Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; Problem P17.6.
We are now in a position to revisit the system with sinusoidal coordinates. To review, we had a 2-d system with coordinates and defined in terms of the usual rectangular coordinates and by
The metric for this system is
We looked at an object with a velocity given by where is a constant. Clearly the acceleration of the object is zero, but if we calculate the velocity components in the system, we get
so although , since varies with time.
To get the ‘true’ acceleration, we need to find the actual differential and divide this by . We’ve seen how to do this when we defined the Christoffel symbols:
To calculate , we need the for the system, which we can calculate in the usual way using the geodesic equation. We start with
Since is independent of only derivatives with respect to are non-zero. Consider first ; then we get
Now for :
We would like to compare these equations with the equation involving the Christoffel symbols:
However, because the metric here is not diagonal, we get second derivatives of more than one coordinate in each equation. We can eliminate by multiplying 10 by and subtracting it from 9. This gives the convenient result:
From this we conclude that
from which we conclude
We can now evaluate 6 to find:
Thus by taking the proper derivative using the Christoffel symbols, the differentials of both components of velocity are zero, so the acceleration is zero in the system as well.