Off[General::"spell1"]

This is some matterial that could be a helpfull companion to the paper, 
"An all-purpose metric for the exterior of any kind of rotating neutron star", by G. Pappas and T. A. Apostolatos.

Multipole Moments for the two-soliton

\(This\ algorithm\ generates\ the\ moments\ m\_n\ as\ defined\ in\),  G . Fodor,C . Hoenselaers,and Z . Perjés, J . Math . Phys . 30, 2252 (1989).
To calculate the relativistic moments of the spacetime, \(one\ should\ use\ these\ m\_n\ and\ calculate\ the\ appropriate\ quantities\ P\_n . 
The\ first\ moments\ for\ the\ two - 
        soliton\ \ are\), (Subscript[P, 0]) = (Subscript[m, 0]) = M, (Subscript[P, 1]) = (Subscript[m, 1]) = i a M, (Subscript[P, 2]) = (Subscript[m, 2]) = -((a^2) - k) M, etc.

multipoleOrder = 6;
P3 = (z - M - I a) (z + I b) - 
          k;
Q3 = (z + M - I a) (z + I b) - 
          k;
Ε3 = (P3/Q3);
ξ3 = 
        z ((1 - Ε3)/(1 + Ε3));
z = 1/x;
multipoleseriesTwoSoliton = 
        Simplify[
          Series[ξ3, {x, 0, 
              multipoleOrder}]];
FullSimplify[CoefficientList[multipoleseriesTwoSoliton, x]] // 
      MatrixForm

Metric functions for the two-soliton

The metric functions can be expressed using the following quantities:

\(\(A = \(-16\)\ d\ k\ M\^2\ \((\(r\_-\)\ \(r\_+\) + \(R\_-\)\ \(R\_+\))\) - \((\(-4\)\ I\ \((a\^2\ b - a\ \((b\^2 + d + k)\) + 
                      b\ \((d + k - 
                            M\^2)\))\)\ \(κ\_-\)\ \((\(R\_-\)\ \(r\_+\) - \(r\_-\)\ \(R\_+\))\) + \((4\ \((a - b)\)\^2\ \((a\ b - d - k)\) - 
                      4\ \((\((a - 2\ b)\)\ b - 
                            d)\)\ M\^2)\)\ \((\(R\_-\)\ \(r\_+\) + \(r\_-\)\ \(R\_+\))\))\)\ κ\_+\%2 + κ\_-\%2\ \((\((4\ \((a - b)\)\^2\ \((a\ b + d - k)\) - 
                      4\ \((\((a - 2\ b)\)\ b + 
                            d)\)\ M\^2)\)\ \((\(r\_-\)\ \(R\_-\) + \(r\_+\)\ \(R\_+\))\) - 4\ I\ \((\((a - b)\)\ \((a\ b + d - k)\) - 
                      b\ M\^2)\)\ \((\(r\_-\)\ \(R\_-\) - \(r\_+\)\ \(R\_+\))\)\ \(κ\_+\))\);\)\)
\(\(B = \(κ\_-\)\ \(κ\_+\)\ \((\(-2\)\ I\ M\ \((\(R\_-\) - \(R\_+\))\)\ \((\(-\(κ\_-\)\) + \(κ\_+\))\)\ \((\(-a\^2\)\ b + 2\ a\ k + b\ \((b\^2 - 2\ k + M\^2)\) - 
                    b\ \(κ\_-\)\ \(κ\_+\))\) + 
              2\ I\ M\ \((\(r\_-\) - \(r\_+\))\)\ \((\(κ\_-\) + \(κ\_+\))\)\ \((\(-a\^2\)\ b + 2\ a\ k + 
                    b\ \((b\^2 - 2\ k + M\^2)\) + 
                    b\ \(κ\_-\)\ \(κ\_+\))\) + 
              1\/2\ \((8\ d\ M\ \((\(-a\^2\) + b\^2 + 
                          M\^2)\)\ \((\(-\(r\_-\)\) + \(R\_-\) - \(r\_+\) + \(R\_+\))\) - 
                    8\ d\ M\ \(κ\_-\)\ \((\(r\_-\) + \(R\_-\) + \(r\_+\) + \(R\_+\))\)\ \(κ\_+\))\))\);\)\)

From A and B we define the Ernst potential:  ℰ(ρ, z) = ((A - B)/(A + B)), and the metric function: \(TraditionalForm\`f(ρ, 
          z) = \(\(\(A\ \(A\^*\) - 
                B\ \(B\^*\)\)\/\(\((A + 
                    B)\)\ \((\(A\^*\) + \(B\^*\))\)\)\)\(.\)\)\)

