Apprentice to the Obvious' blog
http://www.davidwoolsey.com/AttO/AttO_blog/AttO_blog.html
This is David Woolsey's weblog containing some of his observations and thoughts about the world. The world of science, the physical world, human behavior, and other things. It'll be mostly about science, technology, and engineering, but there will be a few uncontrolled excursions into other things as well. You have been warned.<br/><br/>I am a long time student and practitioner of physics. I make do with the "stone knives and bear skins" of an undergraduate degree in late twentieth century physics from U.C. Berkeley. That, and a whole lot of on-the-job experience. I will sometimes lack the mathematical tools to be exact in what I will write about. I will sometimes be off target or even somewhat wrong. However, in the past I have made many useful observations that others have missed but I have failed to communicate my observations. This weblog will hopefully address that failure to communicate.<br/><br/>Why the name? I had at first thought to call this web log something like "Master of the Obvious" but that would be "getting ahead of myself" just a bit. I really like those moments when something yet unthought comes sharply into focus as an obvious truth of the way the world works. I like being able to look back and identify the moments when my thought processes abruptly changed when thinking something through in detail, or by intuition, or by a "bolt out of the blue". Those are the true moments of bifurcation and change in my path through life. I'll get around to sharing some of them with you here.<br/><br/>Why the content? It is important that there be investigations into speculative science, both in terms of performing it and in terms of investigating the performance of those experiments. It is very important that the results of speculative science be properly looked into for flaws so that quality control standards are enforced on its investigators. Otherwise valuable lines of speculation may well descend to the level of junk science through lack of proper "adult supervision" (that means you, professional scientists). Most ventures into the speculative should rightfully be expected to lead to dead ends or other failures. That is just the way it is. On the other hand, the whole world of high tech venture capital investment behavior is run from a "template" that expects 90% of the startup businesses to fail. Yet, those 90% failure odds does not scare away the investors because the remaining 10% sometimes succeed wildly. I would hope for a similar set of "investment" behaviors by at least some of the scientific community about speculative ideas in science and engineering because sometimes investing a bit of time and attention to look into what seems crazy might just have a payoff. At the very least, the payoff might come in the form of the joy of discovering why something /doesn't work/, after all.<br/><br/>Tools? I'll be using Apple's obsolete iWeb application to construct the weblog pages. We'll see how long that lasts before I turn to something else such as going back to using the Alpha text editor.<br/>iWeb 3.0.4http://www.davidwoolsey.com/AttO/AttO_blog/AttO_blog_files/139_3951.jpgApprentice to the Obvious' blog
http://www.davidwoolsey.com/AttO/AttO_blog/AttO_blog.html
Black Holes and Transverse Tidal Effects, a short essay on some thoughts
http://www.davidwoolsey.com/AttO/AttO_blog/Entries/2016/12/25_Black_Holes_and_Transverse_Tidal_Effects,_a_short_essay_on_some_thoughts.html
294611d9-69b3-4de1-97af-ef98ac2d6e93Sun, 25 Dec 2016 12:59:44 -0800<a href="http://www.davidwoolsey.com/AttO/AttO_blog/Entries/2016/12/25_Black_Holes_and_Transverse_Tidal_Effects,_a_short_essay_on_some_thoughts_files/Transverse_Blueshifted_Radiation-Fig1.png"><img src="http://www.davidwoolsey.com/AttO/AttO_blog/Media/object033_1.png" style="float:left; padding-right:10px; padding-bottom:10px; width:182px; height:138px;"/></a>Okay, it had to happen eventually, this article is about one of my own speculative sets of thoughts. Hopefully it isn't too far fetched or krankish. And, dear readers, do keep in mind this is a weblog, not a peer reviewed article. Please do not treat or site the contents as in any way authoritative on the subject of gravitation and Hawking radiation. The content is meant to convey a set of insights that might be useful for further development.<br/><br/>History. It all began a week or so before the UCB Physics department Colloquium on Black Hole Firewalls given in 2013 by Raphael Bousso (<a href="http://physics.berkeley.edu/black-holes-and-firewalls-can-we-take-nothing-for-granted">http://physics.berkeley.edu/black-holes-and-firewalls-can-we-take-nothing-for-granted</a>). I decided to think up a question or two to ask after the colloquium about black hole firewalls so that I might make an intelligent introduction for additional conversations about gravitation at a later date. As you can see from the webcast of the lecture, I was unable to ask the questions I wanted to ask at the end of his lecture. Furthermore, I was unable to effectively follow up with future conversations about the ideas with anyone who knew enough about the subject matter that they could be of help in fleshing it out and making the real (and very difficult) calculations that might lead to actual predictions. In the end I would up writing the thoughts out in some detail in an essay, from which this weblog entry is derived. But I'm getting ahead of myself, its all in the article below. This is just the introduction.<br/><br/>"Terms". Since the questions and ideas I'll present below were constructed as tools to "open doors" to future conversations, I would appreciate it if you, the intelligent reader with a comprehensive knowledge of General Relativity, do not simply expropriate them as entirely your own. That sort of theft would not be helpful in "opening doors" to future conversations for anyone. Instead, if there turns out to be material of insight and value here, please give me some credit for having come up with the thoughts, incomplete as they are, in whatever work you derive from my own. To put it in more direct terms, if I find papers written in the near future that flesh out and claim credit for these ideas without some credit to me for having had the insights, then it'll guarantee that I won't be working with those authors on any future works. This is only the third weblog entry I've posted, there are many more "in the queue" that you might want the opportunity to collaborate with me on – especially the ideas that are not conversation starting "throwaway" ones like this one.<br/><br/>Why now? The present impulse to put this article up on the interweb comes from two relatively recent events in gravitational astronomy (how's that for a cool new field!). First, we had the LIGO results a year ago, demonstrating the existence and detection of gravity waves from compact object mergers. Then we have the recent Nature article titled <a href="http://www.nature.com/news/ligo-black-hole-echoes-hint-at-general-relativity-breakdown-1.21135#comment-3044303675">LIGO black hole echoes hint at general-relativity breakdown</a> (<a href="http://www.nature.com/news/ligo-black-hole-echoes-hint-at-general-relativity-breakdown-1.21135#comment-3044303675">http://www.nature.com/news/ligo-black-hole-echoes-hint-at-general-relativity-breakdown-1.21135#comment-3044303675</a>) together with all its referenced links. So, it seems we can not only detect gravity waves now, but we may be able to detect hints of the effects I've tried to predict in the article below. Thus, now would be a good time to get this set of ideas out to be "hammered on" and worked out. And, since there may be a method of experimentally detecting Hawking radiation, and Stephen Hawking is getting kind'a old now, further delay would be counterproductive.