# Ask a Physicist Answers

My 7 year old son Ben (who is sitting next to me now) has recently become interested in how small things can be. He is not convinced that nothing can be measurably smaller than a Planck length because whatever is Planck sized can always be divided into something smaller.

He thinks perhaps Planck lengths can be divided into energy beams that then become infinite.

—Question submitted by Ben from Norwalk, CT

The Planck length is incredibly small — less than a billionth of a trillionth of a trillionth of an inch.**Image Credit:** UNSW

Let's first look at what is meant by the “Planck length.” Our current best theory of gravity is Einstein's general theory of relativity. According to this theory, the gravitational force is due to a kind of “bending” space and time in four dimensions in the presence of a mass.

#### Competing Ideas

To better understand this, imagine a big rubber sheet with a bowling ball in the middle. The bowling ball warps the rubber sheet, so that if a marble is now placed on the sheet, it will follow the warping and roll toward the bowling ball, as if there were a kind of force present. You can liken that force the bowling ball creates to the force of gravity that planets and stars exert on other celestial bodies.

On the other hand, our best theory of physics at very small distances (the size of an atom or smaller) is called quantum field theory. It describes forces in a completely different way: It says that forces are due the exchange of small subatomic particles. So we have two ways of thinking about gravity, but which is correct?

Is it caused by the warping of space and time, as described by Einstein, or is it caused by the exchange of subatomic particles? We don't know the answer to that yet. But the Planck length gives us some idea of where to start looking for the answer.

Einstein's general theory of relativity is characterized by two constants that appear in its equations: the gravitational constant, G (which determines the strength of the gravitational force), and the speed of light, c. The equations of quantum field theory contain the speed of light c, along with a constant called Planck's constant ħ (pronounced “h-bar”), which determines the energies at which the effects of the theory become important.

#### A Decillionth of an Inch

Physicist Max Planck showed that there is a unique way of combining these three constants (G, c, and ħ) by multiplications and divisions that results in a quantity that has units of length. That combination (√(ħG/c^{3})) is called the Planck length.

Physicists presume that any new theory of “quantum gravity” that would encompass both general relativity and quantum field theory (and thus include the successful predictions of both) would have to be able to explain phenomena that are at about the size of the Planck length. The Planck length is an unimaginably small distance: it's about half of a decillionth (a billionth of a trillionth of a trillionth) of an inch.

It's possible to measure distances down to about the size of the wavelengths of particles used to make the measurements. Higher energy particles have shorter wavelengths, allowing us to study shorter distances. The energy required to make measurements at the Planck length (the Planck energy) is a whopping 12 octillion electron volts—far beyond the capacity of today's particle accelerators.

With the equipment we have available today, the smallest objects we know about are particles called leptons (which include the electron), and heavier particles called quarks, which combine in triplets to form particles like protons and neutrons.

As far as we can determine today, both leptons and quarks appear to be point-like particles. Nevertheless, it's not inconceivable that they have some internal structure that we can't yet measure. Physicists have a number of theories and speculations in this area, but we really don't know for certain if there is any meaningful “smallest” distance.

**Answered by:**

David G. Simpson, PhD

Research Physicist

NASA Goddard Space Flight Center