\(\(L = 
        4\ I\ d\ \((a + I\ M)\)\ M\ \((\((a - b)\)\^2\ \((\((a + b)\)\^2 - 4\ k)\) - 2\ \((a\^2 - b\^2 - 2\ k)\)\ M\^2 + 
                M\^4)\)\ \((\(r\_-\) + \(R\_-\) + \(r\_+\) + \(R\_+\))\) + 
          4\ I\ \((\(-I\)\ b\^4\ M\^2 + I\ b\^2\ M\^2\ \((d + 2\ k - 
                      M\^2)\) + I\ M\^2\ \((2\ k\ \((d + k)\) + 
                      d\ M\^2)\) + a\^4\ b\ \((M - z)\) + 
                a\^3\ \((b\^2 - d - k)\)\ \((M - z)\) + 
                b\^3\ \((d + k + M\^2)\)\ \((M - z)\) - 
                b\ \((4\ k\ \((d + k)\) - \((d + k)\)\ M\^2 - M\^4)\)\ \((M - 
                      z)\) + a\ \((\(-4\)\ I\ b\ k\ M\^2 + 
                      b\^2\ \((d + 5\ k - M\^2)\)\ \((M - z)\) + \((d + 
                            k)\)\ \((4\ k + M\^2)\)\ \((M - z)\) + 
                      b\^4\ \((\(-M\) + z)\))\) + 
                a\^2\ \((I\ b\^2\ M\^2 - I\ d\ M\^2 + 
                      b\^3\ \((\(-M\) + z)\) + 
                      b\ \((d + 5\ k + 2\ M\^2)\)\ \((\(-M\) + 
                            z)\))\))\)\ \(κ\_-\)\ \((\(-\(R\_-\)\)\ \(r\_+\) + \(r\_-\)\ \(R\_+\))\) + 
          4\ \((I\ b\^5\ M\^2 - I\ b\^3\ \((3\ k\ M\^2 - 2\ M\^4)\) + I\ b\ M\^2\ \((\(-2\)\ k\ \((d + k)\) + 
                      3\ k\ M\^2 + M\^4)\) + 
                a\^3\ \((\(-I\)\ k\ M\^2 - 2\ b\^3\ \((M - z)\) - 
                      2\ b\ \((2\ k + M\^2)\)\ \((M - z)\))\) + 
                a\^2\ \((\(-2\)\ I\ b\^3\ M\^2 - I\ b\ \((3\ k\ M\^2 + 2\ M\^4)\) + 
                      2\ b\^2\ \((d + 5\ k)\)\ \((M - 
                            z)\) + \((4\ k\ \((d + k)\) + \((2\ d + 
                                  k)\)\ M\^2)\)\ \((M - z)\))\) + 
                a\ \((7\ I\ b\^2\ k\ M\^2 + I\ k\ M\^2\ \((2\ \((d + k)\) + M\^2)\) + b\^5\ \((M - z)\) - 
                      2\ b\^3\ \((2\ k - M\^2)\)\ \((M - z)\) - 
                      b\ \((8\ k\ \((d + k)\) - 4\ k\ M\^2 - M\^4)\)\ \((M - 
                            z)\))\) + a\^5\ b\ \((M - z)\) - 
                b\^4\ \((d + k)\)\ \((M - z)\) - 
                M\^2\ \((2\ k\ \((d + k)\) + d\ M\^2)\)\ \((M - z)\) + 
                b\^2\ \((4\ k\ \((d + k)\) - \((2\ d + 
                            3\ k)\)\ M\^2)\)\ \((M - z)\) + 
                a\^4\ \((\(-M\)\ \((d + k - I\ b\ M)\) + \((d + 
                            k)\)\ z)\))\)\ \((\(R\_-\)\ \(r\_+\) + \(r\_-\)\ \(R\_+\))\) - 
          16\ d\ k\ M\^2\ \((I\ a - I\ b + M - 
                z)\)\ \((\(r\_-\)\ \(r\_+\) + \(R\_-\)\ \(R\_+\))\) - 
          4\ \((I\ b\^5\ M\^2 - I\ b\^3\ \((3\ k\ M\^2 - 2\ M\^4)\) + I\ b\ M\^2\ \((2\ \((d - k)\)\ k + 
