|Subject: The basic basics of "gravitational
waves", if any
Date: Thu, 13 Jan 2005 19:22:03 +0200
From: Dimi Chakalov <firstname.lastname@example.org>
To: Richard H Price <email@example.com>
CC: Angelo Loinger <Angelo.Loinger@mi.infn.it>,
Jorge Pullin <firstname.lastname@example.org>,
Eanna Flanagan <email@example.com>,
Scott Hughes <firstname.lastname@example.org>,
Tim Smith <email@example.com>,
Joseph H Taylor <firstname.lastname@example.org>,
Jacqueline N Hewitt <email@example.com>,
Karel Kuchar <firstname.lastname@example.org>
The authors of a recent paper on the so-called GWs, Éanna Flanagan and Scott Hughes, stated the following: "We note that Richard Price requested this article very nicely." [Ref. 1]
May I offer my comments on their efforts. I believe you and Jorge Pullin are editors of the January 2005 issue of the New Journal of Physics, "Spacetime 100 Years Later", and the article you've kindly requested will appear there.
This whole story is not about "skeptics (and crackpots)"
[Ref. 1]. It's about Einstein's legacy (and a couple of
billion dollars, taxpayers'
My comments will be very brief, but I am ready to expand
Very briefly, I believe Hermann Weyl and Angelo Loinger have demonstrated, with agonizing clarity, that the so-called gravitational waves cannot exist. Please see
Angelo Loinger, On the origin of the notion of GW et cetera,
You can read there Hermann Weyl's "How far can one get with a linear field theory of gravitation in flat space-time?", Amer. J. Math. 66 (1944) 591.
I'm afraid the authors of [Ref. 1] are not aware of this article by H. Weyl nor have read any paper by A. Loinger.
As to the famous PSR1913+16 and the decay of its angular momentum and energy, please see the exercise on p. 4 in
Angelo Loinger, On PSR1913+16, astro-ph/0002267 v1,
What are the FACTS supporting the GWs? If you look at
Another illustration are the audio files made by Scott Hughes,
They do sound nice, as an illustration. You can hear also the sound file by John Cramer,
All this will remain an illustration until somebody actually detects the hypothetical "sinusoidal amplitude" of the "quadrupole radiation" of the mythical GWs [Ref. 3].
So, why are people obsessed with these GR waves? They have even managed to convince NASA and the European Space Agency to launch a satellite, LISA, into orbit in 2013 (just a couple of billion dollars, taxpayers' money).
NB: Please look at just one of the many vital issues in [Ref. 1]:
"Assuming that h_ab --> 0 as r --> [inf], we define the
We know from textbooks, such as that by Jim Hartle,
that these GR waves might become "observable" only by the requirement 'r --> [inf]'.
That is, only if we instruct LIGO and LISA to detect the "sinusoidal amplitude" of some "quadrupole radiation" of some "isolated system" "in the limit when the system becomes completely spatially isolated from everything else", as explained eloquently by Roger Penrose,
Do you really believe LIGO or LISA can catch such an animal? It's a bona fide Perennial, and is totally unobservable. We cannot have TWO distinguishable states of this Perennial to define a finite spacetime interval between two successive perturbations of the spacetime metric of this Perennial. Thus, we cannot compare the curvature generated by the GWs to the background curvature [Ref. 1].
We can speculate only about ONE state at 'r --> [inf]': "Computation of these variables at a point requires knowledge of the metric perturbation h_ab everywhere." [Ref. 1]
This is the nature of "non-locality" of gravitational energy in Einstein's GR: if you want to squeeze it into a "point", you can only define it with respect to 'everything else in the universe at r --> [inf]'. Hence your "reference object" cannot MOVE. It's frozen, like Karel Kuchar's Perennials. I believe you know what is a Perennial, since you work with Karel Kuchar,
I regret that have to be brief, but, again, I'll be more than happy to elaborate on all crucial details in [Ref. 1]. If you are interested, please read first the papers by Angelo Loinger mentioned above, and come back to me with your questions.
Also, I believe have a very simple illustration of the illusion of these waves at
Nothing wrong will happen to Einstein's GR if you fail to detect your "waves". More at
BTW my kids liked your entry on gravitation in Microsoft Encarta, and I tried to elaborate, but couldn't match your style and clarity, as you can see at
Perhaps you and your colleagues can help.
[Ref. 1] Eanna Flanagan and Scott Hughes,
The basics of gravitational wave theory, gr-qc/0501041 v1,
p. 4: "2. The basic basics: Gravitational waves in linearized gravity
"The most natural starting point for any discussion of GWs is linearized gravity.
