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In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a shorter proper time between two events than a non-accelerated (inertial) clock between the same events. The twins paradox is an example of this. Two-dimensional analogy of space-time distortion described in General Relativity. ...
A pocket watch, a device used to keep time There are two distinct views on the meaning of time. ...
A clock (from the Latin cloca, bell) is an instrument for measuring time. ...
In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...
This article or section is in need of attention from an expert on the subject. ...
In contrast, coordinate time can be applied to events that occur a distance from an observer. In special relativity, coordinate time is reckoned relative only to inertial observers, whereas proper time can be measured by accelerated observers too. Coordinate time is the interval of time independent of relativistic time dilation. ...
The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...
In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space. In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ...
For other uses, see Curve (disambiguation). ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
By convention, proper time is usually represented by the Greek letter τ to distinguish it from coordinate time represented by t or T.
Mathematical Formalism
The formal definition of proper time involves describing the path through spacetime that repesents a clock, observer, or test particle, and the metric structure of that spacetime. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...
In Special Relativity In special relativity, proper time can be defined as
,
where v(t) is the coordinate speed at coordinate time t, and x, y and z are orthogonal spatial coordinates. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
Space has been an interest for philosophers and scientists for much of human history. ...
If t, x, y and z are all parameterised by a parameter λ, this can be written as
.
In differential form it can be written as the path integral
,
where P is the path of the clock in spacetime.
To make things even easier, inertial motion 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 In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...
This article or section is in need of attention from an expert on the subject. ...
,
where Δ means "the change in" between two events. In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ...
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 as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system xμ and equipped with a corresponding metric tensor gμν, the proper time experienced in moving between two events along a timelike path P is given by the line integral For more technical Wiki articles on tensors, see the section later in this article. ...
In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...
In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...
In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...
In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ...
This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...
where
Derivation For any spacetime, there is an incremental invariant interval ds between events with an incremental coordinate separation dxμ 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 . In the context of special relativity, space-like separated points (or events) in spacetime have a spacetime interval less than 0 (see sign convention). ...
In physics, the adjective light-like refers to a contour in spacetime in the context of special relativity whose proper length vanishes. ...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
Taking the square root of each side of the line element gives the above definition of . After that, take the path integral of each side to get as described by the first equation.
Derivation for Special Relativity In special relativity spacetime is mapped with a four-vector coordinate system xμ = (t,x,y,z) where The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...
In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ...
- t is a temporal coordinate and
- x,y, and z are orthogonal spatial coordinates.
This spacetime and mapping are described with the Minkowski metric: A pocket watch, a device used to keep time There are two distinct views on the meaning of time. ...
In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
Space has been an interest for philosophers and scientists for much of human history. ...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
(Note: The +--- metric signature is used in this article so that will always be positive definite for timelike paths.) The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...
In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ...
In special relativity, the proper time equation becomes
,
as above.
Examples in Special Relativity
Example 1: The twin "paradox" For a twin "paradox" 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 x = y = z = 0 for 10 years of coordinate time. The proper time for A is then This article or section is in need of attention from an expert on the subject. ...
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 light-years, 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 light-years, 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 ΔT, 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 ω 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 T is the current coordinate time. When r and ω 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 T1 and T2, 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 equations, not just the Minkowski metric. Because inertial motion in curved spacetimes lacks the simple expression it has in SR, the path integral form of the proper time equation must always be used. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...
This article or section is in need of attention from an expert on the subject. ...
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 θ = arctan(x / y) − ωt. 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 at-rest object of counter-rotating in the view of the rotating observer. Now the proper time equation becomes
.
So for the inertial at-rest observer, coordinate time and proper time are once again found to pass at the same rate, as expected and required for the internal self-consistency of relativity theory.
Example 4: The Schwarzschild solution - Time on Planet Earth The Schwarzschild solution has an incremental proper time equation of Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ...
,
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 in radians.
- is a longitudinal coordinate, analogous to the latitude on the Earth's surface but independent of the Earth's rotation. This is also given in radians.
- m is the geometrized mass of a central massive object, being m=MG/c2,
- M is the mass of the object,
- G is the gravitational constant.
To demonstrate the use of the proper time relationship, several sub-examples involving the Earth will be used here. The use of the Schwarzschild solution for the Earth is not entirely correct for the following reasons: North Pole Scenery When not otherwise qualified, the term North Pole usually refers to the Geographic North Pole – the northernmost point on the surface of the Earth, where the Earths axis of rotation intersects the Earths surface. ...
Some common angles, measured in radians. ...
A sphere rotating around its axis. ...
According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...
- Due to its rotation, the Earth is an oblate spheroid instead of being a true sphere. 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 Earth, kg, meaning that 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 Oblate also refers to a member of the Roman Catholic religious order of the Missionary Oblates of Mary Immaculate, or in some cases to a lay or religious person who has officially associated himself (or herself) with a monastic community such as the Benedictines for reasons of personal enrichment without...
A sphere is a perfectly symmetrical geometrical object. ...
In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ...
Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ...
.
At the equator, the radius of the Earth is meters. In addition, the rotation of the Earth needs to be taken into account. This imparts on an observer an angular velocity of of divided by the sidereal period of the Earth's rotation, seconds. So . The proper time equation then produces World map showing the equator in red For other uses, see Equator (disambiguation). ...
Sidereal time is time measured by the apparent diurnal motion of the vernal equinox, which is very close to, but not identical with, the motion of stars. ...
.
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.
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