- #36
ShayanJ
Gold Member
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The discussion remembers me something I read in a lecture note of general relativity so I want to mention it here both because it may be useful and also because to find out the connection I feel! But I can't give a reference because its actually a lecture note in my own language.
In that lecture note, at first it is argued that gravity should be a massless spin2 field. Then the free Fierz-Pauli Lagrangian is given:
[itex]
L=\frac 1 2 (h_{\mu \ ,\lambda}^\lambda h^{\mu \nu}_{\ \ ,\nu}-h_{\mu \lambda}^{\ \ ,\mu}h^{,\lambda})+\frac 1 4(h_{,\mu}h^{,\mu}-h_{\mu\nu}^{\ \ ,\lambda} h^{\mu\nu}_{\ \ ,\lambda})
[/itex]
(There may be some typing errors in the lecture notes so maybe this isn't right completely!)
Anyway, the EL equations coming from that Lagrangian are [itex] D^{\alpha \beta}_{\mu\nu} h_{\alpha \beta}=0 [/itex], where [itex] D^{\alpha\beta}_{\mu \nu}=(\eta_{\mu}^{\alpha}\eta_{\nu}^{\beta}-\eta_{\mu\nu}\eta^{\alpha \beta})\partial^\rho \partial_\rho+\eta_{\mu\nu}\partial^{\alpha}\partial^\beta+\eta^{\alpha \beta}\partial_\mu \partial_\nu-\eta_{\mu}^{\beta}\partial^\alpha \partial_\nu-\eta_{\nu}^{\alpha}\partial^\beta \partial_\mu[/itex], which satisfies [itex] \partial^\mu D^{\alpha \beta}_{\mu\nu}=0[/itex].
But when we put the stress-energy tensor of matter in the RHS, we'll have [itex] D^{\alpha \beta}_{\mu\nu} h_{\alpha \beta}=T_{\mu\nu} [/itex] which gives [itex] \partial^\mu T_{\mu\nu}=0 [/itex]. But this is an inconsistency because its not the stress-energy tensor of the matter alone that should be conserved but the total stress-energy tensor. So to solve this inconsistency, people started to add higher order terms to the LHS but it turns out that continuing to no finite order is enough and so an infinite series pops out. Now its Ogievetsky and Polubarinov that sum up this infinite series and show that it actually gives Einstein's Field Equations(1965,Ann.Phys. ,35, 107).
Then, as an easier solution, its proposed to vary the action [itex] S_0=\int [\phi^{\mu\nu}(\partial_\alpha\Gamma_{\mu\nu}^{\alpha}-\partial_\nu\Gamma_{\alpha\mu}^{\alpha})+\eta^{\mu\nu}(\Gamma_{\mu\nu}^{\alpha}\Gamma_{\rho\alpha}^{\rho}-\Gamma_{\beta\mu}^{\alpha}\Gamma_{\alpha\nu}^{\beta})] d^4x [/itex] w.r.t. [itex] \Gamma_{\mu\nu}^{\alpha} [/itex] and [itex] \phi^{\mu\nu} [/itex] which gives an equivalent theory to the mentioned Fierz-Pauli action. Then the Minkowski metric is replaced by a dynamical metric and the variation of the new action w.r.t. the dynamical metric is taken as the stress-energy tensor of gravity. After modifying the action farther for getting a consistent theory, the action takes the form of the Einstein-Hilbert action and so we're led to GR.(This is mentioned as Deser's method!)
I said this because I failed to see the connection between this procedure and the discussion here, so maybe someone can clarify or give some references explaining this procedure farther.(For example in calculating that infinite series, I'm really interested in seeing it but I couldn't find the paper.)
(Here's a link to the lecture note, at least you can see the equations! The things I said start at page 20)
In that lecture note, at first it is argued that gravity should be a massless spin2 field. Then the free Fierz-Pauli Lagrangian is given:
[itex]
L=\frac 1 2 (h_{\mu \ ,\lambda}^\lambda h^{\mu \nu}_{\ \ ,\nu}-h_{\mu \lambda}^{\ \ ,\mu}h^{,\lambda})+\frac 1 4(h_{,\mu}h^{,\mu}-h_{\mu\nu}^{\ \ ,\lambda} h^{\mu\nu}_{\ \ ,\lambda})
[/itex]
(There may be some typing errors in the lecture notes so maybe this isn't right completely!)
Anyway, the EL equations coming from that Lagrangian are [itex] D^{\alpha \beta}_{\mu\nu} h_{\alpha \beta}=0 [/itex], where [itex] D^{\alpha\beta}_{\mu \nu}=(\eta_{\mu}^{\alpha}\eta_{\nu}^{\beta}-\eta_{\mu\nu}\eta^{\alpha \beta})\partial^\rho \partial_\rho+\eta_{\mu\nu}\partial^{\alpha}\partial^\beta+\eta^{\alpha \beta}\partial_\mu \partial_\nu-\eta_{\mu}^{\beta}\partial^\alpha \partial_\nu-\eta_{\nu}^{\alpha}\partial^\beta \partial_\mu[/itex], which satisfies [itex] \partial^\mu D^{\alpha \beta}_{\mu\nu}=0[/itex].
But when we put the stress-energy tensor of matter in the RHS, we'll have [itex] D^{\alpha \beta}_{\mu\nu} h_{\alpha \beta}=T_{\mu\nu} [/itex] which gives [itex] \partial^\mu T_{\mu\nu}=0 [/itex]. But this is an inconsistency because its not the stress-energy tensor of the matter alone that should be conserved but the total stress-energy tensor. So to solve this inconsistency, people started to add higher order terms to the LHS but it turns out that continuing to no finite order is enough and so an infinite series pops out. Now its Ogievetsky and Polubarinov that sum up this infinite series and show that it actually gives Einstein's Field Equations(1965,Ann.Phys. ,35, 107).
Then, as an easier solution, its proposed to vary the action [itex] S_0=\int [\phi^{\mu\nu}(\partial_\alpha\Gamma_{\mu\nu}^{\alpha}-\partial_\nu\Gamma_{\alpha\mu}^{\alpha})+\eta^{\mu\nu}(\Gamma_{\mu\nu}^{\alpha}\Gamma_{\rho\alpha}^{\rho}-\Gamma_{\beta\mu}^{\alpha}\Gamma_{\alpha\nu}^{\beta})] d^4x [/itex] w.r.t. [itex] \Gamma_{\mu\nu}^{\alpha} [/itex] and [itex] \phi^{\mu\nu} [/itex] which gives an equivalent theory to the mentioned Fierz-Pauli action. Then the Minkowski metric is replaced by a dynamical metric and the variation of the new action w.r.t. the dynamical metric is taken as the stress-energy tensor of gravity. After modifying the action farther for getting a consistent theory, the action takes the form of the Einstein-Hilbert action and so we're led to GR.(This is mentioned as Deser's method!)
I said this because I failed to see the connection between this procedure and the discussion here, so maybe someone can clarify or give some references explaining this procedure farther.(For example in calculating that infinite series, I'm really interested in seeing it but I couldn't find the paper.)
(Here's a link to the lecture note, at least you can see the equations! The things I said start at page 20)