An interesting post. I must admit that I have not come across the pion as the intermediate particle giving mass to nucleons, definitely a lack in my knowledge of high energy theory.
Do you have a reference where "Instead, much of the mass is in a background condensate, mathematically analogous to the Higgs field (vev)." is explained in detail? A quick wikipedia search does not get me anywhere.
I think your post is a little confusing because you don't clearly distinguish between the explicit breaking of chiral symmetry due to non-zero quark masses, which is what gives rise to the (small) pion mass, and the spontaneous breaking of the (approximate) chiral symmetry, which is what gives rise to the (large) nucleon mass. So if you imagine taking the quark masses to zero, the pion mass goes to zero, while the nucleon mass doesn't change much. On the other hand, if you increase the temperature (like at RHIC), you can reverse the spontaneous symmetry breaking, but the explicitly broken symmetry remains.
arXiv:1504.02175, "3×10^14 T" 10^10 T is 4×10^25 J/m^3, E/c^2 = 445,000 g/cm^3. u_B = (B^2)/2μ_o, 3×10^14 T, ~twice nuclear density. Vacuum birefringence arXiv:1212.1897 Footnotes become important.
"For me chirality has always been the most puzzling aspect of the standard model. It’s just so uncalled for." First observation, then theory. "But fundamentally, on the level of elementary particles, where would such a distinction come from?" The false vacuum then decay (dilution) had a pseudoscalar component, thus baryogenesis and onward.
"The pions aren’t exactly massless because chiral symmetry isn’t an exact symmetry" Test vacuum chiral divergence at 0.1 nm^3 scales. 1) Chemically and macroscopically identical, single crystal test masses in enantiomorphic space groups: Eötvös experiment; 2) Cryogenic molecular beam divergent rotational temperatures of extreme geometrically chiral molecules' racemate. Look.
I think "... a fruitful research direction about which we will here more in the future" should be "... a fruitful research direction about which we will hear more in the future" ("here" should be "hear").
This is the first time I saw someone say that the pion is the Higgs boson of hadronic mass. Usually they just talk vaguely about QCD. Kudos to Sabine for highlighting the real God particle.
I think the statement: "But at low energies, such as close by chiral symmetry breaking, little can be calculated from first principles. Instead, one works on the level of effective models, such as that based on pions and nucleons rather than quarks and gluons." does not do justice to the results by lattice QCD: it can nowadays handle realistic pion masses, and it does provide remarkable results from first principles of QCD.
Normally I would identify the pions with a decrease in the mass of a nucleus from the constituent nucleons, and the gluons with the increase in the mass of nucleons from the constituent valence quarks.
It seems here you are implying the proton and neutron have increased masses from the sum of the masses of the constituent valence quarks not due to gluons or the fundamental degrees of freedom of QCD as I had always visualized it, but due to some contribution of pion exchange amongst the quarks. Could you give some more details on this to help me understand better? Thanks!
Hi, I sort of wish you would refer to it as pion condensate rather than just pions. Saying pions are responsible for mass of nucleons is technically correct, but it makes it sound like the nucleon mass is just mass of quarks + mass of pions which it is not. The pion condensate is the order parameter for chiral symmetry and when it goes to zero at high temperatures, the mass of nucleons disappear.
I didn't want to make it more confusing as it is already. The pion gives rise to masses of nucleons as much or as little as the Higgs boson (!) gives rise to masses of quarks. You're right of course in that it's not the particle that is relevant, but the condensate (which I did mention). Best,
No, the increased masses of the nucleons do not come from the masses of the valence quarks, that's the whole point, they come from the quark condensate. It's an effective model, and "effective" here isn't just a word but a technical term: There are no quarks and gluons in this description, instead there are pions and nucleons. They are fundamentally composed of quarks and gluons of course, and in principle the effective theory is a low-energy approximation to the full QCD interaction. So the picture you have in mind is on a vague level correct, in practice what one uses is the effective theory. Best,
Sabine what do you think of the idea that dark matter could be composed of pions: The SIMP Miracle (arXiv: 1402.5143) that has been accepted in Physical Review Letters.
I didn't (mean to) imply the mass came from the valence quarks, I said my impression was the gluons cause an 'increase in the mass of nucleons from the constituent valence quarks,' corresponding with what you said.
