Isotropic Coordinates

 

Isotropic coordinates are used when the 3-D subspace of spacetime (sometimes called a time slice) needs to look as Euclidean as possible. The word isotropic means that all 3 spatial dimensions are treated the same. Hence this type of coordinate system is best when the object whose gravity you are modelling has symmetries that do not discriminate between the x, y, or z directions.

 

We want the line element to have the form

 

(19.1)

 

where  is the Euclidean 3-D subspace defined as follows.

 

 

(19.2)

 

We have used Cartesian coordinates (x,y,z) for the 3-D subspace. The isotropy is manifested in the following way. Every coefficient of the squared coordinate terms on the right hand side of (19.2) is equal to the same number (in this case the number 1). The advantage of the isotropic coordinates is that the 3-D subspace part of the line element is invariant under changes of flat space coordinates. For instance we can use spherical polar coordinates as well to represent the 3-D subspace. We get

 

(19.3)  

 

We construct the full 4-D line element by assuming that a constant time slice will be related to the 3-D subspace by a conformal factor. Then since it's conformal in 3-D, angles between 3-D vectors and ratios of 3-D vector lengths are the same for all metrics. We now introduce a new coordinate. We let the conformal factor be defined as

 

(19.4)

 

where  is a new radial coordinates defined as follows:

 

(19.5)

 

 

We want our 4-D isotropic line element to come from the Schwarzschild line element that we obtained earlier.

 

(19.6)  

 

We want the isotropic 4-D metric to be of the form

(19.7)  

 

 

We need to find the conformal factor that we have introduced. This can be done by a straight coordinate transformation into the Schwarzschild metric or just a one-to-one identification technique. We will now use the latter method.

 

By looking at the two line elements we see that we must have

 

(19.8)

 

 

Also we demand that

 

 

 

(19.9)

 

 

Using these last two equations we can derive the following result:

 

(19.11)

 

 

We want

  .

 

Hence we take the + sign in (19.11). Integration gives

 

 

 

(19.12)

 

Then  implies

 

 

(19.13)

 

Hence the final form for the 4-D Isotropic Line Element in General Relativity is as follows:

 

(19.14)  

 

Isotropic coordinates are used for compiling the relativistic astronomical data tables for planets, and all other orbiting objects (including satellites) in solar system studies. The reason for this is that they are excellent coordinates for expansions of gravitational quantities. These expansions facilitate direct comparisons with Newtonian flat space gravity and with alternate theories of gravitation that rival General Relativity.