# Why is energy not an SI base unit?

According to a textbook I have begun to read, there are seven base SI units:

• Length
• Mass
• Time
• Temperature
• Amount of a substance
• Electric current
• Luminous intensity

What I do not understand is, why have these been chosen as the fundamental units? It seems to me that mass, amount, current and luminous intensity could all be expressed with energy. Instead energy is for some reason a derived unit. Why is this?

• I'm not sure how you could express amount (particle number) and current (charge flow) in terms of energy? – jabirali Feb 23 '15 at 4:29
• I was thinking at a particle level, since matter is made of energy, multiple particles together as an amount would still have an associated energy, and charge flow could be measured as how much work the moving charges could do. – Segmented Feb 23 '15 at 4:40
• @Segmented: not really, since two different circuits can be doing different amounts of work with the exact same current (because the resistance is different). You can certainly measure how much work the moving charges do, but current is the rate of movement of charge, not the work it does. So you'd still need another fundamental unit in addition to energy, to use to derive a unit for current (resistance or potential difference). – Steve Jessop Feb 23 '15 at 15:53
• Oh, unless you mean that one amp could be defined as $5.6 \times 10^{-12}$ kg of electrons per second, or something. Converting with $e = mc^2$ I think that's 511kW. I fear this would be confusing, considering that the mass of the electrons isn't what we're interested in ;-) – Steve Jessop Feb 23 '15 at 16:01
• @SteveJessop While not what I said that is indeed along the lines I was thinking =) – Segmented Feb 23 '15 at 16:57

• Temperature
• Amount of a substance
• Luminous intensity

are pretty much bogus fundamental units. The unit temperature is just an expression of the Boltzmann constant (or you could say the converse, that the Boltzmann constant is not fundamental as it is merely an expression of the anthropocentric and arbitrary unit temperature).

The unit energy will be whatever is the unit of force times the unit of length. AJoule is the same as a Newton-Meter, which are already defined in the SI system.

You should read the NIST page on units to get the low-down on it.

In my opinion, electric charge is a more fundamental physical quantity than electric current, but NIST (or more accurately, BIPM) defined the unit current first and then, using the unit current and unit time, they defined the unit charge. I would have sorta defined charge first and then current.

Just like the unit charge (or current) is just another way to express the vacuum permittivity or, alternatively the Coulomb constant and the unit temperature is just another way to express the Boltzmann constant, the unit time, unit length, and unit mass, all three taken together could be just another way to express the speed of light, the Planck constant, and the gravitational constant. But because $G$ is not easy to measure (given independent units of measure) and can never be measured as accurately as we can measure the frequency of "radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom", we will never have $G$ as a defined constant as we do for $c$ and as we will soon for $\hbar$ and perhaps for $\epsilon_0$ and $k_\text{B}$.

But once we define length, time, and mass independently, we cannot define energy independently. The Joule is a "derived unit".

EDIT: so i will try to explain why the candela is bogus. (i had already for the mol.) so there is a sorta arbitrary specification of frequency, then what is the difference between 1 Candela and $\frac{4 \pi}{683} \approx$ 0.0184 watts? bogus base unit.

• Would I be right in assuming that if you instead chose to define length time and energy, mass would be the derived unit? – Segmented Feb 23 '15 at 6:47
• @Segmented Yes. And that's not just purely a curiosity, different branches of physics use very diverse selections of base units and it's not a big deal. When talking about elementary particles, mass and energy aren't distinguished at all - both are measured in eV and mass is just a rest energy. – orion Feb 23 '15 at 13:07
• Even more accurately: the CGPM decided on unit definitions; the BIPM is tasked with the practical implementation and dissipation of those decisions. – Henning Makholm Feb 23 '15 at 15:16
• Temperature is not a bogus dimention. While the Bolzmann factor links temperature and energy (by essentially removing the dimensionless-but-we-gave-it-dimensions quantity of entropy), there simply is no good way to actually link temperature to energy over a large energy range. – DanielSank Feb 23 '15 at 15:57
• @Jefromi Luminous intensity is definitely anthropocentric (it's actually defined in terms of a function modeling the human eye), and "amount of a substance" is essentially dimensionless (Avogadro's Constant is equal to the number 1). That information should really be in the answer, though... – Brilliand Feb 24 '15 at 0:37

why have these been chosen as the fundamental units?

Courtesy of the National Institute of Standards and Technology (NIST), we have some historical context. It basically boils down to wanting to have absolute measurements with respect to units of mass, length, and time.

These few were originally chosen because these form a set of mutually independent dimensions. That is to say, it is from these three that many common (at the time, though still often today) terms could be derived (velocity, acceleration, momentum, energy, etc)--though a few other units require other fundamental units (e.g., temperature).

Instead energy is for some reason a derived unit. Why is this?

Because energy is derived from other more primitive variables. From Wikipedia's article on Fundamental units

Many of these quantities are related to each other by various physical laws, and as a result the units of some of the quantities can be expressed as products (or ratios) of powers of other units (for example, momentum is mass multiplied velocity while velocity is measured in distance divided by time). These relationships are discussed in dimensional analysis. Those that cannot be so expressed can be regarded as "fundamental" in this sense)

So, since energy is the product of a few variables (e.g., $E=\frac12mv^2$ or $E=mg\Delta h$ or what have you), it cannot be a fundamental unit.