                      3\ k\ M\^2 + M\^4)\) + 
                a\^3\ \((\(-I\)\ k\ M\^2 - 2\ b\^3\ \((M - z)\) - 
                      2\ b\ \((2\ k + M\^2)\)\ \((M - z)\))\) + 
                a\ \((7\ I\ b\^2\ k\ M\^2 + I\ k\ M\^2\ \((\(-2\)\ d + 2\ k + M\^2)\) + b\^5\ \((M - z)\) - 
                      2\ b\^3\ \((2\ k - M\^2)\)\ \((M - z)\) + 
                      b\ \((8\ \((d - k)\)\ k + 4\ k\ M\^2 + M\^4)\)\ \((M - 
                            z)\))\) + 
                a\^2\ \((\(-2\)\ I\ b\^3\ M\^2 - I\ b\ \((3\ k\ M\^2 + 2\ M\^4)\) - 
                      2\ b\^2\ \((d - 5\ k)\)\ \((M - 
                            z)\) - \((2\ d\ \((2\ k + M\^2)\) - 
                            k\ \((4\ k + M\^2)\))\)\ \((M - z)\))\) + 
                a\^5\ b\ \((M - z)\) + b\^4\ \((d - k)\)\ \((M - z)\) + 
                M\^2\ \((2\ \((d - k)\)\ k + d\ M\^2)\)\ \((M - z)\) - 
                b\^2\ \((4\ d\ k - 4\ k\^2 - 2\ d\ M\^2 + 
                      3\ k\ M\^2)\)\ \((M - z)\) + 
                a\^4\ \((I\ b\ M\^2 + d\ \((M - z)\) + 
                      k\ \((\(-M\) + 
                            z)\))\))\)\ \((\(r\_-\)\ \(R\_-\) + \(r\_+\)\ \(R\_+\))\) - 
          4\ d\ M\ \((I\ a\^3 + 2\ I\ b\ k - 
                a\^2\ M + b\^2\ M + 
                M\^3 - I\ a\ \((b\^2 + 2\ k + 
                      M\^2)\))\)\ \(κ\_-\)\ \((\(r\_-\) - \(R\_-\) + \(r\_+\) - \(R\_+\))\)\ \(κ\_+\) + 
          4\ I\ \((\(-I\)\ b\^4\ M\^2 - I\ b\^2\ M\^2\ \((d - 2\ k + 
                      M\^2)\) - I\ M\^2\ \((2\ \((d - k)\)\ k + 
                      d\ M\^2)\) + a\^4\ b\ \((M - z)\) + 
                a\^3\ \((b\^2 + d - k)\)\ \((M - z)\) - 
                b\^3\ \((d - k - M\^2)\)\ \((M - z)\) + 
                b\ \((4\ \((d - k)\)\ k - \((d - k)\)\ M\^2 + M\^4)\)\ \((M - 
                      z)\) + a\^2\ \((I\ b\^2\ M\^2 + I\ d\ M\^2 + b\ \((d - 5\ k - 2\ M\^2)\)\ \((M - z)\) + 
                      b\^3\ \((\(-M\) + z)\))\) + 
                a\ \((\(-4\)\ I\ b\ k\ M\^2 - \((d - 
                            k)\)\ \((4\ k + M\^2)\)\ \((M - z)\) + 
                      b\^4\ \((\(-M\) + z)\) + 
                      b\^2\ \((d - 5\ k + M\^2)\)\ \((\(-M\) + 
                            z)\))\))\)\ \((\(r\_-\)\ \(R\_-\) - \(r\_+\)\ \(R\_+\))\)\ \(κ\_+\) + 
          4\ M\ \((\(R\_-\) - \(R\_+\))\)\ \((\(-\((\((a - b)\)\ \((a\ b - 
                              k)\)\ \((\((a + b)\)\ \((a\^2 - 
                                    d)\) + \((\(-3\)\ a + 
                                    b)\)\ k)\) + I\ \((a - 
                              b)\)\ \((b\ \((a + b)\)\ \((a\^2 - 
