It's very helpful to use natural units (or QCD units) when working with particle physics. Set the speed of light, the reduced Planck's constant, and Boltzmann's constant equal to 1. For QCD units also set the mass of the proton equal to 1. Then compute all the other units based on this. (There are several other systems of "natural" units also in use. All of them set the speed of light to 1, but differ in what other units are included.)
A particular advantage to intuition comes with the famous e=mc^2. Since c is 1, and 1^2 is 1, e=m. Energy and mass are completely equivalent. Strong force binding energy in a proton is mass. The only bit that's not straight from binding energy is from the Higgs mechanism, which is where the mass of the electron (and the quarks themselves) comes from. Matt Strassler has an excellent explanation of that: https://profmattstrassler.com/articles-and-posts/particle-ph...
This comment is much better than the article, because it tells that some of the complex relations between conceptual entities in physic, could be presented in a simplified but equivalent way. So it hints that the "problem" presented in the article is only an artifact.
Henry Poincaré wrote exactly this in 1902 in "the science and hypothesis"
Nous sommes acculés à la définition suivante, qui n'est qu'un aveu d'impuissance : les masses sont des coefficients qu'il est commode d'introduire dans les calculs.[0]
I am also happy for the reference to the excellent blog of Matt Strassler
Equalities that involve units aren't straightforward. Not a physicist of any sort but don't all the 1's still fundamentally change the equality since they change the units? I would guess they therefore implicitly change the intuition behind these equations?
I would say it's the opposite. A unit conversion is not fundamental at all, but a matter of arbitrary convention. 0C and 32F are exactly the same temperature. Mass and energy are exactly the same property. c^2 is a constant, so E doesn't even vary as the square of anything - it's just a linear unit conversion. If you follow it further, you find c is only even in there because some other units are defined in terms of it. That is, the ME equivalence doesn't really have anything to do with the speed of information propagation per se - but our chosen unit conventions do.
It's because of people's fondness for legacy and backwards compatibility that we don't use natural units all the time, and still write "e=mc^2". I find this insane, personally, but that's just the way it goes.
I guess I should add that apparently this is still controversial? But in my mind, the only reason for this is that people find meaning in units. In this case, ME being the same property implies duration being exactly the same as length, which implies spacetime is a unified, regular 4D manifold. ME not being the same property implies time is special. I've been fully persuaded that duration is fundamentally exactly the same thing as length, so I've chosen my side, in case there's any serious controversy.
Definitely not. Everything couples to the stress-energy tensor, but not every type of particle has rest/invariant mass. No one really speaks in terms of relativistic mass anymore. (So a photon has energy but no rest mass.)
That's what makes natural units natural. Certain quantities (like mass and energy, or distance and time) are physically related such that the distinction is, in a sense, artificial. The conversion factors between the units used for these pairs of quantities are the fundamental physical constants that get set to unity in a natural system of units.
The distinction between distance and time is not artificial at all. It is encoded in the metric, at the most fundamental level. The existence of symmetries is not the same as equivalence.
Parent is not talking about the "distinction between distance and time" but about the multipliers that come into play into their equations (which are artificial and based on the base units selected).
The distinction between feet and seconds is just as real as the distinction between distance and time, and is much more real than the distinction between feet and meters. Look at the parent's literal words and pretend you did not already know what's going on.
I'm well aware of the practical value and conceptual clarity of natural units. They are just not being explained well in this thread. "What the teacher really meant was..." is not a good defense when the student doesn't understand.
> The distinction between feet and seconds is just as real as the distinction between distance and time
There is a "real" distinction between [light]seconds of distance and seconds of time, but it's a difference in what is being measured, not in how it is quantifiable. Measuring them with the same units doesn't imply that they are perfect substitutes, any more than one could arbitrarily replace an ounce of gold with the same weight of feathers; in practice, they are non-interchangeable enough that gold is even measured with a different type of "ounce". Likewise with spacetime: for most purposes human non-physicists think about, they are completely non-substitutable. So practically, it works well to use a "Troy" sort of system for one. But it's a purely human distinction.
> but it's a difference in what is being measured, not in how it is quantifiable. Measuring them with the same units doesn't imply that they are perfect substitutes
I agree! Please read my comments carefully. I am not arguing against the usefulness or deep conceptual importance of natural units! I am critiquing the terrible explanations being given in this thread.
> Likewise with spacetime: for most purposes human non-physicists think about, they are completely non-substitutable. So practically, it works well to use a "Troy" sort of system for one. But it's a purely human distinction.
