3 ms·

> Virtually every force we experience in everyday life (with the exception of gravity) is electromagnetic in origin. [...] It wasn’t until the arrival of Oliver

> Virtually every force we experience in everyday life (with the exception of gravity) is electromagnetic in origin. [...] It wasn’t until the arrival of Oliver Heaviside, who reformulated and simplified the equations [...] Maxwell's original equations connected light and electricity. Maxwell's original 20 equations had 20 unknowns, using 'quaternion-based notation', which no one understood. Heaviside restated Maxwell's 20 equations into 4 equations using vector calculus. The restatements helped with simplification, but I believe it wasn't without cost. There's a lot that's still unexplained in our modern world, especially with regards to individual humans' experiences. I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about. There ought to be a link between electromagnetism and gravity, we just haven't figured it out yet. This wikipedia article was cited by Bing CoPilot in response to my query. It's above my pay grade, maybe one of you can translate it for me: https://en.wikipedia.org/wiki/Gravitoelectromagnetism https://en.wikipedia.org/wiki/Gravitoelectromagnetism

bfuller 2y ago
> I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about. I'm curious now, would you indulge me? If its woo woo we can just pretend no one is reading :)
- taxicabjesus 2y ago
  Some people are of the philosophical bent that our world is entirely random. But this doesn't commonly match our experience. For example, I often asked people how they met their significant other. Sometimes it was nothing special, some couples had quite a story. I had the sense that I got certain passengers for more than just transportation. Some people were having a rotten day, and I was able to cheer them up. One lady had some time to kill before her bus' departure time, so we went to the 24 hour diner, ordered our own pies and compared notes. When we got to the bus station she said it was the best birthday she'd had in quite a long time. This was a semi-recent comment about the matching algorithm: https://news.ycombinator.com/item?id=34402081 https://news.ycombinator.com/item?id=34402081 The most important thing I learned in my taxi was about substance abuse. This HN poll didn't get any upvotes, but it references some of the diaries I never finished: https://news.ycombinator.com/item?id=39071316 https://news.ycombinator.com/item?id=39071316 Another comment: https://news.ycombinator.com/item?id=25238488 https://news.ycombinator.com/item?id=25238488 If you're so inclined, I'm curious if you've experience is also that our universe is more than random?
  - nathan_compton 2y ago
    The universe clearly isn't completely random. I'm not sure anyone believes that.
  - bfuller 2y ago
    I am the founder of randonautica, so in short yes I have found that our universe is more than random. Thanks for the answer, sometimes I wish a little more intuition was allowed here rather than only analytics.
seanhunter 2y ago
Here is a set of lecture slides on the changing form of Maxwell's equations including the component form (which was apparently Maxwell's very first version), the quaternion form which came second and then Heaviside's version[1] Fun fact about Heaviside (that noone asked for) - he's also the guy who invented the "cover up" method of doing partial fraction decomposition quickly.[2] [1] https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/MaxwellUCL2.pdf https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/Ma... [2] https://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf https://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf
aap_ 2y ago
I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful. He certainly got rid of the quaternions but that seems like a minor difference because Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part. The major difference I found was that Maxwell was expressing things in terms of the scalar and vector potential (which is what you have to do in QED) whereas Heaviside got rid of that and just had an electric and magnetic field instead. I found that you need 7 of Maxwell's equations to derive the 4 Heaviside(?) equations. If you actually wanted to embrace quaternions you could write the famous 4 equations as just two (using natural units): ∇E + dB/dt = -ρ ∇B - dE/dt = J
- segfaultbuserr 2y ago
  > Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part. Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact... Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism. > I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful. In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations. The logic of the paper went like the following. First, he started with a definition of electric current [1]: C = kE D = cE / 4π Γ = C + D in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms. Next, a definition of magnetic current [2]: B = µH G = Ḃ / 4π = µḢ / 4π G' = gH + µḢ / 4π H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k. Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]: curl (H - h) = 4πΓ = 4πkE + cĖ -curl (e - E) = 4πG = 4πgH + µḢ in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0] This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions. This is not a complete summary, as he continued his analysis in a series of publications. A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin. [0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form." [1] Electric Papers, Volume I, Page 429, https://archive.org/details/electricalpapers01heavuoft/page/429 https://archive.org/details/electricalpapers01heavuoft/page/... [2] Page 441: https://archive.org/details/electricalpapers01heavuoft/page/441 https://archive.org/details/electricalpapers01heavuoft/page/... [3] Page 443: https://archive.org/details/electricalpapers01heavuoft/page/443 https://archive.org/details/electricalpapers01heavuoft/page/... [4] Page 449: https://archive.org/details/electricalpapers01heavuoft/page/449 https://archive.org/details/electricalpapers01heavuoft/page/...
teleforce 2y ago
Quaternion was crucial and instrumental tool in Maxwell discovery and the formulation of the electromagnetics (EM) equations. When Terence Tao was asked how come nobody has proof of the Riemann hypothesis, arguably the hardest of the Math problems, and according to him this is because there is no appropriate tools available at the moment to proof it. I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis. Unlike other waves for example sound waves, EM has a unique polarization property. In order to completely and correctly model EM based phenomena quaternion based formulation and representation is necessary. One of the reasons that almost all existing wireless modulation are not utilizing polarization is due to most of the microwave and wireless engineers are not familiar with quaternion. Ironically their biased attitude is not unlike early mathematicians and scientists that were very much opposed to complex number, and it turn out that almost all of the modern wireless modulation for example OFDM are utilizing complex number. For the derivation of the Maxwell’s equations using geometric algebra involving quaternion please check these articles and they can be summarized the into one elegant equation [1][2]. [1] Maxwell’s eight equations as one quaternion equation: https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwell-s-eight-equations-as-one-quaternion https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwe... [2] A derivation of the quaternion Maxwell’s equations using geometric algebra: https://peeterjoot.com/2018/03/05/a-derivation-of-the-quaternion-maxwells-equations-using-geometric-algebra/ https://peeterjoot.com/2018/03/05/a-derivation-of-the-quater...
- nathan_compton 2y ago
  Lay people seem to have this weird obsession with Quaternions and love to suggest that somehow theoretical physicists are missing something because they don't use them. But physicists are almost disgustingly familiar with SU(2) which is isomorphic to the quaternions and easier to work with and understand (quite obviously, in my opinion). It is hard to imagine, from my point of view, that a mere isomorphism stands between physicists and progress, especially given that physicists have long generalized _beyond_ SU(2) and the quaternions in their understanding of fundamental fields. Formulating an SU(3) gauge theory in terms of quaternions would at least be difficult and almost certainly be goofy, if not impossible. As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.
nathan_compton 2y ago
Gravitoelectromagnetism doesn't actually have anything to do with electromagnetism except that certain formal features of the theory of general relativity correspond roughly to the mathematical structures we talk about in electromagnetism, albeit with the proviso that the symmetries underlying the two theories are different.

