Synge's World Function: Explained & Needed

[PLuz]

Hello,

Can anyone explain to me the need of the Synge's World Function defined, e.g. here: http://relativity.livingreviews.org/open?pubNo=lrr-2011-7&page=articlepa4.html

in section 3.1?

Isn't the length of a geodesic connecting two points well defined?

Thank you

 

[Bill_K]

PLuz, Thanks for the reference, that's a very interesting paper!

The world function Ω(x, x') is an important quantity that can be used in relativity to describe the curved geometry. As you point out, it's a function of two points related to the length of the (assumed to be unique) geodesic connecting them.

What use is it? This paper is a good illustration of its use. Quoting Synge, "Ω is a powerful tool for the execution of systematic approximations without abandoning the techniques of tensor calculus." The motion of point particles in a curved spacetime depends in a nonlocal way on the spacetime geometry, so rather than expand the metric in a power series, it's more appropriate to expand the world function.

 

[PLuz]

Thank you for the prompt response. That was a very enlightening answer.

Just out of confusion, and what would be the problem in expanding in terms of powers of the metric?Is it because that would only be well defined for an asymptotically flat spacetime?