## Sunday, December 21, 2014 ... /////

### Discrete spacetimes contradict Unruh effect

Two young Indian men, Golam Mortuza Hossain and Gopal Sardar, wrote a paper about loop quantum gravity and similar "discrete" models of quantum gravity whose mathematical argumentation seems vastly better than that of an average paper about similar subjects:

Absence of Unruh effect in polymer quantization (gr-qc)
Yes, the Unruh effect isn't reproduced by those theories, they say. Backreation sensibly asserted that if the paper is right, it's a way to prove that these theories are dead. Well, it's about the 500th proof that they are wrong, I would say.

These two guys' mathematical and theoretical physical advantage over an average author of "loop quantum gravity" papers seems self-evident. Show me loop quantum gravity papers that actually manipulate with the Mathieu equation, elliptic cosine and sine functions, Riemann zeta function, and even with a simpler mathematical operation in modern physics – the Bogoliubov transformation.

The controversy is bound to persist: Is that paper right? Let me tell you in advance what the obvious answer is. The question is undecidable because the very existence of a nearly flat spacetime – a pre-requisite of any discussion of the Unruh effect – contradicts loop quantum gravity and similar models. So even the very question "what will the accelerating observer see in a nearly empty space" is meaningless because there is no nearly empty space in these theories or observers that could live in it.

Recall that the Unruh radiation is a seemingly thermal radiation of temperature$T={\frac {\hbar a}{2\pi ck_{{\text{B}}}}},$ that an observer accelerating with acceleration $a$ in the empty (locally or globally) Minkowski space will observe. Due to the empty-spacetime (or Rindler wedge) context, the Unruh radiation is simpler than the Hawking radiation even though the Hawking radiation from a black hole, whose essence really boils down to the Unruh effect in some sense, was actually discovered about two years before the 1976 Unruh effect (because Hawking has been really smart and able to solve non-minimal problems directly).

Nevertheless, their argument is rather clever. They do realize that discrete models of the spacetime inevitably break the local Lorentz symmetry and modify especially the behavior of the modes above the Planck energy. And this has consequences even at low energies, despite many people's desire to believe otherwise. In terms of a GR-like description, the trans-Planckian Lorentz symmetry is actually needed for a consistent decoupling of the unphysical polarizations of the gravitons. If you break it and modify the theory, it's too bad.

Mathematically, they argue that the Gaussian wave functions that we normally deal in free field theories – and in the analyses of the Bogoliubov transformation – are replaced by a completely different function, the elliptic sine or cosine function, and the analogy of the Bogoliubov transformation ends up with the conclusion that the vacuum state is not changed at all when we switch to the accelerating frame.

I was explaining the Unruh effect to some high school kids during my recent talk in Northern Bohemia. How is it possible that the emptiness of the spacetime depends on the acceleration? In classical (non-quantum) physics, it cannot depend. Well, it's because the emptiness of the vacuum means $N_a=0$ for all the occupation numbers (number of excitations in harmonic oscillators) counting the particles in different modes of the fields. And these operators $N_a$ change if we switch to an accelerating frame.

More obviously, the vacuum state is an eigenstate of $H$ with the lowest eigenvalue (zero, after you subtract the ground state energy, obviously). But in a differently accelerating reference frame, we have to use a different Hamiltonian because the infinitesimal change of the new time coordinate $t'$ means an infinitesimal change of the old time $t$ – so far, so good – composed with a small boost that changes the velocity (and therefore the slope of the slice) because we are accelerating.

Because $H'=H+ a J_{xt}$, roughly speaking, where the second term is the boost generator, these two Hamiltonians $H,H'$ differ similarly as the operators$\frac{\alpha x^2}{2} + \frac{p^2}{2\alpha}$ for different values of $\alpha$. Classically, regardless of $\alpha$, the "ground state" (point in the phase space) is simply $x=0$, $p=0$. Quantum mechanically, the ground state is a Gaussian wave packet around $x=0$ and $p=0$ but we find out that its width depends on $\alpha$ – and therefore on the acceleration. So the ground state of one observer cannot be the ground state of the other observer. One can be obtained from the other as a squeezed state, and this is a superposition of the ground state and excited states with all possible values of the excitation number. In the Unruh setup, we find out that the average occupation numbers are thermal in some sense.

