The whole curvature tensor vanishes

Once we plug in the solution for the spacetime metric, we find that the curvature of this spacetime is zero. One way of verifying this is to examine the components of the Riemann tensor.

In[29]:=

  
  Table[RiemannR[-i,-j,-k,-l],{i,3},{j,3},{k,3},{l,3}]

Out[29]=

  {{{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
   
     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
   
     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}}, 
   
    {{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
   
     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
   
     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}}, 
   
    {{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
   
     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
   
     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}}}

Up to

But isn't this spacetime is really flat?