User:JModest/Sandbox

• For velocity (special relativity) time dilation see: Time Dilation

Gravitational time dilation is an effect of relativity theory whereby the flow of time differs between two points in a gravitational field or an accelerated reference frame. From an observer's location in the field their own clock will always appear to run normally while identical clocks with less potential in stronger areas of the field will run slower and clocks with more potential in weaker areas of the field will run faster. A clock on the top of a building will have greater gravitational potential and thus run a small fraction slower than a clock at the base of the building. This is more than a mere consequence of observation. Upon uniting the clocks, a difference in time displayed makes evident the base clock experienced less time than the elevated clock. Neither is gravitational time dilation an effect of clocks, but rather any measure of time is dilated including the frequency of light, the rate of radioactive decay, and the time experienced by a person.

With Einstein's development of general relativity it became possible to accurately predict time dilation due to gravitational fields. The accuracy of modern atomic clocks and missions like Gravity Probe A makes possible the verification of time dilation of general relativity accurate to 70 parts per million.

Gravitational time dilation is distinguished from velocity time dilation by the involvement of non-inertial or accelerated reference frames. This is true both as a consequence of gravity in a curved gravitational field or a consequence of acceleration which can give corresponding effects in flat spacetime outside any gravitational field. This is so closely related via the equivalence principle it is convention to refer to all time dilation in accelerated frames as gravitational time dilation.

Inertial Reference Frames

Special relativity postulates the consistency of the speed of light in all inertial reference frames. As physical objects have different relative velocities it is necessary for their relative observations of time to change. This is time dilation in the setting of special relativity. Two observers who have constant relative inertial velocity will observe the other's clock to be dilated by the Lorentz factor, $\gamma \,$. Each will observe the other's time to run slow compared with their own time.

${\mbox{A Observes}}\rightarrow B/\gamma \Rightarrow A=B\Leftarrow A/\gamma \leftarrow {\mbox{B Observes}}\,$

This symmetric situation is characteristic of velocity time dilation.

Non-Inertial Reference Frames

Gravitational Time Dilation	Velocity Time Dilation
General Relativity	Special Relativity
Non-Inertial Frames	Inertial Frames
Asymmetrical	Symmetrical

Gravitational time dilation involves non-inertial frames and asymmetric time dilation. This is illustrated in the example of the twin paradox. The thought experiment consists of two twins who start our thought experiment identical in age. While the first twin stays here on earth, the second twin is an astronaut and decides to take a round-trip journey to a neighboring star. During the trip, the astronaut-twin travels near the speed of light and significant time dilation occurs. When the astronaut-twin returns home he finds his twin has aged significantly more than he has. The earth twin is similarly troubled to find his twin the astronaut is now far younger than himself. Asymmetrical time dilation has taken place.

If the incorrect assumption is made that these twins are confined to inertial frames then a paradox manifests. If both twins see the other's clock slowed by the same factor then why would their final age be different? Or similarly, If the astronaut sees the earth as moving and the homebody sees the astronaut as moving then what is the difference between the two? The answer is acceleration. The astronaut changes his velocity (a vector quantity) which introduces non-inertial motion and asymmetry. This is neither paradoxical nor demonstrates inconsistency in special relativity but rather became insightful in the development of a completely relativistic theory of gravity as is shown below.

Linear Acceleration

Consider a box with constant acceleration situated outside the effect of any gravitational field. The height of the box is h along the direction of acceleration. A clock on the ceiling of the box at point h emits pulses of light which are observed from point 0 on the floor below h. During the light travel time of any given pulse (h/c) the box accelerates by ah/c. The rate that pulses are observed by 0 on the floor is then dilated from the emission rate by the factor $(1+ah/c^{2})$. In other words, the clock on the ceiling runs faster by this factor. The time dilation equation for an accelerated frame is therefore:

$\Delta T=\Delta T_{0}(1+ah/c^{2})\,$

where:

$\Delta T$ is proper time elapsed at h

$\Delta T_{0}$ is the proper time elapsed for the ground observer

$h$ is the distance along the axis of acceleration between observers

$a$ is acceleration

$c$ is the speed of light

A more complete treatment of an accelerated frame can be accomplished using accelerated coordinates.

Rotating Reference Frame

The vertical axis is time and horizontal plane is two dimensional space. The center clock has less potential and runs faster

Consider a rotating massless disk with an observer $0$ in its center. On the rim of the disk is point P which has angular velocity $\omega$ distance r from the center. Due to the rotating reference frame, the clock feels a constant pseudo force analogous to acceleration. The equation for time dilation in a rotating reference frame is:

$\Delta T=\Delta T_{0}{\sqrt {1-r^{2}\omega ^{2}/c^{2}}}\,$

where:

$\Delta T$ is proper time elapsed at rotating point P

$\Delta T_{0}$ is the proper time elapsed for observer at center

$r$ is the distance from center to rotating point P

$\omega$ is angular velocity of P

$c$ is the speed of light

Equivalence Principle

The "peculiar consequence" Einstein noted in 1905 concerning clocks that did not synchronize as introduced above took on a new direction in 1907 when Einstein introduced the modern equivalence principle. He expressed free-fall as inertial motion which at the time was counter to the Newtonian description of gravity as a force acting on bodies that were inertially at rest on the surface of a non-rotating mass. As a thought experiment, Einstein compared the apparent effects of accelerating in a rocket with standing on a massive body. He concluded they were physically equivalent then explored the consequences of combining the equivalence principle with special relativity.

