查看“︁後牛頓形式論”︁的源代码

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在不同的[[引力度规理论]]中，决定时空度规的场方程具有很大的差异。但是，在弱场和慢运动及低能的情况下，几乎所有度规理论的时空度规都具有相同的结构，都可以写成[[闵可夫斯基]][[度规]]加上微擾，并按照由系统的物质变量所定义的各种引力势的幂级数展开。各种度规理论都具有相同形式的度规展开式，它们的区别仅在于展开系数有不同的值。这样，就可以用一个统一的后牛顿理论来描述各种度规理论。这样一个统一的理论称为
[[参数化后牛顿形式体系|参数化后牛顿（PPN）形式体系]]，度规展开式中的展开系数称为PPN参数。

== 坐标系 ==
采用近整体[[罗伦茲]]坐标系，其中的坐标为<math>\left( t,{{x}^{1}},{{x}^{2}},{{x}^{3}} \right)</math>。始终用3维[[欧几里得]]矢量记号。所有的坐标任意性（“规范自由度”）已用对标准的PPN规范专门化的坐标除去。

== 物质变量 ==

*（1）<math>\rho =</math>在与引力作用着的物质瞬时共动局部自由降落系中测量的静质量密度。
*（2）<math>{{v}^{i}}=\frac{d{{x}^{i}}}{dt}=</math>物质的坐标速度。
*（3）<math>{{w}^{i}}=</math>PPN坐标系（相对于宇宙平均静系）的坐标速度。
*（4）<math>p=</math>在与物质瞬时共动的自由降落系中测量的压强。
*（5）<math>\pi =</math>单位静质量的内能，它包括所有形式的非静质量、非引力的能量，即压力能和热能。

== PPN参数 ==
<math>\gamma ,\beta ,\xi ,{{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}},{{\zeta }_{1}},{{\zeta }_{2}},{{\zeta }_{3}},{{\zeta }_{4}}</math>

== 度规势 ==

*1、<math>U=\int{\frac{\rho ({x}')}{\left| \vec{x}-{\vec{x}}' \right|}}{{d}^{3}}{x}'</math>

*2、<math>{{U}_{ij}}=\int{\frac{\rho ({\vec{x}}')}{{\left| \vec{x}-{\vec{x}}' \right|}^3}({{x}_{i}}-{{x}_{i}}^{\prime })({{x}_{j}}-{{x}_{j}}^{\prime }){{d}^{3}}{x}'}</math>

*3、<math>{{\Phi }_{w}}=\int{\frac{\rho ({\vec{x}}')\rho ({\vec{x}}'')}{{{\left| \vec{x}-{\vec{x}}' \right|}^{3}}}\left( \left( \vec{x}-{\vec{x}}' \right)\cdot \left( \frac{\left( {\vec{x}}'-{\vec{x}}'' \right)}{\left| \vec{x}-{\vec{x}}'' \right|}-\frac{\left( \vec{x}-{\vec{x}}'' \right)}{\left| {\vec{x}}'-{\vec{x}}'' \right|} \right) \right){{d}^{3}}{x}'{{d}^{3}}{x}''}</math>

*4、<math>A=\int{\frac{\rho ({\vec{x}}'){{\left( {\vec{v}}'\centerdot (\vec{x}-{\vec{x}}') \right)}^{2}}}{{{\left| \vec{x}-{\vec{x}}' \right|}^{3}}}}{{d}^{3}}{x}'</math>

*5、<math>{{\Phi }_{1}}=\int{\frac{\rho ({\vec{x}'}){{{{v}'}}^{2}}}{\left| \vec{x}-{\vec{x}}' \right|}}{{d}^{3}}{x}'</math>

*5、<math>{{\Phi }_{1}}=\int{\frac{\rho ({\vec{x}'}){{{{v}'}}^{2}}}{\left| \vec{x}-{\vec{x}}' \right|}}{{d}^{3}}{x}'</math>


*6、<math>{{\Phi }_{2}}=\int{\frac{\rho ({\vec{x}}')U({\vec{x}}')}{\left| \vec{x}-{\vec{x}}' \right|}}{{d}^{3}}{x}'</math>

*7、<math>{{\Phi }_{3}}=\int{\frac{\rho ({\vec{x}}')\pi ({\vec{x}}')}{\left| \vec{x}-{\vec{x}}' \right|}}{{d}^{3}}{x}'</math>	

*8、<math>{{\Phi }_{4}}=\int{\frac{p({\vec{x}}')}{\left| \vec{x}-{\vec{x}}' \right|}}{{d}^{3}}{x}'</math>

