by Otto E. Rossler, Faculty of Mathematics and Science, University of Tübingen, 72076 Tübingen, Germany
When you are doing this picturing job, you will directly get close to Einstein’s heart. He only did not yet have this special sentinel available in 1907. Noether’s ultra-hard result of 1917 came 2 years in the wake of Einstein’s opus maximum that is being celebrated this year.
I need your kind help to improve on the following finding: “Noether’s Theorem + Einstein Equivalence Principle = c-global.” I have 5 steps to offer so far, the sixth would be your initiative.
(i) If true, this result amounts to a revolution in physics. For it removes an inconsistency that was reluctantly accepted by Einstein in 1907 in the absence of Noether’s theorem: the embarrassing conclusion that c is reduced downstairs in a constantly accelerating long rocketship in outer space described by special relativity. This drawback found in the equivalenve principle let Einstein fall silent on gravitation for 3 ½ years and retarded progress on general relativity afterwards. Two years in the latter’s wake came Emmy Noether’s “global conservation of angular momentum in nature.” Her formal result can be visualized geometrically:
Take a frictionless bicycle wheel suspended from its hub and lower it and pull it back up again in gravity.
Everything is pre-specified if this simple sentinel is pictured in the mind. Firstly, the rotation rate of this “clock” will go down reversibly like that of any other clock that is hauled downwards. Secondly, since angular momentum is conserved, the other two components besides rotation rate (mass and radius) cannot go both unchanged. It becomes a rewarding game to figure out what is bound to happen in this simple gedanken experiment.
(ii) The conserved angular momentum obeys a simple formula if the wheel has a constant horizontal (or vertical) orientation which orientation will be automatically preserved. The one-liner is given as Eq. (8.32) in Tipler’s big textbook, for example, but Madame du Chatelet could already have written it down in the 18th century:
L = ω m r^2
Since this expression is hard to remember by heart, the dialect word L’hombre (Spanish for “man”) can be helpful as a bridge. L is the conserved angular momentum, ω is the rotation rate, m the mass and r the radius of our horizontally rotating frictionless bicycle wheel.
If ω is halved (as is approximately valid on the surface of a neutron star with its almost unit-redshift): what about m and r , the two other components of the conserved L down there?
I propose that m is halved and r is doubled. The halved mass is the key. It follows from the halved frequency (and hence energy) of any photon emitted down there. These photons look non-reduced in their frequency locally. They remain locally transformable as usual into massive particles in accordance with quantum mechanics’ creation and annihilation operators. Thus if a sufficiently sturdy PET scan could be lowered onto the neutron star, it would still work there. The locally normal-appearing half-mass atoms possess a doubled Bohr radius (and hence size) according to the laws of quantum mechanics. Both facts, taken together, yield L’ = ½ ω ½ m (2r)^2 = L , in conformity with the above equation.
But this result of a doubled radius r of the halved-rotation-speed wheel downstairs, is ostensibly at variance with a well-known fact implicit in the theory which underlies the constantly accelerating Einstein rocketship: special relativity. The latter requires that light rays that connect points on a stationary solid object with the same points on the same object while the latter is moving away at constant orientation, travel along parallel lines. This railway tracks principle of special relativity demands that the doubled radius of the horizontally rotating wheel found valid downstairs must be optically masked when viewed from above. So our wheel indeed looks non-enlarged from above even though its radius r has doubled!
(iii) To check on this, let your Noether wheel for once rotate vertically rather than horizontall. Then the doubled radius will remain optically masked in the horizontal direction, but not so in the vertical direction: You now get a 2:1 vertical ellipse on the neutron star when looking down on the wheel from above.
The optical contraction of all horizontal directions, valid downstairs on our wheel, implies that when you look down from above, transversally moving light will be seen to “creep” at half speed on the surface of the Neutron star. This is what Einstein effectively found in 1907. Thus everything appears to be consistent.
But: does light really “creep” down there? We see that the answer is no. For the distance travelled downstairs is doubled compared to above as the optically compressed wheel teaches us. Hence c remains constant in spite of its apparent creeping. This new information was unavailable in 1907 owing to the absence of the Noether-wheel.
