A Sloan Volume <100 Mpc slice of a 1,000 Mpc volume at a redshift of 0 and bias of 1 using CDM initial conditions in a critical density Universe. The color scale represents a factor of 3000 in density. We also have a model with a bias of 2. These models were simulated with 47 million particles on the Pittsburgh Supercomputer Center CRAY T3D. The dynamic range of this simulation (at a redshift of 2) can best be seen in this mosaic, where each panel is a magnification by a factor of 2. For a better view of this simulation, we have a VRML and an Inventor description of a 80 Mpc subcube at a redshift of 0 and a bias of 1.4. The colors of the particles represent a factor of 1 million in density. (from the N-body Shop at the University of Washington: RIGHTS RESERVED BY THEIR AUTHORS).
ALL RIGHTS RESERVED, except as noted by Dr Kenneth Michael Beck
(Revised 02JUN19 and 22AUG19)
When I was a driver with lyft – a share ride company, I had many clients/passengers who wanted to chat about science and technology. I carried a copy of GRAVITATION in the front seat, so we had a reference if they want to read it or peruse it. Not a surprise, since this is Seattle – home of Amazon Cloud Computing; Microsoft; now Google/Google Maps(?). It’s enough to say, this is the center of most engineering done by these companies. And most companies that deliver product, live by a set of co-ordinates on maps and GPS, whether you believe it is by satellite or cellphone. GPS is based in part on the Theory of General Relativity.
Yet, despite the range of available topics, one short story of and by Einstein from 1949, always ‘exploded their minds,’ in their own words. Please don’t let this story ‘explode your mind?’
At the end of the 19th Century, the mathematician Georg Reimann understood what no other person understood. That Euclidean Geometry had failed, its days were numbered, and at the end of the day it was dead wrong. Gauss had struggled his whole life with that form and conception of co-ordinates and succeeded only in surfaces and field-theory, but to free our conception of Spacetime and Gravity we had to free ourselves, not just of Euclidean Geometry, but even of the co-ordinate system and our thinking about its meaning, as Riemann showed us. Einstein summarized all this in 1949…
“Now it came to me…the independence of the gravitational acceleration from the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation.
“This happened in 1908. Why were SEVEN YEARS required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that co-ordinates must have an immediate metric meaning*.” – A. Einstein (1949) as quoted by Kip Thorne, et al in GRAVITATION (pp5).
Later, Einstein remembered Gauss, but moved rapidly past his limited thought process to Reimann,
“[In 1912] I suddenly realized that Gauss’s theory of surfaces holds the key for unlocking this mystery. I realized that Gauss’s surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss’s theory was contained in the geometry course given by Geiser when I was a student… I realized that the foundations of geometry have physical significance. My dear friend the mathematician Grossmann was there when I returned from Prague to Zürich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann’s theory [Pais’s italics], namely, whether the invariants of the line element could completely determine the quantities I had been looking for?”
The breakthrough had come in Einstein’s thinking and only 2 years later he published his first, incomplete, work on General Relativity. With the right mathematician – Emma Noether – he rapidly moved forward.
*”metric meaning” – having or being a system or standard of measurement; a unit of measure.
If for two data sets (e.g., hexadecimal encryptions) which maybe reduced (conserved) to two binary data sets, one located in the reference frame of the observer and the other data set in a unspecified frame, [This is just Emma Noether’s famous test for symmetry in the 4D spacetime of E =mc^2, thus validating our right to ask the question. Without this validation, we have no right to even ask] How do we know which data set precedes the other?
Does it matter? We can attempt to “link” the two, either by a commonality of location with a third reference frame or a common reference frame, or just simply apply an arbitrary time stamp to each as agreed.
In most situations that humans find themselves here on Earth, this suffices.
In the singular work, GRAVITATION, Nobel Prize recipient Kip Thorne gives a basic definition of time at the outset in the practical, pragmatic sense of General Relativity, in terms of a “good clock,”
“Good clocks make spacetime trajectories of free particles look straight.” *
That is, good clocks make particles move simply, in straight paths, on their natural world-lines. Time is not definable outside of the other three space dimensions. They are linked by “c”, the proportionality constant between space dimensions and time. Sometimes confused with the speed of light, which has the same value.
Trying to understand a rich topic such as time, sometimes averting one’s attention and watching science-fantasy helps.
