# Why do constants have dimensions?

I am just a beginner in dimensional analysis, and I see that $G$, the universal gravitational constant, is independent of everything. Speed, for example, depends on distance and time, but $G$ does not depend upon anything.

Then why is $G$'s dimensions not $M^0 L^0 T ^0$, as it is not dependent on $M$, $L$ or $T$?

While the constant itself is not dependent on anything else, its effect on the universe is dependent. For example, $G$ is a sort of ratio that relates the force between two objects at a given distance and of two certain masses. $G$ does not affect the electrical force between two charged particles - that's Coulomb's constant; it does not affect the time it takes a massless particle to travel a certain distance - that's the "speed of light," $c$. So really, in a way, it's not quite independent of everything, because in order to have $G$ you have to have masses, distances and forces; otherwise the number itself is useless.

If the gravitational constant were dimensionless, in fact, it would also be meaningless because there would be no way to keep it actually constant. Since it is a ratio, you have to determine what things it is a ratio between; in this case, again, those values are mass, distance and gravitational "force." If there were no units, you couldn't usefully calculate how gravity affects objects; you'd have a number, but no way to use it - really, no way even to calculate the constant to begin with, now that I think about it. While there are meaningful dimensionless quantities, famously the fine structure constant and (more mundanely) percentages for example, they can only arise naturally in cases where the things you are comparing are of the same nature. For example, "how many meters are in a light-year" is a dimensionless quantity, because it is a ratio of two units of distance. Comparing this to $G$ it's easy to see why the constant is not dimensionless; it compares units of different natures, specifically force, distance and mass, none of which can be converted into any of the others unless by a constant such as $G$ which, well, that's pretty much the whole reason $G$ exists at all.

• Why certain numbers like Reynold No. do not require dimensions? – Anubhav Goel Apr 10 '16 at 9:15
• @AnubhavGoel In particular the Reynolds number of a system is a ratio of two kinds of forces, so the force units cancel out and you're left with just the ratio. The physical quantities are still there, the "dimensionlessness" of it is a mathematical result. For example with the Reynolds number, you still have to use fluids with dimensional viscosity, and obstacles with dimensional size to calculate a meaningful dimensionless Reynolds number. – Asher Apr 10 '16 at 13:19
• From wiki""These constants cannot be eliminated by any choice of a system of units. Such constants include: α, the fine structure constant , the coupling constant for the electromagnetic interaction (≈1/137.036). Also the square of the electron charge, expressed in Planck units, which defines the scale of charge of elementary particles with charge. μ or β, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron (≈1836.15). – Anubhav Goel Apr 10 '16 at 13:57
• You wrote, "If the gravitational constant were dimensionless, in fact, it would also be meaningless because there would be no way to keep it actually constant. " – Anubhav Goel Apr 10 '16 at 13:58
• From above wiki extract we don't need dimensions to keep constants constant. – Anubhav Goel Apr 10 '16 at 13:59

There is one essential thing you have to keep in mind

there is something important about physical formulas ...... think of the OHMS law R=V/I as you all know resistance of something is influenced just by its internal construction like the length,substance it is made from...etc since R is not influenced by I(current) and V(voltage) if you double current of the electric circuit , voltage will be doubled too. in another point if view I can say that since current and voltage are related to each other the result of their division which is resistance is neither related to current nor to voltage. the exact thing occurs for F(force) and m(mass) since if we are speaking for earth for example F=m.g you can easily see that m( mass) is omitted and G the constant of gravitation is just related to the mass of earth and its radius(r) which are always specific numbers.

Newton's constant, $G$, was introduced in his law of gravitation, $$F = G \frac{M m}{r^2}$$ In other words, the constant may be expressed as $$G = F \frac{r^2}{M m}$$ and the dimension of $G$ may be deduced in the same manner as the dimension of velocity in $v=d/t$. Of course, if Newton's law of gravitation holds, all estimates of $G$ from this inverted formula should agree.

• This doesn't answer the question - it simply motivates the units of G. – DilithiumMatrix Apr 9 '16 at 17:58
• The question is why is $G$ dimensionful? Why isn't this an answer if it motivates the units of G? – innisfree Apr 10 '16 at 0:15
• +1 I dont see anything wrong in this answer... – AccidentalFourierTransform Apr 10 '16 at 16:19

G is actually not independent of everything. It does depend on factors like density of space, rate of universe expansion etc. Its just scientists have not found this dependence.

For time there have been many constants which were later were found to be dependent.

For example at time of Kepler

$T^2 =k R^3$

Later,

k was found to depend on Mass(by Newton) of our sun, and new constant G was defined.

Its not far when G would break to a number of factors.

Speed doesn’t depend on distance and time. We measure it by measuring distance and time. We should understand the difference between physics and mathematics. We use mathematics because we want to live better in nature. Speed is a abstract concept that we created it. We never can discover how the nature works or what are the rules nature based on? About G, we defined it and so it will have dimensions based on our contracts. It doesn’t depend on its dimensions, but it is measured by them.

## protected by Qmechanic♦Apr 10 '16 at 11:11

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