- Gravitational time dilation
**Gravitational time dilation**is the effect of time passing at different rates in regions of differentgravitational potential ; the higher the local distortion ofspacetime due to gravity, the slower time passes.Albert Einstein originally predicted this effect in histheory of relativity and it has since been confirmed bytests 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 experiments are "extremely" small, with differences being measured in

nanoseconds .Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of

special relativity in accelerated frames of reference. Ingeneral relativity , it is considered to be difference in the passage ofproper time at different positions as described by a metric tensor of spacetime. The existence of gravitational time dilation was first confirmed directly by thePound-Rebka experiment .**Definition**Clock s which are far from massive bodies (or at highergravitational potential s) run faster, and clocks close to massive bodies (or at lower gravitational potentials) run slower. This is because gravitational time dilation is manifested in acceleratedframes of reference or, by virtue of theequivalence principle , in the gravitational field of massive objects.It can also be manifested by any other kind of accelerated reference frame such as an accelerating

dragster orspace shuttle . Spinning objects such asmerry-go-round s andferris wheel s are subjected to gravitational time dilation as an effect of their angular spin.This is supported by the

general theory of relativity due to theequivalence principle that states that all accelerated reference frames possess a gravitational field. According to general relativity,inertial mass andgravitational mass are the same. Not all gravitational fields are "curved" or "spherical"; some are flat as in the case of an acceleratingdragster orspace shuttle . Any kind of g-load contributes to gravitational time dilation.* In an accelerated box, the equation with respect to an arbitrary base observer is $T\_d\; =\; 1\; +\; gh/c^2$, where

** $T\_d$ is the "total" time dilation at a distant position,

** $g$ is the acceleration of the box as measured by the base observer, and

** $h$ is the "vertical" distance between the observers.

* On a rotating disk when the base observer is located at the center of the disk and co-rotating with it (which makes their view of spacetime non-inertial), the equation is $T\_d\; =\; sqrt\{1\; -\; r^2\; omega^2/c^2\}$, where

** $r$ is the distance from the center of the disk (which is the location of the base observer), and

** $omega$ is the angular velocity of the disk.: (It is no accident that in an inertial frame of reference this becomes the familiar velocity time dilation $sqrt\{1\; -\; v^2/c^2\}$ ).**Outside a non-rotating sphere**A common equation used to determine gravitational time dilation is derived from the

Schwarzschild metric , which describesspacetime in the vicinity of a non-rotating massive spherically-symmetric object. The equation is:$t\_0\; =\; t\_f\; sqrt\{1\; -\; frac\{2GM\}\{rc^2\; =\; t\_f\; sqrt\{1\; -\; frac\{r\_0\}\{r$, where

* $t\_0$ is the

proper time between events A and B for a slow-ticking observer within the gravitational field,

* $t\_f$ is the proper time between events A and B for a fast-ticking observer distant from the massive object (and therefore outside of the gravitational field),

* $G$ is thegravitational constant ,

* $M$ is themass of the object creating the gravitational field,

* $r$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate),

* $c$ is thespeed of light , and

* $r\_0\; =\; 2GM/c^2$ is the called theSchwarzschild Radius of M. If a mass collapses so that its surface lies at less than this radial coordinate (or in other words covers an area of less than $4\; pi\; G^2\; M^2\; /\; c^4$), then the object exists within ablack hole .**Inside a non-rotating sphere**The equation above is only valid outside the non-rotating massive spherically-symmetric object. Inside the sphere the equation is:

$t\_0\; =\; t\_f\; sqrt\{1\; -\; frac\{2G(frac\{r\_i\}\{R\})^3M\}\{r\_i\; c^2\; =\; t\_f\; sqrt\{1\; -\; r\_i^2\; frac\{r\_0\}\{R^3$, where

* $r\_i$ is the distance of a point on the inside of the original sphere to the center of that sphere,

