# Least and most uncertain Physical constants **URL:** \ **Category:** Factual Questions\ **Created:** [July 21, 2014, 10:14pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696 "2014-07-21T22:14:01Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\ **Post date:** [July 21, 2014, 10:14pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/1 "2014-07-21T22:14:01Z")
Wikipedia lists the physical constants and their uncertainties here : > **[Physical constant](https://en.m.wikipedia.org/wiki/Physical_constant)** > > A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement. > There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum c, the gravitational constant G, the Planck consta... My question is which of these physical constants (excluding the ones that have a defined value) have the least and most uncertainty in their value on percent basis ? Bonus question : why ? ---
**Author:** ![eltro102](https://avatars.discourse-cdn.com/v4/letter/e/7cd45c/32.png) [@eltro102](https://boards.straightdope.com/u/eltro102)\ **Post date:** [July 21, 2014, 10:27pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/2 "2014-07-21T22:27:38Z")
The relative standard uncertainty of a measurement is the uncertainty divided by the measurement (absolute value) So the most uncertain measurement would be the one with the largest relative standard uncertainty This is the weak mixing angle - with a rsu of 9.5 × 10^−3 The least uncertain is the Rydberd constant - with a rsu of 5.0 × 10^−12 ---
**Author:** ![ticker](https://avatars.discourse-cdn.com/v4/letter/t/d07c76/32.png) [@ticker](https://boards.straightdope.com/u/ticker)\ **Post date:** [July 21, 2014, 11:03pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/3 "2014-07-21T23:03:53Z")
Isn’t the speed of light known to be exactly 299,792,458 metres/second since the metre itself is defined as he distance light travels in 1/299,792,458 seconds? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 21, 2014, 11:31pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/4 "2014-07-21T23:31:32Z")
Yes, but that just shifts the burden onto uncertainty in the value of the meter. The experiments are still the same, and still have error in them. G, meanwhile (Newton’s constant) also has a great deal of uncertainty, at about 10^-4, and is perhaps more relevant to us than the weak mixing angle. Or perhaps not: G by itself is very seldom put to practical use; it’s almost always seen multiplied by the mass of some astronomical body such as the Earth or Sun. But that product is generally quite well known (implying a corresponding uncertainty in the mass by itself). ---
**Author:** ![eltro102](https://avatars.discourse-cdn.com/v4/letter/e/7cd45c/32.png) [@eltro102](https://boards.straightdope.com/u/eltro102)\ **Post date:** [July 21, 2014, 11:39pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/5 "2014-07-21T23:39:48Z")
> [@Chronos](#): > > Yes, but that just shifts the burden onto uncertainty in the value of the meter. The experiments are still the same, and still have error in them. But isn’t the metre based off the second? And isn’t the second based off counting? I’m sure we’ve got counting down to pretty high accuracy, by now. ---
**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\ **Post date:** [July 22, 2014, 12:50am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/6 "2014-07-22T00:50:26Z")
Thank you for the answers.ticker and altro - I specifically excluded defined physical constants. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 22, 2014, 1:57am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/7 "2014-07-22T01:57:36Z")
> [@](#): > > But isn’t the metre based off the second? Yes, exactly my point. There are experiments we used to perform to measure the speed of light. These experiments inherently have some error in them. Nowadays, we’ve defined the value of the speed of light, but we still perform those old experiments, or ones like them. Nowadays we’re performing those experiments to determine the meter. But they still have error in them. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [July 22, 2014, 3:13am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/8 "2014-07-22T03:13:55Z")
I know the [Hubble Constant](http://en.wikipedia.org/wiki/Hubble's_law#Determining_the_Hubble_constant) isn’t a “physical constant” in a strict sense, but when I was in college, I was taught it was between 50 and 100 (km/s/Mpc). Now it’s more like “about 70”. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [July 22, 2014, 9:57pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/9 "2014-07-22T21:57:09Z")
As a child I had a 1970s Guinness Book of Records and I’m sure I remember reading in the section about “biggest numbers” that there was some insanely huge number which was the upper bound for some constant or other, and included a note that the actual value of said constant had actually been found to be 3, or 12, or something extremely modest. But I can’t recall the details. Anyone? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 22, 2014, 11:05pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/10 "2014-07-22T23:05:23Z")
That would be Graham’s Number, and the suspected (not proven) actual value is 6. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 22, 2014, 11:27pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/11 "2014-07-22T23:27:08Z")
The lower bound on the actual value has now been sharpened to 13. (The upper bound has also been significantly sharpened, but nevermind that) Anyway, it should be noted that both Graham’s number and the value it bounds are mathematical constants related to a mathematical problem, not physical ones. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 22, 2014, 11:50pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/12 "2014-07-22T23:50:08Z")
(It’s also worth noting that [Graham’s number](http://en.wikipedia.org/wiki/Graham's_number) was never the tightest upper bound known (for the solution to the problem which originally motivated its consideration); it was just something Ronald Graham found easier to explain to Martin Gardner than the tighter upper bound he knew. [That having been said, the tighter upper bound is still quite large. But, then, it’s not hard to come up with mathematical problems with a large range of uncertainty in their answer. For example: How many digits of sqrt(2) precede any occurrence of the string \? Infinite uncertainty!]) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 23, 2014, 1:03am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/13 "2014-07-23T01:03:42Z")
Also, I don’t think anyone ever really had any good reason to suspect the value was 6 (or, now, 13); I think it was just the lowest value that hadn’t yet been ruled out. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 23, 2014, 2:49am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/14 "2014-07-23T02:49:05Z")