Please find a concise presentation of the Unruh and Hawking effect somewhere if you have never encountered those and if the appetizer above is insufficient. But let us return to the more advanced stuff now.

One way to describe the problem of the discrete theories of the spacetime is to say that they predict a wrong degree of entanglement between adjacent spacetime region. (The glue-entanglement minirevolution in string theory links entanglement with the topology of the spacetime, and because the discrete spacetime is a different i.e. wrong topology of the spacetime, you will predict different i.e. wrong forms of entanglement, too.)

If you assume that small adjacent "strips" of the spacetime are not entangled at all, you may say that the ground state of the whole system will be the tensor product of the ground states of these strips. And this tensor product will be unique and independent of the acceleration, and therefore the Unruh effect will go away. That is pretty much what is happening according to these two Indian authors.

The authors remain neutral on whether the derivation (assuming it is correct) is good news or bad news for loop quantum gravity and friends – although they probably do feel that it is negative news. Sabine Hossenfelder sensibly says that it would be bad news. In reality, however, it is completely neutral news because loop quantum gravity and friends have been totally and irreversibly dead for something like 20 years so a collision with another bullet doesn't make any difference whatsoever.

Some responses to comments by other people. Fizz wrote in the Backreaction comment thread:
I don't work in LQG myself, but LQG deals with diffeomorphism invariant states, both spatial and timelike. So, an LQG state would be in a superposition over all possible coordinate systems, including Cartesian and Rindler plus a whole lot of other coordinate systems. You just can't fix the coordinate system in LQG.
I agree with that. Ideally, if you assume that a theory of quantum gravity (not necessarily a discrete or sick one) predicts realistic (nearly empty and other) spacetimes, they still come in a superposition of all conceivable coordinate systems that are related by diffeomorphism transformations to others. So there simply won't be any obvious way to fix a coordinate system in the bulk in any theory of quantum gravity. One would have to find correlations of quantities with something physical, like in the treatments based on the Wheeler-DeWitt equations. I don't want to go into those things – especially because they have never led to any non-obvious, mathematically refined results yet.

So a correct calculation of these things in a microscopic theory would have to be more complex than the Indian paper. But I think that the paper is morally right and the trans-Planckian modifications of physics would still leave imprints on long-distance phenomena such as the Unruh effect.

In the case of loop quantum gravity and friends, it is impossible not only to fix coordinate systems. It is impossible to fix the background to be a nearly empty space, too, because such configurations don't exist in the theory or at least are not any special relatively to the generic states of a crumbled space and other pathologies.

Our Giotis wrote:
Yes, basically what I’m saying is that it is not consistent to take the $L_{\rm min}\to 0$ limit of the box (and thus $\delta \to 0$) with polymer quantization since polymer presupposes a discrete space. The fact that $\delta\to 0$ is the root cause of their null result. I think more or less my claim is similar to your argument if I understood it correctly.
I sympathize with Giotis' call to be careful about the order of limits and similar things but at the end, I think he is wrong that one can't be careful. The point is that if we assume that the theory predicts a nearly flat space somewhere – and I have already said that it is not the case of loop quantum gravity which is the "main" reason why it's a dead theory – we may interpret the variables such as $\delta$ in terms of the proper distances in the spacetime that we should get.

At the end, Giotis' criticism wants to say that they are making $\delta$ shorter than the Planck length and it's bad. But the actual relevant meaning of $\delta \to 0$ is the condition that $\delta$ is smaller than the inverse acceleration length scale $c^2/a$ which is much longer than the Planck scale. The problem is still there.

If one knew in advance that the discrete theory reproduces an effective quantum field theory (with the Gaussians etc.) at long distances, Giotis' objection would be a necessary condition for a loophole to exist. However, he hasn't actually shown that the loophole is realized in the discrete theory. The loophole is actually not realized and the wave functionals that should be Gaussian are actually not Gaussian.

