Physicists occasionally use a system of natural units, called Planck units, which are calculated based on fundamental constants of nature like Planck’s constant, the gravitational constant, and the speed of light.

Planck’s constant comes up often in discussing quantum physics. In fact, if you were to perform the mathematics of quantum physics, you’d find that little h variable all over the place. Physicists have even found that you can define a set of quantities in terms of Planck’s constant and other fundamental constants, such as the speed of light, the gravitational constant, and the charge of an electron.

These Planck units come in a variety of forms. There is a Planck charge and a Planck temperature, and you can use various Planck units to derive other units such as the Planck momentum, Planck pressure, and Planck force . . . well, you get the idea.

For the purposes of the discussion of string theory, only a few Planck units are relevant. They are created by combining the gravitational constant, the speed of light, and Planck’s constant, which makes them the natural units to use when talking about phenomena that involve those three constants, such as quantum gravity. The exact values aren’t important, but here are the general scales of the relevant Planck units:

Planck length: 10-35 meters (if a hydrogen atom were as big as our galaxy, the Planck length would be the size of a human hair)

Planck time: 10-43 seconds (the time light takes to travel the Planck length — a very, very short period of time)

Planck mass: 10-8 kilograms (about the same as a large bacteria or very small insect)

Planck energy: 1028 electronvolts (roughly equivalent to a ton of TNT explosive)

Keep in mind that the exponents represent the number of zeroes, so the Planck energy is a 1 followed by 28 zeroes, in electronvolts. The most powerful particle accelerator on Earth, the Large Hadron Collider that came online briefly in 2008 can produce energy only in the realm of TeV — that is, a 1 followed by 12 zeroes, in electronvolts.

The negative exponents, in turn, represent the number of decimal places in very small numbers, so the Planck time has 42 zeroes between the decimal point and the first non-zero digit. It’s a very small amount of time!

Some of these units were first proposed in 1899 by Max Planck himself, before either relativity or quantum physics. Such proposals for natural units — units based on fundamental constants of nature — had been made at least as far back as 1881. Planck’s constant makes its first appearance in the physicist’s 1899 paper. The constant would later show up in his paper on the quantum solution to the ultraviolet catastrophe.

Planck units can be calculated in relation to each other. For example, it takes exactly the Planck time for light to travel the Planck length. The Planck energy is calculated by taking the Planck mass and applying Einstein’s E = mc2 (meaning that the Planck mass and Planck energy are basically two ways of writing the same value).

In quantum physics and cosmology, these Planck units sneak up all the time. Planck mass represents the amount of mass needed to be crammed into the Planck length to create a black hole. A field in quantum gravity theory would be expected to have a vacuum energy with a density roughly equal to one Planck energy per cubic Planck length — in other words, it’s 1 Planck unit of energy density.

**Why are these quantities so important to string theory?**

The Planck length represents the distance where the smoothness of relativity’s space-time and the quantum nature of reality begin to rub up against each other. This is the quantum foam I explain in topic 2. It’s the distance where the two theories each, in their own way, fall apart. Gravity explodes to become incredibly powerful, while quantum fluctuations and vacuum energy run rampant. This is the realm where a theory of quantum gravity, such as string theory, is needed to explain what’s going on.

**Planck units and Zeno’s paradox**

If the Planck length represents the shortest distance allowed in nature, it could be used to solve the ancient Greek puzzle called Zeno’s paradox. Here is the paradox:

You want to cross a river, so you get in your boat. To reach the other side, you must cross half the river. Then you must cross half of what’s left. Now cross half of what’s left. No matter how close you get to the other side of the river, you will always have to cover half that distance, so it will take you forever to get across the river, because you have to cross an infinite number of halves.

The traditional way to solve this problem is with calculus, where you can show that even though there are an infinite number of halves, it’s possible to cross them all in a finite amount of time. (Unfortunately for generations of stymied philosophers, calculus was invented by Newton and Leibnitz 2,000 years after Zeno posed his problem.)

As it turns out, during my sophomore year I solved Zeno’s paradox in my calculus course the same semester that I learned about Planck units in my modern physics course. It occurred to me that if the Planck length were really the shortest distance allowed by nature, the quantum of distance, it offered a physical resolution to the paradox.

In my view, when your distance from the opposite shore reaches the Planck length, you can’t go half anymore. Your only options are to go the whole Planck length or go nowhere. In essence, I pictured you “slipping” along that last tiny little bit of space without ever actually cutting the distance in half.

When I first came up with this idea as an undergraduate physics major, I was extremely impressed with myself. I have since learned that I’m not the only person to have come up with this connection between Planck length and Zeno’s paradox. Despite that, I’m still somewhat impressed with myself.

In some sense, these units are sometimes considered to be quantum quantities of time and space, and perhaps some of the other quantities as well. Mass and energy clearly come in smaller scales, but time and distance don’t seem to get much smaller than the Planck time and Planck length. Quantum fluctuations, due to the uncertainty principle, become so great that it becomes meaningless to even talk about something smaller. (See the nearby sidebar “Planck units and Zeno’s paradox.”)

In most string theories, the length of the strings (or length of compactified extra space dimensions) are calculated to be roughly the size of the Planck length. The problem with this is that the Planck length and the Planck energy are connected through the uncertainty principle, which means that to explore the Planck length — the possible length of a string in string theory — with precision, you’d introduce an uncertainty in energy equal to the Planck energy.

This is an energy 16 orders of magnitude (add 16 zeroes!) more powerful than the newest, most powerful particle accelerator on Earth can reach. Exploring such small distances requires a vast amount of energy, far more energy than we can produce with present technology.