LIGO – Hanford, WA with Rattlesnake Mountain to the left. Photo courtesy: USDOE.
FROM BLOCKCHAINS TO STRESS-ENERGY TENSORS
Blockchains are the perfect 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, is it? Again it took “SEVEN YEARS” for Einstein in an earlier era to change his viewpoint.
What the heck is this? Except space being curved, not spacetime. According to this view, there should be a plethora of LEO satellites in the potential well below the South Pole, right? Orbiting in smaller and smaller orbits until…plunk? They are all stuck. An appeal to a “two-dimension illustration” does not save you, as it should be an analog to reality. This is NOT how it works.
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. [Do not let the complex word-play of Wikipedia “play” with your possibly “exploded brain.” The greats of gravitation and General Relativity mentioned in the Wikipedia; Misner; Wheeler; and Thorne, were ALL authors of the same book, GRAVITATION.]
So said Prof. Dr. Emmy Noether, the greatest mathematician of the 20th Century. Do you doubt her? Einstein did not. Einstein defended her. 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 co-ordinates, using Einstein and later, Thorne’s warning against over-rating and using co-ordinates
Example 1.12. (from GRAVITATION pp39)
Note: 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 (units are: cm/second), or how we convert from one dimension to another. Now, in General Relativity, we see another constant. This is G/c2 = the proportionality constant of space to mass = 0.742 x 10 -28 centimeter/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.
*A Short Intro On Prof. Dr. Emmy Noether
She did not receive a position as a tenured faculty in Germany, because she was female. Nor did she receive any pay in the new Democratic Socialist university system now in place in Germany. In fact, she was only allowed to teach for free (So much for equal pay for equal work, Democratic Socialists)!
Without it she had to work unpaid. Einstein, and Hilbert (her PhD advisor) protested this outright sexism. In a letter to this patriarchal social democratic clique in Göttingen’s faculty Einstein wrote,
“Yesterday I received from Miss Noether a very interesting paper on invariant forms. I am impressed that one can comprehend these matters from so general a viewpoint. It would not have done you, the old guard at Göttingen, any harm had you been taught a thing or two by her.”
Later, after her death from breast cancer as a tenured faculty member at Bryn Mawr in Pennsylvania, he wrote,
“Born into a Jewish family distinguished for the love of learning, Emmy Noether, who, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators.”
And before the anti-Jewish violence by the Nazi state began just before the 1936 Olympic Games in Berlin, Emmy Noether never questioned the veracity of the two or three young S.A. members (Nazi Brownshirts), who wore their uniforms into class. When Hitler ordered the doors locked to her at the university in 1933, because she was “a dangerous Jewess (sic),” these stormtroopers were invited by Emmy Noether, along with their fellow students to ‘secret classes’ she continued to conduct, in cafes and city parks, even into her own home. Prof. Dr. Herman Weyl, reflecting on this period, said that,
“Emmy was so strong, so optimistic, like none of us men were. We were willing to commit suicide, as I was. Yet then I thought of what she had been through her whole life never giving up, and I grew hopeful.”
Weyl emigrated to America, as did Noether in 1934 when she received the offer of tenure, aided by Einstein at Princeton. For her part, Emmy grimacing at her naiveté once in America. The Nuremberg Laws and horrific state-sponsored violence against Jews started in 1935.
She had witnessed no real difference in her lifetime between how the Weimar Republic or the Nazis acted towards women; only their common patriarchy. That should be a lesson for all of us.
This info about Emmy Noether is included in the book by Prof. Dr. Talithia Williams.
That is Emmy Noether’s photo top left with the Bill Nye ‘bow tie.’