No! Space and time are not interchangeable in any universal sense even if they are universally linked by a symmetry. A space-like interval and a time-like interval cannot be interchanged with each other by a Lorentz transformation. The distinction between space-like and time-like intervals is observer independent.
Would you agree that if there was a unit for 299,792,458 meters called 'q' then the "speed of light" could be expressed as 1 q/s? Then e = m*c^2 is e = m x 1 or e = m (but the units of q^2/s^2 are still there, just the arithmetic is simplified)
I am not arguing against natural units. I use them daily, and they are more than just practically useful. I am critiquing bad explanations for their interpretation and justification.
"are physically related such that, in a sense..." I didn't say they were equivalent, I said they were connected. You can't transform a time-like interval into a space-like one, but you can transform one time-like interval into another such that the separation on time increase and the separation in space decreases.
There's only so much clarity you can fit into a HN comment.
In the sense that GP states, yes, but I disagree with GP: "E=m" is wrong even if c=1 in the units you're working with since c is not unitless; it has speed units. You can't have a kilogram of energy. You need to multiply a mass by something with squared-speed units to get something with energy units.
Consider the more mundane example of heat and work. The standard unit of heat is the calorie, which is the heat needed to increase the temperature of one gram of water by one degree celsius under standard conditions. The standard unit of work is the joule, which is the work needed to exert a force of one newton over a distance of one meter.
James Joule discovered that heat and work can be related by:
h = kw
Where k is a constant equal to 4.184 calories/joule.
But really, this isn't a relationship between heat and work. This is a statement of equivalence. The constant of 4.184cal/J just tells you how the two units differ. They both describe the same property. Really, h = w. The version with k is only needed if you insist on using the traditional, different units on the left and right sides.
The same is true of e = mc^2. This does not tell you how two different fundamental properties are related, it merely tells you how two units, traditionally considered separate, actually measure the same underlying property. Really, e = m. The c^2 is only needed if you insist on using J on one side and kg on the other.
That is to say, you absolutely can have a kilogram of energy. If you have an object with a mass of one kilogram, you do have a kilogram of energy. Physicists and engineers don't typically use that unit for energy, but there's no reason they couldn't, and likewise no reason they couldn't use joules for mass.
Thank you for the nice explanation, despite my somewhat rude and uninformed posts! I had some trouble accepting it, but the heat/work example made it "click" for me, once my brain got past its initial stubborn reaction of "but of course heat and work are both energy, whereas mass is clearly not!"
So any time two physical quantities are related by a constant factor -- even if said constant factor has what we currently think of as "units" (and even particularly weird units like squared speed) -- they are fundamentally equivalent. I'm still mulling this over, but it's a pretty fascinating mental revelation for me.
Nice. I thought the heat/work thing was an interesting angle on it and I'm glad it worked.
Edit: I also want to say that I didn't find the comment I replied to rude in any way. It's direct and straightforward, but that's pretty common around here.
I won't say you're wrong since I stopped taking physics in high school, but this makes absolutely no sense to me. How does defining the Planck speed unit to be the speed of light suddenly make c a unitless constant? It's not "c=1" (except in informal shorthand), it's "c = 1 Planck speed unit". It seems like all sorts of things would "break" very quickly when you start dropping units.
Maybe someone has a link that goes into more depth on why all of this is okay?
The value of c is arbitrary and depends on one's choice of coordinates. However the presence of c in the line element of a spacetime like ours is mandatory. There is, however, some conceptual value in setting c to some value other than 1, which I'll return to in the last paragraph below.
When using a pseudo-Riemannian manifold one uses a metric signature where (keeping it simple, cf. "metric signature" on wikipedia) coordinates on orthogonal dimensions of can take one sign, or the opposite sign and a constant multiplier. In the Lorentzian case there will be one dimension taking one sign, and one or more taking the other; the choice of whether the solitary dimension takes a + or - sign is a matter of convention or preference. Conventionally the solitary dimension also takes the constant multiplier and is called the timelike dimension. A Lorentzian signature (minuses, 1) or (1, plusses) guarantees that one can describe paths through the manifold as null, timelike, or spacelike, and this gives one a causal structure.
Our universe can be well represented by a Lorentzian manifold (3, 1) or (1, 3), and this has been tested to exquisite precision. It does not tell us the value of the constant c, but we can determine that from tests of causal relations, the boundaries of which will be null. Alternatively, a massless pointlike object will always travel on null geodesics.
One runs into c being set to unity in systems geometrized units in relativity often; it's very handy to have mark off coordinates as e.g. "-seconds" vs "c seconds" (-,+,+,+ aka (1,3)) as long as one doesn't mess up one's dimensional analyses (which is unfortunately easy).