• These formulas are relationships between the variables but do not necessarily imply which one is dependant. I could isolate for any of E,M,v,g, or h in the examples you provide and say that the rest had to be fundamental. – Segmented Feb 23 '15 at 4:52
• Your comment reflects directly to the first half of the answer: historicity. If the scientists of the early 1800's had been concerned with energy (instead of mass, length & time), they may have taken that as fundamental. That isn't what happened, so we are left with life as it is. – Kyle Kanos Feb 23 '15 at 13:35

One Joule (the unit of energy) equals one $\$ kg m$^2$/s$^2$. So, you see, a unit of energy can be expressed in terms of units of mass, distance, and time. The people who chose the SI units could have made, for example, the Joule an SI basic-unit and could have defined the unit of mass in terms of distance, time, and energy (kg = Js$^2$/m$^2$). But then we would still have the same number of fundamental units. So the choice of which SI units to define as basic units is somewhat arbitrary. Though, I think that energy was not chosen as an SI basic-unit because it just doesn't seem as fundamental as time, mass, or distance.

• I feel you are correct, since they are just ratios. However I'm not sure that, as you say, there would be the same number of units if energy was instead chosen... For example, why would you need a separate unit for luminous intensity if energy was chosen as a base unit? Surely you could express that in terms of energy instead...? – Segmented Feb 23 '15 at 4:48
• Luminous intensity cannot be expressed in terms of energy. I don't know exactly why this is, because I'm only an an amature physicist and haven't actualy studied luminous intensity, but I can still answer your question. Making energy a basic SI unit cannot solve any problems, because energy can already be expressed in terms the basic SI units of mass, time, and distance. So if a unit can be expressed in terms of energy, It can also be expressed in terms of mass, time, and distance by making the substitution, J=kg*m^2/s^2. – Sheepman Feb 23 '15 at 5:06
• True, I suppose what I'm arguing here - that the number of base units could be reduced - would not change, as you say, because of choosing energy vs mass. In the end I believe you could end up with either mass time and distance or energy time and distance since mass and energy are related. – Segmented Feb 23 '15 at 5:15
• Actually, the number of base units can be reduced if you want; it is quite arbitrary that the SI system has 7 base units. – jabirali Feb 23 '15 at 5:45
• @SamuelMontgomery, Luminous intensity can most certainly be expressed in terms of power (energy and time) and area (length$^2$). it is a superfluous base unit. – robert bristow-johnson Feb 23 '15 at 6:18

Purely historical and convenience reasons, people standardized measures that were more obvious from real life (remember, quantum mechanics and relativity didn't exist when SI was drafted).

It's actually arbitrary how many quantities you define as fundamental (and associate them with base units), and it doesn't matter which ones are they.

In quantum mechanics energy is actually used as a base quantity (measured in $eV$). They say you effectively set $c=\hbar=1$ and then mass, energy, momentum, frequency and temperature are all measured in eV. Time and distance are also sometimes measured in $eV^{-1}$ instead of seconds & meters if that's more convenient. Velocities are nondimensional (fractions of the speed of light).

Each unit you define as a base unit comes with an associated "natural constant" that wouldn't be needed, if you unified two units of measurement. If you measure temperature in units of energy, the need for Boltzmann constant (the conversion factor) disappears. Similarly, space and time can be measured in the same units if you standardize the conversion velocity (usually the speed of light) to be equal to $1$. Then, you have $\hbar$ that converts between units of frequency and energy (and other related pairs). The unit electric charge connects amperes with the mechanical units.

In the purest form, there would be only one physical unit, and the only natural constants would be the dimensionless strengths of the elementary forces, and relative rest masses of elementary particles - these are the only things that must be given, not derived (so far - hopefully we'll get to the grand unification theory someday).

• In addition to the "coupling constants" that are the expression of your "dimensionless strengths and rest masses", systems of natural units tend to have factors of π in them that can't be eliminated. – zwol Feb 23 '15 at 14:41
• I didn't count π as a "natural constant". Of course it's there. So is 2 and $\sqrt5$ and so on... I'm talking about the constants that come from the physical world. – orion Feb 23 '15 at 15:01

The whole notion of "fundamental units" is bogus, physics can be formulated in a purely dimensionless way. Given any set of equations, you are free to define new variables by e.g. introducing arbitrary scaling constants. This then allows you to study certain scaling limits of the theory. E.g. starting from special relativity (formulated in natural units), you can introduce a dimensionless parameter c to zoom in into the low velocity limit of the theory (you scale all the velocities but such that in terms of the new variables the values the velocities take remain constant as you scale the real velocities down to zero).

Exactly at the scaling limit, certain equations relating physical variables will become singular. Unable to relate these quantities to each other, stupid physicists living at approximately that scaling limit may think that they are fundamentally incompatible and need to be measured in units with incompatible dimensions.

Dimensional analysis as taught in most textbooks is wrong in the sense of "not even wrong". While formally correct, the argument made is not correct, because in natural units you can always relate any physical quantity to another. So, why are the results then still correct? The way the period of a pendulum depends on its length L and the local gravitational acceleration can only be found if you assume that hbar, G and c will not appear in the equations in SI units. But, from what I pointed out above, this must follow from an appropriate scaling argument. So, it's that scaling argument that is the only physically correct argument, not the given "dimensional calculus" argument.

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