of the spacetime and the symmetries of de Sitter space are generally not very transparent from the typical coordinate choices.

By definition, symmetry transformations are those that leave the metric tensor invariant. In other words, they are maps $f: M \to M$ such that the pullback of the metric is the metric itself $f^*g = g$. After making any such transformation, reintroducing coordinates in the same way as before will of course lead to the same coordinate form for the line element.

In general, the $n$-dimensional de Sitter space $dS_n$ can be embedded into $n+1$-dimensional Minkowski space as Minkowski equivalent of a sphere in Euclidean space and the symmetry transformations may be inferred from there.

 

[strangerep]

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

The group of transformations is $SO(4,1)$. There is extensive literature concerning de Sitter (special) relativity, which may interest you.

 

[Orodruin]

Just to point out that this

The group of transformations is $SO(4,1)$.

is what follows from this

In general, the n-dimensional de Sitter space dSn can be embedded into n+1-dimensional Minkowski space as Minkowski equivalent of a sphere in Euclidean space and the symmetry transformations may be inferred from there.

The SO(n,1) transformations are the Lorentz transformations in the embedding Minkowski space.