Choosing the coordinate system for a fast neutron star is awkward

Let a faraway observer see the speed of light inside the neutron star as v << c, where c is the speed of light in the vacuum.

Let the neutron star move faster than v relative to our observer.

                                         o   observer B

                                         |

                                        /\

                 o                     ● --------> 

                 |           neutron star

                /\

         observer

Let us try to define a coordinate system which is "static" relative to the observer. Let us have another observer B inside the neutron star. Let he be at a position x at a coordinate time t, and at x + dx at t + dt.

Let the metric say that he moved a distance ds and the the proper time interval was dτ.

We have x + dx, t + dt within the light cone from x, t.

The "frame dragging" inside the neutron star is so strong that B has to move. Thus, x, t is not within the light cone from x, t.

We cannot define a sensible metric with these coordinates.

The obvious solution is to let the neutron star to drag along the coordinate system along with it, like the rubber string coordinates of our previous blog post.

But if we try to keep most of the coordinate system static, we end up with a very awkward coordinate system.

The coordinate system for an accelerating neutron star

Suppose that our neutron star is initially static and we have a nice static coordinate system. Then we accelerate the neutron star to a speed larger than v.

Can we still use the the static coordinate system? Apparently, no. We have to switch to a different coordinate system. We see that an accelerating object can "break" a coordinate system.