# Talk:Natural units

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## In particle physics/Gaußian units: 4πε₀=1, μ₀=1, c=1

Well, that is impossible. ;) --Chricho ∀ (talk) 22:49, 12 February 2012 (UTC)

Because c2 = 1/(ε0μ0)
Just granpa (talk) 23:10, 12 February 2012 (UTC)
It's grossly inaccurate to say that "In Gaussian units, 4πε₀=μ₀=c=1". See Gaussian units. The differences between Gaussian and SI units cannot be summarized by saying what the values are of ε₀ and μ₀ and c. It's far more complicated than that, and much better to just link to the other article (Gaussian units) than to try to explain it here. I just cleared out this and many other misleading oversimplifications of electromagnetism quantities. --Steve (talk) 02:21, 13 February 2012 (UTC)
That edit was a major improvement. What may still be a little understated relates to this confusion, especially the assumption (from SI) that c2ε0μ0 = 1. Along with the normalization choices in a system of units, there are some complications, such as choices in the form of equations. I guess this could also be treated as a normalization, in the sense that SI clearly makes the normalisation/assumption expressed in this equation that some of the other systems do not make. I would think that mention of this is appropriate in this article, as it serves to confuse many. — Quondum 11:02, 13 February 2012 (UTC)
I added a little note about that. I think it's better to describe this as a specific complication of electromagnetic units, rather than try to fit everything into a grand framework, which this article does far too much already :-/ --Steve (talk) 14:02, 13 February 2012 (UTC)
I agree with the special treatment only in the EM case. I've extended it, since I feel the specific case mentioned is confusing to anyone first trying to figure it all out, and an early pointer on this aspect may save such a person a lot of trouble (even though it is possibly misplaced in this specific article: revert it if you feel so). — Quondum 15:36, 13 February 2012 (UTC)
Thank you, it is much better that way. Indeed, I was confused by ${\displaystyle \varepsilon _{0}\mu _{0}c^{2}=1}$, which was even used in the table. --Chricho ∀ (talk) 10:04, 28 May 2012 (UTC)
You know, Steve, I don't think this edit is so good. It plops in some interpretation of the differences between rationalized and non-rationalized units that is from a particular POV. Someone else may very well say the opposite, that non-rationalized units are more complicated than rationalized. Putting the 4π in the denominator with the r2 in an inverse-square law is more natural (and simpler) because it connects the denominator of the inverse-square law directly to the surface area of a sphere in 3-space and to the concept of flux density which gets integrated with Gauss's law to yield the contained charge. It's pretty flimsy to claim that that
${\displaystyle F=G{\frac {Mm}{r^{2}}}}$
is less complicated than
${\displaystyle F=G'{\frac {Mm}{4\pi r^{2}}}}$ where ${\displaystyle G'=4\pi G}$
just because the latter has this extra 4π in the denominator. Rationalized units are conceptually less complicated than non-rationalized units. 70.109.178.237 (talk) 03:53, 28 May 2012 (UTC)
You misread what I wrote. The "complication" is the fact that the symbolic law of physics is different depending on the system of units. The "complication" is not the factor of 4pi. I will add a few words to make it clearer. --Steve (talk) 13:18, 28 May 2012 (UTC)
Doesn't look any better. And I don't think I am misreading you. The main difference between rationalized and unrationalized units is that one puts a 4π in the denominator with the r2 in an inverse-square law (where it belongs in my opinion) and the other doesn't. (There may remain a scaling constant in the law if the units are not "natural" in some sense.) The complication that electrostatic cgs units have is that charge (or current) is not considered to be a fundamental quantity like it is in SI. So they do to charge like what nearly every unit system does with force, they infer the dimension of something new (force or electric charge) out of what exists (time, length, mass). In cgs electric charge becomes dimensionally force1/2 × length. And it isn't true, except for convention that "Newton's law is F = ma in every system of units." We could define our unit of force to be whatever force is needed to compress some prototype spring (stored in Sèvres) by 1 cm, and then Newton's 2nd law would be F = kma where k would be a constant that would be experimentally determined and listed at the NIST site. And k would be dimensionally (force × time2)/(mass × length). It would not be dimensionless and force would not be the same thing as (mass × length/time2). That is what the complication is about regarding cgs electrostatic units. It is not about "rationalized" vs. "unrationalized". The complication regarding electrostatic units is that some people have defined the Coulomb constant to be the dimensionless 1 and others have defined charge as a unique dimension of physical quantity (so with them, generally the Coulomb constant is not dimensionless). But natural units usually whack the Coulomb constant anyway. 70.109.178.237 (talk) 00:56, 29 May 2012 (UTC)
