I have a problem with understanding tensors... the geometric.
yeah, tell me about it! and, maybe unfortunately, you'll find there is a lot more mathematics to understand...Considering there are only limited exact solutions, I think, to the ten Einstein equations, making physical interpretations is not obvious...if it were, when Einstein developed the equations, he would have found solutions himself and science would not have argued about their meaning for a decade or more. Keep in mind Einstein somehow intuitively understood the physical nature of gravity, and with help from friends found the mathematics to fit...he did NOT derive his fundamental understanding from mathematics to the physcial...nor did anytone else at that time.
If you are studying via formal schooling, taking the time and effort to learn the math
is likely desirable and necessary. If its for hobby and self learning, be prepared for a
long effort. Check some threads here and see the difficulty 'experts' have in reaching specific interpretational agreements and communicating them regarding different aspects of the mathematics.
I only recently discovered in these foums that there is not even an agreed upon definition for a gravitational field...sure it's a 'curvature', but exactly how do you measure it?? There is no single metric [measurement]that takes precedent!
Wikipedia summarizes solutions to the Einstein Field Equations like this:
To get physical results, we can either turn to numerical methods; try to find exact solutions by imposing symmetries; or try middle-ground approaches such as perturbation methods or linear approximations of the Einstein tensor.
See here and decide what you think:
http://en.wikipedia.org/wiki/Exact_solutions_of_Einstein's_field_equations
Also here's an interesting perspective from Einstein online:
"In part, gravity is an observer artefact: it can be made to vanish by going into free fall. Most of the gravity that we experience here on Earth when we see objects falling to the ground is of this type, which we might call "relative gravity". The remainder of gravity, "intrinsic gravity", if you will, manifests itself in tidal forces, and is associated with a specific property of geometry: The curvature of spacetime."
http://www.einstein-online.info/spotlights/background_independence/?set_language=en