* $R$ is the radius of the original sphere, and

* $M$ is the mass of the original sphere with radius $R$.If one is inside the sphere, the sphere can be split in two parts: a hollow sphere above and a solid sphere below. One is weightless anywhere in the interior of a uniform hollow sphere. With respect to one's gravitational potential, it is as if the hollow sphere is not there [

] [Shell theorem ] . What is left is the solid sphere below, and its mass is:Gauss's law for gravity $M\_i\; =\; V\_i\; ho\; =\; frac\{4\}\{3\}pi\; r\_i^3\; ho\; =\; frac\{4\}\{3\}pi\; r\_i^3frac\{M\}\{V\}\; =\; frac\{4\}\{3\}pi\; r\_i^3frac\{M\}\{frac\{4\}\{3\}pi\; R^3\}\; =\; frac\{r\_i^3\}\{R^3\}M$, where

* $r\_i$, $R$ and $M$ are the same as described above,

* $V$ is the volume of the original sphere with radius $R$,

* $M\_i$ is the mass of a sphere with radius $r\_i$,

* $V\_i$ is the volume of a sphere with radius $r\_i$, and

* $ho$ is the (uniform) density of any part of the sphere.The implication is that the gravitational time dilation reaches its maximum at the surface of the non-rotating massive spherically-symmetric object, and that the gravitational time dilation reaches its minimum at the center of the sphere.

**Circular orbits**In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than $frac\{3\}\{2\}\; cdot\; r\_0$. The formula for a clock at rest is given above; for a clock in a circular orbit, the formula is instead

$t\_0\; =\; t\_f\; sqrt\{1\; -\; frac\{r\_0\}\{r$

**Important things to stress*** According to General Relativity, gravitational time dilation is copresent with the existence of an

accelerated reference frame .* The speed of light in a locale is always equal to "c" according to the observer who is there. The stationary observer's perspective corresponds to the local

proper time . Every infinitesimal region of space time may have its ownproper time that corresponds to the gravitational time dilation there, where electromagnetic radiation and matter may be equally affected, since they are made of the same essence (as shown in many tests involving the famous equation $E=mc^2$). Such regions are significant whether or not they are occupied by an observer. A time delay is measured for signals that bend near the sun, headed towards Venus, and bounce back to earth along more or less a similar path. There is no violation of the speed of light in this sense, as long as an observer is forced to observe only the photons which intercept the observing faculties and not the ones that go passing by in the depths of more (or even less) gravitational time dilation.::If a distant observer is able to track the light in a remote, distant locale which intercepts a time dilated observer nearer to a more massive body, he sees that both the distant light and that distant time dilated observer have a slower proper time clock than other light which is coming nearby him, which intercept him, at "c", like all other light he "really" can observe. When the other, distant light intercepts the distant observer, it will come at c from the distant observer's perspective.**Experimental confirmation**Gravitational time dilation has been experimentally measured using

atomic clock s on airplanes. The clocks that traveled aboard the airplanes upon return were slightly fast with respect to clocks on the ground. The effect is significant enough that theGlobal Positioning System needs to correct for its effect on clocks aboard artificial satellites, providing a further experimental confirmation of the effect.cite book |author= Richard Wolfson |title=Simply Einstein |url=http://books.google.com/books?id=OUJWKdlFKeQC&pg=PA216&dq=%22gravitational+time+dilation+%22&lr=&as_brr=0&sig=ACfU3U0_wc8IuNJdGCLnsaO-SyqXYaRapw

page=p. 216 |isbn=0393051544 |publisher=W W Norton & Co. |year=2003 ]Gravitational time dilation has also been confirmed by the

Pound-Rebka experiment , observations of the spectra of thewhite dwarf Sirius B and experiments with time signals sent to and fromViking 1 Mars lander.**ee also***

Gravitational redshift **References*** Einstein, Albert. "Relativity : the Special and General Theory by Albert Einstein."

__Project Gutenberg__.

* Einstein, Albert. "The effect of gravity on light" (1911), translated and reprinted in**The Principle of Relativity**

* Nave, C.R. "Gravity and the Photon."__Hyperphysics__.

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