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G. Smith
G. Smith
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In Spacetime and Geometry, Dr. CarrolCarroll provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The Lagrangian is, for reference, $$\mathcal{L}=\frac{1}{2}\left[\left(\partial_\alpha h^{\alpha\beta}\right)\left(\partial_{\beta}h\right)-\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_{\rho}h^{\alpha}_{\;\,\sigma}\right)+\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_\beta h_{\rho\sigma}\right)-\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h\right)\left(\partial_\beta h\right)\right]$$

This, as can be verified, produces the Einstein tensor when varied. Now, later on, Dr. CarrolCarroll notes that by treating $h_{\mu\nu}$ as a field propagating over Minkowski spacetime, adding coupling to matter in the Lagrangian, and by requiring it to couple to its own energy-momentum tensor/matter energy-momentum tensor, General Relativity is restored.

The part I am confused about is as follows: how is one to add coupling to matter in the Lagrangian? I assume it's constructed from factors of $h_{\mu\nu}$ and not its derivative, but I'm not sure how to do this. Any assistance would be much appreciated.

In Spacetime and Geometry, Dr. Carrol provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The Lagrangian is, for reference, $$\mathcal{L}=\frac{1}{2}\left[\left(\partial_\alpha h^{\alpha\beta}\right)\left(\partial_{\beta}h\right)-\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_{\rho}h^{\alpha}_{\;\,\sigma}\right)+\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_\beta h_{\rho\sigma}\right)-\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h\right)\left(\partial_\beta h\right)\right]$$

This, as can be verified, produces the Einstein tensor when varied. Now, later on, Dr. Carrol notes that by treating $h_{\mu\nu}$ as a field propagating over Minkowski spacetime, adding coupling to matter in the Lagrangian, and by requiring it to couple to its own energy-momentum tensor/matter energy-momentum tensor, General Relativity is restored.

The part I am confused about is as follows: how is one to add coupling to matter in the Lagrangian? I assume it's constructed from factors of $h_{\mu\nu}$ and not its derivative, but I'm not sure how to do this. Any assistance would be much appreciated.

In Spacetime and Geometry, Dr. Carroll provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The Lagrangian is, for reference, $$\mathcal{L}=\frac{1}{2}\left[\left(\partial_\alpha h^{\alpha\beta}\right)\left(\partial_{\beta}h\right)-\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_{\rho}h^{\alpha}_{\;\,\sigma}\right)+\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_\beta h_{\rho\sigma}\right)-\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h\right)\left(\partial_\beta h\right)\right]$$

This, as can be verified, produces the Einstein tensor when varied. Now, later on, Dr. Carroll notes that by treating $h_{\mu\nu}$ as a field propagating over Minkowski spacetime, adding coupling to matter in the Lagrangian, and by requiring it to couple to its own energy-momentum tensor/matter energy-momentum tensor, General Relativity is restored.

The part I am confused about is as follows: how is one to add coupling to matter in the Lagrangian? I assume it's constructed from factors of $h_{\mu\nu}$ and not its derivative, but I'm not sure how to do this. Any assistance would be much appreciated.

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John Dumancic
John Dumancic
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How does one add matter coupling terms to the linearized Lagrangian for General Relativity?

In Spacetime and Geometry, Dr. Carrol provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The Lagrangian is, for reference, $$\mathcal{L}=\frac{1}{2}\left[\left(\partial_\alpha h^{\alpha\beta}\right)\left(\partial_{\beta}h\right)-\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_{\rho}h^{\alpha}_{\;\,\sigma}\right)+\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_\beta h_{\rho\sigma}\right)-\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h\right)\left(\partial_\beta h\right)\right]$$

This, as can be verified, produces the Einstein tensor when varied. Now, later on, Dr. Carrol notes that by treating $h_{\mu\nu}$ as a field propagating over Minkowski spacetime, adding coupling to matter in the Lagrangian, and by requiring it to couple to its own energy-momentum tensor/matter energy-momentum tensor, General Relativity is restored.

The part I am confused about is as follows: how is one to add coupling to matter in the Lagrangian? I assume it's constructed from factors of $h_{\mu\nu}$ and not its derivative, but I'm not sure how to do this. Any assistance would be much appreciated.