Black Holes and Transverse Tidal Effects,
a short essay on some thoughts
Black Holes and Transverse Tidal Effects,
a short essay on some thoughts
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.
History. It all began a week or so before the UCB Physics department Colloquium on Black Hole Firewalls given in 2013 by Raphael Bousso (http://physics.berkeley.edu/black-holes-and-firewalls-can-we-take-nothing-for-granted). 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.
"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.
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 LIGO black hole echoes hint at general-relativity breakdown (http://www.nature.com/news/ligo-black-hole-echoes-hint-at-general-relativity-breakdown-1.21135#comment-3044303675) 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.
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.
Preface
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.
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.
Motivations
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.
1.The existence of something very much like a black hole firewall (that would in fact be demanded by the Equivalence Principle).
2.The resolution of the paradox of stationary vs. infalling observers w.r.t. Hawking radiation and black hole evaporation.
3.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.
4.Existence of a high viscosity, high density photon field in the near vicinity of black hole event horizons.
5.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!
6.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.
7.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.
8.I'm sure there is more that I haven't thought of yet, so let me say "etc."
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.
Effects of tidal accelerations on horizon radiation
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.
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:
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.
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.
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.
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.
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.
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).
Tidal accelerations about gravitational central potentials
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).
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.
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.
For small displacements along the radial direction from the gravitating body, the radial component of the tidal acceleration goes by
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
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.
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:
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.
The equation for the transverse component of the tidal force is something like
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).
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.
The idea
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:
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.
A thought experiment
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.
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.
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
(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.)
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.
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.
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.
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.
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.
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.
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.
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.
Some additional thoughts
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?
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.)
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.
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.
Implications?
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 https://en.wikipedia.org/wiki/Optical_tweezers), 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?
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”.
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.
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
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.
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.
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.
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:
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
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.
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:
•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.
•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.
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.
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.
Afterword
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.
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.
The approximations I saw were:
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.
2.The use of restricted, local patches of spacetime "of radius ~M", which will again exclude tidal effects by their very construction.
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.
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.
More recently it has been pointed out to me that the wave equation used in Hawking's 1975 paper must have been
and that the potential term must have been
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”).
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.
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.
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.
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.
Sunday, December 25, 2016
by David Woolsey