                                    d)\) - \((a\^2 + 3\ a\ b - 2\ d)\)\ k + 
                              2\ k\^2)\)\ M - \((2\ a\^3\ b - a\^2\ k + 
                              k\ \((2\ b\^2 - d + k)\) - 
                              a\ b\ \((b\^2 + d + 
                                    3\ k)\))\)\ M\^2 - I\ \((2\ a\^2\ b - a\ k - b\ \((b\^2 + d + 2\ k)\))\)\ M\^3 + 
                        a\ b\ M\^4 + I\ b\ M\^5)\)\)\ \(κ\_-\) + \((\((a - b)\)\ \((a\ b - 
                            k)\)\ \((\((a + b)\)\ \((a\^2 + 
                                  d)\) + \((\(-3\)\ a + 
                                  b)\)\ k)\) + I\ \((a - 
                            b)\)\ \((b\ \((a + b)\)\ \((a\^2 + 
                                  d)\) - \((a\^2 + 3\ a\ b + 2\ d)\)\ k + 
                            2\ k\^2)\)\ M - \((2\ a\^3\ b - a\^2\ k + 
                            k\ \((2\ b\^2 + d + k)\) - 
                            a\ b\ \((b\^2 - d + 
                                  3\ k)\))\)\ M\^2 + I\ \((\(-2\)\ a\^2\ b + b\^3 - b\ d + a\ k + 2\ b\ k)\)\ M\^3 + 
                      a\ b\ M\^4 + I\ b\ M\^5)\)\ \(κ\_+\))\) + 
          4\ M\ \((\(r\_-\) - \(r\_+\))\)\ \((\((\((a - b)\)\ \((a\ b - 
                            k)\)\ \((\((a + b)\)\ \((a\^2 - 
                                  d)\) + \((\(-3\)\ a + 
                                  b)\)\ k)\) + I\ \((a - 
                            b)\)\ \((b\ \((a + b)\)\ \((a\^2 - 
                                  d)\) - \((a\^2 + 3\ a\ b - 2\ d)\)\ k + 
                            2\ k\^2)\)\ M - \((2\ a\^3\ b - a\^2\ k + 
                            k\ \((2\ b\^2 - d + k)\) - 
                            a\ b\ \((b\^2 + d + 
                                  3\ k)\))\)\ M\^2 - I\ \((2\ a\^2\ b - a\ k - b\ \((b\^2 + d + 2\ k)\))\)\ M\^3 + 
                      a\ b\ M\^4 + I\ b\ M\^5)\)\ \(κ\_-\) + \((\((a - b)\)\ \((a\ b - 
                            k)\)\ \((\((a + b)\)\ \((a\^2 + 
                                  d)\) + \((\(-3\)\ a + 
                                  b)\)\ k)\) + I\ \((a - 
                            b)\)\ \((b\ \((a + b)\)\ \((a\^2 + 
                                  d)\) - \((a\^2 + 3\ a\ b + 2\ d)\)\ k + 
                            2\ k\^2)\)\ M - \((2\ a\^3\ b - a\^2\ k + 
                            k\ \((2\ b\^2 + d + k)\) - 
                            a\ b\ \((b\^2 - d + 
                                  3\ k)\))\)\ M\^2 + I\ \((\(-2\)\ a\^2\ b + b\^3 - b\ d + a\ k + 2\ b\ k)\)\ M\^3 + 
                      a\ b\ M\^4 + I\ b\ M\^5)\)\ \(κ\_+\))\);\)\)

