In
relativityThe 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....
,
proper time is the elapsed
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
between two
eventsIn physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...
as measured by a
clockA clock is an instrument used to indicate, keep, and coordinate time. The word clock is derived ultimately from the Celtic words clagan and clocca meaning "bell". A silent instrument missing such a mechanism has traditionally been known as a timepiece...
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 events than that measured by a nonaccelerated (inertial) clock between the same two events. The
twin paradoxIn physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a highspeed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...
is an example of this effect.
In terms of fourdimensional
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, proper time is analogous to
arc lengthDetermining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
in threedimensional (
EuclideanIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
) space.
By convention, proper time is usually represented by the Greek letter τ (
tauTau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.The name in English is pronounced , but in modern Greek it is...
) to distinguish it from coordinate time represented by
t or
T.
By contrast,
coordinate timeIn the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many coordinate systems, an event is specified by one time coordinate and three spatial coordinates...
is the time between two events as measured by a distant observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, the time is measured using the observer's clock and the observer's definition of simultaneity.
A Euclidean geometrical analogy is that coordinate time is like distance measured with a straight vertical
rulerA ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
, whereas proper time is like distance measured with a
tape measureA tape measure or measuring tape is a flexible form of ruler. It consists of a ribbon of cloth, plastic, fiber glass, or metal strip with linearmeasurement markings. It is a common measuring tool. Its flexibility allows for a measure of great length to be easily carried in pocket or toolkit and...
. If the tape measure is taut and vertical it measures the same as the ruler, but if the tape measure is not taut, or taut but not vertical, it will not measure the same as the ruler.
The concept of proper time was introduced by
Hermann MinkowskiHermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity. Life and work :Hermann Minkowski was born...
in 1908.
Mathematical formalism
The formal definition of proper time involves describing the path through
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
that represents a clock, observer, or test particle, and the
metric structureIn general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
of that spacetime. Proper time is the
pseudoRiemannianIn differential geometry, a pseudoRiemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudoRiemannian manifold is that on a pseudoRiemannian manifold the...
arc length of
world lineIn physics, the world line of an object is the unique path of that object as it travels through 4dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
s in fourdimensional spacetime.
From the mathematical point of view, coordinate time is assumed to be predefined and we require an expression for proper time as a function of coordinate time. From the experimental point of view, proper time is what is measured experimentally and then coordinate time is calculated from the proper time of some inertial clocks.
\int \sqrt {1  \frac{1}{c^2} \left [ \left (\frac{dx}{dt}\right)^2 + \left (\frac{dy}{dt}\right)^2 + \left ( \frac{dz}{dt}\right)^2 \right] } \,dt,
where is the coordinate speed at coordinate time , and , , and are Cartesian
spatialSpace is the boundless, threedimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless fourdimensional continuum...
coordinates.
If , , , and are all parameterised by a
parameterParameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
, this can be written as

In differential form it can be written as the line integral

where is the path of the clock in spacetime.
To make things even easier, inertial
motionIn physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...
in special relativity is where the spatial coordinates change at a constant rate with respect to the temporal coordinate. This further simplifies the proper time equation to

where Δ means "the change in" between two events.
The special relativity equations are special cases of the general case that follows.
In general relativity
Using tensor calculus, proper time is more rigorously defined in
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
as follows: Given a spacetime which is a
pseudoRiemannian manifoldIn differential geometry, a pseudoRiemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudoRiemannian manifold is that on a pseudoRiemannian manifold the...
mapped with a
coordinate systemIn 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...
and equipped with a corresponding
metric tensorIn general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
, the proper time
experienced in moving between two events along a timelike path
P is given by the
line integralIn mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

where

(Note: the Einstein summation convention is used in the above. The expression
A_{μ}B^{μ} is shorthand for , and the
μ in
B^{μ} denotes an index, not a power.)
Derivation
For any spacetime, there is an incremental invariant interval
ds between events with an incremental coordinate separation
dx^{μ} of

This is referred to as the line element of the spacetime.
s may be spacelike, lightlike, or timelike. Spacelike paths cannot be physically traveled (as they require moving faster than light). Lightlike paths can only be followed by light beams, for which there is no passage of proper time. Only timelike paths can be traveled by massive objects, in which case the invariant interval becomes the proper time
. So for our purposes
.
Taking the square root of each side of the line element gives the above definition of
. After that, take the line integral of each side to get
as described by the first equation.
Derivation for special relativity
In
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
spacetime is mapped with a
fourvectorIn the theory of relativity, a fourvector is a vector in a fourdimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the fourvector name tacitly assumes that its components refer to a standard basis...
coordinate system
where
 t is a temporal
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
coordinate and
 x, y, and z are orthogonal spatial
Space is the boundless, threedimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless fourdimensional continuum...
coordinates.
This spacetime and mapping are described with the Minkowski metric:

(Note: The + metric signatureThe signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
is used in this article so that will always be positivePositive is a property of positivity and may refer to: Mathematics and science :* Converging lens or positive lens, in optics* Plus sign, the sign "+" used to indicate a positive number* Positive , a polarity of electrical charge...
definite for timelike paths.)
In special relativity, the proper time equation becomes

as above.
Example 1: The twin "paradox"
For a twin "paradox"In physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a highspeed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...
scenario, let there be an observer A who moves between the coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that A stays at for 10 years of coordinate time. The proper time for A is then

