Graviton Polarizations

[spaghetti3451]

Homework Statement

We will treat the graviton as a symmetric $2$-index tensor field. It couples to a current $T_{\mu\nu}$ also symmetric in its two indices, which satisfies the conservation law $\partial_{\mu}T_{\mu\nu}=0$.

(a) Assume the Lagrangian is $\mathcal{L}=-\frac{1}{2}h_{\mu\nu}\Box h_{\mu\nu} + \frac{1}{M_{\text{Pl}}}h_{\mu\nu}T_{\mu\nu}.$ Solve $h_{\mu\nu}$'s equations of motion, and substitute back to find an interaction like $T_{\mu\nu}\frac{1}{k^{2}}T_{\mu\nu}$.

(b) Write out the $10$ terms in the interaction $T_{\mu\nu}\frac{1}{k^{2}}T_{\mu\nu}$ explicitly in terms of $T_{00}, T_{01},$ etc.

(c) Use current conservation to solve for $T_{\mu 1}$ in terms of $T_{\mu 0}, \omega$ and $\kappa$. Substitute into simplify the interaction. How many causally propagating degrees of freedom are there?

(d) Add to the interaction another term of the form $cT_{\mu\mu}\frac{1}{k^{2}}T_{\nu\nu}$. What value of $c$ can reduce the number of propagating modes? How many are there now?

Homework Equations

The Attempt at a Solution

(a) $\frac{\partial \mathcal{L}}{\partial h_{\mu\nu}}=-\frac{1}{2}\Box h_{\mu\nu}+\frac{1}{M_{\text{Pl}}}T_{\mu\nu}$

I'm having trouble finding $\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})}$:

$\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})} = \frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}\Big( -\frac{1}{2}h_{\mu\nu}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu} \Big) = \frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}\Big( -\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}\partial_{\sigma}\partial_{\rho}h_{\mu\nu} \Big)=-\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}\partial_{\sigma}\frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}(\partial_{\rho}h_{\mu\nu})$

 

[Fred Wright]

I think you are going about this problem the hard way. This model for the graviton is clearly an harmonic oscillator. You know from your studies of Lagrangian dynamics of the electro-magnetic field that the solution for the propagating E field, is E(r,t) = E₀e^i(k⋅r-ωt). I propose you generalize this to the graviton field by positing the solution h_μν = A_μe^ikνx_ν, where A_μ is a constant vector independent of x and t. I suggest you expand this solution in a 4x4 matrix and use the fact that the matrix is symmetrical in its indices to make it a constant, say A₀, multiplying the complex exponential terms and thus reduce the 16 terms to 10 terms.

 

[spaghetti3451]

I think you are going about this problem the hard way. This model for the graviton is clearly an harmonic oscillator.

How did you figure out that this model for the graviton is a harmonic oscillator?

 

[Fred Wright]

How did you figure out that this model for the graviton is a harmonic oscillator?

By looking at the form of the Lagrangian and the hints given in the problem.

 

[spaghetti3451]

By looking at the form of the Lagrangian and the hints given in the problem.

I am very new to field theory, so I still cannot figure out from the form of the Lagrangian and the hints in the problem that the graviton is modeled as a harmonic oscillator. It's best if I follow the precise instructions in the problem, rather than taking ingenious shortcuts.

Would you be able to point out my mistakes in part (a)?

 

[Fred Wright]

I would expand the quad operator to its explicit form then operate on all elements of h_μν. Using the Lagrangian formalism for each element of h_μν will give differential equations for that element from which you will get constants of motion and guess the solution (i.e. a constant multiplying a complex exponential).

 

[spaghetti3451]

Would you be able to provide the first few lines of the solution so that I could work out the remaining steps?

 

[nrqed]

Homework Statement

We will treat the graviton as a symmetric $2$-index tensor field. It couples to a current $T_{\mu\nu}$ also symmetric in its two indices, which satisfies the conservation law $\partial_{\mu}T_{\mu\nu}=0$.

(a) Assume the Lagrangian is $\mathcal{L}=-\frac{1}{2}h_{\mu\nu}\Box h_{\mu\nu} + \frac{1}{M_{\text{Pl}}}h_{\mu\nu}T_{\mu\nu}.$ Solve $h_{\mu\nu}$'s equations of motion, and substitute back to find an interaction like $T_{\mu\nu}\frac{1}{k^{2}}T_{\mu\nu}$.

(b) Write out the $10$ terms in the interaction $T_{\mu\nu}\frac{1}{k^{2}}T_{\mu\nu}$ explicitly in terms of $T_{00}, T_{01},$ etc.

(c) Use current conservation to solve for $T_{\mu 1}$ in terms of $T_{\mu 0}, \omega$ and $\kappa$. Substitute into simplify the interaction. How many causally propagating degrees of freedom are there?

(d) Add to the interaction another term of the form $cT_{\mu\mu}\frac{1}{k^{2}}T_{\nu\nu}$. What value of $c$ can reduce the number of propagating modes? How many are there now?

Homework Equations

The Attempt at a Solution

(a) $\frac{\partial \mathcal{L}}{\partial h_{\mu\nu}}=-\frac{1}{2}\Box h_{\mu\nu}+\frac{1}{M_{\text{Pl}}}T_{\mu\nu}$

I'm having trouble finding $\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})}$:

$\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})} = \frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}\Big( -\frac{1}{2}h_{\mu\nu}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu} \Big) = \frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}\Big( -\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}\partial_{\sigma}\partial_{\rho}h_{\mu\nu} \Big)=-\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}\partial_{\sigma}\frac{\partial}{\partial (\partial_{\gamma}h_{\alpha\beta})}(\partial_{\rho}h_{\mu\nu})$

I would first do an integration by parts on the term $h_{\mu\nu}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu}$ to bring it in the form (after dropping the total derivative) $$ - (\partial_{\sigma} h_{\mu\nu}) \partial^{\sigma}h_{\mu\nu} = - g^{\sigma \delta} (\partial_{\sigma} h_{\mu\nu}) \partial_{\delta} h_{\mu\nu} $$ and then I would calculate $\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})}$. This will be simple to do now.