This is one of those…times. Why were the Time Lords so powerful in the Dr. Who series from BBC? As Dr. Who, herself embodied? Not because they could go backwards or forwards in time, which gained them very little, as they tried to stay on the same “time-line” (whatever THAT was) and “not change history.” It was, because they controlled time and made motion “look simple.”
Challenge Question: Who controls your time? In your frame-of-reference?
Someone or something does. Time-lords? (just kidding, I hope!) GPS, GNSS, or GLONASS constellation of satellites, Grandpa’s giant clock, an XY Oracle(tm) network? You really believe you are independent of them? Prove to yourself who controls time in your frame of reference.
An Issue of Causality
I bet you rely on GPS for your time control. So I will give this as my example, in words. What if I told you that without any fancy tricks and without spoofing, etc.; even in a state of complete ignorance on my part to what I was doing, I could initiate an event (t=0) using a legitimate GPS time sequence at the base of the Himalayas or any massive mountain range and it will appear to occur after an event occurring everywhere else on the surface of the Earth at (t=1).
That is called an issue of causality. Two locales on the surface of the Earth that “appear” to be in the exact same spacetime frame-of-reference, are in fact not.
Yet the GPS time-signal “assumes” that all receivers on the Earth’s surface are in the same spacetime frame-of-reference.
One reason for this occurring here, is that GPS corrections for special and general relativity are taken on the satellites before the signal is sent to Earth. It is built-in to the time sequence. All this is public knowledge. Everyone can and should know how GPS works.
‘Banks to the Moon. Bonds to Mars?’
The reason it might be “Banks” is that financial institutions, unhindered, could manipulate or by the same token, fall prey to, the mass difference between Earth and Moon to effect sudden, catastrophic financial event reversals. How?
We know GPS signals may reach the Moon. NASA has even navigated by them successfully in inter-Lunar space.
In the third in the series, “The Study of Spacetime and Gravity” I raised the issue of NASA proposing or being “conned” or a “spoof” on using Earth-based GPS to navigate to the lunar surface from orbit, as illustrated below in “Beyond Space Service Volume…
Is NASA Goddard for real or is this a spoof, or “Fake-News”?
I quantitatively demonstrated in one dimension this idea was not feasible…
I think I have a far better idea than relying on the GPS for position of space probes and The Gateway Space Station orbiting the Moon.
While in orbit around Earth, the GPS atomic cesium clocks are set-up to run +38,700 nanoseconds (nS) faster than similar ones on Earth.
Ge + Se = corrected time, where Ge is the general relativity correction of 45,900 nS and Se is the special relativity time correction of -7,200 nS slower for speeding around Earth.
The earth approximate mass is = 597.2 x 10^22 kg
It is from the proximity of this massive object, Earth, that general relativity can be thought to function for us in one frame-of-reference, as opposed to the frame-of-reference of the GPS satellites. However, there is subtlety here. On Earth.
Earth is 81.28 times more massive than the Moon (Mm = 7.348 x 10^22 kg).
In one-dimension, it is straight forward to see there will be different time-delays involved with Moon orbit.
For Earth and near-Earth orbit
Se = -7,200 nS
Ge= +45,900 nS
Delay T = +38,700 nS.
For the Moon and near-Lunar orbit…
Gm = 45,900 nS/81.28 = +565 nS
The fixed Delay T of +38,700 nS from the GPS, or GNSS, or GLONASS, or any constellation of atomic clocks will not work accurately near the Moon,
Delay T = +565 nS + Sm
Sm = the special relativity correction for satellites orbiting the Earth as seen from Earth, and now seen in another frame of reference, the Moon.
Their is a sign change in time delay. This is a simple one-dimensional representation of the metric tensor, Guv, which holds all the information on spacetime, mass variation, and curvature. Guv separates for us, future from past. In four-dimensional calculations of the tensor, it may have both positive and negative reversals.
My better idea for NASA is to solve the 4D metric tensor numerically (bringing to mind Euler’s Method from Hidden Figures) taking into account all topology on the lunar surface, and any known assymmtries under the surface, BEFORE making any pronouncement on how we can use GPS all the way into Lunar orbit.
Or even more simply, just have a separate LPS (Lunar Positioning System) in orbit around the Moon, made up of two or three CubeSats, like JPL designed so eloquently for communications recently for the Mars InSight probe.
References hold their own copyrights
*[Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald. Gravitation (Page 26). Princeton University Press, Princeton, New Jersey, (1971).]
**[Huth, John Edward. The Lost Art of Finding Our Way (Page 25). Belknap Press of Harvard University Press, Cambridge, Massachusetts, (2013).]