\(\(Ε = 
        4\ d\ M\ \((\((a - b)\)\^2\ \((\((a + b)\)\^2 - 4\ k)\) - 
                2\ \((a\^2 - b\^2 - 2\ k)\)\ M\^2 + 
                M\^4)\)\ \((I\ a + M - 
                z)\)\ \((\(r\_-\) + \(R\_-\) + \(r\_+\) + \(R\_+\))\) - 
          4\ I\ M\ \((\((a - b)\)\ \((\((a + b)\)\^2 - 
                      4\ k)\)\ \((a\ b - d - 
                      k)\) - I\ \((\((a - b)\)\ \((a + 
                            b)\)\ \((b\^2 - d)\) + 
                      2\ \((\(-2\)\ a\ b + b\^2 + d)\)\ k + 
                      2\ k\^2)\)\ M - \((a + b)\)\ \((2\ a\ b - b\^2 - d - 
                      k)\)\ M\^2 + I\ \((b\^2 - d)\)\ M\^3 + 
                b\ M\^4)\)\ \(κ\_-\)\ \((\(-\(R\_-\)\)\ \(r\_+\) + \(r\_-\)\ \(R\_+\))\) + 
          4\ M\ \((\(-\((a - b)\)\^2\)\ \((\((a + b)\)\^2 - 
                      4\ k)\)\ \((a\ b - d - k)\) + I\ \((a - 
                      b)\)\ \((\((a - b)\)\ b\ \((a + b)\)\^2 - \((a\^2 + 
                            4\ a\ b - 3\ b\^2 - 2\ d)\)\ k + 
                      2\ k\^2)\)\ M + \((2\ a\^3\ b + 2\ k\ \((d + k)\) - 
                      a\^2\ \((2\ d + k)\) + b\^2\ \((2\ d + 3\ k)\) - 
                      2\ a\ \((b\^3 + 
                            2\ b\ k)\))\)\ M\^2 - I\ \((2\ a\^2\ b - 2\ b\^3 - a\ k - 3\ b\ k)\)\ M\^3 - \((a\ b - 
                      d)\)\ M\^4 + I\ b\ M\^5)\)\ \((\(R\_-\)\ \(r\_+\) + \(r\_-\)\ \(R\_+\))\) + 
          16\ d\ k\ M\^2\ \((\(-I\)\ a + I\ b + 
                M)\)\ \((\(r\_-\)\ \(r\_+\) + \(R\_-\)\ \(R\_+\))\) + 
          4\ M\ \((\((a - b)\)\^2\ \((\((a + b)\)\^2 - 4\ k)\)\ \((a\ b + d - 
                      k)\) - I\ \((a - 
                      b)\)\ \((\((a - b)\)\ b\ \((a + b)\)\^2 - \((a\^2 + 
                            4\ a\ b - 3\ b\^2 + 2\ d)\)\ k + 
                      2\ k\^2)\)\ M - \((2\ \((a - b)\)\ \((a + 
                            b)\)\ \((a\ b + d)\) - \((a\^2 + 4\ a\ b - 
                            3\ b\^2 + 2\ d)\)\ k + 
                      2\ k\^2)\)\ M\^2 + I\ \((2\ a\^2\ b - 
                      2\ b\^3 - a\ k - 3\ b\ k)\)\ M\^3 + \((a\ b + 
                      d)\)\ M\^4 - I\ b\ M\^5)\)\ \((\(r\_-\)\ \(R\_-\) + \(r\_+\)\ \(R\_+\))\) + 
          4\ d\ M\ \((\(-I\)\ a\^3 - 2\ I\ b\ k + 
                b\^2\ M + 
                M\^3 + I\ a\ \((b\^2 + 2\ k + M\^2)\) - \((b\^2 + 
                      M\^2)\)\ z + 
                a\^2\ \((\(-M\) + 
                      z)\))\)\ \(κ\_-\)\ \((\(r\_-\) - \(R\_-\) + \(r\_+\) - \(R\_+\))\)\ \(κ\_+\) - 
          4\ I\ M\ \((\((a - b)\)\ \((\((a + b)\)\^2 - 
                      4\ k)\)\ \((a\ b + d - 
                      k)\) - I\ \((\((a - b)\)\ \((a + 
                            b)\)\ \((b\^2 + d)\) - 
                      2\ \((2\ a\ b - b\^2 + d)\)\ k + 2\ k\^2)\)\ M - \((a + 
                      b)\)\ \((2\ a\ b - b\^2 + d - 
                      k)\)\ M\^2 + I\ \((b\^2 + d)\)\ M\^3 + 
                b\ M\^4)\)\ \((\(r\_-\)\ \(R\_-\) - \(r\_+\)\ \(R\_+\))\)\ \(κ\_+\) - 
          4\ M\ \((\(R\_-\) - \(R\_+\))\)\ \((\((a\^5\ b + \((d - 
                            k)\)\ k\ M\^2 + 
                      b\^2\ k\ \((\(-d\) + k - 2\ M\^2)\) - 
                      a\^4\ \((k + I\ b\ \((M - z)\))\) - 
                      a\^3\ \((b\^3 + 
                            b\ \((d + 3\ k + 
                                  2\ M\^2)\) - I\ k\ \((M - 
                                  z)\))\) + 
                      a\ \((b\^3\ \((d - k + M\^2)\) + 
                            b\ \((\(-4\)\ k\^2 + \((d + 3\ k)\)\ M\^2 + 
                                  M\^4)\) - 
                            3\ I\ b\^2\ k\ \((M - 
                                  z)\) - I\ k\ \((2\ \((d + 
                                        k)\) + M\^2)\)\ \((M - z)\))\) + 
                      a\^2\ \((5\ b\^2\ k + 
                            k\ \((d + 3\ k + 
                                  M\^2)\) + I\ b\^3\ \((M - 
                                  z)\) + I\ b\ \((d + 
                                  2\ \((k + M\^2)\))\)\ \((M - 
                                  z)\))\) - I\ b\^3\ \((d + 
                            M\^2)\)\ \((M - 
