# Proper time

In relativity, proper 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 a Lorentz scalar.[1] The proper time interval between two events on a world line is the change in proper time. This interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. The proper time interval between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect.[2]

The dark blue vertical line represents an inertial observer measuring a coordinate time interval t between events E1 and E2. The red curve represents a clock measuring its proper time interval τ between the same two events.

In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space. By convention, proper time is usually represented by the Greek letter τ (tau) to distinguish it from coordinate time represented by t.

By contrast, coordinate time is the time between two events as measured by an observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in special relativity, the time is measured using the observer's clock and the observer's definition of simultaneity.

The concept of proper time was introduced by Hermann Minkowski in 1908,[3] and is a feature of Minkowski diagrams.

## Mathematical formalism

The formal definition of proper time involves describing the path through spacetime that represents a clock, observer, or test particle, and the metric structure of that spacetime. Proper time is the pseudo-Riemannian arc length of world lines in four-dimensional 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.

Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks. The same formalism for spacelike paths leads to a measurement of proper distance rather than proper time. For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is identically zero. Instead an arbitrary and physically irrelevant affine parameter unrelated to time must be introduced.[4][5][6][7][8][9]

### In special relativity

Let the Minkowski metric be defined by

and define

for arbitrary Lorentz frames.

Consider an infinitesimal interval

expressed in any Lorentz frame and here assumed timelike, separating points on a trajectory of a particle (think clock). The same interval can be expressed in coordinates such that at each moment, the particle is at rest. Such a frame is called an instantaneous rest frame, denoted here by the coordinates ${\displaystyle (c\tau ,x_{\tau },y_{\tau },z_{\tau })}$ for each instants. Due to the invariance of the interval (instantaneous rest frames taken at different times are related by Lorentz transformations) one may write

since in the instantaneous rest frame, the particle or the frame itself is at rest, i.e., ${\displaystyle dx_{\tau }=dy_{\tau }=dz_{\tau }=0}$. Since the interval is assumed timelike, one may take the square root of the above expression;[10]

or

Given this differential expression for τ, the proper time interval is defined as

Here P is the worldline from some initial event to some final event with the ordering of the events fixed by the requirement that the final event occurs later according to the clock than the initial event.

Using (1) and again the invariance of the interval, one may write[11]

where v(t) is the coordinate speed at coordinate time t, and x(t), y(t), and z(t) are space coordinates. The first expression is manifestly Lorentz invariant. They are all Lorentz invariant, since proper time and proper time intervals are coordinate-independent by definition.

If t, x, y, z, are parameterised by a parameter λ, this can be written as

If the motion of the particle is constant, the expression simplifies to

where Δ means the change in coordinates between the initial and final events. The definition in special relativity generalizes straightforwardly to general relativity as follows below.

### In general relativity

Proper time is defined in general relativity as follows: Given a pseudo-Riemannian manifold with a local coordinates xμ and equipped with a metric tensor gμν, the proper time interval Δτ between two events along a timelike path P is given by the line integral[12]

${\displaystyle \Delta \tau =\int _{P}\,d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{\mu \nu }\;dx^{\mu }\;dx^{\nu }}}.}$

(4)

This expression is, as it should be, invariant under coordinate changes. It reduces (in appropriate coordinates) to the expression of special relativity in flat spacetime.

In the same way that coordinates can be chosen such that x1, x2, x3 = const in special relativity, this can be done in general relativity too. Then, in these coordinates,[13]

${\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{00}}}dx^{0}.}$

This expression generalizes definition (2) and can be taken as the definition. Then using invariance of the interval, equation (4) follows from it in the same way (3) follows from (2), except that here arbitrary coordinate changes are allowed.

