Dimension (physics)

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measurement (such as meters and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.

Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., meters and feet, gallons and liters, seconds and years. Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.

Any physically meaningful equation, or inequality, must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.

The concept of physical dimension, and of dimensional analysis, was introduced by Joseph Fourier in 1822.^[1]^: 42

Formulation

The Buckingham π theorem describes how every physically meaningful equation involving $n$ variables can be equivalently rewritten as an equation of $n - m$ dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or physical constants of nature.^[1]^: 43 This may give insight into the fundamental properties of the system, as illustrated in the examples below.

The dimension of a physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally rational) power. The dimension of a physical quantity is more fundamental than some scale or unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. Natural units, being based on only universal constants, may be thought of as being "less arbitrary".

There are many possible choices of base physical dimensions. The SI standard selects the following dimensions and corresponding dimension symbols:

time (T), length (L), mass (M), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J).

The symbols are by convention usually written in roman sans serif typeface.^[2] Mathematically, the dimension of the quantity $Q$ is given by

\operatorname {dim} Q={\mathsf {T}}^{a}{\mathsf {L}}^{b}{\mathsf {M}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}

where $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a linearly independent basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge, since Q = TI.

A quantity that has only $b \neq 0$ (with all other exponents zero) is known as a geometric quantity. A quantity that has only both $a \neq 0$ and $b \neq 0$ is known as a kinematic quantity. A quantity that has only all of $a \neq 0$ , $b \neq 0$ , and $c \neq 0$ is known as a dynamic quantity.^[3] A quantity that has all exponents null is said to have dimension one.^[2]

The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.

There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,^[4] although this does not invalidate the usefulness of dimensional analysis.