*9、<math>{{V}_{i}}=\int{\frac{\rho ({\vec{x}}'){{{v}_{i}}'}}{\left| \vec{x}-{\vec{x}}' \right|}{{d}^{3}}{x}'}</math>

*10、<math>{{W}_{i}}=\int{\frac{\rho ({\vec{x}}')\left( {\vec{v}}'\centerdot (\vec{x}-{\vec{x}}') \right)\left( {{x}_{i}}-{{x}_{i}}^{\prime } \right)}{{{\left| \vec{x}-{\vec{x}}' \right|}^{3}}}{{d}^{3}}{x}'}</math>

== 度规 ==
*<math>\begin{align}
 & {{g}_{00}}=-1+2U-2\beta {{U}^{2}}-2\xi {{\Phi }_{w}} \\ 
 & \mathop{{}}_{{}}+(2\gamma +2+{{\alpha }_{3}}+{{\zeta }_{1}}-2\xi ){{\Phi }_{1}} \\ 
 & \mathop{{}}_{{}}+2(3\gamma -2\beta +1+{{\zeta }_{2}}+\xi ){{\Phi }_{2}} \\ 
 & \mathop{{}}_{{}}\text{+}2(1+{{\zeta }_{3}}){{\Phi }_{3}} \\ 
 & \mathop{{}}_{{}}+2(3\gamma +3{{\zeta }_{4}}-2\xi ){{\Phi }_{4}} \\ 
 & \mathop{{}}_{{}}-({{\zeta }_{1}}-2\xi )A-({{\alpha }_{1}}-{{\alpha }_{2}}-{{\alpha }_{3}}){{w}^{2}}U \\ 
 & \mathop{{}}_{{}}-{{\alpha }_{2}}{{w}^{i}}{{w}^{j}}{{U}_{ij}}+(2{{\alpha }_{3}}-{{\alpha }_{1}}){{w}^{i}}{{V}_{i}}+O\left( {{\varepsilon }^{3}} \right) \\ 
\end{align}</math>

*<math>\begin{align}
  & {{g}_{0i}}=-\frac{1}{2}\left( 4\gamma +3+{{\alpha }_{1}}-{{\alpha }_{2}}+{{\zeta }_{1}}-2\xi  \right){{V}_{i}} \\ 
 & -\frac{1}{2}\left( 1+{{\alpha }_{2}}-{{\zeta }_{1}}+2\xi  \right){{W}_{i}}-\frac{1}{2}\left( {{\alpha }_{1}}-2{{\alpha }_{2}} \right){{w}^{i}}U-{{\alpha }_{2}}{{w}^{i}}{{U}_{ij}}+O\left( {{\varepsilon }^{5/2}} \right) \\ 
\end{align}</math>

*<math>\begin{align}{{g}_{ij}}=(1+2\gamma U){{\delta }_{ij}}+O\left( {{\varepsilon }^{2}} \right)\end{align}</math>

== 应力能量张量（理想流体） ==
*<math>{{T}^{00}}=\rho \left( 1+\pi +{{v}^{2}}+2U \right)</math>
*<math>{{T}^{0i}}=\rho \left( 1+\pi +{{v}^{2}}+2U+\frac{p}{\rho } \right){{v}^{i}}</math>
*<math>{{T}^{ij}}=\rho {{v}^{i}}{{v}^{j}}\left( 1+\pi +{{v}^{2}}+2U+\frac{p}{\rho } \right)+p{{\delta }^{ij}}\left( 1-2\gamma U \right)</math>

== 运动方程 ==
*(1)受应力的物质：<math>{{T}^{\mu \nu }}_{;\nu }=0</math>
*(2)检验物体：<math>\frac{{{d}^{2}}{{x}^{\mu }}}{d{{\lambda }^{2}}}+\Gamma _{\alpha \beta }^{\mu }{{u}^{\alpha }}{{u}^{\beta }}=0</math>
*(3)Maxswell方程组：<math>{{F}^{\mu \nu }}_{;\nu }={{\mu }_{0}}{{J}^{\mu }},{{F}_{\mu \nu }}={{A}_{\nu ;\mu }}-{{A}_{\mu ;\nu }}</math>