## Examples in special relativity

### Example 1: The twin "paradox"

For a twin "paradox" scenario, let there be an observer A who moves between the A-coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that A stays at ${\displaystyle x=y=z=0}$ for 10 years of A-coordinate time. The proper time interval for A between the two events is then

So 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 A-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 another 5 years of A-coordinate time to (10 years, 0, 0, 0). For each leg of the trip, the proper time interval can be calculated using A-coordinates, and is given by

So the total proper time for observer B to go from (0,0,0,0) to (5 years, 4.33 light-years, 0, 0) and then 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 ${\displaystyle \Delta T}$, the proper time interval 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 (${\displaystyle d\tau }$) 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 ${\displaystyle \omega }$ 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 ${\displaystyle (T,\;\,r\cos(\omega T),\;\,r\sin(\omega T),\;\,0)}$, where ${\displaystyle T}$ is the current coordinate time. When r and ${\displaystyle \omega }$ are constant, ${\displaystyle dx=-r\omega \sin(\omega T)\;dT}$ and ${\displaystyle dy=r\omega \cos(\omega T)\;dT}$. 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 ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$, the proper time experienced will be

as v= for a rotating observer. This result is the same as for the linear motion example, 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 one 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 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 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.[14]

### 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),
• ɸ is a co-latitudinal coordinate, the angular separation from the north pole in radians.
• θ is a longitudinal coordinate, analogous to the longitude on the Earth's surface but independent of the Earth's rotation. This is also given in radians.
• 1=m is the geometrized mass of the Earth, m = GM/c2,

To demonstrate the use of the proper time relationship, several sub-examples involving the Earth will be used here.

For the Earth, M = 5.9742 × 1024 kg, meaning that m = 4.4354 × 10−3 m. When standing on the north pole, we can assume ${\displaystyle dr=d\theta =d\phi =0}$ (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 ${\displaystyle d\tau =dt\,{\sqrt {1-2m/r}}}$. Then using the polar radius of the Earth as the radial coordinate (or ${\displaystyle r=6,356,752}$ meters), we find that

At the equator, 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 ${\displaystyle \ d\theta /dt}$ of 2π divided by the sidereal period of the Earth's rotation, 86162.4 seconds. So ${\displaystyle d\theta =7.2923\times 10^{-5}\,dt}$. The proper time equation then produces

From a non-relativistic point of view this should have been the same as the previous result. This example demonstrates how the proper time equation is used, even though the Earth rotates and hence is not spherically symmetric as assumed by the Schwarzschild solution. To describe the effects of rotation more accurately the Kerr metric may be used.

## Footnotes

1. ^ Zwiebach 2004, p. 25
2. ^ Hawley, John F.; Holcomb, J Katherine A. (2005). Foundations of Modern Cosmology (illustrated ed.). Oxford University Press. p. 204. ISBN 978-0-19-853096-1. Extract of page 204
3. ^ Minkowski 1908, pp. 53–111
4. ^ Lovelock & Rund 1989, pp. 256
5. ^ Weinberg 1972, pp. 76
6. ^ Poisson 2004, pp. 7
7. ^ Landau & Lifshitz 1975, p. 245
8. ^ Some authors include lightlike intervals in the definition of proper time, and also include the spacelike proper distances as imaginary proper times e.g Lawden 2012, pp. 17, 116
9. ^
10. ^ Zwiebach 2004, p. 25
11. ^ Foster & Nightingale 1978, p. 56
12. ^ Foster & Nightingale 1978, p. 57
13. ^ Landau & Lifshitz 1975, p. 251
14. ^ Cook 2004, pp. 214–219

## References

Barycentric Coordinate Time

Barycentric Coordinate Time (TCB, from the French Temps-coordonnée barycentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to orbits of planets, asteroids, comets, and interplanetary spacecraft in the Solar system. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the barycenter of the Solar system: that is, a clock that performs exactly the same movements as the Solar system but is outside the system's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Sun and the rest of the system.

TCB was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the problematic 1976 definition of Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCB is defined in the context of the general theory of relativity. The relationships between TCB and other relativistic time scales are defined with fully general relativistic metrics.

Because the reference frame for TCB is not influenced by the gravitational potential caused by the Solar system, TCB ticks faster than clocks on the surface of the Earth by 1.550505 × 10−8 (about 490 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCB differ from the traditional values of physical constants (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales). Adapting the large body of existing software to change from TDB to TCB is an ongoing task, and as of 2002 many calculations continue to use TDB in some form.

Time coordinates on the TCB scale are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCB was set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TCB instant 1977-01-01T00:00:32.184 exactly corresponds to the International Atomic Time (TAI) instant 1977-01-01T00:00:00.000 exactly, at the geocenter. This is also the instant at which TAI introduced corrections for gravitational time dilation.

