[Richard777 (OP)]

A “binary system” of acceleration may be represented as two vectors of acceleration (each 4D) acting upon a common point and diverging by an angle of action (“system angle”). The binary system may also be represented as a rank two field tensor or “system tensor”, which is a product of the vectors.

The system changes, so that each vector changes direction. The magnitude of each vector remains unchanged. A space-time “change function” is represented by a 4D “change vector” (not Nabla).

The system change is represented as the dot product of the change vector and the system tensor. The system change is not divergence. The result of the change is a new vector of acceleration, having “special characteristics” which contain information relating to the original system (“history”).

The special characteristics are defined by “vector geometry”. The resulting vector (4D) has a geometry which includes the intersection of three plane surfaces. Edges and angles relating to the surfaces retain the scalar magnitudes and the angle of action of the original system.

The Schwarzschild metric may be simply obtained from the resulting vector.
The math is attached.

[evan_au]

Can a field vector contain the “history” of a previous system?

I think you are asking "If I take a snapshot of some system at a particular point in time, can I extrapolate backwards to discover the past history of the system (and similarly, can I extrapolate forwards to predict the future of the system)?"

Apparently, Newton believed in a deterministic universe, so he would have said "Yes".

But we now know that, at the quantum scale, it is impossible to measure everything with unlimited accuracy
- In fact, the very act of measuring quantum variables changes their values (and their future)

At larger scales, it is still very difficult to get accurate measurements
- Indeed, some factors which influenced the system in the past may be beyond our reach and take measurements

And in many real systems, even tiny measurement errors grow exponentially over (extrapolated) time, so that the errors soon overwhelm any useful information about the past (or future).

So, in theory, you can extract the past from the present, but in practice, it has limited accuracy.

[evan_au]

Can a field vector contain the “history” of a previous system?

Whether there is a unique history depends on how many paths there are to reach the observed state.

Take a swinging pendulum (subject to some air resistance): The observed state (time, height, velocity) may have been reached via multiple paths:
- A recent push with slightly higher amplitude
- A push with much larger amplitude, a long time ago
- And multiple states in-between.

[TommyJ]

In physics divergence can be used to produce a scalar field from vector field. The divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point
As an example of air as it is heated. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region.