Question: When Planck's Constant was set to a specific value, in order to define the kg, through the equation h x f = e = m x c^2 and since massive objects warp space time, was the length of a second also changed, and if so by how much?

Jose Eliel Camargo Molina answered on 14 Mar 2019:

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Great question.

Like you wrote in your question, Planck’s constant appears in the relationship of the energy and frequency of a photon, but it also appears everywhere in quantum mechanics, like in the uncertainty principle.

It would be great to work in units that do not depend on objects that change, or in things that depend on the accuracy of our machines. So this is what scientists are trying to do.

The second is an arbitrary unit we came up with and it allows us to measure time in some unit. But you might as well come up with something completely different, as long as you find a way to define it.

Like how “inches” might have been related to the length of a thumb, or feet to the length of well, a foot.

Scientists try to find more and more robust ways of defining those units of measurement because for example people’s thumbs and feet are never the same lengths.

The most robust way we have found to define the second is the time it takes for 9 192 631 770 oscillations of the radiation corresponding to the transition between two levels of the cesium atom.

It sounds pretty crazy, right? But that is because it is the closest duration to what we used to define a second as long time ago: You take a sundial and measure how long a day is, then you divide that into 24, then into 60 and then into 60 again. What a mess!

So now we have defined a second. A Cesium atom will always take the same time in doing those oscillations, so we are good so far! It will not change. It takes the same no matter what we fix Planck’s constant value to.

Now onto the meter! how do we do it? Well, we can fix the value of the speed of light to a certain value: 299,792,458 m/s.

Now that it is fixed, we have linked the meter to the second. Because we have to define our meter such that using the second we just defined, light travels at exactly that speed I wrote above.

However, it is a bit more difficult for the Kilogram, which the most robust way is to define it using the fixed value for Planck’s constant. Why? Well, the value of Planck’s constant depends on several things, its units are:

kg⋅m2⋅s−1.

We have defined the second and the meter, so then from Planck’s constant, we can get the kg!

So the second is already fixed by measurement, the same goes for the meter which is defined using a fixed value for the speed of light.

Then if we fix Planck’s constant it will only affect the definition of the Kilogram, but nothing else!

But once we fix Planck’s constant, then the Kilogram is also fixed and linked to our definitions of the meter and the second!!

That is part of the motivation to do it, as then we will have units that will not change with time and do not depend on physical objects.