As a concrete example, the line element for (1,3) flat spacetime using Cartesian coordinates is dS^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2. Compare the formula for Euclidean distance in the (Euclidean flat) plane between points p = (p1,p2) and q = (q1,q2) for (x,y) coordinates: d(p,q) = sqrt((q1-p1)^2 + (q2-p2)^2), which we could rewrite as dx = q1-p1, dy = q2-p2, ds = sqrt(dx^2 + dy^2) or ds^2 = dx^2 + dy^2. In Euclidean 3-space, we add another axis: ds^2 = dx^2 + dy^2 + dz^2. In Lorentzian 4-spacetime, we have to change the sign, so ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2. Note that we are not restricted to use any particular unit of distance or system of units; dx could be in metres, miles, astronomical units, light-years, gigaparsecs or practically anything else, while dt could be in seconds or fortnights or any other handy unit of time. When setting c to unity, we do need to choose appropriate units for dx (and dy and dz ...) vs dt. In SI units, that's seconds and light-seconds, but we could use another system if we wanted.
Notably we aren't restricted to Cartesian coordinates, however if we were to change to some other system of coordinates (e.g. polar ones) the line element would need to reflect that. For example, in the Lorentzian 4-spacetime case, we would write ds^2 = dr^2 + r^2 * dtheta^2 + r^2sin^2(theta)dphi^2 -c^2dt^2.
Finally, when using SI units (for example), one can see very clearly that a path taken by an object moving much slower than the speed of light is totally dominated by the amount of time between starting point and finishing point, because the value of c is large. Using the (+,-,-,-) metric signature [ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2], a large c helps make it clear that lightlike paths through spacetime are shorter, and purely timelike paths (where dx=0, dy=0, dz=0, dt != 0) have extremized length, since we don't subtract anything from cdt. This is the root of the explanation of the twin paradox in flat spacetime: the twin moving quickly compared to the speed of light takes a shorter path between together1 = (x1,y1,z1,t1) and together2 = (x2,y2,z2,t1) than the twin moving slowly compared to the speed of light. In the extreme, twin A holds x=const,y=const,z=const, whereas twin B's x coordinate is only equal to twin A's at the start and end of the journey. This holds up under any system of coordinates; we could consider r=const vs changing r in spherical coordinates, for instance.
A particular advantage to intuition comes with the famous e=mc^2. Since c is 1, and 1^2 is 1, e=m.
That is clever, but you've already adjusted the mass to take into account the fact that you are now setting c=1. This reminds me of computer programming where, for the sake of an easy-to-read clarity, I decide to take one line of code and make it two. Basically, instead of doing this:
e=mc^2
You have decided to make it two lines of code:
m=mc^2
e=m
I often do that, especially if I'm working with junior devs and I want the code to be as easy as possible for them to read (90% of the time).
It's clever, but everyone should remember the assumption that allows this: the mass has already been adjusted to reflect the reality that of c=1.
Except you would otherwise have at least one of c, hbar, boltzman’s constant etc. in virtually every equation and now you don’t. You get a lot more mileage out of it than just simplifying one line.
Yea physicists are in a weird spot when it comes to inertia. Everyone’s heard of it but it’s not real. To say something has mass is to say it requires a force to move it which is what we call inertia. And also the E=m equivalence isn’t as simple as may be believed. The famous equivalence is cutting out a piece that is important when talking about photons.
For example, I said E=M. But that's for objects at rest. For an object in motion it's E^2=m^2c^4+p^2c^2. In natural units that turns into E=m+p. The energy of a photon is hf, where h is Planck's constant and f is the frequency. That energy is entirely the photon's momentum. And that's without getting into the issues of invariant mass or relativistic mass increases...
That's one of those things that, if you ask me, should fall outside the scope of a character set. As I see it, they are just the usual numbers, and restyling, resizing and positioning them should be a concern of a "text" layer, above the "character" layer.
It should. But Unicode's mandate included incorporating common existing character sets, and ISO8859-1 had ² and ³, and it all when down from there, so that now you can write 𝘌 = 𝘮𝘤² which isn't the same as E = mc².
A number of early computer systems used teletypes for I/O, and had programming languages in which exponentiation was expressed by moving a half line up, and array subscripts by moving a half line down.
I seem to recall the designer of ascii stating that it was a mistake, and they should've gone with prefix coding instead (following-is-capital,a->A;following is number,<byte>->67 or something along those lines - basically escape codes for everything, I suppose). Instead we got ascii evolved to really, really, really big ascii (utf-32, and the variable encoding utf-8).
But yeah, having a "next-is-superscript"-token would probably be cleaner than individual code points for every little glyph that might be a superscript...