<br/><br/>I guess that, now that I've done a couple of (fairly brutal) critiques of other people's work, it is my turn to have stuff thrown at me. Try to be nice.<br/><br/><br/><br/><br/><br/>Preface<br/><br/>The ideas presented here ought to seem like a completely obvious set, though you may need to be patient in order to slog through my reasoning. It has been a common practice in the world of black holes, event horizons, and General Relativity to dismiss any effects of tidal accelerations by simply scaling the mass of the black hole up, thus reducing the local effects of gravity gradients to inconsequential magnitude. Indeed, the practice of ignoring tidal accelerations runs so deep that when I tried to ask a limited form of the central question presented here of UCB's Prof. Bousso at his colloquium in 2013, he dismissed the relevance of tides before I could even finish setting up the question.<br/><br/>The fact that I've tried to present this before, and had such poor luck in getting an understanding of the ideas across, is why I what follows is probably written in an overly detailed way. Please bear with me.<br/><br/><br/>Motivations<br/><br/>Here is a short list of some possible, non-trivial items of theory and observation that will follow from an affirmative answer to the blueshift question I pose in this text.<br/><br/> 1. The existence of something very much like a black hole firewall (that would in fact be demanded by the Equivalence Principle).<br/><br/> 1. The resolution of the paradox of stationary vs. infalling observers w.r.t. Hawking radiation and black hole evaporation.<br/><br/> 1. Modification to the spacetime metric close to black hole event horizons to account for the energy bound up in the "thermal atmosphere" of (trapped) Hawking radiation with high angular momentum.<br/><br/> 1. Existence of a high viscosity, high density photon field in the near vicinity of black hole event horizons.<br/><br/> 1. Small modification of the expected gravity wave spectrum from matter falling into black holes – small modification to last instant of "ring down". This one is experimentally testable!<br/><br/> 1. Experimental test of Hawking radiation via the detection of gravity wave spectra conforming to the new modified metric and high viscosity "atmosphere" near black holes.<br/><br/> 1. Experimental test of whether black holes are "black" vs "very dark brown" – which is slang for whether black holes form in finite time vs the classical, asymptotic infinitely slow collapse.<br/><br/> 1. I'm sure there is more that I haven't thought of yet, so let me say "etc."<br/><br/>Items 4, 5, and 6 are of special note for experimentalists detecting gravity waves. Item 6 ought to be of interest to Stephen Hawking himself since it is the only (known?) way to experimentally detect Hawking radiation (by its effects on infalling matter via item 4). Motivations now established, let us begin.<br/><br/><br/>Effects of tidal accelerations on horizon radiation<br/><br/>Hawking radiation, as seen from far away from the source, is emitted from just above the black hole event horizon. The radiation starts out as Planck-scale photons (E ∼ 108 J) but is strongly redshifted on its way to observers far from the hole. Hawking radiation from a 1 Solar mass black hole is at a temperature of T ∼ 10-7 K, as seen from afar. The redshift comes from the fact that the "surface" emitting the radiation is itself in radial free fall with a speed of just about c , away from observers distant from the hole.<br/><br/>If we take on the point of view of an observer freefalling into the hole, then the event horizon will seem to "fall away" from us as we approach the Schwarzschild radius (where the event horizon is seen to be by observers far from the hole). The best description I know of why this is so is this:<br/><br/>Imagine an observer falling into the black hole accompanied by a series of test particles in front and behind him. For simplicity we'll consider that the infalling observer has no angular momentum with respect to the hole and so falls along a simple radial path (with respect to the hole). The test particles are arranged to fall along the same radial path.<br/><br/>As our observer falls he notices no local gravity, but he does notice that the test particles in front and behind him are accelerating away from him. The ones farther away have a larger acceleration. This is the radial component of the tidal force.<br/><br/>Now, if the test particles are emitting radiation at a known energy it will be more and more redshifted (as seen by him) as the particles accelerate away from him to higher and higher relative recessional speed. Here's the key to why Hawking radiation is seen to still be very low energy even for an in falling observer: the radiation is always seen to be coming from a surface (an event horizon) which is in free fall away from the observer. Thus the radiation is always seen to be redshifted to a very low energy.<br/><br/>The situation is caused by the Equivalence Principle: the event horizon itself can not be a special place that an observer can detect traveling through. If we imagine a black hole massive enough then we can neglect tidal forces because the gravity at the Schwarzschild event horizon can be made arbitrarily small while the radius of the hole is made arbitrarily large. Thus, the reasoning is that there can't be anything special about the event horizon at the Schwarzschild radius. Or so the “usual story” goes.<br/><br/>The problem with the "usual story" is that it does not treat the measurement of tidal forces the same way as other black hole features. Tidal forces are usually dismissed on the basis that over a given length, the force can be made to go to as low a value as we want by proposing a massive enough black hole, with a slowly enough changing local gravity field. However, most of the other important features of black holes are described in terms of multiples of the Schwarzschild radius (or rS, because I'm getting tired of typing that), so that they are mass scale insensitive. It is this mixture of mental models that I believe has caused the ideas here to be overlooked or misunderstood.<br/><br/>What I think I have found is that by treating tidal forces as acting over a scaled distance – radial distances measured in multiples of rS and circumferential distances measured as the angle around the hole's center – we come to a very different picture of what happens to an infalling observer than the current paradigm would have us conclude. Before we move on to a cartoon of the novel idea, let's first have a review of what causes tidal accelerations (please be patient).<br/><br/><br/>Tidal accelerations about gravitational central potentials<br/><br/>We start with the observation that the radial tidal force causes a redshift of the spacetime ahead and behind an in falling observer, and the transverse tidal force should cause a blueshift of the spacetime to the sides (transverse to the line of fall) of that observer. To understand why, consider a cloud of test particles all around a freefalling observer in a gravitating central potential (for example, the Schwarzschild potential).<br/><br/>Consider particles “in front and behind” along the test observer's radial line of freefall into a black hole, and consider test particles about him, transverse to his line of freefall. As our observer freefalls toward the hole, the test particles along his radial line of freefall will accelerate away from him because the gravitational field strength is changing as a function of radius from the center of the hole. Conversely, as our observer falls toward the central potential the test particles transverse to his line of freefall, those to his “left and right”, will be seen to be accelerating toward him. This is the transverse tidal acceleration and it happens because particles in freefall along different radial paths will have a component of their acceleration relative to each other that is parallel and a component that is perpendicular to their radial paths toward the central potential.