I thought pions were associated with the interactions between nucleons, while the gluons were associated with what held the hadrons together, and therefore increased the mass from the sum of the masses of the valence quarks.
Now if we point toward effective theories formed from the goldstone bosons of the flavor symmetries can't we talk about from where in the full theory the quark condensate, which breaks the symmetry, comes from? Is it reasonable to associate this with the gluons? Has lattice QCD been able to do anything toward this end?
Thanks for the reference, but I don't know the paper. First thing that comes to my mind is that pions are unstable, so not sure how this should work, but I'll have a look at this. Best,
As Sky pointed out in a comment above, you're mistaken in thinking that the pions (the particles) increase the mass of the nucleons. Note how carefully I vaguely said they "generate" it. The pion plays the same role as the Higgs particle (they are both bosons). The particle is an excitation over a background, but it's the (coupling to the) background (the condensate) that is responsible for the mass (now nonvanishing). Yes, the pions (the particles) mediate the remaining interaction.
It's really the same thing, take any explanation of the Higgs (you know, rumors propagating through a crowd, waves on the ocean, what have you), just replace "Higgs boson" with "pion" and "Higgs field" with "quark condensate".
What I was trying to say is that I do not know how this condensate is related to the energy contained in gluons. To begin with it seems strange that it would be gluons and not sea quarks. Either way, I don't see anything that would technically capture this popular image. The lattice people add up patiently everything from first principles, they don't use the effective model, that would be rather pointless. Best,
I hope I'm not out of line pointing out a subtlety in your discussion. The masses of the low-lying baryons can be understood as generated from the quark-antiquark condensate, in an effective description (sorry for my inelegant statement, which you summarized more clearly). While this accounts for most of the mass of the visible universe, the masses of heavy hadrons are not produced this way.
But the mechanism you describe is not adequate for most strongly-interacting particles. These include very excited states of nucleons (higher on their Regge trajectories), non-pseudoscalar mesons, hadrons with heavy quarks, and global resonances (assuming they exist). These masses are all related to the QCD scale by some other mechanism, which is not yet understood. Only some understanding of confinement and the mass gap in QCD will explain these.
Sorry for nitpicking. Thanks for your public service - most of my nonscientist friends are surprised when I tell them the Higgs is not responsible for all the mass in the Universe.
Thanks for presenting this. I often shock my nonscientist friends when I tell them most of the mass in the Universe is not from the Higgs.
Having said that, there are limits to your statement. The quark-antiquark-condensate mechanism gives masses only for low-lying baryons (which make up most of the mass in the Universe), but not other strongly-interacting particles. By these I mean heavier baryons, non-pseudoscalar mesons and glueball resonances (assuming these exist).
The other masses will probably only be understood through the mechanism of quark confinement (which continues to elude everyone).
Sorry to nitpick and thanks again for your elegant review of the topic.
http://hyperphysics.phy-astr.gsu.edu/hbase/particles/haddia.html It is like turning one baseball bat from a whole 20-foot tree trunk. Sawdust. http://www.curtismeyer.com/material/lecture.pdf If this was designed by a supreme being, and it was hired, Personnel needs a performance review.
Until somebody has a better way to fill the Periodic Table, I suppose we're stuck. Getting to where we are was terrifically wasteful, expensive, and messy. I like to think there is a loophole allowing some serious traveling to fresh shores. There should be a bonus round for getting this far and still having dreams of doing better. (Grant funding - quirt of the Gods)
Hi Bee, Pion model may be ok as a temporary intermediate model, but as a fundamental theory of masses it is going backwards.We already know about quarks and gluons.So in principle, everything should be explained in terms of quarks and gluons.What is your thought on this?
It's an effective model. It's a low energy approximation to QCD. I don't understand your comment. It *is* quarks and gluons, just in a simpler description, roughly speaking. Best,
Hi Bee, My complaint is that in old days, before knowing about quarks and gluons, people used to talk about nuclear forces coming from pion exchanges etc.I suppose they are assuming quark gluon condensate as pions to simplify the calculation. But as a fundamental theory of masses, it does not impress me. There are already numerical lattice gauge calculations which reproduce masses of many hadrons from quarks and gluons. So I am not sure what advance in our fundamental understanding of masses can come from these effective models.