Again, I am saying that the complication is not the 4pi, which I personally like in Coulomb's law just as much as you do. The complication is that 4pi might or might not be there depending on the system of units.
Can you make mechanical units as complicated and inconsistent as electromagnetic units are? Yes, if you are sadistic! You can torture students of physics by inventing a new system of units where force is a fundamental quantity, etc. etc. Thank heavens, no one has ever done that to my knowledge! Therefore, in the world we live in, with the unit systems that are actually in use in the world, mechanical units are "simple". There is no unit system I've ever heard of where F=ma is not the correct equation to use.
Many readers looking at this article will have already had the experience of using the same formula, like "F=ma" or "distance = velocity * time", in various different units like meters and centimeters (and inches in USA). They will naturally draw on this experience when thinking about how unit conversions work in general. Therefore it is helpful to point out that the simplicity they might have expected from experience with mechanical equations will not be present in the more complicated world of electromagnetism equations. You seem concerned that readers will get the impression that electromagnetism units are more complicated because there is something fundamentally different about electromagnetism units. I'm happy to add some text clarifying that EM units are complicated because of human reasons (history and sociology) not physics reasons.
Out of the multitude of differences between Gaussian and SI units, I agree with you that rationalized versus nonrationalized is not a particularly important difference. Charge-units being fundamental in SI but not Gaussian would be a more important difference. BUT, this is not a section comparing Gaussian and SI units! It is comparing Gaussian and Lorentz-Heaviside units! In BOTH of those systems, charge is not considered to be a fundamental quantity. In fact, I believe the only difference between them is rationalized versus nonrationalized. That's why it's used as an example! --Steve (talk) 18:18, 30 May 2012 (UTC)
Okay, if we agree (and other interested editors) that the complication regarding rationalized units is not because the 4π in the denominator with the r2, and that the complication regarding electrostatic cgs units is because the Coulomb constant, 1/4πε0=1 which forces electric charge to take on a derived dimension of force1/2 × length rather than being some unique dimension of physical quantity, then we should state it as so in the article. If you disagree with this essential assessment of what complicates EM units, then we are not in agreement here about what the facts are. If we are in agreement about the facts, then we should write it differently in article. Since Gaussian units#"Rationalized" unit systems seems to be the only Wikipedia discussion of what is meant by "rationalized", I tried to spell this out there. But the issue of a derived dimension for charge (because 4πε0=1) is not the same issue as rationalized units vs. unrationalized. And I cannot tell that from what is in this article now. 70.109.176.173 (talk) 20:11, 30 May 2012 (UTC)
I tried to rewrite again. I guess I had structured the section a bit strangely, with the last paragraph being redundant with the others. Maybe that helped cause confusion.
The fact that electric charge has a derived dimension is true in both Gaussian and Lorentz-Heaviside. So again, if we're contrasting these two systems, the fact that charge has a derived dimension is not one of the differences. But...when I think about it, it is a relevant fact for the article anyway so I put it in at the top before introducing the two variations.
I don't understand why you think it is unclear or misleading to say that the two EM unit systems are very inconsistent because there are two natural-units versions of Coulomb's law but only one version of Newton's laws. But whatever. I deleted that and tried a different approach: I described how to get either Gaussian or L-H units starting from SI. I think that is an equally good way to convey the information. --Steve (talk) 14:19, 31 May 2012 (UTC)

## Dimensional consistency and natural units.

In this edit [1], another IP added

"... but it creates more confusion (see below Greater ambiguity paragraph) because the equality is dimensionally inconsistent as mass and momentum have different dimensions or qualities and therefore cannot be summed to give the dimension of energy."

