Uncapable of working through the tensors of the EFE's or visualizing the concepts upon which they are based as expressed in that manner, I have begun reverse engineering GR in an attempt to determine the basic concepts upon which it is based to find the foundation of its mathematical logic and reason. I want to see them and relate them one by one. At this point, however, I am stuck, still requiring some bits of information in order to work it out fully, so I will post what I have so far so that perhaps the members here can help to fill in the blanks. The values for time dilation and length contraction are what I am trying to find overall, depending upon some initial condition for the coordinate system used, but by bypassing the EFE's or if they are basically the same thing, just by taking each part of what can be deduced logically and putting those together algebraicly. We can start with some basic assumptions, 1. SR is valid locally 2. The speed of light is measured at c locally 3. Shells of identical attributes are spherical according to an observer at infinity 4. Mass is invariant Add to this list as many as possible. I might be taking some minor assumptions for granted that may be important. Then we have a list of assumptions about the field and the motion of a particle through the field, such as A. Gravitational flux strength - The field of flux of gravity flows outward from the center. As the flux lines cross a shell, its strength is inversely proportional to the area of the surface of the shell, the number of flux lines being constant, but the strength diminishing with the number per area. A distant observer measures the area to be 4 pi r^2, and with a local tangent contraction of L_t, the local observer will measure the surface area to be (4 pi r^2) / L_t^2. In the radial direction, each flux line can be visualized as a beam of particles crossing the surface, and the strength is proportional to the frequency at which they cross, which is inversely proportional to the local time dilation z at that surface. The strength of gravity, the locally measured acceleration, then, is a' = F [L_t^2 / (4 pi r^2)] [1 / z], where F is some constant. By taking the low gravity approximation where L_t = z = 1 and a = - G M / r^2, we find F = - 4 pi G M, so that gives us a' = - G M L_t^2 / (r^2 z). Hopefully this accurately describes the field of flux and I have explained it well enough. B. Energy conservation - Not so much conservation though, really, more a relation, but for a photon falling radially through the field, it works out to f_r z_r = f_s z_s That is, the frequency at any point in the field is the same according to a distant observer, with each successive pulse travelling through the field in the same time between any two points, so the rate of reception at r must be the same as the rate of emission at s according to the distant observer. The locally measured frequency, then, is just inversely proportional to the local time dilation z. Relating this to the energy of a photon, we have (h f_r) z_r = (h f_s) z_s E_r z_r = E_s z_s And if we relate this to a massive particle as it passes each location in the same way as we did for a photon, we have [m c^2 / sqrt(1 - (v'_r/c)^2)] z_r = [m c^2 / sqrt(1 - (v'_s/c)^2)] z_s z_r / sqrt(1 - (v'_r/c)^2) = z_s / sqrt(1 - (v'_s/c)^2) z / sqrt(1 - (v'/c)^2) = K where K is a constant depending upon the conditions of radial freefall. For a massive particle falling from infinity, for instance, with initial conditions z = 1 and v' = 0, then K = 1. C. Conservation of momentum - Here we just assume the locally measured angular momentum is constant for all shells, so that m v'_t r' is constant. r' would be the inferred distance to the origin, what the distant observer says would be measured if a length contracted ruler at r were extended with the same radial contraction L all the way to the center. For less ambiguity, however, we can restate that to say m v'_t (r / L) is a constant. Since m is invariant, we can reduce that to read P = v'_t r / L = constant. Please add to this list. Okay, so now to define invariants. Here I am defining invariants as quantities that remain the same regardless of the coordinate system. If we transform one GR coordinate system to another, we are changing the positions of the shells. However, the locally measured acceleration at that shell will remain the same, so a' is an invariant in this sense. The time dilation z at that shell will also remain the same, so z is an invariant. L and L_t change as the positions of shells and the ends of a local ruler change according to a distant observer, so those are not invariants. We have already assumed m to be an invariant, although it may be possible to keep it invaraint as an initial condition anyway while letting other factors vary. v' is the locally measured radial speed of a particle which will be invariant upon freefalling from some other shell no matter how we re-position them, while v that the distant observer measures is v = z L v', dependent upon L, so of course is not invariant. Let's get started. Classically, the locally measured acceleration would be a' = d(v'^2) / (2 dr'). With local SR, with change in speed decreasing as one approaches c, however, it can be demonstrated that a' = d(v^2) / [2 dr' (1 - (v'/c)^2)]. While the former equation would still be the instantaneous coordinate acceleration that a local observer would measure, if we define a' instead only as the acceleration of a particle falling from rest at r, the latter equation finds a' as defined as such regardless of the initial measured speed of the particle. So from assumption B, for a particle falling from infinity with K = 1, that becomes a' = d(v'^2) / [2 dr' (1 - (v'/c)^2)] a' = c^2 d(1 - z^2) L / (2 dr z^2) a' = c^2 [-2 dz z] L / (2 dr z^2) a' = - c^2 dz L / (z dr) Or likewise, we can find it with a' = d(v') / [dt' (1 - (v'/c)^2)] a' = c d(sqrt(1 - z^2)) / [z dt z^2] a' = c [- dz z / sqrt(1 - z^2)] / [z^3 (dr / v)] a' = - c dz (z L v') / [sqrt(1 - z^2) z^2 dr] a' = - c^2 dz L / (z dr) We can now relate assumptions A and B as a' = - c^2 dz L / (z dr) = - G M L_t^2 / (r^2 z) c^2 (dz / dr) L = G M L_t^2 / r^2 This is an important step, I think. It relates the time dilation and radial and tangent length contractions independent of the coordinate system involved. z is a function of r here, though, so let's arbitrarily set z = sqrt(1 - 2 G M / (r c^2)) as a coordinate choice, although many are possible. Then we get c^2 [2 G M / (2 r^2 c^2 sqrt(1 - 2 G M / r))] L = G M L_t^2 / r^2 L / sqrt(1 - 2 G M / r) = L_t^2 L = L_t^2 sqrt(1 - 2 G M / r) But we can only have one coordinate choice and the rest must be derived, so I'm not sure how to go further to derive one of the other two and then the last. Other coordinate choices will give other relations for other coordinate systems. This one is for Schwarzschild, of course, but since we are only allowed one coordinate choice, I am at a standstill. For example, we could perhaps set z = L as a coordinate choice and find the relation to L_t, but then we don't know what z and L are, only that they are equal. And other coordinate choices also give z = L with different L_t, such as z = L = 1 / sqrt(1 + 2 G M / r) for instance, which is another valid GR coordinate system. We might set L_t = 1 but are still left with just a relation between z and L, and we can't set L_t = 1 and z = L as two coordinate choices as far as I can see. I need some other piece of information. Any ideas?