"We will refer to h_ab as the metric perturbation; as
we will see, it encapsulates GWs, but contains additional, non-radiative
degrees of freedom as well. In linearized gravity, the smallness of the
perturbation means that we only keep terms which are linear in h_ab --
higher order terms are discarded.
pp. 7-9: "2.2. Global spacetimes with matter sources
"We now return to the more general and realistic situation in which the stress-energy tensor is non-zero. We continue to assume that the linearized Einstein equations are valid everywhere in spacetime, and that we consider asymptotically flat solutions only.
"In this context, the metric perturbation h_ab contains (i) gauge degrees of freedom; (ii) physical, radiative degrees of freedom; and (iii) physical, non-radiative degrees of freedom tied to the matter sources. Because of the presence of the physical, non-radiative degrees of freedom, it is not possible in general to write the metric perturbation in TT gauge. However, the metric perturbation can be split up uniquely into various pieces that correspond to the degrees of freedom (i), (ii) and (iii), and the radiative degrees of freedom correspond to a piece of the metric perturbation that satisfies the TT gauge conditions, the so-called TT piece.
"This aspect of linearized theory is obscured by the standard, Lorentz gauge formulation (2.15) of the linearized Einstein equations. There, all the components of h_ab appear to be radiative, since all the components obey wave equations. In this subsection, we describe a formulation of linearized theory which focuses on gauge invariant observables. In particular, we will see that only the TT part of the metric obeys a wave equation in all gauges. We show that the non-TT parts of the metric can be gathered into a set of gauge invariant functions; these functions are governed by Poisson equations rather than wave equations. This shows that the non-TT pieces of the metric do not exhibit radiative degrees of freedom.
"Although one can always choose gauges like Lorentz gauge in which the non-radiative parts of the metric obey wave equations and thus *appear* to be radiative, this appearance is gauge artifact.
"Such gauge choices, although useful for calculations, can cause one to mistake purely gauge modes for truly physical radiation.
"Interestingly, the first analysis contrasting physical
radiative degrees of freedom from purely coordinate modes appears to have
been performed by Eddington in 1922 .
"Eddington was somewhat suspicious of Einstein’s analysis , as Einstein chose a gauge in which all metric functions propagated with the speed of light. Though the entire metric appeared to be radiative (by construction), Einstein found that *only* the "transverse-transverse" pieces of the metric carried energy. Eddington wrote:
"Weyl has classified plane gravitational waves into three types, viz. (1) longitudinal-longitudinal; (2) longitudinal-transverse; (3) transverse-transverse. The present investigation leads to the conclusion that transverse-transverse waves are propagated with the speed of light *in all systems of co-ordinates*. Waves of the first and second types have no fixed velocity -- a result which rouses suspicion as to their objective existence. Einstein had also become suspicious of these waves (in so far as they occur in his special co-ordinate system) for another reason, because he found that they convey no energy. They are not objective, and (like absolute velocity) are not detectable by any conceivable experiment. They are merely sinuosities in the co-ordinate system, and the only speed of propagation relevant to them is "the speed of thought."
"... It is evidently a great convenience in analysis to have all waves, both physical and spurious, travelling with one velocity; but it is liable to obscure physical ideas by mixing them up so completely. The chief new point in the present discussion is that when unrestricted co-ordinates are allowed the genuine waves continue to travel with the velocity of light and the spurious waves cease to have any fixed velocity.
"Unfortunately, Eddington’s wry dismissal of unphysical modes as propagating with "the speed of thought" is often taken by skeptics (and crackpots) as applying to all gravitational perturbations. Eddington in fact showed quite the opposite. We do so now using somewhat more modern notation; our presentation is essentially the flat spacetime limit of Bardeen’s  gauge-invariant cosmological perturbation formalism. A similar treatment can be found in lecture notes by Bertschinger .
"We begin by defining the decomposition of the metric
perturbation h_ab, in any gauge, into a number of irreducible pieces. Assuming
that h_ab --> 0 as r --> [inf], we define the quantities (...) together
with the constraints (...) and boundary conditions (...) as r --> [inf].
p. 11, Eq (2.65): "Notice that only the metric components hTT_ij obey a wave-like equation.
"The other variables [X1], [X2] and [X3], are determined by Poisson-type equations. Indeed, in a purely vacuum spacetime, the field equations reduce to five Laplace equations and a wave equation (...).