> [@](#): > > For example: How many digits of sqrt(2) precede any occurrence of the string \? Infinite uncertainty!]) That’s a little different, in that it’s not actually known that sqrt(2) is normal, and thus that the question has any (finite) answer at all. And if you really want to get down to it, there are plenty of problems in physics for which that’s true, too. I mean, I could argue that the inverse of the photon mass has infinite uncertainty. Although, come to think of it… It’s definitely known that the lightest neutrino has a nonzero mass, but there is no known lower bound for it. So I suppose that the inverse of the lightest neutrino mass _is_ known to be a number, but also has infinite uncertainty. ---
**Author:** ![Pasta](https://avatars.discourse-cdn.com/v4/letter/p/ecccb3/32.png) [@Pasta](https://boards.straightdope.com/u/Pasta)\ **Post date:** [July 23, 2014, 3:21am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/15 "2014-07-23T03:21:11Z")
> [@Chronos](#): > > It’s definitely known that the lightest neutrino has a nonzero mass, but there is no known lower bound for it. A nitpick: the lightest neutrino can have zero mass. The second lightest can’t. (Of course, your broader point is unaffected by this nit.) > [@am77494](#): > > My question is which of these physical constants (excluding the ones that have a defined value) have the least and most uncertainty in their value on percent basis ? For those exact constants, **eltro102** is correct that the best answer to that question comes from the “relative standard uncertainty” column. More generally, though, it’s not answerable since you have freedom in what you call your constants. As an example perhaps less extreme than **Chronos** ’s, consider some physical constant that is an angle _t_ that has to lie between 0 degrees and 180 degrees. Let’s then say that we measure _t_ very precisely to be 90.00 +/- 0.01 degrees. That means we know the value of _t_ to one part in 11,000. But what if the angle _t_ shows up in our equations inside of a sin() function? We could just as well have defined a quantity _x_ equal to sin(_t_), and called _x_ our fundamental constant. The range of _x_ that our same measurement allows is _x_ = 1 +/- 0.000000015. (An easy, if only statistically approximate, way to get this is just to calculate sin(90.01) and sin(89.99).) That’s about one part in 7 billion. So, do we know this physical constant to 0.01% precision or 0.0000015% precision? A harmless transformation of the variable makes all the difference. > [@](#): > > Bonus question : why ? For the reasons above, there’s no universal answer to this (or, rather, no universal way to decide which parameters you’re asking about). However, for any particular measurement or constant you might choose, there will definitely be something interesting to say about why the uncertainty is what it is. The story will likely involve the details of the experiment’s design and its challenges. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 23, 2014, 5:06am UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/16 "2014-07-23T05:06:44Z")
> [@Chronos](#): > > That’s a little different, in that it’s not actually known that sqrt(2) is normal, and thus that the question has any (finite) answer at all. Yes, that’s fair. A better example is the question of what the third digit of sqrt(2) in base [your favorite number both impressively large as a range of uncertainty and suitably large that no one has calculated anything about this third digit] is. ---
**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\ **Post date:** [July 23, 2014, 7:33pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/17 "2014-07-23T19:33:35Z")
> [@Chronos](#): > > So I suppose that the inverse of the lightest neutrino mass _is_ known to be a number, but also has infinite uncertainty. Say Dm/m (I do not know how to type equations on SDMB but read the D as the capital Delta) is the uncertainty in the mass and x = 1/m ==\> Dx/x = [-(1/m^2). Dm/m] So the relative uncertainty in x will be more for smaller masses than larger masses. I don’t see how you get to infinity. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 23, 2014, 8:01pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/18 "2014-07-23T20:01:17Z")
Suppose m = 0. Or, rather, suppose the range of possible lightest neutrino masses extends arbitrarily close to zero. [Note also that you are reasoning with an infinitesimal difference operator, rather than a discrete one] ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 23, 2014, 8:14pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/19 "2014-07-23T20:14:59Z")
Mind you, if we were proper Bayesians, we would model all our ignorance as probability distributions, and likely want our measurement of uncertainty to take these relative probabilities into account. Accordingly, instead of simply looking at the span of possible values, we might consider the standard deviation or such things. So, depending on our probabilistic model of the inverse neutrino mass, we might still claim the uncertainty in it to be finite (but we could always still reparametrize it some other way into infinity). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 23, 2014, 11:31pm UTC](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696/20 "2014-07-23T23:31:37Z")
> [@](#): > > Quoth **Pasta** : > > A nitpick: the lightest neutrino can have zero mass. The second lightest can’t. (Of course, your broader point is unaffected by this nit.) If it were massless, how could it take part in oscillations? Neutrino oscillation has a characteristic frequency to it, but massless objects are timeless. But assuming there’s an answer to that, this isn’t a nit, and it does affect my point: The second-lightest neutrino does have a lower bound to its mass, and so its inverse would not have infinite uncertainty. > [@](#): > > More generally, though, it’s not answerable since you have freedom in what you call your constants. As an example perhaps less extreme than Chronos’s, consider some physical constant that is an angle t that has to lie between 0 degrees and 180 degrees. Let’s then say that we measure t very precisely to be 90.00 +/- 0.01 degrees. That means we know the value of t to one part in 11,000. > > But what if the angle t shows up in our equations inside of a sin() function? We could just as well have defined a quantity x equal to sin(t), and called x our fundamental constant. The range of x that our same measurement allows is x = 1 +/- 0.000000015. (An easy, if only statistically approximate, way to get this is just to calculate sin(90.01) and sin(89.99).) That’s about one part in 7 billion. Alternately, suppose that the quantity of interest is neither t nor sin(t), but tan(t)? Though one then gets back to the issue that the quantity of interest might not itself be finite. [Next page](https://boards.straightdope.com/t/least-and-most-uncertain-physical-constants/693696.md?page=2)