Tag Info

Hot answers tagged

50

Dear Michael, first of all, the question you are asking is very important and you may master it completely. Dimensionful constants are those that have units - like $c, \hbar, G$, or even $k_{\rm Boltzmann}$ or $\epsilon_0$ in SI. The units - such as meter; kilogram; second; Ampere; kelvin - have been chosen partially arbitrarily. They're results of random ...


45

Something I posted on reddit answers this question quite well, I think: "Rational" and "irrational" are properties of numbers. Quantities with units aren't numbers, so they're neither rational nor irrational. A quantity with units is the product of a number and something else (the unit) that isn't a number. By choosing the unit you use to express a ...


41

To formalize dushya's comment as an answer: Since the kilogram is an arbitrary, man-made unit, the actual numerical value of the proton mass in kilograms is meaningless (i.e. it's as good as its value in pounds, ounces, stones, solar masses, $\textrm{MT}/c^2$, etc.). The true fundamental constants of nature are dimensionless: they have the same value in ...


20

The view of most physicists is that asking "How can it be that the speed of light is constant?" is similar to asking "How can it be that things don't always go in the direction of the force on them?" or "How can it be that quantum-mechanical predictions involve probability?" The usual answer is that these things simply are. There is no deeper, more ...


19

Yes, Avogadro's constant is a redundant artifact from the era in the history of chemistry in which people didn't know how many atoms there were in a macroscopic amount of a material and it is indeed legitimate to set Avogadro's constant equal to one and abandon the awkward obsolete unit "mole" along the way. This $N_A=1$ is equivalent to $$ 1\,\,{\rm mole} = ...


15

$G$ is just a constant of proportionality to get the units right (so that when $m_1$ and $m_2$ are in kilograms and $r$ is in meters you get a force in Newtons rather than wingdingalings or something really weird). Indeed cosmologists like to work in a system of units where $G = c = 1 \text{ (dimensionless)}$, and particle physicists like to work in units ...


15

There was a proposal in 2006 trying to define NA as an exact number[1,2]: $$ N_A^* = 84\;446\;888^3 = 6.022\;141\;410\;704\;090\;840\;990\;72 \times 10^{23} $$ the problem? This value is incorrect, as the currently most accurate result is[3] $$ N_A = 6.022\;140\;84(18) \times 10^{23} $$ i.e. $N_A^*$ is now 3 s.d. away from $N_A$. As I have commented, if ...


15

The infinitesimal length interval between two events in spacetime $ds$ is defined by $$ds^2=c^2 dt^2 - dx^2 - dy^2 - dz^2$$ The creature is dimensionally consistent, because time is multiplied with a speed. You can think of $(t,x,y,z)$ as the four coordinates of spacetime $(x^0,x^1,x^2,x^3)$ and $c$ appears naturally in the equations. However, the usual ...


13

The expression $(\hbar G/c^3)^{1/2}$ is the unique product of powers of $\hbar, G,c$, three most universal dimensionful constants, that has the unit of length. Because the constants $\hbar, G,c$ describe the fundamental processes of quantum mechanics, gravity, and special relativity, respectively, the length scale obtained in this way expresses the typical ...


12

The problem is that you want your unit definitions to be realizable - so specifying "1 mol is long number molecules, 1 gram is 1/12 of the mass of one mol of $C_{12}$" is nice for your thought process, but as long as there is no practical way to count molecules at such scales to a precision of better than $10^{-9}$ (which I think is the precision of the ...


12

You've seen the speed of light quoted as roughly $3*10^8\, \text{m/s}$, so the speed of light is very fast compared to one meter and one second. This is roughly a human walking speed, so your question could be interpreted as asking why light is few hundred million times faster than a walking speed. The speed people walk is rather anthropocentric, though. ...


12

Your guess is correct. After electroweak symmetry breaking, the coupling constant for the residual $U(1)_\textrm{EM}$ gauge group can be written as a function of the couplings of the broken $SU(2)_L \times U(1)_\textrm{Y}$ gauge groups: $$ \alpha = \frac{1}{4\pi}\frac{g^2 g\prime^2}{g^2+g\prime^2} = \frac{e^2}{4\pi} $$ These couplings, however, are running ...


12

Since in the limit of weak gravitational fields, Newtonian gravitation should be recovered, it is not surprising that the constant $G$ appears also in Einstein's equations. Using only the tools of differential geometry we can only determine Einstein's field equations up to an unknown constant $\kappa$: $$G_{\mu\nu} = \kappa T_{\mu\nu}.$$ That this equation ...