                            z)\) + I\ b\ \((2\ k\ \((d + 
                                  k)\) - \((d + 2\ k)\)\ M\^2 - 
                            M\^4)\)\ \((M - 
                            z)\))\)\ \(κ\_-\) + \((\(-a\^5\)\ b + 
                      k\ \((d + k)\)\ M\^2 - b\^2\ k\ \((d + k - 2\ M\^2)\) + 
                      a\^4\ \((k + I\ b\ \((M - z)\))\) + 
                      a\^3\ \((b\^3 + 
                            b\ \((\(-d\) + 3\ k + 
                                  2\ M\^2)\) - I\ k\ \((M - 
                                  z)\))\) + 
                      a\ \((b\^3\ \((d + k - M\^2)\) + 
                            b\ \((d\ M\^2 + \((k - M\^2)\)\ \((4\ k + 
                                        M\^2)\))\) + 
                            3\ I\ b\^2\ k\ \((M - 
                                  z)\) + I\ k\ \((\(-2\)\ d + 
                                  2\ k + M\^2)\)\ \((M - z)\))\) + 
                      a\^2\ \((\(-5\)\ b\^2\ k + 
                            k\ \((d - 3\ k - 
                                  M\^2)\) - I\ b\^3\ \((M - 
                                  z)\) + I\ b\ \((d - 
                                  2\ \((k + M\^2)\))\)\ \((M - 
                                  z)\))\) - I\ b\^3\ \((d - 
                            M\^2)\)\ \((M - 
                            z)\) + I\ b\ \((2\ \((d - 
                                  k)\)\ k - \((d - 2\ k)\)\ M\^2 + 
                            M\^4)\)\ \((M - z)\))\)\ \(κ\_+\))\) + 
          4\ M\ \((\(r\_-\) - \(r\_+\))\)\ \((\((a\^5\ b + \((d - 
                            k)\)\ k\ M\^2 + 
                      b\^2\ k\ \((\(-d\) + k - 2\ M\^2)\) - 
                      a\^4\ \((k + I\ b\ \((M - z)\))\) - 
                      a\^3\ \((b\^3 + 
                            b\ \((d + 3\ k + 
                                  2\ M\^2)\) - I\ k\ \((M - 
                                  z)\))\) + 
                      a\ \((b\^3\ \((d - k + M\^2)\) + 
                            b\ \((\(-4\)\ k\^2 + \((d + 3\ k)\)\ M\^2 + 
                                  M\^4)\) - 
                            3\ I\ b\^2\ k\ \((M - 
                                  z)\) - I\ k\ \((2\ \((d + 
                                        k)\) + M\^2)\)\ \((M - z)\))\) + 
                      a\^2\ \((5\ b\^2\ k + 
                            k\ \((d + 3\ k + 
                                  M\^2)\) + I\ b\^3\ \((M - 
                                  z)\) + I\ b\ \((d + 
                                  2\ \((k + M\^2)\))\)\ \((M - 
                                  z)\))\) - I\ b\^3\ \((d + 
                            M\^2)\)\ \((M - 
                            z)\) + I\ b\ \((2\ k\ \((d + 
                                  k)\) - \((d + 2\ k)\)\ M\^2 - 
                            M\^4)\)\ \((M - 
                            z)\))\)\ \(κ\_-\) + \((a\^5\ b - 
                      k\ \((d + k)\)\ M\^2 + b\^2\ k\ \((d + k - 2\ M\^2)\) - 
                      a\^4\ \((k + I\ b\ \((M - z)\))\) - 
                      a\^3\ \((b\^3 + 
                            b\ \((\(-d\) + 3\ k + 
                                  2\ M\^2)\) - I\ k\ \((M - 
                                  z)\))\) + 
                      a\^2\ \((5\ b\^2\ k + 
                            k\ \((\(-d\) + 3\ k + 
                                  M\^2)\) + I\ b\^3\ \((M - 
                                  z)\) - I\ b\ \((d - 
                                  2\ \((k + M\^2)\))\)\ \((M - 
                                  z)\))\) + I\ b\^3\ \((d - 
                            M\^2)\)\ \((M - 
                            z)\) - I\ b\ \((2\ \((d - 
                                  k)\)\ k - \((d - 2\ k)\)\ M\^2 + 
                            M\^4)\)\ \((M - z)\) + 
                      a\ \((\(-b\^3\)\ \((d + k - M\^2)\) + 
                            b\ \((\(-4\)\ k\^2 - \((d - 3\ k)\)\ M\^2 + 
                                  M\^4)\) - 
                            3\ I\ b\^2\ k\ \((M - 
                                  z)\) + I\ k\ \((\(-2\)\ d + 
                                  2\ k + M\^2)\)\ \((\(-M\) + 
                                  z)\))\))\)\ \(κ\_+\))\);\)\)