The newly retrieved global constancy of c in the equivalence principle comes not really as a surprise since the equivalence principle is based on special relativity with its constant c. This fact explains why Einstein fell silent on gravitation for more thanthree years after feeling forced to conclude that c is non-constant in the equivalence principle.
(iv) The retrieved globally constant c has an important implication: The vertical distance to the surface of the neutron star has increased. That is, the indentation into the curved “cloth of spacetime” has deepened. In other words, the famous empirical Shapiro-time-delay is complemented by a matching new Shapiro space dilation.
The stronger the gravitational pull, the deeper the trough. Therefore, the new globally constant c implies that the distance down to the “horizon” (surface) of a black hole is as infinite from above as the temporal distance for light going down or coming up is known to be since Oppenheimer and Snyder’s 1939 paper.
Hence black holes are never finished in finite outer time! At this point, I hear you ask: But is it not a well-known fact that an astronaut can fall onto (and into) a stellar black hole in finite time, as Oppenheimer and Snyder showed and as we all could witness in Kip Thorne’s carefully researched science fiction blockbuster movie Interstellar?
(v) The answer is a final Noetherian point: all clocks of the falling astronaut get infinitely slowed eventually, so that infinitely much outer time has elapsed on her to be hoped-for arrival down there, provided the universe will still exist by then. As to our lowered wheel, its rotation rate becomes zero on the horizon while the tangential velocity of the rim stays invariant as the wheel’s diameter approaches — invisibly-to-above — infinity (Sanayei effect).
My dear readers: what did we learn from points (i) to (v)? The Noether wheel teaches us several new things:
First, there exists no Hawking radiation by virtue of the new infinite distance of the horizon valid from without.
Second, general relativity must be re-scaled so that it no longer masks the global–c constraint. The Noether wheel thus entails that a new simpler-appearing, re-scaled version of general relativity exists – predictably without any remaining incompatibility with quantum mechanics. The holy grail of unification is therefore within reach: a bonanza for young physicists in the making.
Third, the often heard claim that angular momentum were conserved in general relativity in its present form is falsified by the example of the Noether wheel because the latter brings-in a previously lacking, in the limit of the horizon unbounded, size change as an intrinsic element of the theory.
Am I allowed to add a Footnote to this bonanza?
The recaptured c–global forms an apparently non-ignorable argument in favor of the renewal of a 7 years old safety report: specifically the so-called “LSAG” of the famous LHC–experiment near Geneva. The latter experiment is apart from its other goals designed with the aim in mind to produce miniature black holes down on earth. The Noether wheel’s c–global implies as we saw that black holes cannot Hawking evaporate since nothing can disappear behind a not yet existing horizon. The miniature black holes will rather grow exponentially inside earth in accord with a conference paper published in 2008.
This “dark implication” of the Noether wheel is the reason why I so publicly address you – the young generation – on Lifeboat today: because time is pressing. You may know that CERN has announced to double its (unprecedented in the history of the universe) almost stationary center-of-mass collision energy on a privileged celestial body (earth) in the hope to create Hawking-evaporating black holes on it. As we saw, such pre-Noetherian collision experiments are now scientifically outdated.
Not only a blemish, though: In light of the above Noether-wheel based result of c-global, any attempt at producing miniature black holes down on earth constitutes a “crime against humanity” if you understand what I mean. Are you – the youngest and therefore most open-minded citizens of our planet – able to provide help according to your own judgment? That is, can you perhaps think of a good idea how to persuade CERN to kindly respond to the Noether-based criticism of their announced doubling of their symmetric collision energy? CERN announced to start symmetric collisions in early June. And it in addition decided to non-renew its 7 years old — pre-Noetherian — Safety Report. Every person of course readily understands that it is humanly impossible to respond quickly to surprise evidence (like that of an iceberg named Noether being headed on a collision course) when you are the captain of an only slowly maneuverable ocean liner. Hence my question to you, dear young readers: Do you have any idea how the spotted iceberg can be brought to the attention of the captain?
I have a constructive proposal in mind: There is a female captain elected to take office next year. Would it make sense to try and contact her? Perhaps she sees – besides her being duty-bound – a way to call for a “brief thinking pause devoted to evaluating a formal Noetherian result” before the announced start of collisions in June gets its final okay? Who amongst you would support this kind request?