Compute Physical Coordinates of an Event in a Curved Spacetime

[TheNewBie2019]

TL;DR

I would like to know how to compute the physical distance along the coordinate axes of a curved spacetime in order to obtain the actual coordinates of an event as measured by an observer which lives in the curved spacetime in question.

Hi everyone, this is my first post on PhysicsForums. Thank you so much in advance for your help!
My question is the following. Let us suppose we have an event A in a curved spacetime which, for definiteness, is the spacetime curved by the bodies of the solar system. Adopting a coordinate system centered at the barycenter of the solar system, the metric can be written as (I know that this is not the whole PN metric, but this will suffice for my purpose):

g00=1-2U/c², g0i=gi0=0, gij=-δij(1+2U/c²),

where U is the Newtonian gravitational potential of the bodies of the solar system.

Now, if we want to compute the spacetime interval from the origin to A, we need to integrate ds=√ (g_μνdx^μdx_ν) from 0 to A. Moreover, since ds is invariant under coordinate transformations, this distance will be the same in any coordinate system. Am I correct so far?

But, what if we want the physical distance from the origin to the coordinate, say, xⁱ_A of event A along the axis xⁱ? My answer is that we still need to integrate the ds, which now will not be the same as before but simply given by ds=√(g_ii)dxⁱ (no summation on repeated indices). Also, this will need to be intagrated from the origin to xⁱ_A. Of course, the result will not be invariant under coordinate transformation because the coordinate of A along an axis will be different after a coordinate transformation. Am I correct about this?

Basically, when we have a curved spacetime we cannot compute the physical coordinates of an event by simply taking the value of the adopted coordinates at the event like we do in Newtonian physics. We have to take into account that the spacetime is curved and compute the physical coordinate of an event along an axis as I did above by including the component of the metric tensor relative to the axis in question. Also, the physical distances computed in this way along the three spatial axes will represent the components of the position vector locating the event A with respect to the origin as seen by an observer living in the curved spacetime. Am I right?

As an example, consider a 2-sphere of radius r, whose metric tensor is:

gθθ=r², gθφ=gφθ=0, gφφθ=r²sin²θ,

where θ is the colatitude and φ the longitude. The physical distance along the θ axis of a point at the equator is not π/2, which is simply the value of the θ coordinate of the point. Instead, we need to integrate the ds along the θ axis from 0 to π/2. The ds along the θ axis is ds=rdθ, because dφ=0 along the θ axis. So, integrating ds=rdθ from 0 to π/2 gives rπ/2, which is, correctly, the physical value of the coordinate along the θ axis of the point on the equator as measured by an observer living on the sphere.

Again, thank you very much for your help and your time!

 

[Nugatory]

But, what if we want the physical distance from the origin to the coordinate, say, of event A along the axis xi?

Assuming that you have chosen your coordinates so that you have one timelike and three spacelike coordinates...
The coordinates of event A are $(T,x^1_A,x^2_A,x^3_A)$. Take $i=1$ for definiteness, and the corresponding event on the $x^i$ axis will be $(T,x^1_A,0,0)$. The distance you are looking for is the spacetime interval between that event and the event $(T,0,0,0)$ (location of the spatial origin at time $T$) and you compute that with the line integral you would use to compute any other spacetime interval.