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 r ...

, scalars (or scalar quantities) are

physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...

that are unaffected by changes to a

vector space basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...

(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 multi ...

, 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 eleme ...

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...

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, ...

speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...

, and the magnitude of physical vectors 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 transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...

s 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 In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similar ...

transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Tran ...

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 can ...

. For example, the magnitude (or length) of an electric field vector is calculated as the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

of its

absolute square In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the squar ...

(the

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

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 fo ...

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 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 a ...

and multiplication defined as

scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...

, 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 tensor ...

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

spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\col ...

s, and

tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

Units

Like other

, a physical

quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...

of scalar is also typically expressed by a

numerical value A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

and a

physical unit 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 multi ...

, not merely a number, to provide its physical meaning. It may be regarded as the

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

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 In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...

), 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 measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...

: 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

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 and e ...

, and

electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...

at a point inside a medium. The

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 cannot be described using a scalar, since force has both direction and

magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...

; 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 Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...

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 The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

, 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 a ...

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 (a 3-vector) to comprise a relativistic

4-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 a ...

. Similarly, energy density must be combined with momentum density and

into the stress–energy tensor. Examples of scalar quantities in relativity include

spacetime interval In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

(e.g.,

proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...

and

proper length Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on ...

Relationship with the mathematical concept

Scalar field

Units

Classical scalars

Relativistic scalars

Pseudoscalar

See also

Notes

References