I have already predicted that all these comments are likely to remain controversial because loop quantum gravity isn't a well-defined theory and it doesn't agree with anything that it should agree with. So there will always exist "camps" in this bad science that choose their beliefs according to the desire to make their previous papers look as good as possible. For example, one might argue that the loop quantum gravity community hasn't even agreed with basic claims such as the conclusion that this theory violates the Lorentz symmetry at the fundamental level. These Indian guys aren't completely stupid so they of course agree that such a discrete theory does inevitably violate the Lorentz symmetry and most of their formulae are rather specific descriptions of the character of the violation.

But this question, whether there is the Unruh effect, is in some sense more complex than the existence of the Lorentz violation, so chances are that it will remain even more controversial. This situation is nothing like the research in string theory or any well-defined science where every paper (with some major conclusions) is rather quickly seen to be either right or wrong by pretty much everybody.

I won't spend any more time with these things because the question is ill-defined but I do think that the Indian men's conclusion is right. It's just one specific consequence of my (and not only my, of course) observations that similar discrete theories of the spacetime are in conflict with rather basic properties of effective field theory in the (nearly) empty space. Whether or not their calculations are exactly right according to the "semi-credible" standards of rigor that exist in loop quantum gravity, I am pretty sure that the conclusion is morally right.

As far as I can see, there are many heuristic ways to derive similar conclusions. For example, the pixelated structure of the spacetime near the Planck scale really means that the "right direction of the spatial slice" across this foam isn't well-defined locally but only determined with the accuracy $\Delta a\sim l_{\rm Pl}$. This inaccuracy prevents you from a sufficiently precise discrimination between the two differently accelerating frames at the level of the microstates.

#### snail feedback (58) :

Years ago, when I was a graduate student looking at some of the LQG papers in my spare time, part of this topic was one of the many reasons why I quickly abandoned all hope for the research program. Right from the getgo, they claim to produce an exact local quantum of area observable, yet the very next sentence claim to only admit completely diffeomorphism invariant states by construction (as one should for a theory of gravity). The fact that no one seemed to spot the obvious contradiction was troubling. At the very least, such a procedure ought to wreak havoc with their vacuum.

Was the Unruh effect actually observed in experiments? You cant really say that absence of it in a theory is a falsification if its has not been empirically proven to exist yet.

Well, wouldn't it still mean LQG breaks either QM or SR at relatively low energies? Maybe not a full falsification but certainly not what LQG researchers wanted their theory to be.

LQG breaks SR by construction, you don't need the Unruh effect for that.

So, this is an obviously correct blog post affirming a trivial debunking of an obviously incorrect theory. Nice, in a way, but kind of redundant.

Your question and your statement show that you know very little about physics and that you are lazy. Out of simple courtesy you should, at the very least, read the abstract of the new Indian paper before wasting our time.

Whether they turn out to be right or wrong eventually, my congratulations to Hossain and Sardar for producing such a work from India. Their institute is one of the newer institutes in India and it bodes well for future of India that such research activity is supported.
I have a slight feeling (I am not sure) that if Unruh effect does not exist, there may be problem with equivalence principle. Does any one agree with me!?

You mean that the observer sitting near the horizon of a black hole (thus in accelerated frame) would also see the Unruh effect as Hawking radiation?

I think, with that statement most people will agree. But I was speculating about more general equivalence of acceleration and gravity. So basically I am wondering about how important quantum effects are for Unruh effect. It probably cannot be understood as purely classical GR effect.

I can't imagine there is any classical equivalent of Unruh effect.

The derivation starts with quantized free field and looks at vacuum states connected via Bogoliubov transformation.

Is there any classical field theory that can describe classical particle creation? If there is, I haven't heard of any.

Bill Gates looks remarkably like Phil Lesh. I'm reminded of this every time I see a clip of the Grateful Dead. They are both college dropout, Gates dropped out of Harvard while Lesh dropped out of Berkeley.

