Vacuum Permeability (Can π Be Redefined by Measurement?)

The 2019 revision of the International System of Units (SI) changed the vacuum permeability, \(\mu_0\), from the formerly exact defined value of \(4\pi\times10^{-7}\ \mathrm{N/A^2}\) to a quantity whose recommended value is updated on the basis of experimental results.

This change was part of the 2019 SI revision, which fixed the Planck constant \(h\) exactly to establish a mass standard independent of the International Prototype of the Kilogram. More directly, the fact that \(\mu_0\) is now determined from measurement is also related to the redefinition of the ampere through a fixed elementary charge \(e\).

The CODATA 2022 recommended value of vacuum permeability is

What consequences does this revision have for physics and mathematics?

Since the 2019 SI revision, the speed of light in vacuum \(c\), the elementary charge \(e\), and the Planck constant \(h\) have had the following exact numerical values:

These defined values carry no measurement uncertainty.

Meanwhile, the fine-structure constant \(\alpha\) and vacuum permeability \(\mu_0\) satisfy the following relationship:

Therefore,

Since \(c\), \(e\), and \(h\) are all exact by definition, the numerical value of \(\mu_0\) is determined by measurements of the fine-structure constant \(\alpha\).

Thus, if more precise measurements lead to a revision of the recommended value of the fine-structure constant, the recommended value of vacuum permeability will also be revised.

Historically, the present role of \(\mu_0\) in SI originates in electromagnetic unit systems developed during the nineteenth century. In the early twentieth century, a system incorporating a fourth electrical base unit was proposed to unify the MKS system of Newtonian mechanics (metre, kilogram, second) with practical electrical units. Following a 1946 decision by the International Committee for Weights and Measures (CIPM), an exact value for vacuum permeability based on the definition of the ampere was adopted in 1948.

According to Maxwell's equations, the speed of light in vacuum \(c\), vacuum permittivity \(\varepsilon_0\), and vacuum permeability \(\mu_0\) are related by

Consequently, vacuum permittivity can be expressed as

Where do the factors \(4\pi\) and \(10^{-7}\) in vacuum permeability come from?

The factor \(4\pi\) originates in the geometry of a sphere in three-dimensional Euclidean space. A sphere of radius \(r\) has a surface area of \(4\pi r^2\), and its total solid angle is \(4\pi\) steradians. This geometrical factor appears in descriptions of radial electric fields from point charges and other inverse-square fields from point sources.

By contrast, the factor \(10^{-7}\) is not a fundamental constant of nature. It reflects a historical choice of scale made to reconcile mechanical and practical electrical units. It arose through the conversion between historical CGS electromagnetic units and the MKS system, together with the adoption of the ampere as a practical unit of electric current.

Thus, the former value

was established through a historical choice of units incorporating the geometric factor \(4\pi\) and the unit-conversion factor \(10^{-7}\).

Changing a previously exact defined value of \(\mu_0\) into a quantity revised on the basis of measurements can feel almost as if the mathematical constant \(\pi\) itself were being redefined by experiment. Of course, \(\pi\) does not actually change. But why should a value originally specified using a geometrical factor and a unit-conversion factor now be treated as an experimentally determined quantity?

The Space-Time Unit System (ST unit system) is an attempt to reconsider the historical origins and roles of vacuum permeability \(\mu_0\), vacuum permittivity \(\varepsilon_0\), and the gravitational constant \(G\), and to construct a more coherent unit system without contradicting established physical laws (Ref).

In the ST unit system, neither the kilogram nor the coulomb is a base unit; both are treated as derived units based on space and time. To avoid confusion with the physical constants used in SI, the corresponding constants in the ST unit system are distinguished by the subscript \(0\).

In ST natural units, the reference length is the classical electron radius, and the reference time is the time required for light to travel a distance equal to the classical electron radius. Under this normalization, the numerical values of the speed of light, elementary charge, and electron rest mass are all unity, while the reduced Planck constant \(\hbar\) corresponds to the reciprocal of the fine-structure constant. Furthermore, ST natural units use space (S) and time (T) as their fundamental units instead of the SI metre (m) and second (s).

In ST natural units, vacuum permeability is represented by the dimensionless geometrical constant \(4\pi\), while vacuum permittivity is expressed as \(1/(4\pi c_0^2)\), determined by the geometrical factor and the speed of light. Since these quantities are not treated as independent physical constants, no separate physical constant symbols are assigned to them in ST natural units. By contrast, the gravitational constant is represented by an extremely small dimensionless interaction constant, \(\beta_0\).

Thus, in the ST unit system, the fine-structure constant and the ST gravitational constant are treated as two dimensionless interaction constants determined by measurement, while the reduced Planck constant is derived from the measured fine-structure constant.

The current SI fixes the speed of light, elementary charge, and Planck constant \(h\) as exact defined values. By contrast, the ST unit system takes the speed of light, elementary charge, and electron rest mass as its reference quantities and determines the reduced Planck constant \(\hbar\) from measurements.

Accordingly, in the ST unit system, there is no need to revise vacuum permeability in response to experimental results. Furthermore, because the mass reference is the electron rest mass, the system does not depend on the International Prototype of the Kilogram.

The fundamental interaction constants updated from measurements in this framework are the two dimensionless quantities—the fine-structure constant and the ST gravitational constant—and the value of the reduced Planck constant \(\hbar\) follows from the measured fine-structure constant.