<br/><br/>If we were to put the entire ensemble of observer and test particles inside a closed room (traditionally an elevator), that was likewise also in freefall, there would be no way for the observer to determine whether he was falling into the gravitational field of a compact object (a black hole), or whether he was in a universe that was expanding in one direction and contracting in directions perpendicular to that. The freefalling observer would only see that along one axis objects removed from him would accelerate away in proportion to their displacement, while along the other directions, they would accelerate towards him in proportion to their displacement. This setup is true in the limit of “small rooms” – and local acceleration effects. If the size of the room is not small in comparison to the distance to the compact source of gravity then there will be a few telltale signs our observer might use to suspect that he's in a gravity well instead of a strange universe with an anisotropic Hubble constant.<br/><br/>For small displacements along the radial direction from the gravitating body, the radial component of the tidal acceleration goes by<br/><br/><br/><br/>Where R is the radial distance from the gravitating body, and Δr is the separation of the test particles along the radial distance. However, when the separation, Δr, along the radial is not small in comparison to the radial distance, R, then one should use the difference between the local accelerations<br/><br/><br/><br/>which clearly grows faster in magnitude for displacements towards the gravitating mass than for equal displacements away from it. So, for an extensive enough observer and test particle ensemble (all enclosed in a big enough “room”), the observer might suspect that he's near a compact source of gravity (with the inverse square law in play), rather than in a universe with a strange Hubble constant. However, the Equivalence Principle would not allow our observer to conclusively know which one is the “true” external situation without actually looking outside the room. All of the signals sent from or between test particles in his freefalling “room” would behave just as if there were an anisotropic Hubble constant at work.<br/><br/>For small displacements the magnitude of the transverse tidal acceleration is 1/2 the magnitude of the radial tidal acceleration, in the direction transverse to the radial from the gravitating source:<br/><br/><br/><br/>Where ΔC is a distance along the circumference of a circle of radius R (and transverse to the radial direction). The transverse tidal acceleration scales by the inverse radius from the point of gravitational attraction. And it also scales by the inverse square distance from the gravitating source to the freefalling observer. Thus, the total scaling is by the inverse cube of the radius – over a given distance transverse to the radial acceleration. However, as with the case for the radial tidal acceleration, that is only true for small displacements. Over larger displacements, where ΔC is not small compared to R, the above approximation fails because, while the radial tidal acceleration is due to the gradient of the (inverse square) acceleration law, the transverse tidal acceleration is actually due to the angular separation of the radial paths of the infalling test particles.<br/>The equation for the transverse component of the tidal force is something like<br/><br/><br/><br/>Where glocal(r) = -GM / r2 and φ is the angle (in spherical coord.) separating the freefall radials of the test particles (with the coord. center at the center of the black hole). The approximation of the transverse tidal acceleration for small displacements comes from the fact that the sine of a small angle is approximately equal to the transverse displacement divided by the axial displacement (φ ≃ ΔC/R).<br/><br/>I contend that the correct way to look at the transverse tidal acceleration is that it is the component of the local acceleration felt between two freefalling test particles, falling along different radials, that is, not parallel to each other. Thus, I think, a "cleaner" way of thinking about the transverse tidal acceleration is that it scales with angle (φ) around the gravitating body, not by linear distance between test particles, concluding our review of tidal accelerations.<br/><br/><br/>The idea<br/><br/>While it may seem reasonable to propose that tidal accelerations near event horizons can always be “scaled away” by conducting our thought experiments in the vicinity of very massive black holes, the proposal is in fact a deceptive error. While local tidal acceleration (say, at the event horizon) can be made small by increasing the mass of the black hole, increasing the mass of the black hole also increases the path lengths over which the tidal effects are integrated:<br/><br/><br/><br/>Test particles under transverse tidal acceleration along some circumferential displacement will have a kinetic energy increase proportional to their mass, times the tidal acceleration, times the displacement. (Clarification: kinetic energy relative to other freefalling test particles would increase because we are considering accelerations relative to other test particles.) Thus, in the case of the transverse tidal acceleration near black hole event horizons, one can not simply “scale away” the effects of tides by resorting to arbitrarily massive holes because the characteristic distances over which the accelerations take place themselves scale proportional to the mass of the hole, which cancels the scaling of the accelerations inversely to the mass of the hole. The energy effects of tidal accelerations (integrated over distance) are thus insensitive to changes in the mass of the black hole and can not be “scaled away”. All of which leads to some interesting consequences.<br/><br/><br/>A thought experiment<br/><br/>Imagine a freefalling observer heading towards the event horizon (at rS) and imagine a test particle, say a photon of Hawking radiation, that is also freefalling along with the observer, but along a different radial. (Here we are simply noting that the photon is traveling along a geodesic path and that at a particular point, where the radial mentioned intersects the geodesic, the photon is falling, just like the observer.) The observer sees that the test particle is accelerating towards him due to the transverse tide. Assume 1) observer is falling along a radial with no significant angular momentum, and 2) the Hawking photon is in a closed, “terminal” orbit (one that would begin and end on the event horizon) with orbital angular momentum such that, after emission, it follows a curved path from the event horizon to the observer.<br/><br/>Imagine a diagram of the black hole set up so that rS is the face of a clock, with "12 o'clock" at the diagram's top, and the observer outside the hole's horizon and falling towards it along the “3 o'clock” radial path. Imagine that the test particle of Hawking radiation that is on an intercept path from the horizon to observer was emitted from high noon and is following a (curved looking) geodesic from “noon” to intercept the observer.<br/><br/>From the perspective of the free falling observer, the Hawking photon will be blueshifted by the transverse tidal acceleration along its intercept path because both are accelerating along converging radial paths. The total integral of acceleration over the photon's path will (I think?) be something like the relativistic form of<br/><br/><br/><br/>(With the caveat that the photon is actually following a geodesic – a straight line in its freefalling frame – which may lead to something very different from the equation above! Hence, my need for help with this stuff.)