11

$G$ is not exactly larger than $h$ by a factor of $10^{23}$ in SI units, as you are probably aware (just making sure). There is also no expected numerical relationship between the two that has a physical interpretation. You have to understand that these constants are mostly just due to our (to some extent) arbitrary choices of units. These are, of course, ...


10

SPEED OF LIGHT: This is a very interesting question. Going through the foundations of electromagnetism and the theory that led to Maxwell’s equations, there is an interesting element that can grab your attention. You can see that the speed of light is not as abstract and mysterious as it appears to be, but only if you look from a different perspective. I ...


9

It's all in how you define the units. $E = mc^2$ in nice MKSA units; but then change energy into BTUs and you'll need the ever-lovable "fudge factor" in there. People spent a lot (well, some) of time developing self-consistent sets of units largely to keep equations simple, tho' as Rijul pointed out, assigning ugly numbers to known constants hides a ...


8

In actual fact, the relative speed rule does not apply, ever. The relativistically correct speed addition rule is the following: $$s=\frac{v+u}{1+\frac{vu}{c^2}}$$ When $\frac{vu}{c^2}$ is close to zero (in other words when the velocities invloved are much less than the speed of light, then the correct formula reduces to the Galilean version $s=u+v$. ...


8

Only dimensionless quantities are important. They are just pure numbers and there can't be any ambiguity about their value. This is not so with dimensionful quantities. E.g. if I tell you my speed $v$ relative to you is $0.5\, \rm speedons$ that doesn't give you much information as I have a freedom to define my $\rm speedon$ units any way I want. Only way I ...


8

Using fundamental physical constants try to construct expression which unit is legth. So using dimensional analysis, we have a data of: $G = m^3 \cdot kg^{-1} \cdot s^{-2}$, $c = m \cdot s^{-1}$ and $\hbar = J \cdot s = kg \cdot m^2 \cdot s^{-1}$. Than we are to construct length $l = m$ in the following way: $$l = G^a c^b \hbar^d = m^{3a + b+d} \cdot ...


8

Pure convention. There is no reason alternative conventions couldn't be used, apart from the need to avoid confusion. Newton introduced the constant to make the force law simple, whereas the electrostatic definition with the $4\pi$ is designed to make Poisson's equation (one of the equations for the electric field) look simple. You can write a Poisson ...


8

Although you might not like to hear it, the answer really DOES lie in the definition of $\mu_0$ (and $c$). $\mu_0$ is defined to be exactly $4\pi *10^{-7}\ \text{H m}^{-1}$. Similarly, $c$ is defined as exactly $299792458\ \text{ms}^{-1}$. It immediately follows from the relation $$\epsilon_0=\frac{1}{\mu_0 c^2}$$ that $\epsilon_0$ also has no uncertainty. ...


8

It depends on the unit you want to express it. If you choose c/100 as the speed unit, c will be expressed with a rational number. If you choose c/π, you'll have an irrational one. That depends on measure, not on nature.


8

The short answer is that it is simply not possible to design a "one size fits all" unit system. The staggeringly large range of mass, time and length scales that appear in the Universe prevent this. The Planck unit system you mentioned is mainly useful for people who will never touch an experimental apparatus. The vast majority of scientists and engineers do ...


7

I'll (arguably artificially) restrict my answer to physics, avoiding the creationist side of the equation. The question of whether physical "constants" are actually constant has been studied by physicist, and some cosmological theories predict such variations. As you say, a variation of c would change many things in physics. For example $c$ is a parameter ...


7

Your last couple of comments, about units, are incredibly important. It only makes sense to compare two things if they have compatible units. So, to use your example, it doesn't really make sense to talk about the size of Planck's constant relative to the electron charge, but it does make sense to talk about the mass of the muon relative to the mass of the ...


7

CODATA is a group that compares and combines all the most accurate experimental measurements of fundamental constants to give recommendations for best-guess values that should be used. They periodically update their values as new experiments are done. You are seeing that some wikipedia pages use old (not-updated) CODATA recommendations, while others have ...


7

There is no proof that fundamental constants are constant. Indeed I've seen claims that string theory allows varying constants, though I've also seen comments (I think Lubos Motl blogged on this a while back) that such arguments are wrong. There are lots and lots of publications measuring fundamental constants and review articles of such measurements. ...



Only top voted, non community-wiki answers of a minimum length are eligible