Editing Curved spacetime (section)

== Curvature of time ==
[[File:Einstein's argument suggesting gravitational redshift.svg|thumb|Figure 5–3. Einstein's argument suggesting gravitational redshift]]
In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a ''global'' inertial frame. In small enough regions of spacetime, ''local'' inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.<ref name="Schutz1985">{{cite book| last1=Schutz |first1= Bernard F. |title=A first course in general relativity|date=1985|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0-521-27703-5|page=26}}</ref>{{rp|118–126}}

Years before publication of the general theory in 1916, Einstein used the equivalence principle to predict the existence of gravitational redshift in the following [[thought experiment]]: (i) Assume that a tower of height ''h'' (Fig.&nbsp;5-3) has been constructed. (ii) Drop a particle of rest mass ''m'' from the top of the tower. It falls freely with acceleration ''g'', reaching the ground with velocity {{math|1=''v'' = (2''gh'')<sup>1/2</sup>}}, so that its total energy ''E'', as measured by an observer on the ground, is {{tmath|1=m + {\tfrac{1}{2} m v^2} / {c^2} = m + {m g h} / {c^2} }} (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon ''E{{'}}'' back into a particle of rest mass ''m''{{'}}.<ref name="Schutz1985" />{{rp|118–126}}

It must be that {{math|1=''m'' = ''m''{{'}}}}, since otherwise one would be able to construct a [[perpetual motion]] device. We therefore predict that {{math|1=''E''{{'}} = ''m''}}, so that
: <math>\frac{E'}{E} = \frac{h \nu \, '}{ h \nu} = \frac{m}{m + \frac{mgh}{c^2}} = 1 - \frac{gh}{c^2}</math>

A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by [[Pound–Rebka experiment|Pound & Rebka (1959)]] and later by Pound & Snider (1964).<ref>{{cite web |last1=Mester |first1=John |title=Experimental Tests of General Relativity |url=https://luth2.obspm.fr/IHP06/lectures/mester-vinet/IHP-2GravRedshift.pdf |publisher=Laboratoire Univers et Théories |access-date=9 June 2017 |archive-url=https://web.archive.org/web/20170318085505/https://luth2.obspm.fr/IHP06/lectures/mester-vinet/IHP-2GravRedshift.pdf |archive-date=18 March 2017 |url-status=dead }}</ref>

Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side by side with the ground clock.<ref name="Schutz">{{cite book |last1=Schutz |first1=Bernard |title=Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity |date=2004 |publisher=[[Cambridge University Press]] |location=Cambridge |isbn=0-521-45506-5 |edition=Reprint |url=https://books.google.com/books?id=P_T0xxhDcsIC |access-date=24 May 2017 |language=en |archive-date=17 January 2023 |archive-url=https://web.archive.org/web/20230117023501/https://books.google.com/books?id=P_T0xxhDcsIC |url-status=live }}</ref>{{rp|16–18}} For a 1&nbsp;km tower, the discrepancy would amount to about 9.4&nbsp;nanoseconds per day, easily measurable with modern instrumentation.

Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the ''space'' component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. ''Any'' theory of gravity will predict gravitational time dilation if it respects the principle of equivalence.<ref name="Schutz" />{{rp|16}} This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the "[[Newtonian limit]]" (i.e. the particles are moving slowly, the gravitational field is weak, and the field is static), curvature of time alone is sufficient to derive Newton's law of gravity.<ref name="Carroll">{{cite arXiv|last1=Carroll|first1=Sean M.|title=Lecture Notes on General Relativity|date=2 December 1997|eprint=gr-qc/9712019}}</ref>{{rp|101–106}}

Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time ''and'' curved space. Given ''G'' as the gravitational constant, ''M'' as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance ''r'' from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:<ref name="Schutz" />{{rp|229–232}}
: <math>\Delta s^2 = \left( 1 - \frac{2GM}{c^2 r} \right) (c \Delta t)^2 - \, (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2 </math>