Maybe in an ideal world, sure, but in this world, nobody is going to add those features to the text layer as long as there are perfectly good Unicode characters right there in the character layer. So we may as well use them.
No. Mass doesn't "determine" the speed of time (a lovely phrase, BTW). It is the observer's motion through space that does that. However, for an observer to move through time at a speed faster than 0 it has to move through space at a speed slower than c, and to do that it needs to have mass. A massless particle can only transfer energy if it moves at the speed of light. Massless particles moving slower than the speed of light cannot transfer energy, and so cannot be detected experimentally, and so do not exist.
I love reading about quantum mechanics at all different levels, and have read and appreciated articles well beyond what I could properly comprehend in fullness; but man was this article all over the place.
I've never seen an author use metaphors or similies so poorly before. To draw parallels between something hard to grasp and something made up and vaguely defined in an effort to explain the former is.. ill advised at best. Then there's the fact that the author spends forever to explain basic chemistry then jumps into color charge in such a way that anyone not already intimately familiar with at least the terminology of quantum physics would never be able to understand, then the author jumps from topic to topic seemingly in a race to drop references to as many different concepts as possible without actually explaining any of them, almost like a student writing an essay then going back and swapping words with a thesaurus to seem better informed.
Even the science aside, the writing itself is rather atrocious. The author "answers" mysteries he never even asked or previously posed, and expects readers to already know what he's trying to say so he can refer to that in his explanations of why he said it.
Then the author has a tendency to jump from field to field, converting apples into oranges with the help of a long-dead scientist only so he can add them together in the most basic way and then convert them back to apples again. But he got to prove that he knows of Avagdro, so obviously there's that.
Usually Nautilus articles are written much better than this. If you value your sanity or actually care to understand the topic discussed, do yourself a favor and look elsewhere.
I respectfully disagree. He is imprecise, metaphorical, and offers entertainment, rather than knowlege.
And jet, he knows his reader very well, and he delivers important bits of knowlege in very digestable form. Sure, this is not a substitution for reading a book.
But his target audience will never read that book, nor perform any research on their own.
Now i firmly remeber what QCD exist and firmly believe I don't know what it is about. Isn't it good, by itself?
Exactly...and even though he spends so much time discussing basic chemistry as if he is talking to an average joe, he jumps into use the term "mole" without explaining it.
For some reason people are still caught up in finding some difference between mass and energy.
There is no difference. They are two words that mean the same thing.
There are different kinds of mass-energy, some types are easy to convert into others, some types move at the speed of light, some don't. But there is nothing distinguishing mass from energy.
It was a revelation to see photons attracted by gravity - but once you realize it's energy that has a gravitational field [not just the particles we call mass], it would be surprising if photons were not attracted by gravity. (Although photons moving only at the speed of light have different equations governing their motion under acceleration.)
Now, all that said,
There is a distinction between things that only move at the speed of light, and things that never do. But the words energy and mass [as commonly used] do not properly fit those two categories.
Meanwhile, in real physics, there's a well-defined difference between the norm of a vector and one of its components. Mass is Lorentz-invariant, energy is not.
That's not completely accurate. Chemical energy and binding energy are also Lorentz-invariant.
The only type of energy is not Lorentz-invariant is velocity energy, so you are putting your distinction in the wrong place.
On top of that, there are other violated invariants.
The weight of a lump of iron near a magnetar is greater than the weight of the same lump of iron near an identically massing neutron start.
This is because the potential energy of the iron is greater near the magnetic field, so its mass (as seem by the magnetar) is greater, and so is the gravitational attraction between them.
This means you can't just say "No velocity, the mass is identical", it's not - the extra potential energy means extra mass.
Or in other words there is no such thing as mass as distinguishable from energy.
I haven't thought about it carefully, but since magnetic fields don't do work, my initial reaction to your example of iron near a magnetar is surprise. But, I also expect that the iron magnetizes in a strong field. Wouldn't that be lowering its energy (compared to an unmagnetized lump in a background field), however?
Certainly, but both are encoded in the energy-momentum tensor and for better or worse that's usually considered to be what generates the metric (or perturbations thereof, if you want to do things that way).
In Newtonian mechanics, kinetic energy is rotationally but not Galilean invariant, sure. But in GR in a local inertial frame the pressure T^{ij}, i=j, i!=0 sure looks like kinetic energy \gamma m(v^i)^2. [1]
Should we really strongly distinguish between the pressure and T^00 just because in a local inertial frame the latter looks like \gamma mc^2 ? Sharpen the question by considering vastly different frames of reference.