These quantities that are represented in whatever system of natural units are dimensionless. E.g. in Planck units, the mass term is really the mass divided by Planck mass and the momentum and energy are also scaled by the reciprocals of their respective Planck units. I guess my reversion of the edit needed more explanation. 70.109.178.237 (talk) 01:14, 28 May 2012 (UTC)

I understand the phrase "whatever system of natural units are dimensionless" to assert that Dimensional analysis is simply a pseudo algebra created by our choice of measurement. I believe the situation is different and related to the edit by User:86.125.179.179.
If we are measuring mass in Planck units of mass, energy in Planck units of energy and speed in Planck units of speed, I believe mass, energy, and velocity are still distinct concepts and measurements. Surely we don't really mean that mass, energy, and velocity are identical and indistinguishable. (There would however be a certain beauty in saying that all equations reduce to 1 = 1.)
I believe we still mean that a unit of (Planck) mass is related to (can be transformed to) a unit of (Planck) energy by a function of (Planck) velocity. I understand User:86.125.179.179 to be asserting that the Planck normalization of equations produces an appearance of simplicity which ignores selected unitary functions. I believe we still must recognize that dimensional analysis reflects valid concepts that can help validate our equations of existence.
SBaker43 (talk) 04:16, 28 May 2012 (UTC)
Okay, I am saying that neither
${\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}\,}$
nor
${\displaystyle \left({\frac {E}{E_{P}}}\right)^{2}=\left({\frac {p}{m_{P}c}}\right)^{2}+\left({\frac {m}{m_{P}}}\right)^{2}\,}$
are dimensionally inconsistent.
Physical law that says
${\displaystyle E=mc^{2}\,}$
or
${\displaystyle {\frac {E}{E_{P}}}={\frac {m}{m_{P}}}\,}$
isn't saying "this much mass is that much energy" or that "this mass is that energy", is it? Maybe it is. Maybe not. What is the correct interpretation?
Or does
${\displaystyle E=k_{B}T\,}$
or
${\displaystyle {\frac {E}{E_{P}}}={\frac {T}{T_{P}}}\,}$
say that this much energy (per particle) is that much temperature? Is temperature simply another way of saying what the per particle energy content is?
Is Newton's law of motion relating two different kinds of physical quantity or are they the same?
${\displaystyle F={\frac {dp}{dt}}}$
Is force the very same thing as the time derivative of momentum or is force only proportional to the rate of change of momentum and the two physical quantities are different "stuff"?
Physical law relates different quantities of different kinds of physical quantity together. 70.109.178.237 (talk) 05:14, 28 May 2012 (UTC)

## Normalization of α

The section "Choosing constants to normalize" contains the sentence "In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to 1, because it is a dimensionless number." First thing, I agree with the section as a whole. In particular, I agree that it is "not possible to simultaneously normalize all four of the constants...". I'm not arguing about that. Taken in context perhaps the sentence is okay, but it seems to assert a particular idea on its own that is not correct.

There is no reason that α cannot be redefined to be 1 any more than the other constants, and certainly not because it is dimensionless. For example, define ${\displaystyle \alpha '=\alpha /137={\frac {k_{\mathrm {e} }e^{2}}{\hbar c137}}}$ and likewise ${\displaystyle e^{\prime }=e/{\sqrt {137}}}$. This entails scaling the primary electric charge. This is always allowable in defining a new system and the value of e could be scaled in any of the unit systems. There may be some confusion between basic numerical scaling and redefining an entity with different dimensionality (by absorption/multiplication of various constants) as what happens when defining natural unit systems. Both may produce different numerical values for certain entities; however, they are not the same operation, neither are they mutually exclusive.

