How Does Graviton Polarization Affect Tensor Field Interactions?

• spaghetti3451
In summary: I would first do an integration by parts on the term ##h_{\mu\nu}\partial_{\sigma}\partial^{\sigma}h_{\mu\nu}##Okay, so would the result be:##\frac{\partial \mathcal{L}}{\partial (\partial_{\gamma}h_{\alpha\beta})} = -\frac{1}{2}h_{\mu\nu}\eta^{\sigma\rho}(\delta^{\gamma}_{\sigma}\delta^{\alpha}_{\rho}+\delta^{\gamma}_{\rho}\delta^{\alpha}_{\sigma}) = -h^{\gamma\alpha}##Yes, that looks right. Now
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?

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 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) = E0ei(k⋅r-ωt). I propose you generalize this to the graviton field by positing the solution hμν = Aμeikνxν, where Aμ 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.

Fred Wright said:
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?

failexam said:
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.

Fred Wright said:
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)?

I would expand the quad operator to its explicit form then operate on all elements of hμν. Using the Lagrangian formalism for each element of hμν 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).

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

failexam said:

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?

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.

1. What is a graviton polarization?

A graviton polarization refers to the orientation of the spin of a graviton, which is a hypothetical particle believed to be responsible for the force of gravity. It is analogous to the polarization of a photon in the electromagnetic force.

2. How many types of graviton polarizations are there?

There are two main types of graviton polarizations: longitudinal and transverse. Longitudinal polarizations have a spin parallel to the direction of motion, while transverse polarizations have a spin perpendicular to the direction of motion.

3. How do graviton polarizations affect the behavior of gravity?

The polarization of gravitons determines how they interact with other particles and how the force of gravity is transmitted. Transverse polarizations are responsible for the attractive force of gravity, while longitudinal polarizations do not contribute to the force.

4. Can graviton polarizations be observed or measured?

Currently, there is no experimental evidence for the existence of gravitons, so their polarizations cannot be observed or measured directly. However, scientists are actively researching and developing ways to detect and study gravitons.

5. How do graviton polarizations relate to the theory of quantum gravity?

The concept of graviton polarizations is a crucial component of the theory of quantum gravity, which aims to unify the theories of general relativity and quantum mechanics. Understanding the behavior of graviton polarizations is essential for developing a consistent theory of gravity on a quantum level.

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