First, I didn't say there's anything new about nonlinear sigma model, I said the new thing is that we might be able to find a way to measure details of the chiral phase transition. Besides this though you didn't understand my answer, so let me try it again. It's an effective model. It's an approximation that is valid in the limit used. Yes, that simplifies matters, but in a controlled way. It's not in a regime where you are sensitive to the fundamental constituents. Your complaint that it 'isn't fundamental' is equally nonsensical as complaining that architects don't use general relativity when concerned with stability of skyscrapers. Best,
B.
12:09 AM, August 25, 2015
[Image]
Diagram depicting pion exchange
between a proton and a neutron.
Image souce: Wikipedia.
When the discovery of the Higgs boson was confirmed by CERN, physicists cheered and the world cheered with them. Finally they knew what gave mass to matter, or so the headlines said. Except that most of the mass carried by matter doesn’t come courtesy of the Higgs. Rather, it’s a short-lived particle called the pion that generates it.
The pion is the most prevalent meson, composed of a quark and anti-quark, and the reason you missed the headlines announcing its discovery is that it took place already in 1947. But the mechanism by which pions give rise to mass still holds some mysteries, notably nobody really knows at which temperature it happens. In a recent PRL now researchers at RHIC in Brookhaven have taken a big step towards filling in this blank.
Observation of charge asymmetry dependence of pion elliptic flow and the possible chiral magnetic wave in heavy-ion collisions
STAR Collaboration
Phys. Rev. Lett. 114, 252302 (2015) arXiv:1504.02175 [nucl-ex]
There are three variants of pions: one positively charged, one negatively charged, and one neutral. The mechanism of mass generation by pions is mathematically very similar to the Higgs mechanism, but it involves entirely different particles and symmetries, and an energy about one thousandth of that where the Higgs mechanism operates.
In contrast to the Higgs which gives masses to elementary particles, the pion is responsible for generating most of the masses of composite particles that are found in atomic nuclei, protons and neutrons, collectively called nucleons. If we would only add up the masses of the elementary particles – the up and down quarks – that they are made up of, we would get a badly wrong answer. Instead, much of the mass is in a background condensate, mathematically analogous to the Higgs field (vev).
The pion is the Goldstone boson of the “chiral symmetry” of the standard model, a symmetry that relates left-handed with right-handed particles. It is also one of the best examples for technical naturalness. The pion’s mass is suspiciously small, smaller than one would naively expect, and therefore technical naturalness tells us that we ought to find an additional symmetry when the mass is entirely zero. And indeed it is the chiral symmetry that is recovered when the pions’ masses all vanish. The pions aren’t exactly massless because chiral symmetry isn’t an exact symmetry, after all the Higgs does create masses for the quarks, even if they are only small ones.
Mathematically all this is well understood, but the devil is in the details. The breaking of chiral symmetry happens at an energy where the strong nuclear force is strong indeed. This is in contrast to the breaking of electro-weak symmetry that the Higgs participates in, which happens at much higher energies. The peculiar nature of the strong force has it that the interaction is “asymptotically free”, meaning it gets weaker at higher energies. When it’s weak, it is well understood. But at low energies, such as close by chiral symmetry breaking, little can be calculated from first principles. Instead, one works on the level of effective models, such as that based on pions and nucleons rather than quarks and gluons.
We know that quarks cannot float around freely but that they are always bound together to multiples that mostly neutralize the “color charge” that makes quarks attract each other. This requirement of quarks to form bound states at low energies is known as “confinement” and exactly how it comes about is one of the big open questions in theoretical physics. Particle physicists deal with their inability to calculate it by using tables and various models for how the quarks find and bind each other.
The breaking of chiral symmetry which gives mass to nucleons is believed to take place at a temperature close by the transition in which quarks stop being confined. This deconfinement transition has been subject of much interest and lead to some stunning insights about the properties of the plasma that quarks form when no longer confined. In particular this plasma turned out to have much lower viscosity than originally believed, and the transition turned out to be much smoother than expected. Nature is always good for a surprise. But the chiral phase transition hasn’t attracted much attention, at least so far, though maybe this is about to change now.
These properties of nuclear matter cannot be studied in collisions of single highly energetic particles, like proton-proton collisions at the LHC. Instead, one needs to bring together as many quarks as possible, and for this reason one collides heavy nuclei, for example gold or lead nuclei. RHIC at Brookhaven is one of the places where these studies are done. The GSI in Darmstadt, Germany, another one. And the LHC also has a heavy ion program, another run of which is expected to take place later this year.