Next, it's not clear what "less trivial" is supposed to mean. I understand the importance the fine-structure constant in modern physics, but when it comes to the game of normalizing constants and redefining various symbols, I think there is nothing significantly different about the value of alpha. If it is indeed "less trivial" or in any way different than the constants on the other side of the equation or the mass of elementary particles, it needs to be qualified. Also as a matter of pedagogy, "trivial" is overused and should be avoided. There is nothing "trivial" about this subject or even a single example--especially to a student learning these ideas--in choosing and using a well-defined, correct system of units and converting between them. A suggested phrase is "In a more overarching example, ...". --cperk (talk) 17:45, 8 October 2013 (UTC)

I agree. The section is reformulated as to not make any explicit claims. YohanN7 (talk) 14:09, 7 June 2015 (UTC)
Don't share your perspective. It's not just a claim. It's a fact. ${\displaystyle \alpha ={\frac {e^{2}}{(4\pi \epsilon _{0})\hbar c}}}$ cannot be normalized to a predetermined dimensionless number. You can, with choice of units, normalize three of the four dimensionful constants that define α, but not all four. Neither can you, simply with the choice of units, normalize the Proton-to-electron mass ratio, which will always be around 1836 and dimensionless. 71.36.148.25 (talk) 17:20, 9 June 2015 (UTC)
Where does that attitude come from? My perspective is both as an educated physicist, but also as a student continuing to learn. My perspective as the student sees the "sacredness" of alpha as rather arbitrary. I added this section so that there could be an insightful discussion about the "why" of the claims, prompting further edits or discussion to improve the article. But now I'm just supposed to bow to an anonymous contributor that essentially tells me to shut up and just take Wikipedia as "fact"? I showed that it is perfectly possible to mathematically set ${\displaystyle \alpha }$ to 1 without destroying any physics. I did not contradict the statement about normalizing more than three constants since I showed that to do this, another of the constants must be scaled (e.g. electric charge). Other than some historical and practical reasons, there is also no universal sacredness to the numerical value of the electric charge. User 71.36.148.25 essentially just repeated what is said in the article, adding no additional insight or explanation. I said nothing of the proton-to-electron mass ratio, so that isn't relevant unless 71.36.148.25 meant it to explain the reasons for claims about ${\displaystyle \alpha }$. Besides, masses are of the same dimension, so scaling one value scales the other and the ratio remains unchanged. ${\displaystyle \alpha }$ is not a simple ratio of only two values, but multiple values of different dimensions of which the scaling can be shifted from one to another without destroying the physical relationships between the constants. Please don't just restate what is in the article and demand that it is fact. Please share insightful details rather than being the article's dictator. cperk (talk) 00:01, 20 July 2015 (UTC)
Sure, you can, I suppose, redefine 2 to be 1 by redefining it to be ${\displaystyle {\frac {2}{2}}}$. Why not?
Actually, if I were dictator of the world, instead of the current definition of ${\displaystyle \alpha }$, I would define the primary dimensionless value to be ${\displaystyle {\sqrt {4\pi \alpha }}={\frac {e}{\sqrt {\epsilon _{0}\hbar c}}}=0.30282212}$ (and derive the ${\displaystyle {\frac {1}{137.036}}}$ number from that). And that is " a simple ratio of only two values": the ratio of the elementary charge to the natural unit of charge in rationalized Planck units. And it's not 1. (But it's in the ball park.)
And ${\displaystyle e}$ and ${\displaystyle {\sqrt {\epsilon _{0}\hbar c}}}$ are every bit the same dimensions as are ${\displaystyle m_{p}}$ and ${\displaystyle m_{e}}$ the same dimension of stuff. Then, again, we could just redefine ${\displaystyle \mu ={\frac {m_{p}}{m_{e}}}}$ to be ${\displaystyle {\frac {m_{p}}{1836.15267245\cdot m_{e}}}}$ and then that value is 1 too, and we didn't destroy physics. Dunno why we can't do that. 70.109.187.202 (talk) 01:47, 20 July 2015 (UTC)

## Normalizing ${\displaystyle G}$ and not ${\displaystyle 4\pi G}$

Seriously, by removing ${\displaystyle 4\pi }$ in the denominator we neglect the connection between inverse square laws and how the surface area of a sphere relates to its radius.--195.194.89.243 (talk) 12:52, 25 February 2015 (UTC)