<br/><br/>Additionally, the integral of acceleration over distance gives the Hawking photon an exponential blueshift just like the redshift of a photon leaving the vicinity of the event horizon is exponential. As the particle is accelerated it picks up energy (from dE = F⋅ds = ma⋅ds), but since energy has mass, the increment of energy the particle picks up over a distance ds1 also picks up more energy during the next distance interval, ds2. And so on, as exponential growth.<br/><br/>If the Hawking photon follows a sufficiently long path (in terms of angle φ) to intercept the freefalling observer from whatever patch of horizon emitted it, then it will have been accelerated through a total potential comparable in magnitude to the potential from flat space to the region of space at the horizon which emitted it. Let us compare the radial Newtonian potential from rS to infinity with the transverse tidal Newtonian potential.<br/><br/>The Newtonian potential from rS to infinity is Ur = 2GM / rS where rS = 2GM / c2 so that Ur = c2 , which is the total rest energy of matter transported from rS to flat space. The maximum possible (but see below) "Newtonian" potential for a transverse tidal blueshift from all the way around the backside of the hole's horizon is (something like) Ut = πGM / rS = πc2 / 2 , which is larger than the total potential of the matter's rest mass.<br/><br/>However, the crude approximation above does not include the fact that in the curved spacetime near rS , the total angle around the hole, and projected onto the spacetime, is probably less than 2π because of its “funnel-like” structure. Also, due to the exponential nature of the relativistic accelerations (where accelerated mass takes on more mass to accelerate), the Newtonian potential is not the correct thing to use, but the comparison does give an indication that the transverse tidal acceleration could in fact be as significant a blueshift as the event horizon's redshift. The two Newtonian potentials are of similar magnitude, so the relativistic potentials are likely significantly similar (though one is a redshift while the other is a blueshift). Again, I need help with this.<br/><br/>As noted above, due to the scale invariant nature of the product of transverse acceleration and path length, the blueshift factor is the same for any path starting at φ1 and ending at φ2 about a black hole because the acceleration scales inversely with the hole's mass while the length of the path from φ1 to φ2 scales with the mass of the hole (for paths at a given radius relative to the Schwarzschild radius (r / rS)). In other words the proposed blueshift effect seems to be insensitive to the black hole's mass.<br/><br/>In our example, continued from above, as the observer approaches the horizon in freefall he sees a ring of blueshifted Hawking radiation that has been emitted from around the edge of the event horizon (from "over the horizon" of the event horizon). It has been blueshifted more or less dramatically, depending on how far around the hole it has traveled. As I mentioned above, the shift would be exponential with angular distance around the hole. And the radiation is blueshifted by the same factor for any hole. This leads to a multiplication of the Hawking temperature, as seen by a freefalling observer passing close to the event horizon, by some gigantic, but universal factor.<br/><br/>But where is all the energy coming from to amplify the Hawking radiation? It would have to be coming from the energy of the hole itself. As the observer freefalls towards the horizon he is baked by a ring of blueshifted Hawking radiation and the energy in that radiation is the hole evaporating as he falls into it. The (blueshifted) hawking radiation goes to higher and higher temperature as the hole evaporates “under his feet” – and the observer never reaches the horizon or the hole.<br/><br/>And the best part is; The mechanism that demands this blueshifted "ring of fire" is none other than the Equivalence Principle itself because an acceleration, tidal or not, is the same as gravity. This looks a lot like a good solution to the Firewall paradox, because instead of violating the Equivalence Principle, this version of a Firewall is demanded by it.<br/><br/><br/>Some additional thoughts<br/><br/> • Can we still use spherical coordinates in the spacetime near the event horizon? Its clear that they work somewhat near the hole, but is this a valid way to think about the dynamics near the horizon?<br/><br/> • The embedding diagram for a Schwarzschild black hole is somewhat funnel-like near the event horizon. As noted already, that “funnel-like” structure would decrease the total projected angle φ on the spacetime embedding sheet. The total angle around the event horizon is 2π (a circle on the sheet) but the integral of the angles between radial "normals" of the horizon would be less than 2π because on the sheet they are tilted and lie on the surface of the "conic" portion of the embedding sheet. (However, I don't think the total angle is zero because the embedding sheet is not a parallel-walled "tube" at the event horizon like the way the gravitational potential diagram is.)<br/><br/> • It my well be that the perceived transverse tidal acceleration near the event horizon goes to zero because of spacetime curvature issues. This would be the case if a) an observer falling to the event horizon sees the horizon as a flat surface because of the curvature of light rays, and b) gravity follows light paths. Or is this redundant with my caution about the embedding surface – just two ways of looking at the same effect? In which case the horizon does not go all the way flat for an observer falling through it or else the embedding surface would necessarily have parallel sides – which it doesn't at the horizon.<br/><br/> • What is the population of strongly blueshifted Hawking photons? Even if there is a blueshift due to transverse tidal accelerations it still may not count for much if the total energy in the blueshifted radiation is small. Turns out that there may be an answer to this (see below), and that the numbers could be huge, with remarkable implications.<br/><br/><br/>Implications?<br/><br/>If the blueshift actually exists the way I think I'm proposing here, then there is a super-intense field of trapped Hawking radiation near the surface of black holes. Objects falling through this radiation field would be subjected to optical forces, like the way optical tweezers work (see <a href="https://en.wikipedia.org/wiki/Optical_tweezers">https://en.wikipedia.org/wiki/Optical_tweezers</a>), and would be very strongly damped in their motions because the object's orbital motion (relative to the hole) would cause trapped Hawking radiation coming from the direction of the object's angular motion to be blueshifted a little more than radiation coming from behind its angular motion. A small blueshift of a very large number of photons can be a very large force. So, how many photons would their be in the trapped "optical atmosphere" about a black hole?<br/><br/><br/>Figure 2.1 from Leonard Susskind's Black Holes, Information and the String Theory Revolution showing the effects of the centrifugal barrier potential on escaping Hawking photons. Note the term “Region of Thermal Atmosphere”.<br/><br/>I've been reading Leonard Susskind's Black Holes, Information and the String Theory Revolution. On page 27 there is a figure showing the "effective (centrifugal) potential for free scalar field vs Schwarzschild potential". From the figure it is clear that the only Hawking photons that can escape the hole's potential – at all – are those with extremely low angular momentum. Low being less than about 3 or 4 units of orbital angular momentum. For what I just typed here to have any meaning I need to provide some context.