Otto E. Rossler | Abraham-like return to constant c in general relativity: “ℜ-theorem“ demonstrated in Schwarzschild metric

Otto E. Rossler | Abraham-like return to constant c in general relativity: “ℜ-theorem“ demonstrated in Schwarzschild metric

Added in proof: reception, erratum, confirmation

Reception. With several thousand downloads, the above paper is one of the most-read in general relativity. An early all-out criticism by the prestigious Einstein Institute [27] got subsequently replaced by a toned-down version in late July [28]. The latter no longer repeated the claims contained in the first that the Abraham paper contradicted both general relativity and experiment. The only relevant criticism that remained was a prediction: that if the gothic-R theorem were to be extended from the radial Schwarzschild metric to the full metric, it would prove incompatible with celestial mechanics. Even this conditional prediction had already been laid to rest by the successful re-formulation of the full Schwarzschild metric in terms of the gothic-R variable, achieved by an anonymous author signing with “Ich“ [29] (see his Eq.17). No further claim at falsification has been made to my knowledge. All the high-publicity claims at falsification made since by high-ranking institutions (like KET, CERN and two national parliamentary bodies) rely on the authority of Nicolai’s first paper and are, therefore, baseless as far as I can see. So are the ill-fated experimental decisions made in their wake.

Erratum. The merits of the already mentioned anonymous author do not stop here. He also succeeded in finding a first error in the above paper – not in the gothic-R theorem proper but in the added conjecture (erroneously labeled “Q.e.d.“) that the proper infalling time were infinite. This conjecture is false: the proper infalling time is finite [30]. The reasoning is based on the Rindler metric which is a valid approximation to the Schwarzschild metric [5,31]. The new result at first sight comes as a surprise from the point of view of the gothic-R theorem since the infinite distance implicit in the latter cannot be covered in finite proper time by definition – unless luminal observer speeds are involved. This appears absurd at first sight since observers are massive bodies and massive bodies cannot reach luminal speeds in finite time. Surprisingly, Ich’s result goes hand in hand with a corollary that implies exactly this.

Confirmation. The new confirmative corollary follows from the Rindler metric. Since the Rindler metric involves only special relativity, it can be fully understood in terms of a 2-D Minkowski diagram (the familiar x,t frame of special relativity). The Rindler metric – if I may dwell on it a bit more – refers to a long rocket in constant acceleration in outer space, with earth‘s gravity (1 g) reproduced at the tip. The full rocket consists of many segments each carrying its own pair of boosters on the outside. (Picture many solid hollow cylinders pairwise connected by a rubber tube.) In the x,t plane, the trajectories of all segments come to lie in between t = + x (right-hand part of first bisector) and t = – x (second bisector). This quarter of the full Minkowski plane is called the “Rindler wedge.“ Inside the wedge, we have our 1-light-year-long rocket, momentarily located motionless along the x-axis while accelerating all along at at full blast while stretching from x = 0 (bottom) to x = 1 (tip). In the x,t plane, the trajectory of the tip then rises up vertically to gently bear right along a curved line in the form of a half-hyperbola that asymptotically approaches the first bisector to asymptotically reach it at t = x = ∞. The lower (past) part of the same trajectory does the same thing reflected downwards, approaching the second bisector in negative infinite time. (This means, physically speaking, that the constant acceleration is superimposed onto a constant negative, initially at t = – ∞ luminal, speed.) The more inner segments of the rocket (x ⟨ 1 on the x-axis) all do the same thing along proportionally downscaled full hyperbolas having a correspondingly larger constant acceleration each (g/x locally). This principle continues right down to the 90-degree angle at x = 0 (the origin) where the acceleration becomes infinite (g/0). The assumed gradient in accelerations is necessary in order for the rocket to remain connected over time – an accepted if paradoxical fact in special relativity [5]. It follows that the intra-rocket times (“rocket tip times“) T remain definable indefinitely – all along straight lines through the origin. The bundle of these “T times“ ranges, from T = – ∞ at slope – 1, via T = 0 at slope zero, to T = + ∞ at slope = + 1 [5]. All T-times are on the same footing, that is, can each be identified with the x-axis on shifting the initial condition by simply “scrolling up“ or “scrolling down,“ respectively.

Now the two results announced. First, the finite proper infalling time result [30]: The internal observer at the tip of the rocket (at x = 1 and t = 0) lets go of his handle and simply stays put while moving up in time t along the x = 1 vertical. He then simultaneously is “falling“ freely inside the rocket – so as to leave it through an opening in the bottom at t = 1 (1 year) at the point x = t = 1 while the rocket’s bottom departs from him at the speed of light. He at this point has effectively “fallen“ through the whole length of the rocket in 1 year of his proper time.

Second, the new luminal-speeds result: To best see it, we assume for starters that the hole in the tail had been plugged by a trampoline (the asssumption can be dropped later). The coasting passenger – if resilient enough – then bounces back all the way up toward the tip in another year of his proper time. In the Rindler diagram, this rebouncing trajectory is again a straight line: starting at the point x = t = 1, it continues along the first bisector in coincidence with the latter so as to let the jumper re-catch his handle, which contimued along the curved hyperbola of the rocket‘s tip, at x = t = ∞.