Like Gates, Lesh is a bit of a nerd and was regarded as the techo-geek of the band. Lesh is the one playing bass, wearing a preppy button-down shirt, looking nerdy as ever. Jerry Garcia is looking cool as ever in his black t-shirt. He almost always wore a black t-shirt. I believe this is why Steve Jobs made black the focal point of his daily attire. Sorry Steve, you made more money than Jerry Garcia, mountains more, but you never came close to him in terms of talent; sheer, raw talent. Here they are performing China Cat Sunflower, which is one of my favorite Dead tunes:

I think, I should rephrase the question. If one of the two,
Hawking radiation and Unruh radiation is observed, but not the other one then
there will be problem with equivalence principle. Right? If neither is observed
then perhaps one can blame it on the method used to combine GR and QM. So it seems to me that it is vital to observe both! Of course experimental limitations are always there.

This reminds me the case when they tried to quantize the String using polymer and the result was not consistent.

BTW did you understand Sabines' and Rovelli's loophole argument? It has some similarities with my comment but I can't pinpoint it exactly and I'm not willing to spend much time on it. As you say it's a waste of time; LQG does not have a sensible semiclassical approximation to begin with.

Kashyap, if the Unruh effect didn't exist, there could be a problem with many things but I don't think that it has anything to do with the equivalence principle. The equivalence principle is the equivalence between acceleration and gravitational attraction caused by other bodies. The Unruh effect doesn't depend on any gravitational attraction at all.

The equivalence principle translates to the equivalence between the Unruh and Hawking effects, but not to the existence of either separately.

As Tony says, the Unruh (and Hawking) effect is a purely quantum phenomena. The temperature goes like h*a/c*k, so it is proportional to Planck's constant. Classically it is zero. Many things are purely quantum - discrete of the spectra of many observables (nonzero spacing between eigenvalues), tunneling, Pauli's principle, stimulated emission (laser), entanglement (knowledge of relative properties of subsystems without the knowledge of both, in many complementary ways), and so on.

Dear etudiant, your source - because of being an owner of a dropping asset - is much less impartial than he is probably nervous about seeing that it was dropping so much, with no indication of a reversal.

For any
currency to have mechanisms to protect its value, it must have value in the first place.

Dear Giotis, I would agree with the statement that in QFT, the Unruh effect may be computed as a low-energy effect - almost all the quanta are come from the energy scale "a" (acceleration) and everything else can be ignored.

However, in a particular formalism, there may also exist other ways to calculate the same thing. If their is right, and I think it is, it indeed implies a contradiction, as Rovelli says. But this contradiction isn't reducing the value of this paper. It is a contradiction within loop quantum gravity etc. because different methods yield different results to the same questions.

Consider anomalies. They are fully dictated by the low-energy spectrum, and in this sense, they are infrared, long-distance effects. But we are still computing them from uncancellable ultraviolet-divergent loop diagrams, from high energies. This case is analogous. The deviation of the spectrum seen by an accelerating observer from the right EFT-Unruh result is a form of anomaly, and it may be possible - and it is right if this paper is essential right - to show that a given theory has the anomaly by doing calculations at very high energies.

Yes, your country and good people it contains might be urgently needed to (globally) help keep nice fundamental physics going in the future too (it already is the source of nice things), so I rely on you ... ;-)

Cheers

I think Kashyap is (correctly) saying that if Hawking radiation is observed but the Unruh effect does not exist, that would imply the equivalence principle is wrong.

But even for those of you who are older, it must feel like some
mysterious pre-history of the PCs because almost no one bought it.

Maybe, most of us who are older... I had an IMSAI 8080 (1975) on loan for several months about 1980. Going back further, I had actually read the issue of Popular Electronics that announced the Altair (at the time it was published).

Going back even further, to the 1960s, I saw a demo of IBM's idea of a personal computer at a computer show in Boston. You couldn't lift it unless you were strong, and so far as I know (I might be wrong) the only output was in Nixie tubes. I don't think IBM thought of it as a personal computer but rather as something that a corporation or group of engineers or scientists would own. Even at that, the IBM-er who demo'd it to me thought it was a laugh. He said that "obviously" there would never come a time when more than a handful of people would have computers in their homes.

