The Schwarzschild metric represents a four dimensional curved space metric. This metric can be mathematically compared to the metric for surface of a sphere in the three dimensional flat space, where we can write an incremental displacement vector at a point but cannot write a position vector linking two points on the surface. Even though, in the four-dimensional curved space, it is difficult to write a position vector, we can write an incremental displacement vector based on the four dimensional metric. This incremental vector based classical approach will give exactly same results as the tensor analysis in the classical general relativity. But, we face some geometrical difficulties. The main difficulties are:
- It is difficult to write unit vector expressions and draw a geometrical picture of the global or even a local coordinate system.
- The general incremental vector dA is not always path independent but incremental displacement vector ds is always path independent, in the same curved space. The two incremental vectors behave differently in the same space.
In general relativity, the fourth dimension is defined as time and each point in the four-dimensional space-time has to be associated with an event. Therefore, the conventional definition of vector itself requires a change in a curved space-time (C. W Misner, K. S. Thorne, J. A. Wheeler, Gravitation. W. H. Freeman and Company. San Francisco (1973)). The temptation to regard vectors as arrows linking two events itself has to be discarded. The bilocal (point for head and point for tail) version of a vector has to be replaced by a purely local version. Also, the vectors cannot be physically moved around and each vector has to be attached to a specific local event. There is no position vector on a manifold. Points can only be labelled by their coordinates and are not vectors. But, even this definition also raises some concerns. The following preprint on Research Gate lists the concerns with both the bilocal and local versions of the vectors:
Preprint Vector analysis in the curved space-time: The problems, the ...
The analysis dealing with relations between various scalar components of a tensor can bypass these geometrical difficulties. Even though, a vector analysis can give all the desired mathematical results, it has to be avoided as it lacks geometrical support.