But how does one find out whether chiral symmetry is restored together with the deconfinement transition? It’s a tough question that I recall being discussed already when I was an undergraduate student. The idea that emerged over long debates was to make use of the coupling of chiral matter to magnetic fields.
The heavy ions that collide move at almost the speed of light and they are electrically charged. Since moving charges create magnetic fields, this generically causes very strong magnetic fields in the collision region. The charged pions that are produced in large amounts when the nuclei collide couple to the magnetic field. And their coupling depends on whether or not they have masses, ie it depends on whether chiral symmetry was restored or not. And so the idea is that one measures the distribution of charged pions that come out of the collision of the heavy nuclei, and from that one infers whether chiral symmetry was restored and, ideally, what was the transition temperature and the type of phase transition.
So much for the theory. In practice of course it isn’t that easy to find out exactly what gave rise to the measured distribution. And so the recent results have to be taken with a grain of salt: even the title of the paper carefully speaks of a “possible observation” rather than declaring it has been observed. It will certainly take more study to make sure they are really seeing chiral symmetry restoration and not something else. In any case though, I find this an interesting development because it demonstrates that the method works, and I think this will be a fruitful research direction about which we will hear more in the future.
For me chirality has always been the most puzzling aspect of the standard model. It’s just so uncalled for. That molecules come in left-handed and right-handed variants, and biology on Earth settled on mostly the left-handed, ones can be put down as a historical accident – one that might not even have repeated on other planets. But fundamentally, on the level of elementary particles, where would such a distinction come from?
The pion, I think, deserves a little bit more attention.
posted by Sabine Hossenfelder at 4:08 AM on Aug 21, 2015
"The origin of mass. Or, the pion’s PR problem."
30 Comments -
The label on the Wikepdia figure reads "... between a proton and a nucleon" should it not read "... between a proton and a neutron"?
5:35 AM, August 21, 2015
Bob,
Thanks for letting me know, I've fixed the typo!
5:44 AM, August 21, 2015
Hi Bee,
An interesting post. I must admit that I have not come across the pion as the intermediate particle giving mass to nucleons, definitely a lack in my knowledge of high energy theory.
Do you have a reference where "Instead, much of the mass is in a background condensate, mathematically analogous to the Higgs field (vev)." is explained in detail? A quick wikipedia search does not get me anywhere.
Cheers,
Erik
7:56 AM, August 21, 2015
Hi Erik,
This may be a starting point. Best,
B.
8:18 AM, August 21, 2015
I think your post is a little confusing because you don't clearly distinguish between the explicit breaking of chiral symmetry due to non-zero quark masses, which is what gives rise to the (small) pion mass, and the spontaneous breaking of the (approximate) chiral symmetry, which is what gives rise to the (large) nucleon mass. So if you imagine taking the quark masses to zero, the pion mass goes to zero, while the nucleon mass doesn't change much. On the other hand, if you increase the temperature (like at RHIC), you can reverse the spontaneous symmetry breaking, but the explicitly broken symmetry remains.
10:23 AM, August 21, 2015
Right. Well, I did say that it's not an exact symmetry due to the small quark masses.
10:43 AM, August 21, 2015
arXiv:1504.02175, "3×10^14 T" 10^10 T is 4×10^25 J/m^3, E/c^2 = 445,000 g/cm^3. u_B = (B^2)/2μ_o, 3×10^14 T, ~twice nuclear density. Vacuum birefringence arXiv:1212.1897 Footnotes become important.
"For me chirality has always been the most puzzling aspect of the standard model. It’s just so uncalled for." First observation, then theory. "But fundamentally, on the level of elementary particles, where would such a distinction come from?" The false vacuum then decay (dilution) had a pseudoscalar component, thus baryogenesis and onward.
"The pions aren’t exactly massless because chiral symmetry isn’t an exact symmetry" Test vacuum chiral divergence at 0.1 nm^3 scales. 1) Chemically and macroscopically identical, single crystal test masses in enantiomorphic space groups: Eötvös experiment; 2) Cryogenic molecular beam divergent rotational temperatures of extreme geometrically chiral molecules' racemate. Look.
11:45 AM, August 21, 2015
I think "... a fruitful research direction about which we will here more in the future" should be "... a fruitful research direction about which we will hear more in the future" ("here" should be "hear").