I was a mainframe programmer (later switched to PCs), and even in the late 1970s I thought similarly as that IBM-er. I read personal computer magazines and sort of scorned them, since they were explaining algorithms that had been discovered in the 1950s or earlier.

What a mistake! Years later, after Peter Norton (The Norton Utilities) became well known, I realized that he was getting rich with a program that I easily could have written.

"They do realize that discrete models of the spacetime inevitably break the local Lorentz symmetry and modify especially the behavior of the modes above the Planck energy. And this has consequences even at low energies, despite many people's desire to believe otherwise." Good point. Let us assume that my "Wolframian string theory" is complete rubbish. MOND is wrong, or Milgrom is the greatest astrophysicist of his generation. Consider the Milgrom Denial Hypothesis (MDH): The main problem with string theory is that string theorists fail to realize that Milgrom is the Kepler of contemporary cosmology. Is the MDH wrong? No, because if Milgrom's MOND were wrong, then there is no way that Milgrom could have convinced McGaugh and Kroupa.
“The failures of the standard model of cosmology require a new paradigm”, Jan. 2013

This personal contact with the history of computing must be rather rare, right?

Thanks to Jolly Joker too for a comment.

Lubos, off topic: There's a posting "Dualities" on Woit's blog, with comments coming from Polchinski, including Polchinski's criticism of our humble correspondent.

I'm close. For me it was 1981 when I saw the first PC at the party thrown by a rich guy who could afford it but had no idea what to do with it.

Then my college got a donation of 100 PCs from IBM, intended to hook the students who will later work in various industries. The room where PCs were located was mostly empty. I started learning programming in UCSD Pascal, loaded from the floppies. There was no DOS yet. I ended up doing cellular automata simulations and eventually lost desire to continue with the college. I wanted do computer graphics and travel.

When visiting a friend in northern Italy by chance I got introduced to a guy who started electronic publishing company. They were still using mini VAX but intended to add PCs as intelligent terminals. Just for fun I wrote them a program drawing character glyphs and they immediately offered me a great salary, company car and an entire house, rental fees also paid by the company.

I stayed in Italy for about 5 years and when I returned to the US, Windows 3.0 just came out. Again for fun, I wrote a driver for Intel fax board and was offered a licensing deal with the startup that later became a big software house (I forgot the original name, I think it may be Symantec now, or they bought them). It was too much trouble for me to go through all the legal documents and I ignored the offer.
I guess I was always engineer at heart, with little desire to deal with anything business related.

I know what you mean. If only my great grandfather had made enough time between his incessant bouts of extra-marital fornication to fully pursue development of his Wonder-Brass Sprocket, thus making the steam-powered Charles Babbage PC possible a century ahead of time, then I might have been rich enough today to launch a nuclear strike on Brussels from my private island fortress.

Sorry, as for the experiments I can't find anything better than what shows up with simple Google search.

However, as far as theory is concerned, I found the following two posts by Lubos on stackexchange very helpful (be sure to read the comments too):

String theory is an ugly artifice of a theory. Just ridiculous in its Goldberg like construction. It may be correct but if it is I want no part of this Universe or any other. Dumb Universe. So idiotically designed. Winding number? What kind of marijuana has the Universe been smoking?

I suppose so. That's why I said, "most of us who are older." Maybe I should have said, "an overwhelming majority ..." But it's rarer than it should be, because there were tens or hundreds of thousands of mainframe programmers, and yet very few of them learned anything at all about the early microcomputers. That's because very few of them were actually interested in programming. They were not even interested in how the mainframe software worked, and much less in how the machine worked.

I'm 70. I seem to recall your saying that you were older than that. Part of the reason I missed out on opportunities is that I see things as more formidable than they really are, and less likely of success, and since childhood I've been handicapped by a tendency to think that "everyone else" knows more about any given thing than I do.

Your story of the rich guy reminds me of the Christmas Eve I bought a Sinclair computer at a pharmacy for \$39.95, and a cassette recorder to go with it. The recorder was defective, so I returned it. The clerk to whom I returned it was surprised I wasn't also returning the computer. She said, "Are you sure you don't want to return the computer? You can." I said no. She showed me a big stack of computers returned for credit or refund. People were coming in and complaining that they had no idea of what it was for or how to use it, so they wanted their money back.