"This manifestly demonstrates that only the hTT_ij metric
components -- the transverse, traceless degrees of freedom of the metric
perturbation -- characterize the radiative degrees of freedom in the spacetime.
Although it is possible to pick a gauge in which other metric components
appear to be radiative, they will not be: Their radiative character is
an illusion arising due to the choice of gauge or coordinates.
"Although the variables [X1], [X2], [X3], and hTT_ij have the advantage of being gauge invariant, they have the disadvantage of being non-local.
"Computation of these variables at a point requires knowledge
of the metric perturbation h_ab everywhere.
p. 12: "Can we split up the metric perturbation h_ab in
V into radiative and non-radiative pieces? In general, the answer is no:
Within any finite region GWs cannot be distinguished from time-varying
nearzone fields generated by sources outside that region. One way to see
this is to note that in finite regions of space, the decomposition of the
metric into various pieces becomes non-unique, as does the decomposition
of vectors into transverse and longitudinal pieces.
"Within finite regions of space, therefore, GWs cannot
be defined at a fundamental level -- one simply has time-varying gravitational
fields. However, there is a certain limit in which GWs can be approximately
defined in local regions, namely the limit in which the wavelength of the
waves is much smaller than length and time scales characterizing the background
metric. This definition of gravitational radiation is discussed in detail
and in a more general context in Sec. 5. As discussed in that section,
this limit will always be valid when one is sufficiently
far from all radiating sources.
"At the most fundamental level, GWs can only be defined
within the context of an approximation in which the wavelength of the waves
is much smaller than lengthscales characterizing
the background spacetime in which the waves propagate. In this section,
we discuss perturbation theory of curved spacetimes, describe the approximation
in which GWs can be defined, and derive the effective stress tensor which
describes the energy content of GWs. The material in this section draws
on the treatments given in Chapter 35 of Misner, Thorne and Wheeler ,
Sec. 7.5 of Wald , and the review articles [31, 32].
"There are two foundations for the derivation of the energy
and momentum carried by GWs :
p. 42: We thank Richard Price and Jorge Pullin for the
invitation to write this article, and are profoundly grateful to Tim Smith
at the Institute of Physics for patiently and repeatedly extending our
deadline as we wrote this article."
[Ref. 2] J.M. Weisberg and J.H. Taylor,
Relativistic Binary Pulsar
[Ref. 3] Jim Hough, Sheila Rowan, and
B.S. Sathyaprakash, The Search for Gravitational Waves, gr-qc/0501007 v1,
pp. 2-3: "For many years physicists have been facing up to the exacting experimental challenge of searching for gravitational waves. Predicted by General Relativity to be produced by the acceleration of mass , but considered by early relativists to be transformable away at the speed of thought, they have remained an enigma ever since.
"What are gravitational waves? This will be discussed more fully in the next section but they can be thought of as ripples in the curvature of space-time or as tiny fluctuations of the direction of g , the acceleration due to gravity, on Earth.
"Why are we interested in their detection? To some extent
to verify the predictions of General Relativity -- although given the success
of the other predictions of General Relativity being verified, it will
be a major upset if gravitational waves do not exist!
"2. Gravitational Waves
"Postulating that this perturbation might be sinusoidal in nature and working in a coordinate system defined by the trajectories of freely-falling test masses, Einstein’s field equation yields a wave equation of the form
where the amplitude of the wave is related to the perturbation
of the metric which is really the amplitude of the curvature of space-time.
"Gravitational waves are produced when mass undergoes acceleration, and thus are analogous to the electromagnetic waves that are produced when electric charge is accelerated. However the existence of only one sign of mass, together with law of conservation of linear momentum, implies that there is no monopole or dipole gravitational radiation. Quadrupole radiation is possible and the magnitude of [the gravitational wave amplitude] h produced at a distance r from a source is proportional to the second time derivative of the quadrupole moment of the source and inversely proportional to r , while the luminosity of the source is proportional to the ‘square’ of the third time derivative of the quadrupole moment.
"For quadrupole radiation there are two ‘orthogonal’ polarisations
of the wave at 45 degrees to each other, of amplitude h+ and hx, and each
of these is equal in magnitude to twice the strain in space in the relevant
direction. The effect of the two polarisations on a ring of particles is
shown in figure 1, and from this the principle of most gravitational wave
detectors -- looking for changes in the length of mechanical systems such
as bars of aluminium or the arms of Michelson type interferometers -- can
be clearly seen.