That is all. One sees that the two straight legs of the observer’s trip are mutually equivalent (except for orientation in time). For it is possible to “scroll down“ the initial time T when the observer lets go of his handle, all the way down from T = 0 (asumed so far) to T = – ∞. In the new equivalent picture, the observer reaches the trampoline, not at x = t = 1 but rather at x = t = 0 (origin). This symmetric picture reveals that during either half trip (the two being mirror images of each other), an infinite distance in outer space is covered by the observer – in finite proper time! Hence there always exists an appropriately chosen frame in which an infinite distance is being bridged by the falling (or rebouncing) observer in finite proper time.

This new result is surprising since luminal speeds of massive bodies had no place in physics up until now. The reason they are a reality lies in the free choice of frames that is the hallmark of the Rindler metric, the above “scrolling operation.“ For we can always make sure that the “arriving event“ at the bottom of the rocket (which is the horizon [31]) coincides with the origin of the metric (the 0,0 foot point of the Rindler wedge). This point can be reached from inside the wedge (or be left into the wedge, respectively) only along one of the two 45- degree trajectories, that is, along luminal trajectories coincident with one of the wedge‘s boundaries.

This fact – that the origin of the Rindler metric is “nonsingular“ – comes as a surprise. Recall that the bottom of the rocket was factually reached by our first “falling“ observer on his stepping out into the light from the hole in the rocket’s bottom, at x = t = 1. This fact [30] now also means that a luminal speed is accessible to a falling observer or particle inside the Rindler metric. But cannot such a speed only be reached after an infinite period of constant acceleration by definition? This is correct. Amazingly, both seemingly contradictory facts are mutually compatible for once. For the waiting time under permanent constant acceleration inside the rocket, is infinite: The handle (or a companion sitting on the neighboring seat) has to wait upstairs an infinite period of time under constant acceleration, bridging an infinite distance in outer space in the process, before being at last reunited with the back-bouncing, youthful, observer.

More abstractly speaking, the “scrolling operation“ includes the two 45-degree singular limiting cases – with their luminal speeds – as effective nonsingular cases. This mathematical finding is amenable to a deeper (differential-topological) explanation. Here, it suffices to note that such a situation – that the singular limits are nonsingular – is unheard of in physics. This fact gives the Rindler metric and its close relative, the Schwarzschild metric, a unique place in nature.

What does the effective infinite intra-rocket (and extra-rocket) distance found mean? It means that a well-known result possesses a new corollary. So far, it was known that photonic Hawking radiation by definition takes zero proper time to emerge from the horizon after having bridged the whole (infinite) gothic-R distance and now, material Hawking radiation analogously takes a finite proper time to come out. Similarly light takes an infinite external time to come out as we saw in the paper and now, material particles take a longer (twice as long) infinite outside time to come out across the whole gothic-R distance. Again, we only have reproduced a self-evident fact one feels.

Nevertheless the (now trivial) prediction of an infinite “emerging time“ because of an infinite distance to be covered from the horizon, is what gave the above paper its worldwide attention. For this prediction implies that microscopic black holes generated on earth cannot evaporate in finite time – and hence put the planet at risk if earth-bound. In this way, an ethical dimension got suddenly attached to a pure-physics result. This dimension was, interestingly, not seen at the time of writing the paper but was the merit of a relativist colleague who when recommending publication jokingly wondered whether there could not be repercussions on the “LHC“ experiment. Although I had never heard of the latter, the remark eventually triggered a vain attempt at defusing the joke. When it failed, a more serious attempt followed so it almost became a sport to hunt for a more sophisticated argument in order to defuse the joke. Each floundered for a different reason so that a vague hunch of a danger-conserving principle being at work formed – that all the uncanny failures may be non-coincidental. The suspicion turned tangible when the final unsuccessful attempt at giving the all clear had been communicated to CERN in May and published in July [32]: neutron stars seem to possess a special quantum protection against natural, cosmic ray-borne, very fast analogs to any miniblack holes potentially created on earth (superfluidity was the likely culprit). Eventually the idea of a joke played by nature on humankind – that the artificial slowness of human-made analogs could be a curse – befell the whole planet on September 10 when more than 500 newspapers across the globe referred to it in one way or the other. The joke still waits to be defused. Thinking twice (by no longer opposing the safety conference publicly demanded on April 18 [33]) remains an option to date following the felicitous fehlleistung that occurred at CERN on September 20. The whole globe is grateful for the second chance at falsification granted to it. Letting an idea die is always the less costly option according to Karl Popper.

I thank Gerhard Huisken for leading me to Robert Wald’s book, Georg Slotta for references [29,30] and many exchanges that opened my eyes to the Rindler metric, Dieter Fröhlich, Christophe Letellier and Peter Kloeden for discussions and Andy Hilgartner, Artur Schmidt, and Kensei Hiwaki for stimulation. For J.O.R. 12/31/08.

Otto E. Rossler | Abraham-like return to constant c in general relativity: “ℜ-theorem“ demonstrated in Schwarzschild metric