<br/><br/>The initial wavenumber of a Hawking photon at the Schwarzschild radius is about 1035 / m (an escaping Hawking photon gets redshifted from near the Planck energy to an energy corresponding to wavenumber of ∼1/rS when climbing out of the hole's radial potential, as seen by distant observers). If the Schwarzschild radius is, say, 2 km in radius (for a ~1 solar mass hole) then the angular momentum of that Hawking photon could (for a photon with wavevector tangent to the hole's surface) be as high as<br/><br/><br/><br/>Now, if only the photons that are nearly S-wave (0 ≤ ℓ ≤ 5) have any chance to escape, and the allowed orbital angular momentum can go as high as 1040 , then there can be a really huge number of photons trapped near the hole. (The reason why the photons with ℓ >5 can't escape is that their wave vector is not close enough to exactly normal to the hole's surface.) And by "really huge" I mean that given all possible angular momentum states from 0 to 1040 in the "u" direction about the hole, times an equal number of states in the perpendicular "v" direction, we have a total available state space for the photon angular momentum of about 1080.<br/><br/>Actually this 1080 is an incorrect estimate that I made for an earlier draft of this essay. It represents only the available trapped photon states originating at a single patch of the horizon. The more correct number would be found by integrating over the whole surface similar to how the number of phonon states are found in the Debye model but taking the Debye temperature as the Planck temperature. I think. Anyway, the point is that it is a really huge number of trapped photon states in the vicinity of the hole's horizon and the real calculation is unimportant for the purpose of this essay.<br/><br/>Of those 1080 states, all but ~ 10 are states of trapped Hawking photons comprising the “thermal atmosphere” noted on page 27 of Susskind's book. That is why the emission rate of Hawking radiation, as seen from far away from the hole, is so small – most of the available photon states have too little of their momentum normal to the event horizon to escape.<br/><br/>However, if there are 1080 states available for trapped Hawking radiation, then there are likely to be a really huge number of photons in the trapped "atmosphere". Even if the energy of the individual photons is small there can still be a super intense optical field that infalling matter would encounter, near the Schwarzschild radius, on its way into the black hole. For example, even if we treat the photons as having only the energy they'd have after climbing out of the hole's potential (in other words as ordinary Hawking radiation at T ∼ 10-7 K seen from afar), then the total energy in the optical atmosphere could still be huge:<br/><br/><br/><br/>Where Eγ is the energy of a single (escaped, low angular momentum) Hawking photon as seen from far away from the hole, rS is the Schwarzschild radius, and N is the approximate number of photon states available. Putting this in terms of the black hole's mass we get<br/><br/><br/><br/>Which, for a ~1 solar mass black hole, is about as much mass as the hole itself! And, since the number of available trapped photon states, N, scales with the surface area of the event horizon, and the photon energy of each mode, as seen from far from the hole, scales as the inverse of the hole's radius, we have yet another scale invariant quantity. The trapped photon atmosphere is of the same relative mass magnitude independent of the black hole's mass.<br/><br/>Clearly there is a balance struck where the mass of the hole isn't outside the hole, or there would be no event horizon to generate the blueshifted trapped Hawking radiation, but I think you might be getting an idea for the energy scale in spite of the gross approximations I've made. Now, with that said, here are a couple of testable consequences. If there is a substantial portion of the hole's mass trapped outside the horizon, in the form of a trapped particle atmosphere, then:<br/><br/> • There ought to be a modification to the spacetime metric near the exterior of the hole's horizon. This modification of the metric ought to be observable as a change in orbit shape and decay rate, due to a departure from a Schwarzschild solution, during the very end stage of black hole collision/coalescence by gravity wave detectors.<br/><br/> • There ought to be a slight modification of the gravity wave emission that would correspond to the huge drag effects of the trapped atmosphere during the very end stage of black hole collision/coalescence by gravity wave detectors.<br/><br/>Here I might suggest that a better set of calculations be made to show that the trapped radiation is a prediction of extending Hawking's idea to include the interaction between the trapped radiation and the transverse tidal blueshift. If the existence of the trapped radiation is shown to be an unavoidable consequence of Hawking's theory, then there is a method of testing the theory. And from there it doesn't matter whether experimental evidence shows the existence of the trapped radiation or not because progress will thereby result from showing the existence or non-existence of Hawking radiation. Also, if Hawking radiation exists, then there is a need for a Black Hole Firewall, but if Hawking radiation does not exist, then there is no need for a Firewall. Thus, the proposed solution for the Firewall Paradox proposed here is nicely matched to whether it is actually needed or not.<br/><br/>And that's about all I think I know about that. It has been about 20 years since I was any good at manipulating the mathematics required for solving problems in General Relativity. Any help with that would be most appreciated.<br/><br/><br/>Afterword<br/><br/>Since this is a revised version of the essay I have had a bit more time to think and read about the ideas presented. Having now read the 1975 paper by Stephen Hawking titled Particle Creation by Black Holes [hawking1975] I have a few additional observations.<br/><br/>First, I can't see where in the 1975 paper Hawking does anything that would include the transverse tidal acceleration or its effects on the local radiation field. I see several places in the paper where approximations were made such that only radial factors would be included in the calculations that followed.<br/><br/>The approximations I saw were:<br/><br/> 1. The use of Penrose diagrams, which exclude any effects that might arise from paths around the hole because they include only radial and time coordinates.<br/><br/> 1. The use of restricted, local patches of spacetime "of radius ~M", which will again exclude tidal effects by their very construction.<br/><br/>These approximations would effectively exclude what I'm trying to point to in this essay from consideration in Hawking's original work showing black hole radiation.<br/><br/>Section 4 of the paper, "The Back-Reaction on the Metric", contains a description of an observer falling through the event horizon. This looked promising until I noted that the analysis was carried out in a local patch of the spacetime "of radius ~M" at the horizon (see #2 above). The analysis contained some consideration of Hawking modes with angular momentum encountering a centrifugal barrier, but it was only concerned with the modes that had low enough angular momentum to escape to infinity not with (trapped) modes behaving as pointed to here.<br/><br/>More recently it has been pointed out to me that the wave equation used in Hawking's 1975 paper must have been<br/><br/><br/><br/>and that the potential term must have been<br/><br/><br/><br/>But the given wave equation here has no derivatives in directions other than r and t – nothing in the angular directions. Therefore it can not include momentum transport in the spatial directions transverse to the radial direction. This makes sense for an analysis of radiation effects from black holes as seen from far away (in the “far field”), but is not (I think) a complete description of what may be happening close to the hole (in the “near field”).