Editing Gravitational time dilation (section)

==Definition==
[[Clock]]s that are far from massive bodies (or at higher gravitational potentials) run more quickly, and clocks close to massive bodies (or at lower gravitational potentials) run more slowly. For example, considered over the total time-span of Earth (4.6 billion years), a clock set in a geostationary position at an altitude of 9,000 meters above sea level, such as perhaps at the top of [[Mount Everest]] ([[Topographic prominence|prominence]] 8,848{{nbsp}}m), would be about 39 hours ahead of a clock set at sea level.<ref>{{cite book |title=From Atoms to Galaxies: A Conceptual Physics Approach to Scientific Awareness |first1=Sadri |last1=Hassani |publisher=CRC Press |year=2011 |isbn=978-1-4398-0850-4 |page=433 |url=https://books.google.com/books?id=oypZ_a9pqdsC&pg=PA433}} [https://books.google.com/books?id=oypZ_a9pqdsC&pg=PA433 Extract of page 433]</ref><ref>{{cite book |title=How Einstein Created Relativity out of Physics and Astronomy |edition=illustrated |first1=David |last1=Topper |publisher=Springer Science & Business Media |year=2012 |isbn=978-1-4614-4781-8 |page=118 |url=https://books.google.com/books?id=2U6qvi5TlE4C}} [https://books.google.com/books?id=2U6qvi5TlE4C&pg=PA118 Extract of page 118]</ref> This is because gravitational time dilation is manifested in accelerated [[Frame of reference|frames of reference]] or, by virtue of the [[equivalence principle]], in the gravitational field of massive objects.<ref>[https://books.google.com/books?id=MBjkuQAoyZIC&pg=PA28 John A. Auping, ''Proceedings of the International Conference on Two Cosmological Models''], Plaza y Valdes, {{ISBN|9786074025309}}</ref>

According to general relativity, [[Mass#Inertial mass|inertial mass]] and [[gravitational mass]] are the same, and all accelerated reference frames (such as a [[Born coordinates|uniformly rotating reference frame]] with its proper time dilation) are physically equivalent to a gravitational field of the same strength.<ref>[http://www.cathodixx.com/pdfs/RELATIVITY.pdf Johan F Prins, ''On Einstein's Non-Simultaneity, Length-Contraction and Time-Dilation'']</ref>

Consider a family of observers along a straight "vertical" line, each of whom experiences a distinct constant [[g-force]] directed along this line (e.g., a long accelerating spacecraft,<ref>{{cite book |title=Introduction to Relativity: For Physicists and Astronomers |edition=illustrated |first1=John B. |last1=Kogut |publisher=Academic Press |year=2012 |isbn=978-0-08-092408-3 |page=112 |url=https://books.google.com/books?id=9AKPpSxiN4IC}}</ref><ref>{{cite book |title=What Is Relativity?: An Intuitive Introduction to Einstein's Ideas, and Why They Matter |edition=illustrated |first1=Jeffrey |last1=Bennett |publisher=Columbia University Press |year=2014 |isbn=978-0-231-53703-2 |page=120 |url=https://books.google.com/books?id=OiquAgAAQBAJ}} [https://books.google.com/books?id=OiquAgAAQBAJ&pg=PA120 Extract of page 120]</ref> a skyscraper, a shaft on a planet). Let <math>g(h)</math> be the dependence of g-force on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at <math>h=0</math> is

: <math>T_d(h) = \exp\left[\frac{1}{c^2}\int_0^h g(h') dh'\right]</math>

where <math>T_d(h)</math> is the ''total'' time dilation at a distant position <math>h</math>, <math>g(h)</math> is the dependence of g-force on "height" <math>h</math>, <math>c</math> is the [[speed of light]], and <math>\exp</math> denotes [[exponentiation]] by [[E (mathematical constant)|e]].

For simplicity, in a [[Rindler coordinates|Rindler's family of observers]] in a [[Minkowski space|flat spacetime]], the dependence would be

: <math>g(h) = c^2/(H+h)</math><!-- please, check the sign: I am lazy to investigate whether it would be positive or negative -->

with constant <math>H</math>, which yields

: <math>T_d(h) = e^{\ln (H+h) - \ln H} = \tfrac{H+h}H</math>.

On the other hand, when <math>g</math> is nearly constant and <math>gh</math> is much smaller than <math>c^2</math>, the linear "weak field" approximation <math>T_d = 1 + gh/c^2</math> can also be used.

See [[Ehrenfest paradox]] for application of the same formula to a rotating reference frame in flat spacetime.