Cayuga White

Cayuga White is a mid-season ripening wine grape developed from crosses of the hybrids Schuyler and Seyval Blanc at Cornell University's New York State Agricultural Experiment Station in Geneva, New York. It is a hardy vine with some bunch-rot disease resistance. In warmer climates it should be picked at lower sugars to avoid overripe, sometimes labrusca-like, flavors; however this has not been observed in cooler climates such as the Pacific Northwest, where desirable, Riesling-type flavors are tasted in fully ripe Cayuga fruit. Picked at the proper time, it can produce a very nice sparkling wine with good acid balance, structure, and pleasant aromas, or a fruity white wine similar to a Riesling or Viognier. One advantage of Cayuga is that, if harvested unripe (e.g., in a shorter summer in cool climates), it can still make a good wine, albeit one with more green apple flavors in that case.

This grape, when grown on mature vines in fertile soil, can produce astonishing yields. If allowed to set fully on thirty-year-old vines in Aurora, Oregon, Cayuga's yield has been measured at over 13 tons per acre, though in that case a "green harvest" (removing much of the fruit before the final phase of the ripening cycle) is advised, so that the vine can more fully ripen the remaining fruit.

Cayuga is relatively easy to make wine from. In cooler climates, it retains enough acid that a residual sugar level is advised, in order to achieve balance in the palate. Cayuga is grown in small regions in Ohio.

In one informal survey of grapebreeders who grow hybrid grapes and make wine from them, Cayuga was the most-popular answer to the question "Which hybrid grape is the easiest to grow and make good wine from?"

Coordinate time

In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time.

In the special case of an inertial observer in special relativity, by convention the coordinate time at an event is the same as the proper time measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the Einstein synchronisation convention.

Event horizon

The notion of an event horizon (EH) was originally based on escape velocity of light, meaning that light originating from EH boundary could escape, and light originating inside EH could cross it temporarily but would return back. Later a strict definition was introduced as a boundary beyond which events cannot affect any outside observer at all. This strict definition of EH has caused information and firewall paradoxes, therefore Stephen Hawking has supposed an apparent horizon to be used. [1] [2]

In general relativity, an event horizon is a region in spacetime beyond which light cannot totally escape, because the gravitational pull of a massive object becomes so great as to make escape impossible. An event horizon is most commonly associated with black holes, but can, in principle, arise and evolve in exactly flat regions of spacetime if a hollow spherically symmetric thin shell of matter is collapsing in a vacuum spacetime. [3]

Any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon, with its image becoming more and more redshifted as time elapses. This means that the wavelength of the light emitted from the object is getting longer as the object moves away from the observer..

The black hole event horizon is teleological in nature, meaning that we need to know the entire future space-time of the universe to determine the current location of the horizon, which is essentially impossible. Because of the purely theoretical nature of the event horizon boundary, the traveling object doesn't necessarily experience strange effects and does, in fact, pass through the calculatory boundary in a finite amount of proper time. [4]

More specific types of horizon include the related but distinct absolute and apparent horizons found around a black hole. Still other distinct notions include the Cauchy and Killing horizons; the photon spheres and ergospheres of the Kerr solution; particle and cosmological horizons relevant to cosmology; and isolated and dynamical horizons important in current black hole research.

Four-velocity

In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.

Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object is massive, so that its speed is less than the speed of light, the world line may be parametrized by the proper time of the object. The four-velocity is the rate of change of four-position with respect to the proper time along the curve. The velocity, in contrast, is the rate of change of the position in (three-dimensional) space of the object, as seen by an observer, with respect to the observer's time.

The value of the magnitude of an object's four-velocity, i.e. the quantity obtained by applying the metric tensor g to the four-velocity U, that is ||U||2 = U ⋅ U = gμνUνUμ, is always equal to ±c2, where c is the speed of light. Whether the plus or minus sign applies depends on the choice of metric signature. For an object at rest its four-velocity is parallel to the direction of the time coordinate with U0 = c. A four-velocity is thus the normalized future-directed timelike tangent vector to a world line, and is a contravariant vector. Though it is a vector, addition of two four-velocities does not yield a four-velocity: the space of four-velocities is not itself a vector space.

Geocentric Coordinate Time

Geocentric Coordinate Time (TCG - Temps-coordonnée géocentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to precession, nutation, the Moon, and artificial satellites of the Earth. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the center of the Earth: that is, a clock that performs exactly the same movements as the Earth but is outside the Earth's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Earth.