[XXX - Fig 1]
Figure 1. Schematic diagram of how gravitational waves interact with a ring of matter. The ‘quadrupole’ nature of the interaction can be clearly seen, and if the mirrors of the Michelson Interferometer on the right lie on the ring with the beam splitter in the middle, the relative lengths of the two arms will change and thus there will be a changing interference pattern at the output.
p. 17: "Some early relativists were sceptical about the
existence of gravitational waves; however, the 1993 Nobel Prize in Physics
was awarded to Hulse and Taylor for their experimental observations and
subsequent interpretations of the evolution of the orbit of the binary
pulsar PSR 1913+16, the decay of the binary orbit being consistent with
angular momentum and energy being carried away from this system by gravitational
waves. Thus it is now universally accepted that gravitational waves must
exist unless there is something seriously wrong with General Relativity.
There are many significant experiments underway and at the planning stage
and the community is poised now to herald their detection and the start
of a new astronomy."
Subject: Local effects of cosmological
It seems to me that you have completely ignored my email from Thu, 13 Jan 2005 19:22:03 +0200,
I wrote: "My comments will be very brief, but I am ready to expand them and elaborate on *all details* mentioned in [Ref. 1]. I will me *more than happy* to do that, so please don't hesitate to require further explanations."
I haven't heard from you, however.
In your recent paper [Ref. 4], you talked
about "an expansion factor
Why not 'both/and'? I think it all depends on the nature
of time. The
Now, you say that can distinguish between the size of *the same atom* in two or more frozen snapshots from the cosmological time arrow, and came up with an answer in the format 'either/or' [Ref. 4].
What kind of observer would you consider? I guess you need some "ideal observer" who can time the dynamics of the whole universe, only you'd need an absolute reference frame for such "ideal observer". This is a very old puzzle,
The crux of the matter, the way I see it, is in the difference between (i) the shift from t_1 to t_2 in a frozen (Cauchy) surface, and (ii) the shift from t_1 to t_2 along the cosmological time arrow. Your wristwatch, as well as LIGO, AIGO, TAMA, GEO600, VIRGO, LISA, The Big Bang Observer, etc., can read only the first shift (local mode of spacetime). To detect the second shift, you need some "global view" on the "dark" stuff that pushes the whole universe "between" t_1 and t_2 . Only you can't find this "dark" mover in the Hamiltonian formulation of GR: it is what Aristotle and Karel Kuchar called "the Unmoved Mover",
See also the "dark" Unmoved Mover (the 3-D lake) in the context of the so-called GW astronomy at
Briefly, I claim that GW astronomy is a terrible waste of time and money. If you want to waste your time -- fine. The problem is that you're going to waste BILLIONS of U.S. dollars as well.
I will appreciate your professional feedback, and I'm also requesting the professional criticism from all your colleagues.
Just don't keep quiet, please.
[Ref. 4] Richard H. Price, In an expanding
universe, what doesn't
"The simple question will take the form of a simple model:
"In the long-term our atom either ignores the expansion
I tried today to explain the crux of the matter (cf. above) to my 12-year old daughter Kalina, just as I did with the lake metaphor a few days ago. I believe she understood the ideas perfectly well. Here's the explanation of the two "steps" of time, which passed the test of Kalina's brain.
Imagine a reel of snapshots, in which you see an object, X , moving from the left to the right. These snapshots are separated by a dark strip, [---]. Let's label the snapshots with t1, t2, t3, and think of the commas as dark strips, [---]. So, the story will look like this:
If you're a 3-D Flatlander and stay confined into your 3-D lake, you will see only the local mode of spacetime, which is a perfect continuum of the states of X moving from the left to the right. What I mean by 'perfect continuum' is the following: in this local mode of spacetime, the size of the dark strips is zero to you (because of the so-called speed of light -- but I didn't explain this to Kalina), because you can see things only in their past state, as you know from looking at the Sun. In this continual chain of past states, the dark strips are strictly zero: the spacetime itself cannot "move" in its local mode, hence you may develop the wrong impression that you live in some time-symmetric "block universe", and will never discover the origin of the elementary 'timelike displacement' and the principle of locality. All these fancy words mean the following: there is something that "carries" the state at t1 via the gap [---] to t2 , etc., with speed that cannot exceed the speed of light, and this "something" sets the elementary "tick" of the 'reel of snapshots'. Capiche?