Totally agree. Take a look at the Planck units article. The problem, 195, is historical and some understandable inertia in the discipline. Whether it's ${\displaystyle G}$ or ${\displaystyle 4\pi G}$ that is normalized to 1 (by the judicious choice of units) or if it's ${\displaystyle 4\pi \epsilon _{0}}$ or ${\displaystyle \epsilon _{0}}$ that is normalized, the Planck scale is still about the same order of magnitude. So a lot of physicists think "why bother" regarding changing the meaning of something, even if the change is more optimal and more natural. 65.183.156.110 (talk) 04:46, 26 February 2015 (UTC)

revert (Several issues. k_e is not as well known as epsilon_0. In this article, Planck units do not necessarily normalize k_e.) seems a little POV, so I summarily re-reverted it. Look a little closer:

k_e is not as well known as epsilon_0
This seems to be a difficult claim to justify. Looking at the Google Ngram viewer, it is known pretty well.
Whyever not? One could argue that ε0 could be normalized in Planck units, I guess, but the choice of normalization would have to be stated. Consistency with the article Planck units would require normalization of the Coulomb constant, and no justification is given in this article for any other normalization. The text explicitly says: "Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object, or even elementary particle. They only refer to the basic structure of the laws of physics ...", so to introduce a rather arbitrary choice of the elementary charge and the fine structure constant (and then to normalize a complicated expression of these) goes directly against that.

Technically speaking, Planck did not introduce an electromagnetic unit at all. Perhaps we should simply omit the Planck charge from this entry (as well as in Plank units)? A system of units should not be artificially extended by Wikipedia. —Quondum 06:27, 2 March 2015 (UTC)

There apparently are sources for the unit Planck charge. Whether it was created by Planck or named for him later, it seems appropriate to include here. Bcharles (talk) 21:47, 21 March 2015 (UTC)
Gravitation (Misner, Thorne, Wheeler), for example, mentions Planck units for time, length and mass, but makes no mention of charge or temperature. I managed to find an explicit naming in a "reputable" reference (Roger Penrose, The Road to reality), but even here it is rather unsatisfactory, as he gives the normalization choices leading to Planck units, but these choices (G = c = ℏ = kB = 1) do not determine a Planck charge, yet he then concludes what the value of the Planck charge is (about 11.7 proton charges). Several sources say something like "one could define a Planck charge as...". I have no in-principle objection to the idea that charge and temperature could have been considered to be part of the system post-dating Planck, as long as we get this from a reputable source. At least all the sources that do give a value that I looked at were consistent in their choice, not giving the alternate CGS-Gaussian version of Planck units that I reverted. Whatever the origin, Planck charge and temperature seem to have some momentum. —Quondum 03:41, 22 March 2015 (UTC)

## Greater ambiguity?

I am deleting this bullet point under "advantages and disadvantages". There is no reason to presume that mP and lP would be omitted when stating values in Planck units. Bcharles (talk) 22:41, 21 March 2015 (UTC)

This does not really hold: it is reasonable that when using Planck units, many people would omit their explicit mention, and examples are not difficult to find. For example, Penrose says "... in terms of Planck units, gram = 4.7×104", where he omits mP. Nevertheless, the point was made rather fuzzily in the article, and I'm not going to argue that it should be retained. —Quondum 04:08, 22 March 2015 (UTC)

## Same dimension of space and time

In natural units space and time have the same dimension due to c=1. This statement deserves further details.--5.2.200.163 (talk) 13:52, 1 February 2016 (UTC)

## Bohr atom in Heaviside Lorentz Units

The bulk of the NIST CODATA tables can be explained in terms of Bohr's atom, since this provides the units for further calculations, and this is what the CODATA values do. Since the bohr atom throws up a dimensionless constant (fine-structure constant), setting the base to this allows many more units to become simple values. Everything relating to the electron are simple values of π or the reciprocals of 137/π.

One then uses K=137.036 etc, as an exponent-marker like E=10, so as E5 = 10^5 then K^5 means 137.036^5. The tables are set so that the speed of light, the electron mass and the electron charge are set to K-units (eg K4 'm/s'). The unnamed K-units occupy the same space as the given SI unit.

