View source for Physics:Post-Newtonian expansion

{{Short description|Method of approximation in general relativity}}
[[File:GR2bodyparameterspace.png|thumb|330x330px|Diagram of the parameter space of compact binaries with the various approximation schemes and their regions of validity.]]
[[File:PM vs PN expansion EN.svg|thumb|Post-Minkowskian vs. post-Newtonian expansions|330x330px]]
In [[Physics:General relativity|general relativity]], '''post-Newtonian expansions''' ('''PN''' '''expansions''') are used for finding an approximate solution of [[Physics:Einstein field equations|Einstein field equations]] for the [[Physics:Metric tensor (general relativity)|metric tensor]]. The approximations are expanded in small parameters that express orders of deviations from [[Physics:Newton's law of universal gravitation|Newton's law of universal gravitation]]. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of [[Physics:Effective field theory|effective field theories]]. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.

== Expansion in 1/''c''<sup>2</sup> ==
The '''post-Newtonian approximations''' are [[Taylor series|expansions]] in a small parameter, which is the ratio of the velocity of the matter that creates the gravitational field, to the [[Physics:Speed of light|speed of light]], which in this case is more precisely called the ''[[Physics:Speed of gravity|speed of gravity]]''.<ref>{{cite journal |author=Kopeikin, S. |title=The speed of gravity in General Relativity and theoretical interpretation of the Jovian deflection experiment |journal=Classical and Quantum Gravity |year=2004 |volume=21 |issue=13 |pages=3251–3286 |doi=10.1088/0264-9381/21/13/010 |arxiv=gr-qc/0310059 |bibcode=2004CQGra..21.3251K|s2cid=13998000 }}</ref> In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to [[Biography:Isaac Newton|Newton]]'s law of gravity. A systematic study of post-Newtonian expansions within hydrodynamic approximations was developed by [[Biography:Subrahmanyan Chandrasekhar|Subrahmanyan Chandrasekhar]] and his colleagues in the 1960s.<ref>{{cite journal |last=Chandrasekhar |first=S. |year=1965 |title=The post-Newtonian equations of hydrodynamics in General Relativity |journal=The Astrophysical Journal |volume=142 |page=1488|doi=10.1086/148432 |bibcode=1965ApJ...142.1488C }}</ref><ref>{{cite journal |author-link=Subrahmanyan Chandrasekhar |last=Chandrasekhar |first=S. |year=1967 |title=The post-Newtonian effects of General Relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the MacLaurin spheroids |journal=The Astrophysical Journal |volume=147 |page=334|doi=10.1086/149003 |bibcode=1967ApJ...147..334C }}</ref><ref>{{cite journal |author-link=Subrahmanyan Chandrasekhar |last=Chandrasekhar |first=S. |year=1969 |title=Conservation laws in general relativity and in the post-Newtonian approximations |journal=The Astrophysical Journal |volume=158 |page=45|doi=10.1086/150170 |bibcode=1969ApJ...158...45C |doi-access=free }}</ref><ref>{{cite journal |last1=Chandrasekhar |first1=S. |year=1969 |last2=Nutku |first2=Y. |title=The second post-Newtonian equations of hydrodynamics in General Relativity |journal=Relativistic Astrophysics |volume=86|page=55 |doi=10.1086/150171 |bibcode=1969ApJ...158...55C |doi-access=free }}</ref><ref>{{cite journal |author-link=Subrahmanyan Chandrasekhar |last1=Chandrasekhar |first1=S. |year=1970 |last2=Esposito |first2=F.P. |title=The 2½-post-Newtonian equations of hydrodynamics and radiation reaction in General Relativity |journal=The Astrophysical Journal |volume=160 |page=153|doi=10.1086/150414 |bibcode=1970ApJ...160..153C |doi-access=free }}</ref>

== Expansion in ''h'' ==
Another approach is to expand the equations of general relativity in a power series in the deviation of the metric from its value in the absence of gravity.
:<math>h_{\alpha \beta} = g_{\alpha \beta} - \eta_{\alpha \beta} \,.</math>
To this end, one must choose a coordinate system in which the eigenvalues of <math>h_{\alpha \beta} \eta^{\beta \gamma} \,</math> all have absolute values less than 1.