Now, you cannot "look at the future" and see what is going on in the dark strips, hence if you observe effects of Gravitational Waves (GWs), you too will be befuddled by the simple picture you observe in your 3-D lake, and will most certainly conclude that these GWs propagate "inside" your 3-D lake only. Remember how we played Frisbee on the beach last summer? Well, you can certainly claim that your Frisbee travels "inside" your 3-D lake only, but GWs are very different animals, since they originate from the dark strips stacked on a "vertical" line, like a skewer, while you can only see the movement of X on a "horizontal" line, which I call 'local mode of spacetime'.
Got it? 'Yes dad!' -- was her answer. And then she struck back, 'But where did X come from in the first place?' That was totally unexpected. How do I explain quantum mechanics and quantum cosmology? She wouldn't understand why 'unitary evolution' is an oxymoron in quantum cosmology. I tried to answer her question with some simple math.
Suppose X has come from the dark strips as well, some 13.7 billion years ago, as measured with your wristwatch inside your 3-D lake. Let's denote this very special instant with t0 . Good. See what I write below:
(t2 - t0) - (t1 - t0) = t2 - t1 = Y
(t3 - t0) - (t2 - t0) = t3 - t2 = Y
Do you see what happens to t0 ? It disappears (I didn't explain why, because I just don't know), so I can avoid your question. But look at Y : what is it? It is the elementary step/increment of time, and it does include the dark strips, so they are always with you, right from the start some 13.7 billion years ago. Now, you may go back in your room and listen to your rap "music", or whatever you call it. 'Thanks dad!' -- was the answer. Then she added, 'Are you doing this for pleasure or for money?'
Kids are cruel, aren't they?
It is a very strange arrow, since we cannot show its "direction": there is no "extra" (or bulk) 4-D space in which the good old 3-D space could "expand"; see again the balloon metaphor here, courtesy of Ned Wright. Contrary to Tuomo Suntola, there is no evidence that some 4th spatial dimension (not the time dimension of 4-D spacetime) exists in the local mode of spacetime. Notice that the balloon metaphor could be very misleading: on the one hand, we do observe the effects of the "expansion" of space, as inferred from the redshifted Supernovae, but, on the other hand, we can only point to one component of this "expansion" -- the distance between Earth and the receding object, in 3-D space. Is there another, perhaps "dark", component? If yes, can we use it to explain the "axis" of the cosmological time arrow in the global mode of spacetime, as denoted with Z ?
Let's introduce some postulates. First, we take for granted that all points from the expanding 3-D space (local mode) are re-created equal. Each of these points is a fully legitimate "center of the whole universe", in the sense of the famous statement from the Hindu catechism: The One is an unbroken Circle (ring) with no circumference, for the circumference is nowhere and the center is everywhere. There's nowhere where God is not, but there is no direction in 3-D space toward The Beginning either. Where is this "direction", then?
It is hidden in the "vertical" dark strips in the drawing above. That's the missing component of the cosmological time arrow. It covers the whole universe en bloc and is responsible for its self-acting move along the cosmological time arrow. It propels the whole universe. It is the Aristotelian First Cause. And it must be totally "dark", or else there will be some next agent which would drive the whole universe, until we reach the Aristotelian First Cause, and stop there. Hence the latter must be "dark", and we can only introduce it 'by hand': it belongs to the final Weltbild. We can introduce the "dark" First Cause by hand only, since 'the absolute full description of the universe' cannot be self-contained and "self-axiomatic" due to Gödel's theorem.
And now of course you say -- I know all that flaky stuff, where's da math? Hold on, please.
We introduce a second postulate: Every "horizontal" infinitesimal displacement of X (see the ladder again) is being re-created as an asymptotically flat, and perfectly continual, 3-D space called 'local mode of spacetime'. Then comes a third postulate: The "point" of re-creation is a topological transition of a 4-D sphere into 4-D torus, and vice versa. This is the "dark" axis of the "vertical" component of the cosmological time arrow: just link the center of the 4-D sphere with the center of the 4-D torus, and voila! Notice that the "dark" expansion of the 3-D space occurs simultaneously in the two "directions" along the axis, as fixed by the centers of the 4-D sphere and 4-D torus.
And here comes the fourth postulate:
RsRt = 1 (Eq. 1)Here Rs denotes the radius of the 4-D sphere and Rt denotes the width of the donut/torus that you can eat.