TCG was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the ill-defined Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCG is defined in the context of the general theory of relativity. The relationships between TCG and other relativistic time scales are defined with fully general relativistic metrics.

Because the reference frame for TCG is not rotating with the surface of the Earth and not in the gravitational potential of the Earth, TCG ticks faster than clocks on the surface of the Earth by a factor of about 7.0 × 10−10 (about 22 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCG differ from the traditional values of physical constants. (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales.) Adapting the large body of existing software to change from TDB to TCG is a formidable task, and as of 2002 many calculations continue to use TDB in some form.

Time coordinates on the TCG scale are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCG was set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TCG instant 1977-01-01T00:00:32.184 exactly corresponds to TAI instant 1977-01-01T00:00:00.000 exactly. This is also the instant at which TAI introduced corrections for gravitational time dilation.

TCG is a Platonic time scale: a theoretical ideal, not dependent on a particular realisation. For practical purposes, TCG must be realised by actual clocks in the Earth system. Because of the linear relationship between Terrestrial Time (TT) and TCG, the same clocks that realise TT also serve for TCG. See the article on TT for details of the relationship and how TT is realised.

Barycentric Coordinate Time (TCB) is the analog of TCG, used for calculations relating to the solar system beyond Earth orbit. TCG is defined by a different reference frame from TCB, such that they are not linearly related. Over the long term, TCG ticks more slowly than TCB by about 1.6 × 10−8 (about 0.5 seconds per year). In addition there are periodic variations, as Earth moves within the Solar system. When the Earth is at perihelion in January, TCG ticks even more slowly than it does on average, due to gravitational time dilation from being deeper in the Sun's gravity well and also velocity time dilation from moving faster relative to the Sun. At aphelion in July the opposite holds, with TCG ticking faster than it does on average.

Gravitational time dilation

Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The higher the gravitational potential (the farther the clock is from the source of gravitation), the faster time passes. Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.This has been demonstrated by noting that atomic clocks at differing altitudes (and thus different gravitational potential) will eventually show different times. The effects detected in such Earth-bound experiments are extremely small, with differences being measured in nanoseconds. Relative to Earth's age in billions of years, Earth's core is effectively 2.5 years younger than its surface. Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source.

Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of space-time. The existence of gravitational time dilation was first confirmed directly by the Pound–Rebka experiment in 1959.

Intercalation (timekeeping)

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

International Atomic Time

International Atomic Time (TAI, from the French name temps atomique international) is a high-precision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid. It is the principal realisation of Terrestrial Time (with a fixed offset of epoch). It is also the basis for Coordinated Universal Time (UTC), which is used for civil timekeeping all over the Earth's surface. As of 31 December 2016, when another leap second was added, TAI is exactly 37 seconds ahead of UTC. The 37 seconds results from the initial difference of 10 seconds at the start of 1972, plus 27 leap seconds in UTC since 1972.

TAI may be reported using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. TAI in this form was synchronised with Universal Time at the beginning of 1958, and the two have drifted apart ever since, due to the changing motion of the Earth.

Islamic calendar

The Islamic, Muslim, or Hijri calendar (Arabic: التقويم الهجري‎ at-taqwīm al-hijrī) is a lunar calendar consisting of 12 lunar months in a year of 354 or 355 days. It is used to determine the proper days of Islamic holidays and rituals, such as the annual period of fasting and the proper time for the pilgrimage to Mecca. The civil calendar of almost all countries where the religion is predominantly Muslim is the Gregorian calendar. Notable exceptions to this rule are Iran and Afghanistan, which use the Solar Hijri calendar. Rents, wages and similar regular commitments are generally paid by the civil calendar.The Islamic calendar employs the Hijri era whose epoch was established as the Islamic New Year of 622 AD/CE. During that year, Muhammad and his followers migrated from Mecca to Yathrib (now Medina) and established the first Muslim community (ummah), an event commemorated as the Hijra. In the West, dates in this era are usually denoted AH (Latin: Anno Hegirae, "in the year of the Hijra") in parallel with the Christian (AD), Common (CE) and Jewish eras (AM). In Muslim countries, it is also sometimes denoted as H from its Arabic form (سَنة هِجْريّة, abbreviated هـ). In English, years prior to the Hijra are reckoned as BH ("Before the Hijra").The current Islamic year is 1440 AH. In the Gregorian calendar, 1440 AH runs from approximately 11 September 2018 to 30 August 2019.

