Comparison of covariant form of Maxwell equations with Einstein's GR












1














We know, the the vector form of Maxwell equations
begin{align}
vecnablacdotvec{E} &= 4pirho label{Diff I}\
vecnablatimesvec{B} &= dfrac{4pi}{c} vec{j}+dfrac{1}{c}dfrac{partialvec{E}}{partial t} label{Diff IV}\
vecnablatimesvec{E} &= -dfrac{1}{c}dfrac{partialvec{B}}{partial t} label{Diff III}\
vecnablacdotvec{B} &= 0 label{Diff II}
end{align}



The last two of them allow us to introduce the potentials:
begin{align}
vec{E} &= -frac1c frac{partial vec{A}}{partial t} - vecnablaphi\
vec{B} &= vecnablatimesvec A
end{align}

which tells us about gauge invariance of equations.



All four of Maxwell's equations can be written compactly as



begin{align}
partial_{mu}F^{munu} &= frac{4pi}{c}j^{nu} tag{1}\
partial_{[mu}F_{alphabeta]} &= 0;. tag{2}
end{align}



And according to the last one equation, the first one we can rewrite (use preferred gauge) in form:
begin{equation}
Box A^{mu} = -frac{4pi}{c} j^{mu}
end{equation}



Now we consider the Einstein GR equations:
begin{equation}
R_{munu} = 8pi G (T_{munu} - frac12g_{munu}T).
end{equation}



Or in "$Gamma-$field" form (indexes are omitted):
begin{equation}
partial Gamma - partial Gamma + GammaGamma - GammaGamma = 8pi G (T_{munu} - frac12g_{munu}T).
end{equation}



We know, in weak field limit the equations get simply form externally similar to Maxwell ones (so called gravitomagnetism) in 3-vector form, or in covariant form:
begin{equation}
Box h_{munu} = -16pi G (T_{munu} - frac12eta_{munu}T)
end{equation}



Thus, the question: Why for covariant form of Maxwell equations we need two different entities of equations, but for the GR the only one? Or another words, can we write the Einstein GR equations for weak field limit similar to Maxwell equations in field form, not via potentials?










share|cite|improve this question





























    1














    We know, the the vector form of Maxwell equations
    begin{align}
    vecnablacdotvec{E} &= 4pirho label{Diff I}\
    vecnablatimesvec{B} &= dfrac{4pi}{c} vec{j}+dfrac{1}{c}dfrac{partialvec{E}}{partial t} label{Diff IV}\
    vecnablatimesvec{E} &= -dfrac{1}{c}dfrac{partialvec{B}}{partial t} label{Diff III}\
    vecnablacdotvec{B} &= 0 label{Diff II}
    end{align}



    The last two of them allow us to introduce the potentials:
    begin{align}
    vec{E} &= -frac1c frac{partial vec{A}}{partial t} - vecnablaphi\
    vec{B} &= vecnablatimesvec A
    end{align}

    which tells us about gauge invariance of equations.



    All four of Maxwell's equations can be written compactly as



    begin{align}
    partial_{mu}F^{munu} &= frac{4pi}{c}j^{nu} tag{1}\
    partial_{[mu}F_{alphabeta]} &= 0;. tag{2}
    end{align}



    And according to the last one equation, the first one we can rewrite (use preferred gauge) in form:
    begin{equation}
    Box A^{mu} = -frac{4pi}{c} j^{mu}
    end{equation}



    Now we consider the Einstein GR equations:
    begin{equation}
    R_{munu} = 8pi G (T_{munu} - frac12g_{munu}T).
    end{equation}



    Or in "$Gamma-$field" form (indexes are omitted):
    begin{equation}
    partial Gamma - partial Gamma + GammaGamma - GammaGamma = 8pi G (T_{munu} - frac12g_{munu}T).
    end{equation}



    We know, in weak field limit the equations get simply form externally similar to Maxwell ones (so called gravitomagnetism) in 3-vector form, or in covariant form:
    begin{equation}
    Box h_{munu} = -16pi G (T_{munu} - frac12eta_{munu}T)
    end{equation}



