How NOT to get to flat coordinates

The metric we recognize right away as representing flat spacetime in polar coordinates is obtainable from the metric we just calculated by allowing G to vanish from the problem.

In[30]:=

  
  
  Table[Metricg[-i,-j],{i,3},{j,3}]/. G -> 0

Out[30]=

                   2
  {{1, 0, 0}, {0, r , 0}, {0, 0, -1}}

But simply setting G = 0 would mean that the gravitational constant was zero. And G = 0 means no gravity, period, so we wouldn't have advanced our understanding of what gravity does at all.

We can also make flat coordinates by letting M=0:

In[31]:=

  
  Table[Metricg[-i,-j],{i,3},{j,3}]/. M -> 0

Out[31]=

                   2
  {{1, 0, 0}, {0, r , 0}, {0, 0, -1}}

But that means we've changed the problem by taking the source of the gravitational field out of the spacetime. This tells us nothing about what mass does to three-dimensional spacetime. So we have to find a better way to get to flat coordinates, one where we still retain information connected to the presence of mass and gravity (i.e. M and k are still parameters somewhere in the solution).

Up to

But isn't this spacetime is really flat?