On Thursday, August 20, 2020 at 4:04:39 AM UTC-5, rockbr...@gmail.com wrote:   
   > On Thursday, August 13, 2020 at 7:51:55 PM UTC-5, Savin Beniwal wrote:   
   > > First of all, as per my knowledge This dimensional analysis   
   > > does not depend on the system of coordinates neither on the metric tensor   
   as   
   > > g_mn has no dimension.   
      
   > Your choice corresponds to [g_mn] = 1, "no dimension" meaning   
   > "dimensionless" and [x^m] = 1 = [x^n]. That's a special case of [g_mn   
   > dx^m dx^n] = A, [x^m] = [m] and [x^n] = [n] corresponding to the   
   > selection A = 1, [m] = 1 = [n].   
      
   Though the rest of the analysis is fine, you misread what he said here   
   and there was no contradiction. The only thing you can infer from the   
   remark above is what you later inferred, namely that his statement   
   amounts to the convention:   
      
   A = [0]^2, [0] = [1] = ... = [N - 1].   
      
   > But your assumption is wrong: coordinates *do* have dimensions: space-like   
   > coordinates are generally taken with dimension L, time-like with T.   
      
   In fact, you both missed pointing out key examples that show how/why the   
   more general analysis applies. For spherical space-time coordinates, x^0   
   = t, x^1 = r, x^2 = theta, x^3 = phi, you have [0] = T, [1] = L, [2] = 1   
   = [3]. Or ... you could also say that the dimensions are [2] = R = [3]   
   for radians; in which case you would have to go back to the tradition of   
   stating the argument of trig functions with units; the correspondence   
   being that the classical versions (sin_C, cos_C) would be equated to the   
   post-Calculus versions cos x + i sin x = exp(ix) by the conversion   
   sin_C(x) = sin(x/radian), cos_x(x) = cos(x/radian) with 1 degree =   
   pi/180 radian.   
      
   In either case, the units of [g_mn] are not dimensionless! [g_22] =   
   [g_33] = A or A/R^2, which under the convention A = L^2 would equate to   
   L^2 or (L/R)^2.   
      
   You also misidentified the mistake he made, though you're correct in   
   pointing out that his characterization of T_mn is wrong. But that error   
   (under his conventions A = L^2, [0] = ... = [N-1] = L) is harmless.   
      
   The original poster's claim that T_mn has dimensions M/LT^2 is actually   
   what's wrong - when made under the assumption [0] = L. Its dimensions   
   depend on [0]; and M/LT^2 is only valid with [0] = T. Otherwise, it is   
   M/L^2T:   
      
   [#T_mn] = ML^2/T 1/[01...(N-1)] A/[mn]   
      
   becomes under the assumption A = L^2 and [m] = L = [n]:   
      
   [#T_mn] = ML^{2-N}/T.   
      
   so with [g_mn] = A/L^2 and [g] = A^N/[01...(N-1)]^2 and T_mn =   
   #T_mn/|g|^{1/2}, which yields dimensions (under the same assumptions A =   
   L^2, [m] = L = [n]):   
      
   [T_mn] = ML^2/T A^{1-N/2}/[mn] = ML^{2-N}/T   
      
   which reduces to M/L^2T for N = 4.   
      
   Only when [0] = T, instead of [0] = L; with [1] = [2] = ... = [N-1] = L,   
   A = L^2, do you get [T_mn] = M/LT^2, the unit for energy density and   
   pressure.   
      
   That's the mistake he actually made.   
      
   It would also help to see his last claim addressed *directly* - *in*   
   *terms* of the conventions he used!   
      
   The action S = integral 1/(2 kappa) R |g|^{1/2} d^N x   
      
   with the choice A = L^2, [0] = [1] = ... = [N-1] = L has units involving   
   these:   
      
   [S] = ML^2/T,   
   [g^{1/2} d^N x] = A^{N/2} = L^N,   
   [R] = 1/A = 1/L^2,   
   [G'] (your N-dimensional G) = L^{N-1}/MT^2,   
   [c] = L/T.   
      
   Therefore   
      
   ML^2/T = [S] = [R] [|g|^{1/2} d^N x]/[kappa]   
   or   
   ML^2/T = 1/A A^{N/2}/[kappa]   
   or   
   [kappa] = A^{N/2-1}T/ML^2 = L^{N-4}T/M   
   while   
   [G'/c^3] = L^{N-1}/MT^2 (T/L)^3 = L^{N-4}T/M.   
      
   So   
      
   kappa = (number factors) * G'/c^3   
      
   and the verdict is indeed that it's c^3, not c^4 ... even on grounds of   
   his assumptions.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)