bfuller 2y ago

> I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about. I'm curious now, would you indulge me? If its woo woo we can just pretend no one is reading :)

taxicabjesus 2y ago
Some people are of the philosophical bent that our world is entirely random. But this doesn't commonly match our experience. For example, I often asked people how they met their significant other. Sometimes it was nothing special, some couples had quite a story. I had the sense that I got certain passengers for more than just transportation. Some people were having a rotten day, and I was able to cheer them up. One lady had some time to kill before her bus' departure time, so we went to the 24 hour diner, ordered our own pies and compared notes. When we got to the bus station she said it was the best birthday she'd had in quite a long time. This was a semi-recent comment about the matching algorithm: https://news.ycombinator.com/item?id=34402081 https://news.ycombinator.com/item?id=34402081 The most important thing I learned in my taxi was about substance abuse. This HN poll didn't get any upvotes, but it references some of the diaries I never finished: https://news.ycombinator.com/item?id=39071316 https://news.ycombinator.com/item?id=39071316 Another comment: https://news.ycombinator.com/item?id=25238488 https://news.ycombinator.com/item?id=25238488 If you're so inclined, I'm curious if you've experience is also that our universe is more than random?
- nathan_compton 2y ago
  The universe clearly isn't completely random. I'm not sure anyone believes that.
- bfuller 2y ago
  I am the founder of randonautica, so in short yes I have found that our universe is more than random. Thanks for the answer, sometimes I wish a little more intuition was allowed here rather than only analytics.

