I am trying to derive some equations which will let me simulate the motion of a spaceship in different geometries of spacetime. Suppose I know the metric and Christoffel symbols at the location of the spaceship, and I know the force applied on it (for example the spaceship is firing its thrusters). How do I calculate its coordinate acceleration from this information?
What I have tried so far:
As far as I understand the most natural thing to calculate from the force provided by the thrusters, is the proper acceleration of the spaceship. Then the wikipedia page for proper acceleration gives the following equation relating proper acceleration to coordinate acceleration:
$$ A^{\lambda} := \frac{DU^{\lambda}}{d\tau} = \frac{dU^{\lambda}}{d\tau} + \Gamma^{\lambda}\,_{\mu\nu}\,U^{\mu}U^{\nu} $$
One can rearrange the equation (following the Wikipedia page) to
$$ \frac{dU^{\lambda}}{d\tau} = A^{\lambda} - \Gamma^{\lambda}\,_{\mu\nu}\,U^{\mu}U^{\nu} $$
The rightmost term can be evaluated by just expanding the sum and putting in the Christoffel symbols for the spacetime around the black hole (which I can find in a reference). So far everything seems to make sense.
However, finding $A^{\lambda}$ from the force provided by the thrusters is something I can't figure out. Wikipedia says this about the proper acceleration:
... is the object's proper-acceleration 3-vector combined with a null time component as seen from the vantage point of a reference or book-keeper coordinate system in which the object is at rest.
From this I understand that
$$ (A^{x}, A^{y}, A^{z}) = \mathrm{\frac{(vector\; force\; provided\; by\; thrusters)}{(mass\; of\; spaceship)}} $$
in the either the reference of the spaceship, or a stationary reference frame which is instantaneously comoving with the spaceship at a given moment in time (I can't tell which one).
The part about the null time component completely confuses me.
From what I understand of the wikipedia article, the proper acceleration is also equal to the covariant derivative of the coordinate velocity. However, I don't know much differential geometry and the covariant derivative is completely meaningless to me.
Questions:
- How to go from force provided by thrusters (as a classical 3-vector) to proper acceleration as a 4-vector?
- Is it correct to say that the three spacelike components of the 4-vector proper acceleration are equal to the resultant acceleration in the frame of the spaceship? If not, how are they defined?
- What is the timelike component $A^{t}$ equal to and how can one work it out?