1:19 PM, August 21, 2015
This is the first time I saw someone say that the pion is the Higgs boson of hadronic mass. Usually they just talk vaguely about QCD. Kudos to Sabine for highlighting the real God particle.
2:39 PM, August 21, 2015
I think the statement:
"But at low energies, such as close by chiral symmetry breaking, little can be calculated from first principles. Instead, one works on the level of effective models, such as that based on pions and nucleons rather than quarks and gluons."
does not do justice to the results by lattice QCD: it can nowadays handle realistic pion masses, and it does provide remarkable results from first principles of QCD.
6:12 PM, August 21, 2015
Normally I would identify the pions with a decrease in the mass of a nucleus from the constituent nucleons, and the gluons with the increase in the mass of nucleons from the constituent valence quarks.
It seems here you are implying the proton and neutron have increased masses from the sum of the masses of the constituent valence quarks not due to gluons or the fundamental degrees of freedom of QCD as I had always visualized it, but due to some contribution of pion exchange amongst the quarks. Could you give some more details on this to help me understand better? Thanks!
6:12 PM, August 21, 2015
Hi, I sort of wish you would refer to it as pion condensate rather than just pions. Saying pions are responsible for mass of nucleons is technically correct, but it makes it sound like the nucleon mass is just mass of quarks + mass of pions which it is not. The pion condensate is the order parameter for chiral symmetry and when it goes to zero at high temperatures, the mass of nucleons disappear.
9:01 PM, August 21, 2015
Wolbi,
I didn't say nothing can be calculated...
2:18 AM, August 22, 2015
JimV,
Thanks for pointing it out, I've fixed that typo.
2:18 AM, August 22, 2015
Sky,
I didn't want to make it more confusing as it is already. The pion gives rise to masses of nucleons as much or as little as the Higgs boson (!) gives rise to masses of quarks. You're right of course in that it's not the particle that is relevant, but the condensate (which I did mention). Best,
B.
2:21 AM, August 22, 2015
Jim,
No, the increased masses of the nucleons do not come from the masses of the valence quarks, that's the whole point, they come from the quark condensate. It's an effective model, and "effective" here isn't just a word but a technical term: There are no quarks and gluons in this description, instead there are pions and nucleons. They are fundamentally composed of quarks and gluons of course, and in principle the effective theory is a low-energy approximation to the full QCD interaction. So the picture you have in mind is on a vague level correct, in practice what one uses is the effective theory. Best,
B.
2:28 AM, August 22, 2015
Sabine what do you think of the idea that dark matter could be composed of pions: The SIMP Miracle (arXiv: 1402.5143) that has been accepted in Physical Review Letters.
8:19 AM, August 22, 2015
I didn't (mean to) imply the mass came from the valence quarks, I said my impression was the gluons cause an 'increase in the mass of nucleons from the constituent valence quarks,' corresponding with what you said.
I thought pions were associated with the interactions between nucleons, while the gluons were associated with what held the hadrons together, and therefore increased the mass from the sum of the masses of the valence quarks.
Now if we point toward effective theories formed from the goldstone bosons of the flavor symmetries can't we talk about from where in the full theory the quark condensate, which breaks the symmetry, comes from? Is it reasonable to associate this with the gluons? Has lattice QCD been able to do anything toward this end?
10:09 AM, August 22, 2015
Nicolas,
Thanks for the reference, but I don't know the paper. First thing that comes to my mind is that pions are unstable, so not sure how this should work, but I'll have a look at this. Best,
B.
10:54 AM, August 22, 2015
Jim,
As Sky pointed out in a comment above, you're mistaken in thinking that the pions (the particles) increase the mass of the nucleons. Note how carefully I vaguely said they "generate" it. The pion plays the same role as the Higgs particle (they are both bosons). The particle is an excitation over a background, but it's the (coupling to the) background (the condensate) that is responsible for the mass (now nonvanishing). Yes, the pions (the particles) mediate the remaining interaction.
It's really the same thing, take any explanation of the Higgs (you know, rumors propagating through a crowd, waves on the ocean, what have you), just replace "Higgs boson" with "pion" and "Higgs field" with "quark condensate".