<br/><br/>However, the gist of the question posed in my essay is the interaction between a radially infalling traveling wave (or particle, or observer) and a trapped wave/particle in the thermal atmosphere near the event horizon. I am specifically not puzzling here about the waves that escape to infinity that are the Hawking radiation itself.<br/><br/>Further, I would have thought that if there was a contribution to black hole radiation from the effect I'm proposing then there ought to be a difference between Hawking radiation and Unruh radiation. But there is no difference in their forms. So either the contribution from the effect proposed here is identically zero, or it was left out of the calculation. I'm guessing the later.<br/><br/>Lastly, and ideally, if Stephen Hawking missed what I think he may have missed it would be best if the ideas were forwarded to him so that he has first crack at resolving the issues I may have chanced to find. It would be the polite thing to do. However, we do not live in an ideal world and I'm quite willing to accept any help I can get.<br/><br/><br/><br/><br/><br/>Merry 2016 Christmas, Stephen Hawking. Hopefully this article can be the "wrapping paper" for a really excellent christmas present in the form of experimental proof of some of your work.<br/><br/>Critique of Eagleworks Lab's test of a supposedly reactionless thruster
http://www.davidwoolsey.com/AttO/AttO_blog/Entries/2016/12/21_Critique_of_Eagleworks_Labs_test_of_a_supposedly_reactionless_thruster.html
2ac10e7d-3a81-472a-9088-8233b42fd486Wed, 21 Dec 2016 13:22:01 -0800<a href="http://www.davidwoolsey.com/AttO/AttO_blog/Entries/2016/12/21_Critique_of_Eagleworks_Labs_test_of_a_supposedly_reactionless_thruster_files/Newtons_3rd.jpg"><img src="http://www.davidwoolsey.com/AttO/AttO_blog/Media/object000_1.jpg" style="float:left; padding-right:10px; padding-bottom:10px; width:183px; height:137px;"/></a>As I've said on the AttO intro. page, it is important that there be investigations into speculative science, both in terms of performing it and in terms of investigating the performance of those experiments. It is very important that the results of speculative science be properly looked into for flaws so that quality control standards are enforced on its investigators. Otherwise valuable lines of speculation may well descend to the level of junk science through lack of proper "adult supervision". Most ventures into the speculative should rightfully be expected to lead to dead ends or other failures -- and I think the whole category of "reactionless drives" fits this description.<br/><br/>So, getting right to it. Starting a few years ago there have been an ongoing series of stories about a so-called reactionless thruster for space travel circulating on the internet. In fact, there were several stories, about several variations of a reactionless thruster. Some of them were called Q-Drive or Cannae Drive or EmDrive, with the most recent set of stories being about the one at the NASA Eagleworks Lab called the EmDrive. The EmDrive is the one I'll be critiquing here. However, I expect that the critiques apply to all the other variations of the reactionless drives because, as far as I can tell, they are all being tested in the same flawed way.<br/><br/>In August of 2014 there was a set of news stories about the so-called emDrive being tested at a NASA facility called the Eagleworks. The Eagleworks tests were in response to research performed at the Northwestern Polytechnical University (NPU) in Xi’an, Shaanxi, China, on microwave-driven resonant cavity thrusters from 2004 through 2013. The Eagleworks tests studied variations on what they propose use the (supposed) quantum vacuum plasma thrusters (QVPT). In that same month of August 2014 I posted a brief critique of the tests to the Hackers Conference's mailing list after the news items surfaced there. Thus, this weblog article is largely derived from my Hackers list posting and its followups because the issues I cited then are still relevant now.<br/><br/>The news item then was a 2014 Wired story titled 10 questions about Nasa's 'impossible' space drive answered (<a href="http://www.wired.co.uk/news/archive/2014-08/07/10-qs-about-nasa-impossible-drive">http://www.wired.co.uk/news/archive/2014-08/07/10-qs-about-nasa-impossible-drive</a>) and it referenced a paper titled Anomalous Thrust Production from an RF Test Device Measured on a Low-Thrust Torsion Pendulum (<a href="https://ntrs.nasa.gov/search.jsp?R=20140006052">https://ntrs.nasa.gov/search.jsp?R=20140006052</a>) by D. Brady, H. White, P. March, J. Lawrence, and F. Davies. The paper provided details about the test rig and about the objects being tested. The test procedures seemed, at first reading, to be careful and controlled. The more recent paper about the EmDrive that is inspiring me to post is titled Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum (<a href="http://arc.aiaa.org/doi/10.2514/1.B36120">http://arc.aiaa.org/doi/10.2514/1.B36120</a>) by Harold White, Paul March, James Lawrence, Jerry Vera, Andre Sylvester, David Brady, Paul Bailey again seems to be a very careful study.<br/><br/>However, there is a significant physical effect that the vacuum isolation chamber itself is likely causing and that the experimenters are misunderstanding as a "reactionless" thrust result that is present in all of the test setups.<br/><br/>The WikiPedia article on RF resonant cavity thruster (<a href="https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster">https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster</a>) does a good and reasonable job of presenting the history of testing on this class of so-called "reactionless" drive devices. The WikiPedia article even has a subsection titled "Radiation Pressure" that almost details the source of the errors being made in all the tests. Almost, but not quite.<br/><br/>Here are my observations about the Eagleworks test apparatus and procedures together with my suggestion for how to prove me wrong.<br/><br/>Firstly, the Eagleworks have done some COMSOL modeling of the electromagnetic response of the test objects. However, the modeling seems to assume free-space conditions for the test object and I am unconvinced that they are modeling the resonant response of the test objects together with the electromagnetic environment in which they are being tested. Which leads to the next observation.<br/><br/>Secondly, the tests were carried out within an enclosed, conductive chamber. I know from my own (1D) experience with laser tubes and optical resonators that electromagnetic emissions within an enclosed high Q-factor cavity can set up some counterintuitive behaviors. For example, a HeNe laser cavity that is not energized can serve as a very highly reflecting "filter" to an externally applied pulse that has its center frequency tuned to the resonance on the tube -- if the pulse is short enough in duration. The reason for the transient reflective properties of such a laser tube is that the light entering the cavity within the tube takes a while to "ring up" to a strong enough resonance that the quantity of light exiting the far end of the tube becomes significant. This is sort of the inverse example of what I'm going to describe, but it is one that I have direct experience with, so I'm including it. The transmissive properties of the laser's cavity depend on its resonant Q being able to overcome the reflectance of its mirrors. If the cavity is a poor resonator, then the mirrors bounding it are just that -- mirrors. In which case, incoming light is reflected from the laser without passing through it. Perhaps this is a poor example which does not serve to illustrate well. A better example is as follows.