I had actually bought two, one for my 6-year old daughter. I was divorced, and her mother had custody, but I thought I could show her how to play games with it during visits. I talked with her on the phone a few days later (no contact in the meantime), and she said, "I read the book, and I tried it. I understand everything except GOSUB. I can't figure out what it's for."

Bill

What *were* they interested in, was that some kind of a secretary's work?

It wasn't expensive...

GOSUB is indeed the most abstract thing.

It's like GOTO except that the system remembers where it came from - this GOSUB may be at many places - and the command RETURN anywhere in the subroutine returns one to/after theplace where we came from via GOSUB.

Although it's ugly, understand why one may want to do such a thing is the precursor of all the modular programming etc. - not getting it is like to be really confined in the unmodular BASIC programming paradigm.

Are you asking whether the programming was some kind of secretary's work? No.

They were interested in all kinds of things, such as golf, home life, TV, etc. etc. etc. - the same things that millions of non-programmers are interested in.

You seem to be missing the fact that the person in question (my daughter) was 6 years old at the time.

No, I did not miss that, it is absolutely cool, and I was specifically thinking with this fact in mind.

It's in the family then.

Having learned Pascal first, then C, at the young age of 26, I dislike BASIC even to this day.

OK, thanks, I see. I was misled by the fact that you were explaining very basic matters. She didn't really understand *everything* but GOSUB. Sinclair BASIC had math functions, didn't it? She would not have understood them.

"Absolutely cool" - Perhaps you won't find it so cool that 16 years later she was a Ph.D. candidate in philosophy and theology. :-) But she got fed up with it, and she quit. I'd tell you more, but I don't want to run any risk of identifying her.

Yes, I'm familiar with "simulated" recursion. I've used it a number of times.

I'm not sure you appreciate the fact that there are adults - even programmers! - who seem unable to understand more than a very slight amount about programming. I've known programmers who didn't understand formal parameters, pointers, or recursion. They just can't handle abstraction.

The same is true in other fields. The sociologist Charles Murray, I think, has said that many highly intelligent people spend their lives with other highly intelligent people and have no idea of how stupid many people are. However, I know you've said you've spent time with stupid people...

Dear Smoking Frog, thanks but I don't want to become a completely perfect expert in understanding what BASIC functions your daughter understood at every age. ;-)

Oh, but that's not what I'd have told you more about. :-)

I know that your comment was interesting in a multi-dimensional way. Yes, people including professionals are highly limited in too many respects, including their discipline.

Good to quit philosophy, too, but it would even be better to return to something nice.

auto func=[]( int i ){ cout << i; };

func ();

nice:

Btw, the above wouldn't compile since I forgot to pass a parameter to the function!

Should have been like:

func (5);

Oh well, that was just a typo.

Another nice thing is that I just downloaded root software from CERN:

https://root.cern.ch/drupal/

(their server security certificate seems expired or otherwise invalid today) and find it an amazing package that I may eventually use, if time and projects permit.

Since I am on Windows, it will take some extra effort to build, but I think it will be worth it.

You're experimenting with C++ 11? Lambda functions are nice. C# had them since 2007.

Hiya - I was thinking about the Unruh effect as in some sense "dual" to the Hawking effect and associated with horizons, specifically in the context of the AMPS paradox of firewall censorship of spacetimes over horizons.

Specifically I was wondering isn't there an equivalent problem with monogamy of entaglement over cosmic horizons with Unruh radiation, "morally equivalent" to the one with Hawking radiation, that (if you wanted to censor black-hole interiors) would force you to censor the spacetime behind cosmic horizons, i.e. the whole spacetime? Clearly you falling out of my causal patch is not going to make you burn-up or stop exisiting, and this would seem to be a fatal flaw with firewall proposals.

I can't imagine this hasn't been considered by Polchinski et al, it seems too obvious, so I guess it's correct to say that no equivalent to the AMPS paradox holds in this case, because Unruh radiation are not "evaporation products" and isn't in any meaningful sense entangled with what goes over the horizon?

