# I The new definition of kg and the mass-energy equivalence

#### FranzDiCoccio

Summary
Is there any connection between the new definition of the kilogram based on the Planck constant and the mass-energy equivalence?
Hi,
today I stumbled upon a 2016 article in Scientific American about the (then) possibility of re-defining the kilogram through Planck's constant.
The article is really a very quick review of the topic. At some point the author states the following "So for years, physicists have chased an elusive dream: replacing the physical kilogram with a standard inherent in properties of nature such as the speed of light, the wavelength of photons and the Planck constant (also called h-bar), which links the energy a wave carries with its frequency of oscillation. Scientists could use the Planck constant to compare the energy of a wave with Einstein's iconic $E=m c^2$ equation; in that way, they would determine mass solely through the physical constants."

I really do not see the relevance of Einstein's equation in the redefinition of the kilogram.

As far as I understand, the point here is to define a standard for mass. This is now done basically by choosing some fundamental constants of Nature and assuming them as units for their "physical dimension". In this approach, "everyday" physical dimensions (such as mass), which used to be fundamental, become composite physical dimensions that can be obtained from those defined by the constants.

Thus,, maybe oversimplifying, if action is measured in units of $h$, frequency is measured in units of $\Delta \nu_{\rm Cs}$ and speed is defined in units of $c$, then the (now composite) unit of mass corresponds to
$$1 kg = 1.4755214\cdot 10^{40} \frac{h\,\Delta \nu_{\rm Cs}}{c^2}$$

I might be wrong, but I'm under the impression that the author was a bit shoddy in referring to the mass-energy equivalence.
By the way, she was not very precise about Planck constant either. It is not exactly true that the Planck constant is called "h-bar". That is the reduced Planck constant, whose value is $h/2\pi$.

Does anyone know whether the mass-energy relation really played a role in the choice of $h$ for the definition of the mass unit?

I looked into a previous discussion, but I was not able to find a connection.

Thanks a lot
Franz

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#### BvU

Homework Helper
Summary: Is there any connection between the new definition of the kilogram based on the Planck constant and the mass-energy equivalence?

I looked into a previous discussion, but I was not able to find a connection.
Did you look deep enough ? One level deeper (under Kilogram) there is a proposed definition based on $mc^2 = E = h\nu$. Apparently didn't make it.

Your 1.4755214⋅1040 is -- in this context -- a form of rounding that isn't allowed.

#### FranzDiCoccio

Hi BvU,

Did you look deep enough ?
I looked at the article I linked, at the BIMP page and at the thread I linked.

One level deeper (under Kilogram) there is a proposed definition based on $mc^2 = E = h\nu$.
Ok, now I see. I hadn't found that definition. I did not think of looking into Wikipedia too. Now the comment of the author makes more sense, especially because when the article was written a decision had not been made yet.

Apparently didn't make it.
I'd say that both the present definition and the proposal based on $E = mc^2$ are equivalent, for practical purposes.
They ultimately rely on a very precise measurement of $h$.
It would be interesting to know why the phrasing involving the mass-energy equivalence was rejected.

For what it is worth, my feeling is that there is no need of involving the mass-energy equivalence in the whole process. It feels like an "extra step".
I'll try to look deeper into that.

Your 1.4755214⋅1040 is -- in this context -- a form of rounding that isn't allowed.
Uhm, I copied that straight from the BIMP page (see under kilogram). I did not do any rounding.

#### BvU

Homework Helper
Uhm, I copied that straight
Well, you did replace their $\approx$ by an equals sign .
(at least the one here on p 131). Also:

I sense some humour in the term 'straight'

#### FranzDiCoccio

Oh, right, sorry. Did not notice that, you're right.

"The new definition of kg and the mass-energy equivalence"

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