Post-Newtonian expansion

The post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter that creates the gravitational field, to the speed of light, which in this case is more precisely called the speed of gravity.{{cite journal |author=Kopeikin, S. |authorlink=Sergei Kopeikin |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 }} In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity. A systematic study of post-Newtonian expansions within hydrodynamic approximations was developed by Subrahmanyan Chandrasekhar and his colleagues in the 1960s.{{cite journal |author-link=Subrahmanyan Chandrasekhar |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 }}{{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 }}{{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 }}{{cite journal |author1-link=Subrahmanyan Chandrasekhar |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 }}{{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 }}

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.

:$h\_\{\backslash alpha\; \backslash beta\}\; =\; g\_\{\backslash alpha\; \backslash beta\}\; -\; \backslash eta\_\{\backslash alpha\; \backslash beta\}\; \backslash ,.$

To this end, one must choose a coordinate system in which the eigenvalues of $h\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash gamma\}\; \backslash ,$ all have absolute values less than 1.

For example, if one goes one step beyond linearized gravity to get the expansion to the second order in h:

:$g^\{\backslash mu\; \backslash nu\}\; \backslash approx\; \backslash eta^\{\backslash mu\; \backslash nu\}\; -\; \backslash eta^\{\backslash mu\; \backslash alpha\}\; h\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash nu\}\; +\; \backslash eta^\{\backslash mu\; \backslash alpha\}\; h\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash gamma\}\; h\_\{\backslash gamma\; \backslash delta\}\; \backslash eta^\{\backslash delta\; \backslash nu\}\; \backslash ,.$

:$\backslash sqrt\{-\; g\}\; \backslash approx\; 1\; +\; \backslash tfrac12\; h\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash alpha\}\; +\; \backslash tfrac18\; h\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash alpha\}\; h\_\{\backslash gamma\; \backslash delta\}\; \backslash eta^\{\backslash delta\; \backslash gamma\}\; -\; \backslash tfrac14\; h\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash gamma\}\; h\_\{\backslash gamma\; \backslash delta\}\; \backslash eta^\{\backslash delta\; \backslash alpha\}\; \backslash ,.$

Expansions based only on the metric, independently from the speed, are called post-Minkowskian expansions (PM expansions).

The first use of a PN expansion (to first order) was made by 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 general relativistic two-body problem, which includes the emission of gravitational waves.

{{Main|Newtonian gauge}}

In general, the perturbed metric can be written as{{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 }}

:$ds^2\; =a^2\; (\backslash tau)\; \backslash left[(1+2A)d\backslash tau^2\; -\; 2B\_i\; dx^i\; d\backslash tau\; -\; \backslash left(\; \backslash delta\_\{ij\}\; +\; h\_\{ij\}\; \backslash right)dx^i\; dx^j\backslash right]$

where $A$, $B\_i$ and $h\_\{ij\}$ are functions of space and time. $h\_\{ij\}$ can be decomposed as

:$h\_\{ij\}\; =\; 2\; C\; \backslash delta\_\{ij\}\; +\; \backslash partial\_i\; \backslash partial\_j\; E\; -\; \backslash frac\{1\}\{3\}\; \backslash delta\_\{ij\}\; \backslash Box^2\; E\; +\; \backslash partial\_i\; \backslash hat\{E\}\_j\; +\; \backslash partial\_j\; \backslash hat\{E\}\_i\; +\; 2\; \backslash tilde\{E\}\_\{ij\}$

where $\backslash Box$ is the d'Alembert operator, $E$ is a scalar, $\backslash hat\{E\}\_i$ is a vector and $\backslash tilde\{E\}\_\{ij\}$ is a traceless tensor.

Then the Bardeen potentials are defined as

:$\backslash Psi\; \backslash equiv\; A\; +\; H(B-E\text{'}),\; +\; (B\; +E\text{'})\text{'},\; \backslash quad\; \backslash Phi\; \backslash equiv\; -\; C\; -\; H(B-E\text{'})\; +\; \backslash frac\{1\}\{3\}\; \backslash Box\; E$

where $H$ is the Hubble constant and a prime represents differentiation with respect to conformal time $\backslash tau\; \backslash ,$.

Taking $B=E=0$ (i.e. setting $\backslash Phi\backslash equiv-C$ and $\backslash Psi\backslash equiv\; A$), the Newtonian gauge is

:$ds^2\; =a^2(\backslash tau)\; \backslash left[\; (1\; +\; 2\; \backslash Psi)\; d\backslash tau^2\; -\; (1\; -\; 2\; \backslash Phi)\backslash delta\_\{ij\}\; dx^i\; dx^j\backslash right]\backslash ,$.

Note that in the absence of anisotropic stress, $\backslash Phi=\backslash Psi$.

A useful non-linear extension of this is provided by the non-relativistic gravitational fields.

{{Portal|Physics}}

• Coordinate conditions
• Einstein–Infeld–Hoffmann equations
• Linearized gravity
• Parameterized post-Newtonian formalism

{{Reflist|30em}}

• [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|doi-access=free }}
• {{Cite journal |last=Clifford |first=M. Will |orig-date= |title=On the unreasonable effectiveness of thepost-Newtonian approximation ingravitational physics |journal=PNAS |volume=108 |issue=15 |pages=5938–5945 |doi=10.1073/pnas.1103127108 |doi-access=free |pmid=21447714 |pmc=3076827 |publication-date=2011 |arxiv=1102.5192 }}

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