[Richard777 (OP)]

Acceleration may be represented as a 4D vector.

Can the Schwarzschild metric may be simply obtained from the vector components?

[jeffreyH]

The Schwarzschild metric is one of the simpler solutions to the Einstein field equations. It is very useful for developing equations in a well bounded framework. Acceleration is not a straightforward subject in general relativity. A freely falling body feels no force over a well defined and bounded time interval. As long as tidal forces are negligible. The distribution of force throughout the falling body has to be taken into account. Ultimately this leads to a particulate view of gravitational acceleration. A particle can also be represented by a molecule, where charge is neutral. So if you wish to investigate acceleration within the Schwarzschild model it needs to be done at a microscopic level.