==各种度规理论的PPN参数比较==
{|class="wikitable"
|- bgcolor="#eee0e0" style="white-space: nowrap"
!理论!!任意常数或函数!!宇宙匹配参数!!<math>\gamma </math>!!<math>\beta </math>!!<math>\xi </math>!!<math>{{\alpha }_{1}}</math>!!<math>{{\alpha }_{2}}</math>
|-
|bgcolor=#cccccc colspan=8|[[标准理论]]
|-
|[[广义相对论]]||无||无||1||1||0||0||0
|-
|bgcolor=#cccccc colspan=8|[[标量-张量理论]]
|-
|[[Brans-Dicke]]||<math>{{\omega }_{BD}}</math>||<math>{{\phi }_{0}}</math>||<math>\frac{1+{{\omega }_{BD}}}{2+{{\omega }_{BD}}}</math>||1||0||0||0
|-
|[[一般]]||<math>A\left( \varphi  \right),V\left( \varphi  \right)</math>||<math>{{\varphi }_{0}}</math>||<math>\frac{1+\omega }{2+\omega }</math>||<math>1+\Lambda </math>||0||0||0
|-
|bgcolor=#cccccc colspan=8|[[矢量-张量理论]]
|-
|[[无限制]]||<math>\omega ,{{c}_{1}},{{c}_{2}},{{c}_{3}},{{c}_{4}}</math>||<math>u</math>||<math>{\gamma }'</math>||<math>{\beta }'</math>||0||<math>{{\alpha }'_{1}}</math>||<math>{{\alpha }'_{2}}</math>
|-
|[[Einstein-Æther]] ||<math>{{c}_{1}},{{c}_{2}},{{c}_{3}},{{c}_{4}}</math>||无||1||1||0||<math>{{\alpha }'_{1}}</math>||<math>{{\alpha }'_{2}}</math>
|-
|bgcolor=#cccccc colspan=8|[[Rosen理论]]
|-
|[[Rosen’s bimetric]]||无||<math>{{c}_{0}},{{c}_{1}}</math>||1||1||0||0||<math>\frac{{{c}_{0}}}{{{c}_{1}}}-1</math>
|-
|bgcolor=#cccccc colspan=8|[[ECT理论]]
|-
|[[不考虑自旋场]]||<math>{\beta }</math>||无||<math>\frac{1}{1-\beta }</math>||1||0||0||0
|-
|bgcolor=#cccccc colspan=8|[[Nordtvedt理论]]
|-
|[[Will]]||<math>{{c}_{1}}=-1,{{c}_{2}}={{c}_{3}}={{c}_{4}}=0</math>||无||1||1||0||0||<math>\frac{2{{u}^{2}}}{2+{{u}^{2}}}</math>
|-
|[[Hellings]]||<math>\begin{align}
  & {{c}_{1}}=2,{{c}_{2}}=2\omega , \\ 
 & {{c}_{1}}+{{c}_{2}}+{{c}_{3}}=0,{{c}_{4}}=0 \\ 
\end{align}</math>
||无||<math>{{f}_{1}}\left( \omega ,u \right)</math>||<math>{{f}_{2}}\left( \omega ,u \right)</math>||0||<math>{{f}_{3}}\left( \omega ,u \right)</math>||<math>{{f}_{4}}\left( \omega ,u \right)</math>
|}

==参考文献==
# {{Cite book | author = 郑庆璋，崔世治 | title = 广义相对论基本教程 | location = 广州 | publisher = 中山大学出版社 |year= 1991 |month=  |accessdate =| url = | language= zh| quote =}}
# {{Cite book | author = C.M.Will | title = Theory and experiment in gravitational Physics | location =Canbridge  | publisher = Cambridge Uni.,Press |year= 1981 |month=  |accessdate =| url = | language= en| quote =}}
# {{Cite book | author = 秦荣先，阎永廉 | title = 广义相对论与引力理论实验检验 | location =上海  | publisher = 上海科技日报社 |year= 1987 |month=  |accessdate =| url = | language= zh| quote =}}
# {{cite journal en 
 | quotes = 
 | last =Will 
 | first =C.M. 
 | authorlink = 
 | coauthors = 
 | date = 
 | year =2009 
 | month = 
 | title =The Confrontation between General Relativity and Experiment 
 | journal =Living Rev. Relativity 
 | volume = 
 | issue = 
 | pages = 
 | publisher = 
 | location = 
 | issn = 
 | pmid = 
 | doi = 
 | bibcode = 
 | oclc = 
 | id = 
 | url =http://www.livingreviews.org/lrr-2006-3 
 | language =en 
 | format = 
 | accessdate = 
 | laysummary = 
 | laysource = 
 | laydate = 
 | quote = 
 | archive-date =2019-12-10 
 | archive-url =https://web.archive.org/web/20191210094540/http://www.livingreviews.org/lrr-2006-3 
 | dead-url =yes 
 }}

{{廣義相對論}}
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[[Category:物理学史]]
[[Category:广义相对论]]
[[Category:引力理论]]
[[Category:参数化后牛顿形式]]