So we find that being "at rest" in a special relativity coordinate system means that proper time and coordinate time are the same.
Let there now be another observer B who travels in the x direction from (0,0,0,0) for 5 years of coordinate time at 0.866c to (5 years, 4.33 lightyears, 0, 0). Once there, B accelerates, and travels in the other spatial direction for 5 years to (10 years, 0, 0, 0). For each leg of the trip, the proper time is

So the total proper time for observer B to go from (0,0,0,0) to (5 years, 4.33 lightyears, 0, 0) to (10 years, 0, 0, 0) is 5 years. Thus it is shown that the proper time equation incorporates the time dilation effect. In fact, for an object in a SR spacetime traveling with a velocity of v for a time , the proper time experienced is

which is the SR time dilation formula.
Example 2: The rotating disk
An observer rotating around another inertial observer is in an accelerated frame of reference. For such an observer, the incremental () form of the proper time equation is needed, along with a parameterized description of the path being taken, as shown below.
Let there be an observer C on a disk rotating in the xy plane at a coordinate angular rate of and who is at a distance of r from the center of the disk with the center of the disk at x=y=z=0. The path of observer C is given by , where is the current coordinate time. When r and are constant, and . The incremental proper time formula then becomes

So for an observer rotating at a constant distance of r from a given point in spacetime at a constant angular rate of ω between coordinate times and , the proper time experienced will be

As v=rω for a rotating observer, this result is as expected given the time dilation formula above, and shows the general application of the integral form of the proper time formula.
Examples in general relativity
The difference between SR and general relativity (GR) is that in GR you can use any metric which is a solution of the Einstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
, not just the Minkowski metric. Because inertial motion in curved spacetimes lacks the simple expression it has in SR, the line integral form of the proper time equation must always be used.
Example 3: The rotating disk (again)
An appropriate coordinate conversion done against the Minkowski metric creates coordinates where an object on a rotating disk stays in the same spatial coordinate position. The new coordinates are

and

The t and z coordinates remain unchanged. In this new coordinate system, the incremental proper time equation is

With r, θ, and z being constant over time, this simplifies to

which is the same as in Example 2.
Now let there be an object off of the rotating disk and at inertial rest with respect to the center of the disk and at a distance of R from it. This object has a coordinate motion described by dθ = ω dt, which describes the inertially atrest object of counterrotating in the view of the rotating observer. Now the proper time equation becomes

So for the inertial atrest observer, coordinate time and proper time are once again found to pass at the same rate, as expected and required for the internal selfconsistency of relativity theory.
Example 4: The Schwarzschild solution — time on the Earth
The Schwarzschild solution has an incremental proper time equation of

where
 t is time as calibrated with a clock distant from and at inertial rest with respect to the Earth,
 r is a radial coordinate (which is effectively the distance from the Earth's center),
 θ is the latitudinal coordinate, being the angular separation from the north pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...
in radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s.
 is a longitudinal coordinate, analogous to the latitude on the Earth's surface but independent of the Earth's rotation
A rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
. This is also given in radians.
 m is the geometrized mass of a central massive object, being m = MG/c^{2},
 M is the mass of the object,
 G is the gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
.
To demonstrate the use of the proper time relationship, several subexamples involving the Earth will be used here. The use of the Schwarzschild solution for the Earth is not entirely correct for the following reasons:
 Due to its rotation, the Earth is an oblate spheroid instead of being a true sphere
A sphere is a perfectly round geometrical object in threedimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. This results in the gravitational field also being oblate instead of spherical.
 In GR, a rotating object also drags spacetime along with itself. This is described by the Kerr solution. However, the amount of frame dragging that occurs for the Earth is so small that it can be ignored.
For the EarthEarth is the third planet from the Sun, and the densest and fifthlargest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
, M = 5.9742 × 10^{24} kg, meaning that m = 4.4354 × 10^{ −3} m. When standing on the north pole, we can assume (meaning that we are neither moving up or down or along the surface of the Earth). In this case, the Schwarzschild solution proper time equation becomes . Then using the polar radius of the Earth as the radial coordinate (or meters), we find that

At the equatorAn equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
, the radius of the Earth is r = 6,378,137 meters. In addition, the rotation of the Earth needs to be taken into account. This imparts on an observer an angular velocity of of 2π divided by the sidereal periodSidereal time is a timekeeping system astronomers use to keep track of the direction to point their telescopes to view a given star in the night sky...
of the Earth's rotation, 86162.4 seconds. So . The proper time equation then produces

This should have been the same as the previous result, but as noted above the Earth is not spherical as assumed by the Schwarzschild solution. Even so this demonstrates how the proper time equation is used.
See also
 Lorentz transformation
In physics, the Lorentz transformation or LorentzFitzgerald 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...
 Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
 Proper length
In relativistic physics, proper length is an invariant measure of the distance between two spacelikeseparated events, or of the length of a spacelike path within a spacetime....
 Proper acceleration
In relativity theory, proper acceleration is the physical acceleration experienced by an object. It is acceleration relative to a freefall, or inertial, observer who is momentarily at rest relative to the object being measured...
 Proper mass
The 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...
 Proper velocity
In relativity, propervelocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observermeasured...
 Clock hypothesis
The clock hypothesis is an assumption in special relativity. It states that the rate of a clock doesn't depend on its acceleration but only on its instantaneous velocity...
Footnotes