Nambu–Goto action

The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects (for example, cosmic strings). It is the starting point of the analysis of zero-thickness (infinitely thin) string behavior, using the principles of Lagrangian mechanics. Just as the action for a free point particle is proportional to its proper time — i.e., the "length" of its world-line — a relativistic string's action is proportional to the area of the sheet which the string traces as it travels through spacetime.

It is named after Japanese physicists Yoichiro Nambu and Tetsuo Goto.

Proper length

Proper length or rest length refers to 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 the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of simultaneity is dependent on the observer.

A different term, proper distance, provides an invariant measure whose value is the same for all observers.

Proper distance is analogous to proper time. The difference is that the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events (or along a timelike path).

Raychaudhuri equation

In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.

The equation is important as a fundamental lemma for the Penrose-Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass-energy in general relativity, as it is in Newton's theory of gravitation.

The equation was discovered independently by the Indian physicist Amal Kumar Raychaudhuri and the Soviet physicist Lev Landau.

Ritu (Indian season)

Ritu (Sanskrit: ऋतु, Bengali: ঋতু) defines "season" in different ancient Indian calendars used in countries of India, Bangladesh, Nepal and Sri Lanka, and there are six ritus (also transliterated rutu) or seasons. The word is derived from the Vedic Sanskrit word Ṛtú, a fixed or appointed time, especially the proper time for sacrifice (yajna) or ritual in Vedic religion; this in turn comes from the word Ṛta (ऋत), as used in Vedic Sanskrit literally means the "order or course of things". This word is used in nearly all Indian languages.

Terrestrial Time

Terrestrial Time (TT) is a modern astronomical time standard defined by the International Astronomical Union, primarily for time-measurements of astronomical observations made from the surface of Earth.

For example, the Astronomical Almanac uses TT for its tables of positions (ephemerides) of the Sun, Moon and planets as seen from Earth. In this role, TT continues Terrestrial Dynamical Time (TDT or TD), which in turn succeeded ephemeris time (ET). TT shares the original purpose for which ET was designed, to be free of the irregularities in the rotation of Earth.

The unit of TT is the SI second, the definition of which is currently based on the caesium atomic clock, but TT is not itself defined by atomic clocks. It is a theoretical ideal, and real clocks can only approximate it.

TT is distinct from the time scale often used as a basis for civil purposes, Coordinated Universal Time (UTC). TT indirectly underlies UTC, via International Atomic Time (TAI). Because of the historical difference between TAI and ET when TT was introduced, TT is approximately 32.184 s ahead of TAI.

The Proper Time

The Proper Time is a 1960 film starring Tom Laughlin. It is also Laughlin's directorial and screenwriting debut.

Time dilation

According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime, a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. A clock that is under the influence of a stronger gravitational field than an observer's will also be measured to tick slower than the observer's own clock.

Such time dilation has been repeatedly demonstrated, for instance by small disparities in a pair of atomic clocks after one of them is sent on a space trip, or by clocks on the Space Shuttle running slightly slower than reference clocks on Earth, or clocks on GPS and Galileo satellites running slightly faster. Time dilation has also been the subject of science fiction works, as it technically provides the means for forward time travel.

Time travel

Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically using a hypothetical device known as a time machine. Time travel is a widely-recognized concept in philosophy and fiction. The idea of a time machine was popularized by H. G. Wells' 1895 novel The Time Machine.

It is uncertain if time travel to the past is physically possible. Forward time travel, outside the usual sense of the perception of time, is an extensively-observed phenomenon and well-understood within the framework of special relativity and general relativity. However, making one body advance or delay more than a few milliseconds compared to another body is not feasible with current technology. As for backwards time travel, it is possible to find solutions in general relativity that allow for it, but the solutions require conditions that may not be physically possible. Traveling to an arbitrary point in spacetime has a very limited support in theoretical physics, and usually only connected with quantum mechanics or wormholes, also known as Einstein-Rosen bridges.

In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not a paradox in the sense of a logical contradiction.

Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration. General relativity is not necessary to explain the twin paradox; special relativity alone can explain the phenomenon.Time dilation has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment. It was also confirmed in particle accelerators by measuring the time dilation of circulating particle beams.

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