For example, if one goes one step beyond [[Physics:Linearized gravity|linearized gravity]] to get the expansion to the second order in ''h'':
:<math> g^{\mu \nu} \approx \eta^{\mu \nu} - \eta^{\mu \alpha} h_{\alpha \beta} \eta^{\beta \nu} + \eta^{\mu \alpha} h_{\alpha \beta} \eta^{\beta \gamma} h_{\gamma \delta} \eta^{\delta \nu} \,.</math>
:<math> \sqrt{- g} \approx 1 + \tfrac12 h_{\alpha \beta} \eta^{\beta \alpha} + \tfrac18 h_{\alpha \beta} \eta^{\beta \alpha} h_{\gamma \delta} \eta^{\delta \gamma} - \tfrac14 h_{\alpha \beta} \eta^{\beta \gamma} h_{\gamma \delta} \eta^{\delta \alpha} \,.</math>
Expansions based only on the metric, independently from the speed, are called [[Physics:Post-Minkowskian expansion|'''post-Minkowskian expansions''']] ('''PM expansions''').

{| class="wikitable"
|+
| style="background-color:white;" |
!0PN
| style="background-color:white;" |
!1PN
| style="background-color:white;" |
!2PN
| style="background-color:white;" |
!3PN
| style="background-color:white;" |
!4PN
| style="background-color:white;" |
!5PN
| style="background-color:white;" |
!6PN
| style="background-color:white;" |
!7PN
| colspan="3" style="background-color:white;" |
|-
!1PM
|( 1
| +
|<math>v^2</math>
| +
|<math>v^4</math>
| +
|<math>v^6</math>
| +
|<math>v^8</math>
| +
|<math>v^{10}</math>
| +
|<math>v^{12}</math>
| +
|<math>v^{14}</math>
| +
|...)
|<math>G^1</math>
|-
!2PM
|
|
|( 1
| +
|<math>v^2</math>
| +
|<math>v^4</math>
| +
|<math>v^6</math>
| +
|<math>v^8</math>
| +
|<math>v^{10}</math>
| +
|<math>v^{12}</math>
| +
|...)
|<math>G^2</math>
|-
!3PM
|
|
|
|
|( 1
| +
|<math>v^2</math>
| +
|<math>v^4</math>
| +
|<math>v^6</math>
| +
|<math>v^8</math>
| +
|<math>v^{10}</math>
| +
|...)
|<math>G^3</math>
|-
!4PM
|
|
|
|
|
|
|( 1
| +
|<math>v^2</math>
| +
|<math>v^4</math>
| +
|<math>v^6</math>
| +
|<math>v^8</math>
| +
|...)
|<math>G^4</math>
|-
!5PM
|
|
|
|
|
|
|
|
|( 1
| +
|<math>v^2</math>
| +
|<math>v^4</math>
| +
|<math>v^6</math>
| +
|...)
|<math>G^5</math>
|-
!6PM
|
|
|
|
|
|
|
|
|
|
|( 1
| +
|<math>v^2</math>
| +
|<math>v^4</math>
| +
|...)
|<math>G^6</math>
|-
| colspan="19" style="background-color:white;" |Comparison table of powers used for PN and PM approximations in the case of two non-rotating bodies.
0PN corresponds to the case of Newton's theory of gravitation. 0PM (not shown) corresponds to the [[Physics:Minkowski space|Minkowski flat space]].<ref>{{Cite journal|last1=Bern|first1=Zvi|last2=Cheung|first2=Clifford|last3=Roiban|first3=Radu|last4=Shen|first4=Chia-Hsien|last5=Solon|first5=Mikhail P.|last6=Zeng|first6=Mao|date=2019-08-05|title=Black Hole Binary Dynamics from the Double Copy and Effective Theory|journal=Journal of High Energy Physics|volume=2019|issue=10|pages=206|doi=10.1007/JHEP10(2019)206|arxiv=1908.01493|bibcode=2019JHEP...10..206B|s2cid=199442337|issn=1029-8479}}</ref>
|}