Using L and E we also define the metric function: ω(ρ, z)=i\(\(\([\((A + 
                                B)\)\ \(L\^*\) - \((\(A\^*\) + \(B\^*\))\)\ E]\) - \([\((\(A\^*\) + \(B\^*\))\)\ L - \((A + 
                                B)\)\ \(E\^*\)]\)\)\/\(2\ \((A\ \(A\^*\) - 
                          B\ \(B\^*\))\)\)\).

\(\(Ko = 16\ d\ κ\_-\%2\ κ\_+\%2;\)\)

And from (Subscript[K, 0]) we define the metric function: (e^(2 γ(ρ, z)))=\(\(A\ \(A\^*\) - 
                    B\ \(B\^*\)\)\/\(K\_0\ \(K\_0\^*\)\ \ \(R\_+\)\ \(R\_-\)\ \(r\_+\)\ \(r\_-\)\)\).

Metric functions for the Manko et al.

The metric functions of Manko et al. (Manko V. S., Mielke E. W., Sanabria-Gòmez J. D., 2000, Phys. Rev. D, 61, 081501) with parameters M, a and, b expressed in Weyl-Papapetrou coordinates.

δ = ((-(M^2) (b^2))/((M^2) - (((a - b))^2)));
d = (1/4) ((M^2) - (((a - b))^2));
kk = (Sqrt[d + δ]);
{x -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) + (Sqrt[(((kk - z))^2) + (ρ^2)])), 
        y -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) - (Sqrt[(((kk - z))^2) + (ρ^2)]))};
EM = (((4 ((((kk^2) ((x^2) - 1) + δ (1 - (y^2))))^2) + (a - b) ((a - b) (d - δ) - (M^2) b) (((1 - (y^2)))^2)))^2) - 
            16 (kk^2) ((x^2) - 1) (1 - 
                  (y^2)) ((((a - b) ((kk^2) ((x^2) - (y^2)) + 2 δ (y^2)) + (M^2) b (y^2)))^2) /. {x -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) + (Sqrt[(((kk - z))^2) + (ρ^2)])), 
            y -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) - (Sqrt[(((kk - z))^2) + (ρ^2)]))};
DM = (((4 ((((kk^2) (x^2) - δ (y^2)))^2) + 2 kk M x (2 (kk^2) ((x^2) - 1) + (2 δ + a b - (b^2)) (1 - (y^2))) + (a - b) ((a - b) (d - δ) - (M^2) b) ((y^4) - 1) - 4 (d^2)))^2) + 
            4 (y^2) (((2 (kk^2) ((x^2) - 1) (kk x (a - b) - M b) - 2 M b δ (1 - (y^2)) + ((a - b) (d - δ) - (M^2) b) (2 kk x + M) (1 - (y^2))))^2) /. {x -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) + (Sqrt[(((kk - z))^2) + (ρ^2)])), y -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) - (Sqrt[(((kk - z))^2) + (ρ^2)]))};
FM = 
        8 (kk^2) ((x^2) - 
                  1) ((a - b) ((kk^2) ((x^2) - (y^2)) + 
                        2 δ (y^2)) + 
                  (y^2) (M^2) b) (kk M x ((((2 kk x + M))^2) - 
                        2 (y^2) (2  δ + a b - (b^2)) - (a^2) + 
                        (b^2)) - 
                  2 (y^2) (4 δ d - 
                        (M^2) (b^2))) + (4 ((((kk^2) ((x^2) - 1) + δ (1 - (y^2))))^2) + (a - 
                        b) ((a - b) (d - δ) - 
                        (M^2) b) (((1 - (y^2)))^2)) (4 (2 kk M b x + 2 (M^2) b) ((kk^2) ((x^2) - 1) + δ (1 - 
                              (y^2))) + (1 - 
                        (y^2)) ((a - b) ((M^2) (b^2) - 
                              4 δ d) - (4 kk M x + 
                              2 (M^2)) ((a - b) (d - δ) - 
                              (M^2) b))) /. {x -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) + (Sqrt[(((kk - z))^2) + (ρ^2)])), 
            y -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) - (Sqrt[(((kk - z))^2) + (ρ^2)]))};
fM = EM/DM;
ωM = -(1 - (y^2)) FM/
              EM /. {x -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) + (Sqrt[(((kk - z))^2) + (ρ^2)])), 
            y -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) - (Sqrt[(((kk - z))^2) + (ρ^2)]))};
Exp2γM = 
        EM/(16 (kk^8) ((((x^2) - (y^2)))^4)) /. {x -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) + (Sqrt[(((kk - z))^2) + (ρ^2)])), 
            y -> (1/(2 kk)) ((Sqrt[(((kk + z))^2) + (ρ^2)]) - (Sqrt[(((kk - z))^2) + (ρ^2)]))};
Mgtt3[ρ_, z_] = -fM;
Mgtφ3[ρ_, z_] = 
        fM ωM;
Mgφφ3[ρ_, 
          z_] = -fM ω(M^2) + (ρ^2) (fM^(-1));
Mgzz3[ρ_, z_] = (fM^(-1)) Exp2γM;

Hartle-Thorne metric

Hartle - Thorne  from Berti et al.,  2005, MNRAS, 358, 923
  (with a change in sign of (Subscript[g, t])φ term so as to correspond to a rotation parameter defined as j = (J/(M^2))).