There is a thread on Dozensonline->Applications->New Systems of Measure->Other measurement systems->Ku And Ko

KU follows much of modern metric theory, CGS lightly rationalised to SI. If the same light rationalisation is applied to give a HLU theory, then the charge is reduced by a factor of \sqrt{4\pi}. We see that the electron mass is K-2 (so the unit is about the size of a boron atom, and N_a is a number bigger than 1), and the speed of light is K3. The unit of charge is K-1.

KO shows the results of calculating Bohr's atom in HLU theory, from first principles. We still have the electron charge at K0, and the electron mass at K-2, but the speed of light is set to K4. One will notice that lengths and times are increased by a factor of 4π, which represents a change from radius to circumference or surface of a sphere, and from radian-times (\lambdaba and \hbar) to cycle-times (\lambda and h). The value of h changes from 2pi to 1/2.

The tables represent the 'exact values' of the CODATA variables, since the errors have been moved into conversion from this system.

--Wendy.krieger (talk) 11:13, 1 April 2016 (UTC)

## length units

under the flag of "natural units" often dimensionless length units are in use like M~rG=rs/2. These are usually built using Planckunits, or G c° kB ke and h°, I denominate them with a sub-"r":

• m_r=lP*m/mP
• Q_r=lP*Q/QP
• L_r=lP*L/h°
• T_r=lP*T/TP

I think these should be mentioned in the article too. Ra-raisch (talk) 19:26, 8 December 2016 (UTC)

## codata

codata2014 lists (table VI) natural units and atomic units:

• v_nu=c, v_au=α·c
• L_nu=L_au=h/2π
• m_nu=m_au=me
• E_nu=c²me, E_au=E_h=α²c²me
• p_nu¹=c*me, p_au¹=L_au/a_0
• l_nu=L_nu/(me·c), l_au=a_0
• t_nu=L_nu/c²me, t_au=L_au/E_au

further several electric au, based upon Q_au=e and ε_au=e²/(a_0·E_au). Ra-raisch (talk) 12:27, 3 August 2017 (UTC)

Before you change anything, we might appreciate it if you express exactly what you are proposing to change, with both text and with equations and expressions written with the use of ${\displaystyle {\text{LaTeX}}}$. It will be easier to understand exactly what you are saying. Now I am just guessing. 96.237.136.210 (talk) 04:23, 5 August 2017 (UTC)

a.u. units With CODATA 2014 the following "a.u." (atomic units) as well as 15 other derived units were defined:

• charge e
• mass me
• action ℏ
• velocity α·c ≈ c/137 (maximum speed of an electron in Bohr's model with fine structure constant α)
• energy Eh (Hartree energy)
• time ℏ/Eh
• permittivity e²/a0Eh

n.u. units with CODATA 2014 the following "n.u." (natural units) were defined:

• velocity c (vacuum speed of light)
• action ℏ
• mass me
• energy c²me
• momentum c·me
• length ℏ/(c·me) (reduced Compton wavelength of the electron)
• time ℏ/(c²me)

something like that? Following the articel not mentioning these CODATA natural units should be considered exotic? Ra-raisch (talk) 21:32, 20 August 2017 (UTC)

or may-be rather like this Ra-raisch (talk) 10:27, 22 August 2017 (UTC)

Unit Metric value Derivation
1 n.u. of length 386.1×10−15 m ℏ/mec
1 n.u. of mass 9.109×10−31 kg me
1 n.u. of time 1.288×10−21 s ℏ/c²me
1 n.u. of temperature not listed
1 n.u. of electric charge not listed
Unit Metric value Derivation
1 a.u. of length 0.529×10−10 m a0
1 a.u. of mass 9.109×10−31 kg me
1 a.u. of time 2.418×10−17 s ℏ/Eh
1 a.u. of temperature not listed
1 a.u. of electric charge 1.602×10−19 C e
7 natural units ("n.u.") and 18 atomic units ("a.u.") were published in CODATA 2014 Table VII.