    Thus, the question: Why for covariant form of Maxwell equations we need two different entities of equations, but for the GR the only one? Or another words, can we write the Einstein GR equations for weak field limit similar to Maxwell equations in field form, not via potentials?










    share|cite|improve this question



























      1












      1








      1


      0





      We know, the the vector form of Maxwell equations
      begin{align}
      vecnablacdotvec{E} &= 4pirho label{Diff I}\
      vecnablatimesvec{B} &= dfrac{4pi}{c} vec{j}+dfrac{1}{c}dfrac{partialvec{E}}{partial t} label{Diff IV}\
      vecnablatimesvec{E} &= -dfrac{1}{c}dfrac{partialvec{B}}{partial t} label{Diff III}\
      vecnablacdotvec{B} &= 0 label{Diff II}
      end{align}



      The last two of them allow us to introduce the potentials:
      begin{align}
      vec{E} &= -frac1c frac{partial vec{A}}{partial t} - vecnablaphi\
      vec{B} &= vecnablatimesvec A
      end{align}

      which tells us about gauge invariance of equations.



      All four of Maxwell's equations can be written compactly as



      begin{align}
      partial_{mu}F^{munu} &= frac{4pi}{c}j^{nu} tag{1}\
      partial_{[mu}F_{alphabeta]} &= 0;. tag{2}
      end{align}



      And according to the last one equation, the first one we can rewrite (use preferred gauge) in form:
      begin{equation}
      Box A^{mu} = -frac{4pi}{c} j^{mu}
      end{equation}



      Now we consider the Einstein GR equations:
      begin{equation}
      R_{munu} = 8pi G (T_{munu} - frac12g_{munu}T).
      end{equation}



      Or in "$Gamma-$field" form (indexes are omitted):
      begin{equation}
      partial Gamma - partial Gamma + GammaGamma - GammaGamma = 8pi G (T_{munu} - frac12g_{munu}T).
      end{equation}



      We know, in weak field limit the equations get simply form externally similar to Maxwell ones (so called gravitomagnetism) in 3-vector form, or in covariant form:
      begin{equation}
      Box h_{munu} = -16pi G (T_{munu} - frac12eta_{munu}T)
      end{equation}



      Thus, the question: Why for covariant form of Maxwell equations we need two different entities of equations, but for the GR the only one? Or another words, can we write the Einstein GR equations for weak field limit similar to Maxwell equations in field form, not via potentials?










      share|cite|improve this question















      We know, the the vector form of Maxwell equations
      begin{align}
      vecnablacdotvec{E} &= 4pirho label{Diff I}\
      vecnablatimesvec{B} &= dfrac{4pi}{c} vec{j}+dfrac{1}{c}dfrac{partialvec{E}}{partial t} label{Diff IV}\
      vecnablatimesvec{E} &= -dfrac{1}{c}dfrac{partialvec{B}}{partial t} label{Diff III}\
      vecnablacdotvec{B} &= 0 label{Diff II}
      end{align}



      The last two of them allow us to introduce the potentials:
      begin{align}
      vec{E} &= -frac1c frac{partial vec{A}}{partial t} - vecnablaphi\
      vec{B} &= vecnablatimesvec A
      end{align}

      which tells us about gauge invariance of equations.



      All four of Maxwell's equations can be written compactly as



      begin{align}
      partial_{mu}F^{munu} &= frac{4pi}{c}j^{nu} tag{1}\
      partial_{[mu}F_{alphabeta]} &= 0;. tag{2}
      end{align}



      And according to the last one equation, the first one we can rewrite (use preferred gauge) in form:
      begin{equation}
      Box A^{mu} = -frac{4pi}{c} j^{mu}
      end{equation}



      Now we consider the Einstein GR equations:
      begin{equation}
      R_{munu} = 8pi G (T_{munu} - frac12g_{munu}T).
      end{equation}



