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In Spacetime Physics, Taylor and Wheeler discuss a nice story. I will tell a version of it which is a bit improvised (read mutilated). Imagine a town where people didn't know how to build rulers. However, there were two rail lines in the town. One went North-South and the other went East-West. The NS rail line had markings on it each meter whereas the EW rail line had markings on it each foot. So, people invented two notions of distance: an NS distance which they measured in meters, and an EW distance which they measured in feet. However, one curious kid once figured out that if you took a stick, put it along the NS rail line, measured its NS distance, and then rotated it to align it to the EW rail line, its EW length would always turn out to be $3.28$ times its NS length. So, they had this nice formula $L_{EW}=fL_{NS}$ where $f$ was a universal constant of the town, measured to be $3.28\text{ feet}/\text{meter}$. Finally, an insightful kid came along and realized that lengths of sticks remain invariant under all rotations and thus the same stick can be used to define distances along any of the directions. So, he started measuring the NS distance and the EW distance using the same unit, meter. People cried, "Oh! the dimensions won't work out!", "This is just a trick!", "Socialism never works!", and so on. But of course, each of those sentences is wrong. The kid had discovered that the heart of the concept of distances lies in that they are rotationally invariant and this allows us (in fact, forces us) to measure distances in the same units along all directions.

In Spacetime Physics, Taylor and Wheeler discuss a nice story. I will tell a version of it which is a bit improvised (read mutilated). Imagine a town where people didn't know how to build rulers. However, there were two rail lines in the town. One went North-South and the other went East-West. The NS rail line had markings on it each meter whereas the EW rail line had markings on it each foot. So, people invented two notions of distance: an NS distance which they measured in meters, and an EW distance which they measured in feet. However, one curious kid once figured out that if you took a stick, put it along the NS rail line, measured its NS distance, and then rotated it to align it to the EW rail line, its EW length would always turn out to be $3.28$ times its NS length. So, they had this nice formula $L_{EW}=fL_{NS}$ where $f$ was a universal constant of the town, measured to be $3.28\text{ feet}/\text{meter}$. Finally, an insightful kid came along and realized that lengths of sticks remain invariant under all rotations and thus the same stick can be used to define distances along any of the directions. So, he started measuring the NS distance and the EW distance using the same unit, meter. People cried, "Oh! the dimensions won't work out!", "This is just a trick!", and so on. But of course, each of those sentences is wrong. The kid had discovered that the heart of the concept of distances lies in that they are rotationally invariant and this allows us (in fact, forces us) to measure distances in the same units along all directions.

Generally speaking, when there is a fundamental constant of nature which relates two quantities of different units, it's a sign that we should actually measure the two quantities in the same units, making the constant dimensionless. For example, in quantum mechanics, $[x,p]=i\hbar$ allows us to set up a system where $x$ is measured in $[\text{GeV}]^{-1}$, $p$ is measured in $[\text{GeV}]$ as usual (i.e., as usual after setting $c=1$!), and this makes $\hbar=1$. If you don't use natural units, $\hbar$ would have had the dimensions of action (i.e., those of angular momentum).

Generally speaking, when there is a fundamental constant of nature which relates two quantities of different units, it's a sign that we should actually measure the two quantities in the same units, making the constant dimensionless. For example, in quantum mechanics, $[x,p]=i\hbar$ allows us to set up a system where $x$ is measured in $\text{GeV}^{-1}$, $p$ is measured in $\text{GeV}$ as usual (i.e., usual after setting $c=1$!), and this makes $\hbar=1$. If you don't use natural units, $\hbar$ would have had the dimensions of action (i.e., those of angular momentum).

In Spacetime and Physics, Taylor and Wheeler discuss a nice story. I will tell a version of it which is a bit improvised (read mutilated). Imagine a town where people didn't know how to build rulers. However, there were two rail lines in the town. One went North-South and the other went East-West. The NS rail line had markings on it each meter whereas the EW rail line had markings on it each foot. So, people invented two notions of distance: an NS distance which they measured in meters, and an EW distance which they measured in feet. However, one curious kid once figured out that if you took a stick, put it along the NS rail line, measured its NS distance in meters, and then rotated it to align it to the EW rail line, it'd always correspond to $3.28$ feet of the EW distance. So, they had this nice formula $L_{EW}=fL_{NS}$ where $f$ was a universal constant of the town, measured to be $3.28\text{ feet}/\text{meter}$. Finally, an insightful kid came along and realized that lengths of sticks remain invariant under all rotations and thus the same stick can be used to define distances along any of the directions. So, he started measuring the NS distance and the EW distance using the same unit, meter. People cried, "Oh! the dimensions won't work out!", "This is just a trick!", "Socialism never works!", and so on. But of course, each of those sentences is wrong. The kid had discovered that the heart of the concept of distances lies in that they are rotationally invariant and this allows us (in fact, forces us) to measure distances in the same units along all directions.

In Spacetime Physics, Taylor and Wheeler discuss a nice story. I will tell a version of it which is a bit improvised (read mutilated). Imagine a town where people didn't know how to build rulers. However, there were two rail lines in the town. One went North-South and the other went East-West. The NS rail line had markings on it each meter whereas the EW rail line had markings on it each foot. So, people invented two notions of distance: an NS distance which they measured in meters, and an EW distance which they measured in feet. However, one curious kid once figured out that if you took a stick, put it along the NS rail line, measured its NS distance, and then rotated it to align it to the EW rail line, its EW length would always turn out to be $3.28$ times its NS length. So, they had this nice formula $L_{EW}=fL_{NS}$ where $f$ was a universal constant of the town, measured to be $3.28\text{ feet}/\text{meter}$. Finally, an insightful kid came along and realized that lengths of sticks remain invariant under all rotations and thus the same stick can be used to define distances along any of the directions. So, he started measuring the NS distance and the EW distance using the same unit, meter. People cried, "Oh! the dimensions won't work out!", "This is just a trick!", "Socialism never works!", and so on. But of course, each of those sentences is wrong. The kid had discovered that the heart of the concept of distances lies in that they are rotationally invariant and this allows us (in fact, forces us) to measure distances in the same units along all directions.