The transformation to "almost flat" coordinates

We want to define a set of new coordinates so that the metric looks as flat as it really is. We can kind of guess that we have to redefine the radius so that Metricg[-1,-1] in the new coordinates will have the flat polar coordinate value of 1. Here is what we have now:

In[32]:=


  Table[Metricg[-i,-j],{i,3},{j,3}]

Out[32]=

     -8 G M              2 - 8 G M
  {{r      , 0, 0}, {0, r         , 0}, {0, 0, -1}}

MathTensor has a function called Ttransform that will help us.

In[33]:=


  DefineTensor[gp,"gp",{{2,1},1}]

  PermWeight::sym: Symmetries of gp assigned

  PermWeight::def: Object gp defined

Let's just try a simple guess at a new coordinate q, which depends on the old coordinate r in this way:
q[r] = (r^p)/a

In[34]:=


  Ttransform[gp, Metricg[la,lb], {q, theta, t},
{(a q)^(1/p), theta, t}, -1]

  Components assigned to gp

In[35]:=


  PowerExpand[Table[gp[-i,-j],{i,3},{j,3}]]

Out[35]=

     2/p  -2 + 2/p
    a    q                      2 - 8 G M
  {{--------------, 0, 0}, {0, r         , 0}, {0, 0, -1}}
       2  8 G M
      p  r

Now we can start trying to fit the paramters a and p to the combination 2kM that appears in the original coordinate system.

In[36]:=


  f[q_] = PowerExpand[Simplify[gp[-1,-1]/. r -> (a q)^(1/p)]]

Out[36]=

   2/p - (8 G M)/p  -2 + 2/p - (8 G M)/p
  a                q
  --------------------------------------
                     2
                    p

In[37]:=


  prule = Solve[- 2 + 2/p - (8 G M)/p == 0,p]

Out[37]=

  {{p -> 1 - 4 G M}}

In[38]:=


  Simplify[f[q] /. prule]

Out[38]=

         2
        a
  {------------}
              2
   (1 - 4 G M)

So if we set a = p = (1 - 4 G M) then the coefficient of the metric in the new coordinate system will be 1. Since we've chosen a and p so that gp[-1,-1] = 1, let's just set it by hand now to that value.

In[39]:=


  gp[-1,-1] = 1

Out[39]=

Now we have to figure out what this gives us for the angular component gp[-2,-2]:

In[40]:=


  gp[-2,-2]

Out[40]=

   2 - 8 G M
  r

In[41]:=


  PowerExpand[PowerExpand[gp[-2,-2] /. r ->
(a q)^(1/a)] /. a -> 1 - 4 G M]

Out[41]=

             (2 - 8 G M)/(1 - 4 G M)  (2 - 8 G M)/(1 - 4 G M)
  (1 - 4 G M)                        q

A rule application will finish the reduction of the above expression.

In[42]:=


  gp[-2,-2] = PowerExpand[% /. 2 - 8 G M -> 2 (1 - 4 G M)]

Out[42]=

             2  2
  (1 - 4 G M)  q

Now look at the new metric and see whether it looks like ordinary polar coordinates:

In[43]:=

  Table[gp[-i,-j],{i,3},{j,3}]

Out[43]=

                             2  2
  {{1, 0, 0}, {0, (1 - 4 G M)  q , 0}, {0, 0, -1}}

OOPS!!!! What happened? If these were flat polar coordinates we would just have q^2 for the (-2,-2) component, not (1 - 4 G M)^2 q^2.

We have to make one more change -- we have to redefine the angle theta by a factor of a = 1 - 4 G M, so that the new angular coordinate is thp = theta/a. Let's do that in the next section.

Up to

But isn't this spacetime is really flat?