Geodesic Distance & Maximally Symmetric Spacetimes: Why Does it Matter?

[highflyyer]

Any physical quantity $K(t,x,x')$ on a maximally symmetric spacetime only depends on the geodesic distance between the points $x$ and $x'$.

Why is this so?

N.B.:

This statement is different from the statement that

The geodesic distance on any spacetime is invariant under an arbitrary coordinate transformation of that spacetime.

 

[PeterDonis]

Why is this so?

Why do you think it is so? Have you found a proof of it?

 

[highflyyer]

I haven't found a proof of it. I read this in a paper.

This is my understanding of the problem.

The Euclidean plane is a maximally symmetric space with $3$ translation symmetries and $3$ rotation symmetries. Any physical quantity $K(x,y)$ on the Euclidean plane, where $x$ and $y$ are two arbitrary spacetime points, is constrained by the symmetries of the spacetime to depend only on $(x-y)^{2}$. This is because $(x-y)$ is translation invariant and $(x-y)^{2}$ is rotation invariant. Therefore, the physical quantity $K(x,y)$ depends on the Galilean-invariant geodesic distance $(x-y)^{2}$.

 

[highflyyer]

But I am not sure how the dependence changes if we have a Euclidean disk (that is, a plane with a boundary).

My intuition is that the $K(x,y)$ now depends not only on the spacetime points $x$ and $y$, but also on the 'border.' The dependence is such that $K(x,y)$ for the Euclidean disk tends to $(x-y)^2$ as the 'border' tends to infinity.

But I am not able to carry my intuition any further and write down an explicit form for the dependence of $K(x,y)$ for the Euclidean disk.

It would be really helpful if you share some thoughts here.

 

[PeterDonis]

I read this in a paper

What paper? Please give a reference.

 

[highflyyer]

What paper? Please give a reference.

See the final paragraph on page 7 of https://arxiv.org/pdf/0804.1773.pdf.

 

[PeterDonis]

I am not sure how the dependence changes if we have a Euclidean disk (that is, a plane with a boundary).

A plane with a boundary is not maximally symmetric.

 

[PeterDonis]

See the final paragraph on page 7 of https://arxiv.org/pdf/0804.1773.pdf.

Ok, this mentions the proposition but doesn't give a proof. Possibly one of the references in that paper does.

Your reasoning in post #3 seems OK to me.