      Or in "$Gamma-$field" form (indexes are omitted):
      begin{equation}
      partial Gamma - partial Gamma + GammaGamma - GammaGamma = 8pi G (T_{munu} - frac12g_{munu}T).
      end{equation}



      We know, in weak field limit the equations get simply form externally similar to Maxwell ones (so called gravitomagnetism) in 3-vector form, or in covariant form:
      begin{equation}
      Box h_{munu} = -16pi G (T_{munu} - frac12eta_{munu}T)
      end{equation}



      Thus, the question: Why for covariant form of Maxwell equations we need two different entities of equations, but for the GR the only one? Or another words, can we write the Einstein GR equations for weak field limit similar to Maxwell equations in field form, not via potentials?







      general-relativity maxwell-equations gauge-invariance






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      edited 4 hours ago

























      asked 5 hours ago









      Sergio

      887823




      887823






















          2 Answers
          2






          active

          oldest

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          2














          If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $U(1)$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $A_mu$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $F_{munu}$ of gauge covariant derivatives is analogous to the commutator $R_{munurhosigma}$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $R_{mu[nurhosigma]}=0$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.






          share|cite|improve this answer





















          • Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
            – Sergio
            4 hours ago



















          2















          1. The counterpart to $(g,R)$ in GR is $(A,F)$ in E&M.


          2. If you write $F=F(A)$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.


          3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.







          share|cite|improve this answer





















          • Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
            – Sergio
            43 mins ago











          Your Answer





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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2














          If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $U(1)$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $A_mu$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $F_{munu}$ of gauge covariant derivatives is analogous to the commutator $R_{munurhosigma}$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $R_{mu[nurhosigma]}=0$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.






          share|cite|improve this answer





















          • Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
            – Sergio
            4 hours ago
















          2














          If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $U(1)$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $A_mu$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $F_{munu}$ of gauge covariant derivatives is analogous to the commutator $R_{munurhosigma}$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $R_{mu[nurhosigma]}=0$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.






          share|cite|improve this answer





















          • Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
            – Sergio
            4 hours ago














          2












          2








          2






          If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $U(1)$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $A_mu$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $F_{munu}$ of gauge covariant derivatives is analogous to the commutator $R_{munurhosigma}$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $R_{mu[nurhosigma]}=0$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.






          share|cite|improve this answer












          If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $U(1)$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $A_mu$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $F_{munu}$ of gauge covariant derivatives is analogous to the commutator $R_{munurhosigma}$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $R_{mu[nurhosigma]}=0$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 4 hours ago









          J.G.

          9,11921528




          9,11921528












          • Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
            – Sergio
            4 hours ago


















          • Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
            – Sergio
            4 hours ago
















          Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
          – Sergio
          4 hours ago




          Ok, as I understand, the answer, the Bianchi relation is the GR analog to the (2).
          – Sergio
          4 hours ago











          2















          1. The counterpart to $(g,R)$ in GR is $(A,F)$ in E&M.


          2. If you write $F=F(A)$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.


          3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.







          share|cite|improve this answer





















          • Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
            – Sergio
            43 mins ago
















          2















          1. The counterpart to $(g,R)$ in GR is $(A,F)$ in E&M.


          2. If you write $F=F(A)$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.


          3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.







          share|cite|improve this answer





















          • Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
            – Sergio
            43 mins ago














          2












          2








          2







          1. The counterpart to $(g,R)$ in GR is $(A,F)$ in E&M.


          2. If you write $F=F(A)$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.


          3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.







          share|cite|improve this answer













          1. The counterpart to $(g,R)$ in GR is $(A,F)$ in E&M.


          2. If you write $F=F(A)$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.


          3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 4 hours ago









          Qmechanic

          101k121821140




          101k121821140












          • Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
            – Sergio
            43 mins ago


















          • Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
            – Sergio
            43 mins ago
















          Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
          – Sergio
          43 mins ago




          Thank you for reply. Did you know the some articles, where Bianchi relation are violate in manner of existing of graviromagnetic monopole?
          – Sergio
          43 mins ago


















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