Thus, the volume of 3-D space at the instant of re-creation is infinite: it corresponds to the instant of transition of 4-D sphere with infinite radius into a 4-D torus with zero width, in line with Eq. 1. However, this very special instant can exist only in the global mode of spacetime, since it is totally UNphysical: the 4-D sphere is converted down to an absolutely flat infinite Euclidean 3-D space, which has absorbed the 4-D torus into a bare dimensionless point ("the center is everywhere", cf. the first postulate) from that same absolutely flat and infinite 3-D space. Thus, there are two indistinguishable UNphysical states in the global mode, as viewed from the local mode: "inside" the infinitesimal and "outside" the cosmological horizon. These two UNphysical states should show up as some numerically finite but physically unattainable boundaries of the physical world. In the local mode, the universe can only approach them asymptotically, although they are expressed with finite numerical values, such as the Planck length.
Incredible? Well, you never know with the infinitesimal, all sorts of things might "happen" inside it. Once you pass the instant of transition, you're again in "the dark room": you expand on a 4-D torus until you get again to the next instant of transition back to the 4-D sphere. Maybe the transition is smooth, since it happens in the global mode. Last but not least, the sphere-torus transition should consist of four (not two) segments, to resemble the Kruskal-Szekeres diagram (or Ted Newman's H-space) and the alleged conversion of 'space' into 'time' (as in the two "mirror" worlds, material and tachyonic). I don't know (yet). More here and here, philosophical considerations here and here.
All this is needed to design a 3-D space with "blueprints" from the global mode, such as 'big' and 'small', 'inside' and 'outside', 'left' and 'right'. Once we dress all this in math (hopefully by November 2015), we can discuss the nature of time, and (hopefully) go back to Schrödinger's paper from 1931.
As to the nature of space, I hope we will be able to reexamine the Hubble Law of 1929: since the metric "expansion" is not associated with any physical object's velocity, my guess is that we should introduce a new, scale-dependent relativity principle, namely, that things look "large" and "small" with respect to the scale of tables and chairs, but if a physical object increases/decreases its size, the metric of space will increase/decrease accordingly in such a way that a table with length two meters, stretched to the size of a galaxy, will again be "two meters" in its scale-dependent reference frame due to the corresponding alteration of spacetime metric at its scale. Same holds for the direction toward The Small. Thus, the peculiar redshift may be an effect of the changing metric toward The Large, and will not imply that some galaxy is receding from Earth with speed proportional to the distance to it (see R.G. Vishwakarma, Recent Supernovae Ia observations tend to rule out all the cosmologies! astro-ph/0511628 v2, 1 December 2005, p. 10). This is just a wild guess toward the nature of the so-called dark energy which we need to embed in GR from the outset. Notice that the "dimensions" of an atom and a galaxy may be "the same" in the global mode of spacetime, hence all physical processes that happen "inside" such volume of 3-D space can be correlated in their common Holon state. We stay 'in the middle', hence we see, in the local mode of spacetime, a galaxy that is "large" and an atom that is "small". These two "directions" in 3-D space are the crux of the new theory of quantum gravity, which I believe will not be a metric theory but a wave theory of space: two GWs running against each other in the global mode, which create different volumes of 3-D space in the local mode, in a way resembling the cancellation of the two "waves" in John Cramer's interpretation of QM. I know that all this sounds very unclear, but please come again here in November 2015. I too believe that pure thought can grasp reality, as stressed by Albert Einstein.
But where's da math? Too late; sorry.
Napoleon confessed in his memoirs that he removed Laplace from office as Minister of the Interior after only six weeks "because he brought the spirit of the infinitely small into the government" (C.B. Boyer, A History of Mathematics, John Wiley & Sons, New York, 1968, p. 536). I've never been supported by anybody, and nobody's paying any attention to my work either. I can lose nothing, hence I'm free. Most importantly, I don't need Eq. 1 above to practice PHI.
Why would a fish need a bicycle?
Dear Professor Rucker,
I like your "Infinity and the Mind", and particularly your presentation of the Thompson lamp paradox. I gave a link to your book at
I wonder if you could help me with the following exercise. I want to blow up a 3-D sphere, but will work in 2-D, and will imagine the 2-D section of the sphere as a circle (say, a watch). So, I start with a finite diameter of the "watch", and inflate it up to the "point" at which its diameter will be infinite. Suppose it breaks at 12 and 6 at 'infinity', such that we have two parallel lines (1-D Euclidean space) at 9 and 3.
If I "move" in the same "direction", I'd get a 2-D section of a torus, correct?
If true, the torus will be shrinking until it passes also through the 'infinity' state of zero width (?), at which point it will somehow "tear apart", and get back to the initial circle with finite diameter.
I'm afraid I'm trying to invent the wheel here, and will appreciate your help with references and any other materials.