== Uses ==
The first use of a PN expansion (to first order) was made by [[Biography:Albert Einstein|Albert Einstein]] in calculating the perihelion precession of Mercury's orbit. Today, Einstein's calculation is recognized as a common example of applications of PN expansions, solving the [[Physics:Two-body problem in general relativity|general relativistic two-body problem]], which includes the emission of [[Astronomy:Gravitational wave|gravitational wave]]s.

==Newtonian gauge==
{{Main|Physics:Newtonian gauge}}
In general, the perturbed metric can be written as<ref>{{cite web |url=http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf |title=Cosmological Perturbation Theory |page=83,86 |access-date=2016-08-10 |archive-date=2016-08-26 |archive-url=https://web.archive.org/web/20160826054737/http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf |url-status=dead }}</ref> 
:<math>ds^2 =a^2 (\tau) \left[(1+2A)d\tau^2 - 2B_i dx^i d\tau - \left( \delta_{ij} + h_{ij} \right)dx^i dx^j\right]</math>
where <math>A</math>, <math>B_i</math> and <math>h_{ij}</math> are functions of space and time. <math>h_{ij}</math> can be decomposed as

:<math>h_{ij} = 2 C \delta_{ij} + \partial_i \partial_j E - \frac{1}{3} \delta_{ij} \Box^2 E + \partial_i \hat{E}_j + \partial_j \hat{E}_i + 2 \tilde{E}_{ij} </math>
where <math>\Box</math> is the [[D'Alembert operator|d'Alembert operator]], <math>E</math> is a scalar, <math>\hat{E}_i</math> is a vector and <math>\tilde{E}_{ij}</math> is a traceless tensor.
Then the Bardeen potentials are defined as
:<math>\Psi \equiv A + H(B-E'), + (B +E')', \quad \Phi \equiv - C - H(B-E') + \frac{1}{3} \Box E</math>
where <math>H</math> is the [[Astronomy:Hubble's law|Hubble constant]] and a prime represents differentiation with respect to conformal time <math>\tau \,</math>.

Taking <math>B=E=0</math> (i.e. setting <math>\Phi\equiv-C</math> and <math>\Psi\equiv A</math>), the Newtonian gauge is 
:<math>ds^2 =a^2(\tau) \left[ (1 + 2 \Psi) d\tau^2 - (1 - 2 \Phi)\delta_{ij} dx^i dx^j\right]\,</math>.

Note that in the absence of anisotropic stress, <math>\Phi=\Psi</math>.

A useful non-linear extension of this is provided by the [[Physics:Non-Relativistic Gravitational Fields|non-relativistic gravitational fields]].  

==See also==
*[[Physics:Coordinate conditions|Coordinate conditions]]
*[[Einstein–Infeld–Hoffmann equations]]
*[[Physics:Linearized gravity|Linearized gravity]]
*[[Physics:Parameterized post-Newtonian formalism|Parameterized post-Newtonian formalism]]

==References==
{{Reflist|30em}}

==External links==
*[http://www.math.ca/cjm/v1/p209 "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld] {{Webarchive|url=https://web.archive.org/web/20120308133707/http://www.math.ca/cjm/v1/p209 |date=2012-03-08 }}
* {{cite journal |doi=10.12942/lrr-2014-2|pmid=28179846|pmc=5256563|arxiv=1310.1528|bibcode=2014LRR....17....2B|title=Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries|year=2014|last1=Blanchet|first1=Luc|journal=Living Reviews in Relativity|volume=17|issue=1|pages=2}}

{{Isaac Newton}}

{{DEFAULTSORT:Post-Newtonian Expansion}}
[[Category:General relativity]]

{{Sourceattribution|Post-Newtonian expansion}}