physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, scalars (or scalar quantities) are physical quantities that are unaffected by changes to a vector space basis (i.e., a coordinate system transformation). Scalars are often accompanied by

units of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a mul ...

, as in "10 cm". Examples of scalar quantities are

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...

, distance, charge,

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...

speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quantity ...

, and the magnitude of

physical vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...

s in general (such as

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

). A change of a vector space basis changes the description of a vector in terms of the basis used but does not change the vector itself, while a scalar has nothing to do with this change. In classical physics, like Newtonian mechanics, rotations and reflections preserve scalars, while in relativity, Lorentz transformations or space-time translations preserve scalars. The term "scalar" has origin in the multiplication of vectors by a unitless scalar, which is a '' uniform scaling'' transformation.

Relationship with the mathematical concept

A scalar in physics is also a scalar in mathematics, as an element of a mathematical field used to define a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

. For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field for the vector space in which the electric field is described. As the vector space in this example and usual cases in physics is defined over the mathematical field of

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s or

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, the magnitude is also an element of the field, so it is mathematically a scalar. Since the inner product is independent of any vector space basis, the electric field magnitude is also physically a scalar. The mass of an object is unaffected by a change of vector space basis so it is also a physical scalar, described by a real number as an element of the real number field. Since a field is a vector space with addition defined based on vector addition and multiplication defined as scalar multiplication, the mass is also a mathematical scalar.

Scalar field

Since scalars mostly may be treated as special cases of multi-dimensional quantities such as vectors and

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...

s, ''physical scalar fields'' might be regarded as a special case of more general fields, like vector fields, spinor fields, and tensor fields.

Units

Like other physical quantities, a physical quantity of scalar is also typically expressed by a numerical value and a physical unit, not merely a number, to provide its physical meaning. It may be regarded as the product of the number and the unit (e.g., 1 km as a physical distance is the same as 1,000 m). A physical distance does not depend on the length of each base vector of the coordinate system where the base vector length corresponds to the physical distance unit in use. (E.g., 1 m base vector length means the meter unit is used.) A physical distance differs from a metric in the sense that it is not just a real number while the metric is calculated to a real number, but the metric can be converted to the physical distance by converting each base vector length to the corresponding physical unit. Any change of a coordinate system may affect the formula for computing scalars (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalars themselves. Vectors themselves also do not change by a change of a coordinate system, but their descriptions changes (e.g., a change of numbers representing a position vector by rotating a coordinate system in use).

Classical scalars

An example of a scalar quantity is

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...

: the temperature at a given point is a single number. Velocity, on the other hand, is a vector quantity. Other examples of scalar quantities in physics are

, charge,

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

, and electric potential at a point inside a medium. The distance between two points in three-dimensional space is a scalar, but the direction from one of those points to the other is not, since describing a direction requires two physical quantities such as the angle on the horizontal plane and the angle away from that plane.

Force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...

cannot be described using a scalar, since force has both direction and magnitude; however, the magnitude of a force alone can be described with a scalar, for instance the gravitational force acting on a particle is not a scalar, but its magnitude is. The speed of an object is a scalar (e.g., 180 km/h), while its

is not (e.g. 108 km/h northward and 144 km/h westward). Some other examples of scalar quantities in Newtonian mechanics are electric charge and

charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...

Relativistic scalars

In the theory of relativity, one considers changes of coordinate systems that trade space for time. As a consequence, several physical quantities that are scalars in "classical" (non-relativistic) physics need to be combined with other quantities and treated as

four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as ...

s or tensors. For example, the

at a point in a medium, which is a scalar in classical physics, must be combined with the local

current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...

(a 3-vector) to comprise a relativistic 4-vector. Similarly, energy density must be combined with momentum density and

into the stress–energy tensor. Examples of scalar quantities in relativity include electric charge, spacetime interval (e.g., proper time and proper length), and

invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...

Pseudoscalar

Notes

References

* Feynman, Leighton & Sands 1963. * * {{DEFAULTSORT:Scalar (Physics)