seanhunter 2y ago

Here is a set of lecture slides on the changing form of Maxwell's equations including the component form (which was apparently Maxwell's very first version), the quaternion form which came second and then Heaviside's version[1] Fun fact about Heaviside (that noone asked for) - he's also the guy who invented the "cover up" method of doing partial fraction decomposition quickly.[2] [1] https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/MaxwellUCL2.pdf https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/Ma... [2] https://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf https://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf

aap_ 2y ago

I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful. He certainly got rid of the quaternions but that seems like a minor difference because Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part. The major difference I found was that Maxwell was expressing things in terms of the scalar and vector potential (which is what you have to do in QED) whereas Heaviside got rid of that and just had an electric and magnetic field instead. I found that you need 7 of Maxwell's equations to derive the 4 Heaviside(?) equations. If you actually wanted to embrace quaternions you could write the famous 4 equations as just two (using natural units): ∇E + dB/dt = -ρ ∇B - dE/dt = J

segfaultbuserr 2y ago
> Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part. Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact... Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism. > I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful. In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations. The logic of the paper went like the following. First, he started with a definition of electric current [1]: C = kE D = cE / 4π Γ = C + D in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms. Next, a definition of magnetic current [2]: B = µH G = Ḃ / 4π = µḢ / 4π G' = gH + µḢ / 4π H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k. Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]: curl (H - h) = 4πΓ = 4πkE + cĖ -curl (e - E) = 4πG = 4πgH + µḢ in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0] This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions. This is not a complete summary, as he continued his analysis in a series of publications. A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin. [0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form." [1] Electric Papers, Volume I, Page 429, https://archive.org/details/electricalpapers01heavuoft/page/429 https://archive.org/details/electricalpapers01heavuoft/page/... [2] Page 441: https://archive.org/details/electricalpapers01heavuoft/page/441 https://archive.org/details/electricalpapers01heavuoft/page/... [3] Page 443: https://archive.org/details/electricalpapers01heavuoft/page/443 https://archive.org/details/electricalpapers01heavuoft/page/... [4] Page 449: https://archive.org/details/electricalpapers01heavuoft/page/449 https://archive.org/details/electricalpapers01heavuoft/page/...

teleforce 2y ago

Quaternion was crucial and instrumental tool in Maxwell discovery and the formulation of the electromagnetics (EM) equations. When Terence Tao was asked how come nobody has proof of the Riemann hypothesis, arguably the hardest of the Math problems, and according to him this is because there is no appropriate tools available at the moment to proof it. I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis. Unlike other waves for example sound waves, EM has a unique polarization property. In order to completely and correctly model EM based phenomena quaternion based formulation and representation is necessary. One of the reasons that almost all existing wireless modulation are not utilizing polarization is due to most of the microwave and wireless engineers are not familiar with quaternion. Ironically their biased attitude is not unlike early mathematicians and scientists that were very much opposed to complex number, and it turn out that almost all of the modern wireless modulation for example OFDM are utilizing complex number. For the derivation of the Maxwell’s equations using geometric algebra involving quaternion please check these articles and they can be summarized the into one elegant equation [1][2]. [1] Maxwell’s eight equations as one quaternion equation: https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwell-s-eight-equations-as-one-quaternion https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwe... [2] A derivation of the quaternion Maxwell’s equations using geometric algebra: https://peeterjoot.com/2018/03/05/a-derivation-of-the-quaternion-maxwells-equations-using-geometric-algebra/ https://peeterjoot.com/2018/03/05/a-derivation-of-the-quater...

nathan_compton 2y ago
Lay people seem to have this weird obsession with Quaternions and love to suggest that somehow theoretical physicists are missing something because they don't use them. But physicists are almost disgustingly familiar with SU(2) which is isomorphic to the quaternions and easier to work with and understand (quite obviously, in my opinion). It is hard to imagine, from my point of view, that a mere isomorphism stands between physicists and progress, especially given that physicists have long generalized _beyond_ SU(2) and the quaternions in their understanding of fundamental fields. Formulating an SU(3) gauge theory in terms of quaternions would at least be difficult and almost certainly be goofy, if not impossible. As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.

nathan_compton 2y ago

Gravitoelectromagnetism doesn't actually have anything to do with electromagnetism except that certain formal features of the theory of general relativity correspond roughly to the mathematical structures we talk about in electromagnetism, albeit with the proviso that the symmetries underlying the two theories are different.