So I have been working on this lagrangian whose equations of motion I am trying to derive. The original paper contained six equations, which I found by varying the action to corresponding four fields in the lagrangian using Mathematica, namely Zeta, Phi, Alpha, and A(Maxwell field). But one of the equations I cannot recreate (equation 6 in the paper). By varying the maxwell field and the scalar field, I was able to get the corresponding equations. In the paper, they gave 3 equations in the name of Einstein equations, out of which initially I got only two by varying w.r.t to alpha and Zeta. Then a comment suggested I consider varying the Lagrangian explicitly w.r.t to r, which was promoted to Rho(t, r), and varied wrt this new field and then reverted back to r in the final expression. The issue now is From Einstein's equation I get 4 equations (tt, rr, tr, theta theta, phi phi is same as theta theta). But by varying, i have gotten only 3 equations , and the equation which i used from the 4 einstein equation to derive equation 6 in the paper is unfortunately not there in the equation obtained through varying. 1 equation is missing. I have linked the paper https://arxiv.org/abs/2112.07455. Any suggestions would be appreciated. If you want further details, leave them in the answers.
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$\begingroup$ When possible, please provide the free link so all readers can access it - arxiv.org/abs/2112.07455 $\endgroup$– PraharCommented Jun 11, 2023 at 20:49
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$\begingroup$ sorry this is my first time asking a question in stack exchange $\endgroup$– Suriyah R KCommented Jun 13, 2023 at 18:51
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If you wish to obtain equations of motion by varying the Einstein–Hilbert action with respect to metric functions, you should ensure that you have enough variational freedom to obtain all of the equations. In this case I believe the missing equation could be obtained by modifying the last term of line element $(2)$ to be $\rho^2(r,t)d\Omega_2^2$ and varying the action with respect to this new function $\rho$. Once all the equations are obtained we can then set $\rho=r$ (after checking that such choice is compatible with EoM's).
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$\begingroup$ So I tried the method you suggested, changed r - > ρ, and carried out the variational principle. The equation I got was one of the Einstein equations (4 in total tt, tr, rr, theta theta ( phi is the same as theta theta) ). But the number of equations I get via varying the action w.r.t the metric function zeta, alpha, and rho are only 3. Unfortunately, the Einstein equation I used to derive equation 6 in the paper, is not available in these three equations. I need one more equation... $\endgroup$ Commented Jun 13, 2023 at 18:51
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$\begingroup$ @SuriyahRK Einstein equations are obtained by varying the metric. If you have specify a metric component with an explicit coordinate dependence, or a component that cannot be varied independently of others, then you might loose an equation. So another component in (2) that is written explicitly is $g_{rr}\equiv 1$. Replacing it with another function should do the trick. $\endgroup$– A.V.S.Commented Jun 14, 2023 at 16:44
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$\begingroup$ Thank you. Thats exactly what I did,n ad it worked. Thinking about it, I am just applying Einstein's equation in a round about way instead of implicitly varying w.r.t to the whole metric itself , I am renaming each component by a function and varying with it. Anyway thank you very much for the idea!! $\endgroup$ Commented Jun 14, 2023 at 17:21