Am1 = ((15 r (r - 2 Mm) (1 - 3 (u^2)))/(16 (Mm^2))) Log[(r/(r - 2 Mm))];
Am2 = ((15 ((r^2) - 2 (Mm^2)) (3 (u^2) - 1))/(16 (Mm^2))) Log[(r/(r - 2 Mm))];
L1 = 
        80 (Mm^6) + 8 (Mm^4) (r^2) + 10 (Mm^3) (r^3) + 20 (Mm^2) (r^4) - 
          45 Mm (r^5) + 15 (r^6);
p1 = (((8 Mm (r^4) (r - 2 Mm)))^(-1));
W1 = (r - Mm) (16 (Mm^5) + 8 (Mm^4) r - 10 (Mm^2) (r^3) - 
                30 Mm (r^4) + 15 (r^5)) + 
          (u^2) (48 (Mm^6) - 8 (Mm^5) r - 24 (Mm^4) (r^2) - 
                30 (Mm^3) (r^3) - 60 (Mm^2) (r^4) + 135 Mm (r^5) - 
                45 (r^6));
Fm1[Mm_, r_, u_] = -p1 W1 + Am1;
Fm2[Mm_, r_, u_] = 
        5 (r^3) p1 (3 (u^2) - 1) (r - Mm) (2 (Mm^2) + 
                6 Mm r - 3 (r^2)) - Am1;
Gm1[Mm_, r_, u_] = 
        p1 ((L1 - 72 (Mm^5) r) - 
                3 (u^2) (L1 - 56 (Mm^5) r)) - 
          Am1;
Hm1[Mm_, r_, u_] = 
        Am2 + (((8 Mm (r^4)))^(-1)) (1 - 3 (u^2)) (16 (Mm^5) + 
                8 (Mm^4) r - 10 (Mm^2) (r^3) + 15 Mm (r^4) + 
                15 (r^5));
Hm2[Mm_, r_, 
          u_] = -Am2 + (((8 Mm r))^(-1)) 5 (1 - 
                3 (u^2)) (2 (Mm^2) - 3 Mm r - 
                3 (r^2)); 
gttHT[Mm_, jj_, q_, r_, 
          u_] = -(1 - 2 (Mm/r)) (1 + (jj^2) Fm1[Mm, r, u] - 
              q Fm2[Mm, r, u]);
grrHT[Mm_, jj_, q_, r_, 
          u_] = (((1 - 2 (Mm/r)))^(-1)) (1 + (jj^2) Gm1[Mm, r, u] + 
              q Fm2[Mm, r, u]);
gθθHT[Mm_, jj_, q_, r_, u_] = 
        (r^2) (1 + (jj^2) Hm1[Mm, r, u] - 
              q Hm2[Mm, r, u]);
gφφHT[Mm_, jj_, q_, r_, u_] = 
        (r^2) (1 + (jj^2) Hm1[Mm, r, u] - q Hm2[Mm, r, u]) (1 - 
              (u^2));
gtφHT[Mm_, jj_, r_, 
          u_] = -2 ((Mm^2)/r) jj (1 - (u^2));

Using the expressions presented in, Fintan D . Ryan, Phys . Rev . D 52, 
        5707  –5718 (1995), to calculate the moments of the Hartle - 
        Thorne spacetime.

gtt = gttHT[M, j, q, r, u];
gφφ = 
        gφφHT[M, j, q, r, 
          u];
gtφ = gtφHT[M, j, r, u];

Series[gtt, {r, ∞, 4}]
Simplify[
      Series[gφ(φ/((r^2) (1 - (u^2)))), {r, ∞, 4}]]
Simplify[Series[gtφ, {r, ∞, 4}]]

Ω = ((-D[gtφ, r] + (Sqrt[D[gtφ, r]^2 - D[gtt, r] D[gφφ, r]]))/D)[gφφ, r];

FullSimplify[
      Series[(((M Ω))^(1/3)) //. {r -> M/(x^2), 
            u -> 0}, {x, 0, 13}]]

Xpar = Simplify[InverseSeries[%] /. {x -> U}]

Em = ((-gtt - gtφ Ω)/(Sqrt[-gtt - 2 gtφ Ω - gφφ (Ω^2)]));

Eseries = 
      Simplify[Series[
          Normal[Series[
                Em //. {r -> M/(x^2), u -> 0}, {x, 0, 
                  12}]] //. {x -> Xpar}, {U, 0, 12}]]

Series[-(U/3) D[Normal[Eseries], U], {U, 0, 12}]

\(Comparing\ this\ expression\ with\ the\ on\ in\ eq . \ \((17)\)\ in\), Fintan D . Ryan, Phys . Rev . D 52, 
        5707  –5718 (1995), \(one\ sees\ that\ j = \(J\/M\^2\ and\ q = \(\(Q\/M\^3\)\(.\)\)\)\)