The quantization relation Eq.(3) and the uncertainty relations Eq.s(6) and (7) are suffi- cient for our analyses in later sections; we need not discuss detailed quantum mechanics or quantum field theory. We next move on to general relativistic gravity and black holes. Recall that the line element or metric of special relativity gives the spacetime distance between nearby events. It is usually expressed in Lorentz coordinates ( ct, ⃗ x ) as 19 In general relativity gravity is described by allowing spacetime to be warped or distorted, or, more technically correct, curved. Coordinates in general relativity merely label the points in spacetime and do not by themselves give physical distances; for that we need a metric, which relates coordinate intervals to physical distance intervals. 3,20–22 For a weak gravitational field and slowly moving bodies the line element or metric is approximately given by the so-called Newtonian limit Here φ is the Newtonian potential, with the dimensionless quantity φ/c 2 assumed small, and also assumed to go to zero asymptotically at large distances from the source; for example for a point mass M the potential is φ = − GM/r . What this means is that the proper time, or physical clock time, between 2 events at the same space position and separated only by a time coordinate interval cdt is while the physical meter stick distance between 2 events separated only by a space coordinate interval dx is and similarly for the y and z directions. The Newtonian limit Eq.(9) is a quite good approximation in many cases; for example in the solar system the Newtonian potential is greatest at the surface of the sun, where 2 φ/c 2 ≈ 10 − $^{6}$. Thus spacetime in the solar system is extremely close to that of special relativity, or flat. Since we are interested only in order of magnitude estimates, we will make free use of the Newtonian limit Eq.(9) as an approximation. 20–23 6 ds 2 = ( cdt ) 2 − d⃗ x 2 (8) ds 2 = (1 + 2 φ/c $^{2}$)( cdt ) 2 − (1 − 2 φ/c $^{2}$)( d⃗ x ) 2 (9) ds = cdt $^{√}$1 + 2 φ/c 2 (proper time separation) (10a) dx $^{√}$1 − 2 φ/c 2 (space separation) (10b)