Why curved spacetime causes gravitational time dilation Examples / elevator / GPS

General Relativity

By the definitions of coordinate time ($t\,$) and proper time ($\tau \,$) in general relativity, gravitational time dilation is:

$\Delta \tau =\Delta t{\sqrt {\left|g_{00}\right|}}\,$

where $\tau \,$ is proper time, $t\,$ is coordinate time, and $g_{00}\,$ is a component of the metric tensor. This expresses time dilation in terms of an observer outside the gravitational field or at infinite distance from the mass dilating the coordinate of proper time. Coordinate time is then a meaningful point of reference with which to compare the proper time of two or more observers in the field. It will be the convention of this article to express time dilation in this way.

Schwarzschild metric

In the case that the gravitational field is produced by a spherically symmetric mass distribution we can use the Schwarzschild metric where the $g_{00}\,$ component of the metric is:

$g_{00}=1-{\frac {2GM}{c^{2}r}}\,$

Gravitational time dilation in the Schwarzschild metric is then:

$\Delta T=\Delta T_{0}\left(1-{\frac {2GM}{rc^{2}}}\right)^{-1/2}$

where:

$\Delta T$ is coordinate time or proper time of an observer outside the gravitational field

$\Delta T_{0}$ is the slower proper time of an observer inside the gravitational field

G is the gravitational constant

M is mass

r is the distance to center of mass

c is the speed of light

Weak Field Approximation and Newtonian Potential

Solving the 00 component of Einstein's field equations has only been done in the most simple gravitational arrangements such as the Schwarzschild metric above. It often becomes necessary to use parameterized post-Newtonian formalism to make weak field approximations which works well for areas such as the solar system. This method achieves fractional accuracy in the order of $10^{-11}$ in normal gravity situations.

Following the steps for deriving the correspondence principle in the article Einstein Field Equations, the geodesic equation:

${\frac {d^{2}x^{a}}{d\tau ^{2}}}+\Gamma _{bc}^{a}{\frac {dx^{b}}{d\tau }}{\frac {dx^{c}}{d\tau }}=0\,$

when combined with Newtonian theory,

${\frac {d^{2}x^{i}}{d\tau ^{2}}}=-\Delta ^{i}\Phi$

can give:

$g_{00}=-\left(1+{\frac {2\Phi }{c^{2}}}\right)\,$

where $\Phi \,$ is the Newtonian potential. From the definition of time dilation in general relativity,

$\Delta \tau =\Delta t{\sqrt {\left|g_{00}\right|}}\,$,

the post-Newtonian approximation for gravitational time dilation is:

$\Delta T=\Delta T_{0}\left(1+{\frac {2\Phi }{c^{2}}}\right)^{-1/2}\,$

where:

$\Delta T$ is coordinate time or proper time of an observer outside the gravitational field

$\Delta T_{0}$ is the slower proper time of an observer inside the gravitational field

$\Phi$ is the (negative) gravitational potential as calculated below.

c is the speed of light

It is then possible to calculate gravitational time dilation using Newtonian potential $\Phi$ for most normal-gravity situations. If more than one mass is present in the calculation potentials are summed ($\Phi _{TOTAL}=\Phi _{1}+\Phi _{2}...+\Phi _{n}$). All equations below assume mass it uniformly distributed with the function:

Point Mass	Outside Solid Sphere	Inside Solid Sphere
$\Phi =-{\frac {GM}{r}}$	$\Phi =-{\frac {GM}{r}}$	$\Phi =-{\frac {GM}{2R^{3}}}(3R^{2}-r^{2})$
Outside Spherical Shell	Inside Spherical Shell	Center Of Solid Sphere
$\Phi =-{\frac {GM}{r}}$	$\Phi =-{\frac {GM}{R}}$	$\Phi =-{\frac {3GM}{2R}}$

where:

G = gravitational constant

M = mass

r = distance from center of mass to time dilated observer

R = diameter of sphere

Examples

Earth

Assuming the earth is a solid sphere of uniform density with no satellite we can calculate time dilation using the above PPN approximation. The relevant Earth parameters are:

See Also

• General relativity
• Gravitational Redshift
• Time Dilation
• Time
• Hafele-Keating experiment
• Ives-Stilwell experiment
• Trouton-Rankine experiment
• Length contraction and Lorentz-Fitzgerald contraction
• Minkowski space
• Pound-Rebka experiment
• Relativistic Doppler effect and Transverse Doppler effect
• Relativity of simultaneity
• Special relativity
• Twin paradox