Why is there a need for a strong field versus weak field model? is the weak field just a simplified version, or is there a fundamental difference? If you explain by math, I would appreciate the full description of each variable.  But really, I am just looking for a conceptual understanding.  Newton's laws work for an apple falling from a tree branch or the Moon orbiting around the Earth, and so we called them universal laws.  Are the strong field relations "universal laws" that work everywhere, while weak field approximations are simpler but don't extrapolate well into heavy gravity fields?  Or is there something different about the heavy gravity fields that change the laws of nature, so that one can't use strong field equations in a weak field and vice versa?
Thank you in advance.
 A: I'll take "strong (weak) field laws" to mean General Relativity (Newton's Law of Gravity). Newton's Law of gravity defines a universal force between all masses, given by the two mass expression:
$$ F =G\frac{Mm}{r^2}$$
It can be used to define a conservative field (called: gravity) defined by a potential:
$$ \vec g = -\nabla \Phi$$
which can be related to the density of matter:
$$ \nabla^2\Phi = \rho$$
This makes solving basic problems of ballistics, orbits, and gross interplanetary trajectories relatively easy.
Meanwhile, General Relativity is a geometric field theory of spacetime, in which the metric of spacetime, $g_{\mu\nu}$, is the field. The curvature of the metric is related to the stress-energy tensor, $T_{\mu\nu}$.
$$ G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$$
This is actually 10 coupled differential equations. Mass appears in the equation via $T_{tt}$, which is the energy density. The other components are energy flux, momentum change, stress, and pressure.
The 1st solution to these equations (a static Black hole) was found in a few years. The 2nd closed for solution (a rotating black hole) took another 40 years. They are very difficult to solve, and it has only become tractable with the advent of numerical relativity.
They only solve for the gravitation. Solving trajectory requires solving the geodesic equation:
$$\frac{d^2x^{\mu}}{ds^2}+\Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{ds}
\frac{dx^{\beta}}{ds}$$
which is also, not trivial.
The good news is that in simple cases, like on the Earth, the geodesic equation reduces to Newtons Law of gravity.
So the weak field limit of GR is Newton's law of gravity, which are much easier to solve.
Newton's Law fails experimental tests when the field is strong, which is parameterized by the Schwarzschild radius:
$$r_s=\frac{2GM}{c^2} $$
The Schwarzschild radius of the Earth is nine millimeters, which is 700 million times smaller than the radius. For the Sun, it's 3 km, or 233 thousand times smaller than it's actual radius. It's easier to observe relativistic effects near the Sun.
Newton's theory is non-relativistic, so it necessarily fails when trajectories are not small compared with the speed-of-light based on special relativity alone, but there are also velocity dependent gravitation effects, too.
A: Ignoring quantum effects, our current best theory of gravity is General Relativity (GR). GR is sufficient to explain all current observations involving gravity, including (but not limited to) observations involving weak gravitational fields.
However, doing calculations in GR can be quite cumbersome. Particularly in weak field situations, it is standard practice to use Newtonian gravity in place of GR, which is much simpler to work with, and is a very good approximation. Newtonian gravity can be shown to be equivalent to working to leading order in the gravitational field in GR, when the gravitational fields are weak.
A: The weak field/slow speed solutions to Einstein's equations are useful for establishing the Newtonian limit, but pretty much eliminate the possibility of learning anything fun about GR!
You do not have to do both approximations together; there are a few articles that deal with the slow-speed/strong field scanario, which allows you to play with the Ricci tensor and scalar and the Energy-Momentum tensor in a more realistic "GR" environment.
The best example of this that I am aware of is Baez's The Meaning of Einstein's Equation. In particular, the section on The Mathematical Details.
Basically, it is the "high speed" part of GR the "explodes" the mathematics beyond simple analysis; the "high gravity" part is very accessible by comparison.