<br/><br/>I know of persons who have experimented with cell phones enclosed within metal pipes -- both ends sealed by screw on caps -- who have found that the cell phones can still communicate with the outside world. This happens because the EM radiation trapped within the pipe's enclosure "rings up" to a very high intensity such that the rate of loss from the trapped fields balances with the rate of generation of the radiation within the enclosure. A significant portion of the losses are in fact radiation escaping from the non-perfect seals of the enclosure, while the rest are resistive losses of the walls of the pipe. The solution to the high intensity radiation field that causes the EM leakage is the inclusion of an absorber within the enclosure to reduce its Q factor. By reducing the cavity's Q-factor the internal resonances are prevented from building up to the point where a significant RF signal can get out of the shielded chamber (pipe). Conversely, the inclusion of damping material will prevent incoming RF from leaking in in small quantities and building up to a large enough resonant level that the cell phone inside the pipe can "take the call".<br/><br/>Finally, in the case of the EmDrive (and similar), if the RF fields inside the vacuum test enclosure itself are allowed to build up -- because the enclosure acts like an electromagnetic "hall of mirrors" within which the test object is placed -- by a factor of the enclosure's Q, which is probably on the order of 10^4, then we can account for the observed forces produced by the test objects. And here is where the irony lies.<br/><br/>In the WikiPedia article on RF resonant cavity thruster (<a href="https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster">https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster</a>), in the subsection on Radiation Pressure, we have the following material. It is a rationale for the apparently observed thrust by one of the inventors, Roger Shawyer:<br/><br/>Shawyer has suggested thrust is caused by a radiation pressure imbalance between the two faces of the cavity. He gave a presentation on this at the International Astronautical Congress 2014, later publishing it in the peer-reviewed Acta Astronautica. In it he wrote, In an EmDrive engine, microwave energy is converted to mechanical force according to the thrust equation, derived from the basic radiation pressure equation: F = 2 P0 / c. Shawyer's thrust equation, derived from Allen Cullen's equations, is given by:<br/><br/><br/><br/>where F is the force, P0 is the incident power, c is the speed of light, Qu is the unloaded Q factor of the cavity, {lambda}0 is the wavelength of the microwaves in free space, and {lambda}g1 and {lambda}g2 are the wavelengths at the end of the largest and smallest cross-section, respectively.<br/><br/>Shawyer insists the EmDrive is an open system. However, physicists point out that relying only on special relativity, without emitting anything and with no interaction with an outside field or matter, makes his drive a closed system. Since the two end plates are part of the thruster and the microwaves are trapped inside the cavity, standard Einstein–Maxwell equations and the conservation of momentum show no effective thrust can occur due to any force on the cavity caused by internal electromagnetic energy.<br/><br/>Unfortunately, in his analysis here, Shawyer was intending to discuss the interactions of the electromagnetic radiation on the inside of the test object, not that portion of it escaping out of the test object and into the interior of the test chamber itself! He was on the right track, he just did not apply his reasoning to the external cavity that the test objects were encased within. Let us take a whack at that now.<br/><br/>By the equation given above, F = 2 P0 / c, electromagnetic radiation gives a force of 6.7*10^-9 N/W.<br/><br/>The test objects were experiencing a force of about 10^-6 N per Watt of input power, which is about 300 times higher than the expected "raw" thrust one would expect from just the recoil from the EM radiation emitted from the test objects (assuming it all went in one direction).<br/><br/>However, the environment that the test objects are in is an enclosed conductive cavity that will have a complicated structure of high intensity standing waves built up in it by any radiating source placed within. Worse yet, the newest incarnation of the test setup has an automatic tuning circuit that is used to find the frequency of highest power resonant behavior of the test ensemble. Which is to say, if there is any leakage power between the interior of the test object and the interior of the test chamber then the automatic tuner will tune the system "correctly" to match that coupled resonance. None of this cavity coupling is being accounted for because if it were we'd see some form of effective absorber incorporated into the test chamber.<br/><br/>I expect that the "thrust" effects that they are seeing are the result of the intensified microwave radiation field within the test chamber itself. If the test chamber's cavity Q is on the order of several thousand (not unreasonable, given that microwave ovens have a Q something like 50,000) then the resulting electromagnetic field intensity will be correspondingly higher than the few Watts they are inputting into the device. The intense field is acting very much like the fields in an optical tweezer -- pushing the test object around in the gradients of the (complicated) standing wave pattern. Therefore, the resulting "thrust" is simply an electromagnetic force acting between the object under test and the walls of the enclosure.<br/><br/>This assessment of the observed force's origin is buttressed by the reports in the 2014 work that the only time there was no force was in the case where a 50 Ohm resistor was used in place of the test object. That resistor would supply "proper termination" to the RF feed and kill most of the leakage from the test object into the test chamber.<br/><br/>If they were to include a few square feet of absorbing material on the interior walls of the test enclosure I'd bet that the "thrust" effects they are reporting would diminish by about 99%.<br/><br/>Back in 2014 I tried to contact Harold White to make my suggestions for damping the test chamber. I telephoned and left voice mail, I sent email (titled <a href="http://davidwoolsey.com/AttO/local_copies/Of_Space_Drives_and_Stabilized_Lasers.html">Of Space Drives and Stabilized Lasers</a>), I even asked friends at NASA to "hunt him down" and relay what I've presented here. No effect. I've waited for over two years for someone, anyone, to do the test of reducing the cavity Q of the test chamber, or even just suggest doing it. No effect.<br/><br/>Eagleworks: I want to be proven wrong. Do a test run with a proper RF absorber in the test chamber environment and see what happens.<br/>Critique of test of Andrea Rossi's E-Cat Reactor (a supposed LENR / cold fusion device)
http://www.davidwoolsey.com/AttO/AttO_blog/Entries/2014/10/25_Critique_of_test_of_so-called_E-Cat_Reactor_%28a_proposed_LENR___cold_fusion_device%29.html