What I was trying to say is that I do not know how this condensate is related to the energy contained in gluons. To begin with it seems strange that it would be gluons and not sea quarks. Either way, I don't see anything that would technically capture this popular image. The lattice people add up patiently everything from first principles, they don't use the effective model, that would be rather pointless. Best,
B.
11:06 AM, August 22, 2015
Hi Bee,
I hope I'm not out of line pointing out a subtlety in your discussion. The masses of the low-lying baryons can be understood as generated from the quark-antiquark condensate, in an effective description (sorry for my inelegant statement, which you summarized more clearly). While this accounts for most of the mass of the visible universe, the masses of heavy hadrons are not produced this way.
But the mechanism you describe is not adequate for most strongly-interacting particles. These include very excited states of nucleons (higher on their Regge trajectories), non-pseudoscalar mesons, hadrons with heavy quarks, and global resonances (assuming they exist). These masses are all related to the QCD scale by some other mechanism, which is not yet understood. Only some understanding of confinement and the mass gap in QCD will explain these.
Sorry for nitpicking. Thanks for your public service - most of my nonscientist friends are surprised when I tell them the Higgs is not responsible for all the mass in the Universe.
11:45 AM, August 23, 2015
Hi Peter,
Interesting, thanks, I didn't know this! Do you have some reference where I could read up on this? Best,
B.
11:48 AM, August 23, 2015
Bee,
Thanks for presenting this. I often shock my nonscientist friends when I tell them most of the mass in the Universe is not from the Higgs.
Having said that, there are limits to your statement. The quark-antiquark-condensate mechanism gives masses only for low-lying baryons (which make up most of the mass in the Universe), but not other strongly-interacting particles. By these I mean heavier baryons, non-pseudoscalar mesons and glueball resonances (assuming these exist).
The other masses will probably only be understood through the mechanism of quark confinement (which continues to elude everyone).
Sorry to nitpick and thanks again for your elegant review of the topic.
11:52 AM, August 23, 2015
Most reviews on lattice gauge theories (like Mike Creutz' book) discuss it.
I wrote a second version of this comment, because I had trouble publishing the first (entirely my fault).
11:54 AM, August 23, 2015
Any book on lattice gauge theories, e.g., that by Mike Creutz.
11:56 AM, August 23, 2015
http://hyperphysics.phy-astr.gsu.edu/hbase/particles/haddia.html
It is like turning one baseball bat from a whole 20-foot tree trunk. Sawdust.
http://www.curtismeyer.com/material/lecture.pdf
If this was designed by a supreme being, and it was hired, Personnel needs a performance review.
Until somebody has a better way to fill the Periodic Table, I suppose we're stuck. Getting to where we are was terrifically wasteful, expensive, and messy. I like to think there is a loophole allowing some serious traveling to fresh shores. There should be a bonus round for getting this far and still having dreams of doing better. (Grant funding - quirt of the Gods)
8:06 PM, August 23, 2015
Hi Bee,
Pion model may be ok as a temporary intermediate model, but as a fundamental theory of masses it is going backwards.We already know about quarks and gluons.So in principle, everything should be explained in terms of quarks and gluons.What is your thought on this?
9:39 AM, August 24, 2015
Kashyap,
It's an effective model. It's a low energy approximation to QCD. I don't understand your comment. It *is* quarks and gluons, just in a simpler description, roughly speaking. Best,
B.
10:29 AM, August 24, 2015
Hi Bee,
My complaint is that in old days, before knowing about quarks and gluons, people used to talk about nuclear forces coming from pion exchanges etc.I suppose they are assuming quark gluon condensate as pions to simplify the calculation. But as a fundamental theory of masses, it does not impress me. There are already numerical lattice gauge calculations which reproduce masses of many hadrons from quarks and gluons. So I am not sure what advance in our fundamental understanding of masses can come from these effective models.
2:14 PM, August 24, 2015
kashyap,
First, I didn't say there's anything new about nonlinear sigma model, I said the new thing is that we might be able to find a way to measure details of the chiral phase transition. Besides this though you didn't understand my answer, so let me try it again. It's an effective model. It's an approximation that is valid in the limit used. Yes, that simplifies matters, but in a controlled way. It's not in a regime where you are sensitive to the fundamental constituents. Your complaint that it 'isn't fundamental' is equally nonsensical as complaining that architects don't use general relativity when concerned with stability of skyscrapers. Best,
B.
12:09 AM, August 25, 2015