The weak field limit metric setup












1












$begingroup$


Assume that



$$g_{ab}=m_{ab}+h_{ab}$$



where $m_{ab}=text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying.



$1)$ I thought that $text{diag}(1,-1,-1,-1)$ $textbf{was}$ $g_{ab}$???



Now the convariant metric is



$$g^{ab}= m^{ab}-m^{ac}m^{bd}h_{cd}$$



$2)$ I cannot see how the RHS of this has been derived, especially with respect to the minus sign.










share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    Assume that



    $$g_{ab}=m_{ab}+h_{ab}$$



    where $m_{ab}=text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying.



    $1)$ I thought that $text{diag}(1,-1,-1,-1)$ $textbf{was}$ $g_{ab}$???



    Now the convariant metric is



    $$g^{ab}= m^{ab}-m^{ac}m^{bd}h_{cd}$$



    $2)$ I cannot see how the RHS of this has been derived, especially with respect to the minus sign.










    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Assume that



      $$g_{ab}=m_{ab}+h_{ab}$$



      where $m_{ab}=text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying.



      $1)$ I thought that $text{diag}(1,-1,-1,-1)$ $textbf{was}$ $g_{ab}$???



      Now the convariant metric is



      $$g^{ab}= m^{ab}-m^{ac}m^{bd}h_{cd}$$



      $2)$ I cannot see how the RHS of this has been derived, especially with respect to the minus sign.










      share|cite|improve this question











      $endgroup$




      Assume that



      $$g_{ab}=m_{ab}+h_{ab}$$



      where $m_{ab}=text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying.



      $1)$ I thought that $text{diag}(1,-1,-1,-1)$ $textbf{was}$ $g_{ab}$???



      Now the convariant metric is



      $$g^{ab}= m^{ab}-m^{ac}m^{bd}h_{cd}$$



      $2)$ I cannot see how the RHS of this has been derived, especially with respect to the minus sign.







      differential-geometry mathematical-physics tensors general-relativity






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      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 22 '18 at 19:48







      Permian

















      asked Dec 22 '18 at 17:43









      PermianPermian

      2,2681135




      2,2681135






















          1 Answer
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          1












          $begingroup$

          1) That is only when you don't have to worry about gravity. Weak gravitational fields. That is the usual case. When you only need special relativity rather than general relativity.



          2) Take the inverse of both sides of the equation $g_{ab}=m_{ab}+h_{ab}$. Rename the $a$ to $e$ here so can check whether $g_{ea} g^{ab} = delta_e^b$. That is to expand and simplify the following:



          $$
          (m_{ea}+h_{ea}) times (m^{ab}-m^{ac} m^{bd} h_{cd})\
          $$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
            $endgroup$
            – Permian
            Dec 23 '18 at 11:16











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          1 Answer
          1






          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          1) That is only when you don't have to worry about gravity. Weak gravitational fields. That is the usual case. When you only need special relativity rather than general relativity.



          2) Take the inverse of both sides of the equation $g_{ab}=m_{ab}+h_{ab}$. Rename the $a$ to $e$ here so can check whether $g_{ea} g^{ab} = delta_e^b$. That is to expand and simplify the following:



          $$
          (m_{ea}+h_{ea}) times (m^{ab}-m^{ac} m^{bd} h_{cd})\
          $$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
            $endgroup$
            – Permian
            Dec 23 '18 at 11:16
















          1












          $begingroup$

          1) That is only when you don't have to worry about gravity. Weak gravitational fields. That is the usual case. When you only need special relativity rather than general relativity.



          2) Take the inverse of both sides of the equation $g_{ab}=m_{ab}+h_{ab}$. Rename the $a$ to $e$ here so can check whether $g_{ea} g^{ab} = delta_e^b$. That is to expand and simplify the following:



          $$
          (m_{ea}+h_{ea}) times (m^{ab}-m^{ac} m^{bd} h_{cd})\
          $$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
            $endgroup$
            – Permian
            Dec 23 '18 at 11:16














          1












          1








          1





          $begingroup$

          1) That is only when you don't have to worry about gravity. Weak gravitational fields. That is the usual case. When you only need special relativity rather than general relativity.



          2) Take the inverse of both sides of the equation $g_{ab}=m_{ab}+h_{ab}$. Rename the $a$ to $e$ here so can check whether $g_{ea} g^{ab} = delta_e^b$. That is to expand and simplify the following:



          $$
          (m_{ea}+h_{ea}) times (m^{ab}-m^{ac} m^{bd} h_{cd})\
          $$






          share|cite|improve this answer









          $endgroup$



          1) That is only when you don't have to worry about gravity. Weak gravitational fields. That is the usual case. When you only need special relativity rather than general relativity.



          2) Take the inverse of both sides of the equation $g_{ab}=m_{ab}+h_{ab}$. Rename the $a$ to $e$ here so can check whether $g_{ea} g^{ab} = delta_e^b$. That is to expand and simplify the following:



          $$
          (m_{ea}+h_{ea}) times (m^{ab}-m^{ac} m^{bd} h_{cd})\
          $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 22 '18 at 20:01









          AHusainAHusain

          2,8282916




          2,8282916












          • $begingroup$
            How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
            $endgroup$
            – Permian
            Dec 23 '18 at 11:16


















          • $begingroup$
            How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
            $endgroup$
            – Permian
            Dec 23 '18 at 11:16
















          $begingroup$
          How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
          $endgroup$
          – Permian
          Dec 23 '18 at 11:16




          $begingroup$
          How could we see this inverse a priori, rather than just checking (once it has been given) that it is the inverse?
          $endgroup$
          – Permian
          Dec 23 '18 at 11:16


















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