7275beee-b557-435c-bf01-9c7c47a67e3fSat, 25 Oct 2014 21:19:20 -0700<a href="http://www.davidwoolsey.com/AttO/AttO_blog/Entries/2014/10/25_Critique_of_test_of_so-called_E-Cat_Reactor_%28a_proposed_LENR___cold_fusion_device%29_files/e-cat.jpg"><img src="http://www.davidwoolsey.com/AttO/AttO_blog/Media/object012_1.jpg" style="float:left; padding-right:10px; padding-bottom:10px; width:183px; height:137px;"/></a>This entry is about Anrea Rossi's Energy Catalyzer device. It is largely based on emails I wrote about it to the Hackers Conference mailing list back in October of 2014. Take a look at the WikiPedia page for the E-Cat for a brief summary of the device's "life and times" (<a href="https://en.wikipedia.org/wiki/Energy_Catalyzer">https://en.wikipedia.org/wiki/Energy_Catalyzer</a>).<br/><br/>I am a skeptic of the E-Cat device. I believe this is the only reasonable position to start from: it is entirely "on them" to prove beyond doubt that the device does what they claim. I've read through most of the report on the testing of Anrea Rossi's E-Cat Reactor found at the University of Bologna’s AMS Acta Digital Library<br/>(global web link: <a href="http://amsacta.unibo.it/4084/1/LuganoReportSubmit.pdf">http://amsacta.unibo.it/4084/1/LuganoReportSubmit.pdf</a> or local link: <a href="http://www.davidwoolsey.com/AttO/local_copies/LuganoReportSubmit.pdf">LuganoReportSubmit</a>.pdf) and find it at first /seems/ like a reasonable set of tests and report.<br/><br/>I was originally going to look at the energy budget implied by the alleged transmutation of Ni isotopes and see if the energy "measured" was in the same ballpark as what would have been evolved through the alleged nuclear processes. But I ran into something on the way when I thought about how the "measurement" was being done. It was one of those "Ah! its all so obvious now" moments.<br/><br/>The measurement of thermal energy departing the device was done using an IR camera and making the assumption that the emissivity of the hot material is known. If you know the temperature, and the emissivity (as a *function* of wavelength and temperature!) then you can derive the total energy flux from the Stefan-Boltzmann formula (I = epsilon sigma_B T^4, where I is the radiation intensity [W/m^2] epsilon is the emissivity, sigma_B is the Stefan-Boltzmann constant [5.67x10^-8 W/m^2/K^4], and T is the temperature [K].)<br/><br/>However, it turns out that they did not characterize the actual emissivity of the alumina material that the casing of the device itself is made of. Instead they relied on the "fact" that, since they chemically tested the alumina material and found it to be 99% (or better) pure alumina that they then assumed it would have the same emissivity as generic, pure alumina. From page 8 of the report we have:<br/><br/>From the analyses performed on the sample taken from the reactor, we determined that the material constituting the outer shell is 99% pure alumina (Appendix 2); better yet, that impurities, if present, are below the experimental limit of measurement. We therefore retrieved from the literature [3] a discrete-point plot of the emissivity of said material as a function of temperature (Figure 6), and extracted from it the values necessary to reproduce the trend as a continuous line (Plot1).<br/>In making the assumption about the alumina material's emissivity they are making at least two critical errors:<br/><br/> 1. They are assuming that the surface texture of the exterior of the casing is irrelevant to its emissivity. It is not; the texture at optical scale can have a dramatic influence on the surface emissivity -- and the material is *clearly* textured, so its emissivity is not actually known unless measured.<br/><br/> 1. They are assuming that the casing has a uniform emissivity. Specifically, that it has the same emissivity on the side facing the IR camera that it does everywhere else.<br/><br/>Using x-ray crystallography testing to determine the bulk constitution of the material (Al2O3) potentially tells nothing about its surface optical properties in the IR and visible spectrum. However, it is obvious from the photographs of the device (Figure 2 on page 3) that the surface texture of the casing material is markedly different from the industrial alumina used for the tubes holding the wires at either end of the device.<br/><br/><br/>Questions about error item 1: Why is the material used for the device casing of such a different surface texture than for the tubes on either side of the device? Why would someone go to the bother of texturing the surface of these alumina parts? Why would they be different from the "glazed" industrial parts provided to hold the assembly in place on the test stand? Texturing the surface of alumina is not an easy thing to do because it is a very hard and tough material.<br/><br/>Error item 2 is even more "interesting". It may be that the emissivity of the back side of the device, facing away from the IR camera, is much lower than the emissivity of the side facing the camera. If this were the case then the thermal radiation would be forced to exit the device primarily towards the camera, leading to higher temperature reading than would be the case if the emissivity were uniform front/back because of the smaller effective area available for radiant energy loss. If the radiation is then falsely assumed to be isotropic, but is instead much higher in the direction towards the camera, then the calculation for the total radiation budget will be erroneously high.<br/><br/>It would be my contention that, if this is a case of fraud, then the two "errors" presented here could account for the apparent 3.5 times multiple of output power. The "nuclear" aspect of the device is simply a "magician's slight of hand" redirection of attention away from where the manipulation is likely happening.<br/><br/>Note that the test device with fuel was a separate specimen from the dummy device without fuel. The fuel is likely irrelevant, it is the device casings that could be different in significant ways. The fact that Rossi handled the fueling was bad enough, but he was also the one who positioned the fueled device (presumably so that the correct side faced the IR camera).<br/><br/>It would be interesting, and probably simple enough, to catch this guy in the act of fraud -- if that is what he's up to. Get hold of the devices and map the emissivity of the whole surface of each one in the whole relevant optical spectrum. Rossi should not object to this examination because the exterior of the device is not supposed to be the "secret sauce" that causes the device to have such an anomalously high apparent thermal power output.<br/><br/>But, hey, what do I know? I'm just some dude on the interweb.<br/><br/><br/>Epilogue<br/><br/>I waited so long to publish this critique of the E-Cat "reactor" because it occurred to me while writing it that if the mechanism I've proposed here of changing the emissivity of the front and back of the test device is what is causing the reported effect then there is a huge market for it in another application. That being electric heating elements for stovetops and other applications where a directional application of radiant thermal power is a more efficient method of heating.<br/><br/>Notice that on a typical electric stove there is a reflector underneath each of the heating elements. On just about every stove I've seen, that reflector is usually grimed up with carbonized food and does a poor job of reflecting heat. Here is a product solution for increased energy efficiency of ordinary stovetop elements:<br/><br/>Make the bottom of the element itself have a low emissivity in the IR. That would force the top surface of the element to emit more of the thermal radiation -- in the direction of where the food is. An obvious benefit of a design that puts the reflector on the back side of the heating element is that the modified surface is not exposed to mechanical abuse in the course of, for example, cooking.<br/><br/>The engineering problem here is that the thermal expansion of the element might make it difficult to make a coating that stays on the element without flaking off or otherwise degrading. So I sat on the idea and did not publish this weblog ... until I met someone who is founding a company based on flexible high temperature ceramics that would do the job of providing a modified emissivity to the heating elements without flaking off. So I suggested he look into making a product line for efficiency improvements of stovetop heating elements, waited for almost a year, and now I feel that is enough of a head start for him that I can go forward and publish this page.<br/><br/>See, investigating crazy proposals can lead to non-crazy spin-off results!<br/>