Is that broadly speaking correct or am I missing something?

Cheers,

Merry Stringy Christmas!

Liam

The math functions are impenetrable to anyone who has not learned anything about them or even heard of them.

Re abstractions: I think some majority of Cobol and Fortran programmers in the old days thought of variable and data structure declarations as concrete entities; they didn't understand that they were mere maps and could be remapped in different parts of the same program.

Much the same applies to what pointers point to.

I can't believe that my experience with these people is anomalous - I saw this dumbness far too often for that.

I'm just the opposite. I love to see how far I can go in a stupid language.

Merry Christmas. You reminded me of my college job. Having learned IBM 360 assembler at the community college down the street and Fortran and APL for the 1130 machine my school had ,I got a weekend job with the State as an operator for the mainframes. they were putting various records "on line" for police agencies and needed it to run 24/7. A NOVA minicomputer provided a front end to a 360 as a sort of router all written in BASIC. Most of the mainframe programs were written in COBOL. I looked at a lot of these just to understand these other languages and sometimes the senior programmers would be in working and I would ask questions. I had come across some particularly impenetrable COBOL. I realiked that although it followed all the written requirements for variable naming and allocation, it named address pointers as counters , integers as real this or that etc..

Not just experimenting! I use them in all my new and when refactoring old code almost on a daily basis. I am a big fan of functional languages.

My father and I built a Heathkit H8 around 1978 or thereabouts, another 8080 based computer kit.
We always had some ongoing Heathkit project spread out in the shop, TV's, shortwave radios, etc. Wish Heathkit was still around when I was raising my kids

Merry Christmas!

I've known people who did bad things with names (symbols), but never anything as bad as what you've described.

I thought Cobol didn't have pointers until Cobol 2002, but you seem to be talking about a much earlier time.

I was as said never expert in cobol. The pointers I refer to was the usage in the shop for in memory data structures. The memory allocation was fixed at the assembly level. The data would be at fixed addresses pointed to in cobol with a name to those predetermined addresses. The pointer data type and structure did not exist but these were integer addresses. We called it indirect addressing . He named these things like e.g count-adder.
This was 1972 and so decades before OOP and pinters used more dynamically I suppose. I left when I graduated 1974.

Hello professor Lubos, I know this is completely off-topic and i apologize for that, but there is something i have been struggling with when it comes to learning general relativity and i was wondering if you could help me with it , the idea that coordinates you use are just lables for timespace events, then how do you you assign these lables even in principle, how do we know that the four coordinates used in the Schwarzschild solution are r,\theta and \phi, And how does it help to call them that since as i understand they don't have the same metrical meaning in general relativity. Any explanation would be immensely appreciated.

Hi!

You may use *any* four (or "D") functions of the spacetime as your coordinates as long as they are smooth enough, cover the space in a one-to-one way with the right topology, and so on. It's really the *point* - a fundamental postulate - of general relativity that no choice of coordinates is better than another one.

You may also call the coordinates in any way you want. x0,x1,x2,x3 is a good enough notation for any choice of coordinates. The choice r,theta,phi,t etc. indicates that your coordinates are probably analogous to some spherical coordinates for the flat space. That doesn't mean that the space is flat -but it's often spherically symmetric if the coordinates are called in this way.

The coordinate differences can't be directly interpreted as proper times or proper distances. However, the formula for the proper length/time for coordinate changes is given by the simple formula

ds^2 = sum_{i,j} g_{ij} dx^i dx^j

where g_{ij} are components of the metric tensor, a field with 10 components that translates the coordinate differentials to proper lengths by the "generalized Pythagorean" formula above.

So much has changed since those days! The top hackers of today don't even resemble West Coast types:

http://www.zerohedge.com/sites/default/files/images/user5/imageroot/2014/12/kim%20pc.jpg

speaking of the Unruh effect, has anyone here heard of "Modified inertia by a Hubble-scale Casimir effect"...it's somehow connected to Unruh radiation.

I'd like to know if there's any legitimacy to the idea :)