> There is a distinction between things that only move at the speed of light
The real mindbender is when you stop thinking about "things" and only consider the field values at each point in spacetime, where there is a strong relationship between the field values at here-and-now p1 and then-and-elsewhere p2. The "distinction" between field-values-at-p1 and field-values-at-p2 is largely a matter of identifying whether the spacetime interval between p1 and p2 is "lightlike", "null", or "timelike", which can be done in a Lorentzian spacetime like ours.
In a toy case where there is one massless particle in a spacetime, if its identifying non-vacuum field value is at p1, we can show that if p1 and p2 are not lightlike-separated, then the field value at p2 must be the vacuum value. Alternatively, if the particle is non-massles then if p1 and p3 are lightlike separated, then the value at p3 must be the vacuum value. In this way we can constrain where in the spacetime the particle may be found, and this is the core of the Initial Value Formulation of General Relativity[1]). That is the door to modern general relativity, and it's a pity it wasn't invented while Einstein was still alive.
In this view, if "things" move, it is only because we treat a pattern of field values as a "thing" and we decompose the spacetime into a space+time by deliberately making a (very weakly constrained) choice of time-separated spacelike volumes. But the fundamental picture is that the only objects are interacting fields which permeate the whole of spacetime taking on some value at each point, these fields including matter fields (and the energy-momentum tensor) as well as the metric (another tensor field).
> photons attracted by gravity ... energy has a gravitational field ... photons moving only at the speed of light
Here you have, without realizing it, split the various interacting matter fields up so that you can assign a worldline to one particular field value, then split up spacetime and applied coordinates so that you can cut up the worldline into "photon at t1", "photon at t2", "photon at t3". Then you have made a choice of gauge so that you can talk about a "gravitational field" which the photon appears (to an observer making those choices) to feel (and change! although you didn't point that out). These are perfectly reasonable things to do, but it is perhaps worth doing so deliberately and with the understanding that for each such choice there are other options.
> the words energy and mass [as commonly used] do not properly fit "things that only move at the speed of light and things that never do"
Well, if there is an identical field value in a massless field at lightlike-separated p1 and p2 then there will be an identical contribution into the values of energy-momentum tensor at p1 and p2. If we consider the components of the energy-momentum tensor, we can (in a suitable set of local coordinates) treat the 00 component of the tensor as "rest mass" with other components as other forms of energy (e.g. the 11, 22 and 33 components together can be treated as kinetic energy) or momentum (e.g. the non-diagonal components are momentum fluxes). The whole tensor is the source of spacetime curvature, not just the time-time component, which is the closest to what one would in common usage think of as mass.
Again, the crucial thing is that behind the common usage there are a substantial number of important choices that are not made by "nature", but rather by a person describing the relationship of some set of field values at some points in spacetime.
I would rather describe as a very weak, long-ranging, attracting force.
A force of a very special kind, because all other forces are strong, short-ranging (they need to interact, the only not-mechanical force) and repelling.
Indeed, the reason most objects seem to be neutral is because of how strong the electromagnetic force is! Were it weaker it would be much easier to separate charges on longer length / time scales.
The conclusion that energy is more fundamental or easier to have an intuition for has been the case since, what 100 years ago? It certainly was when I studied physics 20 years ago.
But he doesn't need to go into quarks for that. Simply comparing the masses of protons and neutrons (+ electrons) to the mass of an atom shows they're different. There's a little bit less mass due to the lower energy of the bound nucleons. Quarks certainly make the effect more dramatic though.
A more practical objection to using mass is that nobody can agree on what the word means. Does a photon have mass? Yes or no, depending on if you're thinking of relativistic or rest mass. If you call it energy, there's no ambiguity.
If I could recommend he change anything though, it would be the horrible mixture of units - MeV/c^2, atomic mass units, and grams. The author surely has the time to convert them all to the same unit so the reader can easily compare them. That would eliminate the need to explain Avogadro's law. It's a completely redundant complication of chemistry and has nothing to do with the fundamental concepts he's focusing on.
There is a difference though: A hydrogen atom is slightly lighter than it's constituents. That's why it is stable. However, the nucleons are much, much heavier than the sum of the quark masses. Which is strange, because it means that naively, the unbound state should be preferred. But it isn't, instead, you have confinement!
A particular advantage to intuition comes with the famous e=mc^2. Since c is 1, and 1^2 is 1, e=m. Energy and mass are completely equivalent. Strong force binding energy in a proton is mass. The only bit that's not straight from binding energy is from the Higgs mechanism, which is where the mass of the electron (and the quarks themselves) comes from. Matt Strassler has an excellent explanation of that: https://profmattstrassler.com/articles-and-posts/particle-ph...