One step closer… from blockchains to stress-energy tensors

Originally copyrighted and published on 04NOV2019

Blockchains maybe a good set of “test particles” for General Relativity.  That is, in locally accelerated frames of reference, traveling roughly in the same direction.  It is, as a group, a relatively small series of “points” with little mass-energy (energy equivalence).   Thus, we can say they move as “free-particles” in straight-lines through flat spacetime.  They are the relative reference to events all around us where different particles may accelerate differently, perhaps even in opposite directions.  It is from free-particles in motion we can determine the relative curvature of spacetime when compared to those with mass-energy.

What is curved, space or spacetime? This question, so simply ask, has befuddled too many. In the presence of mass-energy, spacetime curves. That is not easily visualized.  Again it took “SEVEN YEARS” for Einstein in an earlier era to transform his viewpoint.


This is an effect of the conservation of mass-energy.  Not just mass; Not just energy; mass-energy from the energy equivalency.  So, not just space; not just time; spacetime.  Geometrodynamics  or spacetime works on mass-energy locally, directing its path.  Mass-energy works on spacetime locally, by curving it.  That is the symmetry simply stated.   “Spacetime curves.”

Noether proved that “that every differentiable symmetry of the action of a physical system has a corresponding conservation law.” Dr. Noether’s paper – Invariante Variationsprobleme – on this topic was delivered by her on 26JUL1918 for her habilitation (or ability to seek a tenured professorship in German universities) and offered a profound insight into how ‘behavior in systems’ is constant.

Where ρ = <<density of the Earth>>, G = Roo = [the trace of the Reimann tensor for Earth] = 4πρ. G is created from the local distribution of matter. In the generalized, simply-formed Einstein Field Equation, G = Einstein tensor, formed from T = stress-energy tensor, G = 8πT Of special notice, we did not define either tensor in coordinates, using Einstein and later, Thorne’s warning against over-rating and using coordinates, until the end of the exercise. Example 1.12. (from GRAVITATION pp39)

In Special Relativity, we have already learned and seen confusion around “c”, whose value is the same as the speed of light, but represents in Einstein’s equations, the proportionality constant of space with the time dimension, and how we convert from one dimension to another.  Now, in General Relativity, we are introduced to another constant, G/c2 , which is the proportionality constant of space to mass = 0.742 x 10 -28 cm/gram,  illustrating how weak the effect of mass is on spacetime.

The Reimann Tensor (left side), where each components is a vector. Off-diagonal components are zero (right-side).  A gravitational wave from a supernova, or colliding black holes, or colliding neutron stars, will show up only in these off-diagonal elements, producing non-zero results and changing the spacetime curvature. (from GRAVITATION pp40-41)


 




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