Scalar (physics)

Encyclopedia

In physics

, a

that is not changed by coordinate system

rotations or translations (in Newtonian mechanics), or by Lorentz transformation

s or space-time translations (in relativity). This is in contrast to a vector. A related concept is a

) flips sign under improper rotation

s.

is expressed as the product

of a numerical value

and a physical unit, not merely a number. The quantity does not depend on the unit (i.e for distance, 1 km is the same as 1000 m), although the number depends on the unit. Thus, following the example of distance, the quantity does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric

in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless.

, charge

, or the temperature

, or 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

cannot be described using a scalar, since force is composed of direction and magnitude

, however, the magnitude of a force alone can be described with a scalar, for instance the gravitation

al 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 velocity

is not (i.e. 180 km/h

Examples of scalar quantities in Newtonian mechanics:

An example of a pseudoscalar is the scalar triple product

(see vector), and thus the signed volume. Another example is magnetic charge (as it is mathematically defined, regardless of whether it actually exists physically).

, 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-dimensional vectors or tensors. For example, the charge density

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. Similarly, energy density

must be combined with momentum density and pressure

into the stress-energy tensor

.

Examples of scalar quantities in relativity:

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a

**scalar**is a simple physical quantityPhysical quantity

A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...

that is not changed by coordinate system

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

rotations or translations (in Newtonian mechanics), or by Lorentz transformation

Lorentz transformation

In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

s or space-time translations (in relativity). This is in contrast to a vector. A related concept is a

**pseudoscalar**, which is invariant under proper rotations but (like a pseudovectorPseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

) flips sign under improper rotation

Improper rotation

In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...

s.

## Physical quantity

A physical quantityQuantity

Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

is expressed as the product

Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...

of a numerical value

Number

A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

and a physical unit, not merely a number. The quantity does not depend on the unit (i.e for distance, 1 km is the same as 1000 m), although the number depends on the unit. Thus, following the example of distance, the quantity does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless.

## Examples

Some examples of scalars include the massMass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

, charge

Charge (physics)

In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...

, or the temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, or electric potential

Electric potential

In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

at a point inside a medium. The distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

between two points in three-dimensional space is a scalar, but the direction

Direction (geometry, geography)

Direction is the information contained in the relative position of one point with respect to another point without the distance information. Directions may be either relative to some indicated reference , or absolute according to some previously agreed upon frame of reference Direction is the...

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

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

cannot be described using a scalar, since force is composed of direction and magnitude

Magnitude (mathematics)

The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

, however, the magnitude of a force alone can be described with a scalar, for instance the gravitation

Gravitation

Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

al force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

acting on a particle is not a scalar, but its magnitude is. The speed

Speed

In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

of an object is a scalar (e.g. 180 km/h), while its velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

is not (i.e. 180 km/h

*north*).Examples of scalar quantities in Newtonian mechanics:

- electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

and charge densityCharge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

An example of a pseudoscalar is the scalar triple product

Triple product

In mathematics, the triple product is a product of three vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product....

(see vector), and thus the signed volume. Another example is magnetic charge (as it is mathematically defined, regardless of whether it actually exists physically).

## Scalars in relativity theory

In the theory of relativityTheory of relativity

The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

, 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

Classical physics

What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

need to be combined with other quantities and treated as four-dimensional vectors or tensors. For example, the charge density

Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

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

Current density

Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

(a 3-vector) to comprise a relativistic 4-vector. Similarly, energy density

Energy density

Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

must be combined with momentum density and pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

into the stress-energy tensor

Stress-energy tensor

The stressâ€“energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

.

Examples of scalar quantities in relativity:

- electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
- spacetime interval (e.g., proper timeProper timeIn relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

and proper lengthProper lengthIn relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....

) - invariant massInvariant massThe invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...

## See also

- Scalar fieldScalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
- Scalar field theoryScalar field theoryIn theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
- Scalar (mathematics)Scalar (mathematics)In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
- Relative scalar
- Euclidean vector