The Einstein approximation: a strange acceleration term

The Einstein approximation formula is not simply about remapping the radial coordinate of the Schwarzschild metric (= bulging coordinates). In bulging coordinates, the metric has off-diagonal elements everywhere but on the x and y coordinate axes. This is demonstrated by the fact that the coordinate lines do not intersect orthogonally anywhere, but only on the coordinate axes.

However, the Einstein approximate metric does not have off-diagonal elements anywhere for a static mass M.

Let us calculate what the Einstein approximate metric says about the acceleration if a test mass m moves tangentially relative to M.

Tangential movement in the Einstein approximate metric

In this post we have set c = 1.

In the Einstein approximation around a spherical mass M, the perturbation to the Minkowski metric η is:

     h =

        4 G *

              1/2 M / r             0                         0

               0                   1/2 M / r                  0

               0                        0            1/2 M / r

We will study a familiar configuration:

                             ●  M

                            /

                          /

                        /  r

                                      R = distance (m, M)

                              • --> v

                            m

   ^ y

   |

    -----> x

https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

We use the geodesic equation to determine the y acceleration of the test mass m. Let us multiply the geodesic equation (the first equation above) by dτ² / dt². Then we can assume that ds in the equation is dt.

Alternatively, we could start from the geodesic equation in the coordinate time (above). Since dy / dt is zero, the second term on the right is zero: we can use dt in the place of ds in the shorter form geodesic equation.

In the geodesic equation, μ = y, and αβ can be any of tt, tx, xt, xx. The combinations tx and xt do not contribute because the metric is diagonal.

For the location of the test mass m:

       dg_tt / dy    = 2 G M * 1 / R²,

       Γ^y_tt          = 1 / g_yy  * -G M / R².

Also,

       dg_xx / dy   =  2 G M * 1 / R²,

       Γ^y_xx          = 1 / g_yy * -G M / R².

We have

       Γ^y_tt * dt / dt * dt / dt      =  Γ^y_tt,

       Γ^y_xx * dx / dt * dx / dt   =  Γ^y_xx * v².

We can assume that g_yy = 1 by setting M small. The y acceleration is

       d²y / dt² = G M / R²

                       + v² G M / R².

The second acceleration term above is a strange extra term which does not appear if we use the Schwarzschild metric and coordinates.

We suspect that the extra term can lead to errors when we calculate gravitomagnetic effects.

Conclusions

The Einstein approximate metric is not obtained from the Schwarzschild metric by simply making the coordinates "bulging". Bulging would introduce off-diagonal elements, but the Einstein approximate metric is diagonal.

Anyway, the Einstein approximate metric introduces a strange radial acceleration term in a tangential movement.

We will next calculate what the Einstein approximation says about a moving cylinder. We know what the correct metric roughly is for a static cylinder. By Lorentz transforming we obtain the metric for a moving cylinder, and can compare to the Einstein approximate metric.