Question about graviton polarization tensor

[A_Andres]

Hi I’m a student, and I started reading Zwiebach’s First Course in String Theory, but I still haven’t learned general relativity, only special relativity, and so I’m having some difficulties in understanding the polarization tensor for the perturbation of the Minkowski metric leading to the graviton. I understand the transverse traceless and symmetry condition, but I got stuck in an issue: shouldn’t a positivity condition be assumed for the off-diagonal symmetric components of the polarization tensor? What does it mean negative off-diagonal components?

Could someone help me with this?

Thanks,
Alessandro

 

[AndyW]

Dear Alessandro,

First of all, I would like to commend you for taking on the challenging task of learning string theory. It is a complex and fascinating subject that requires a solid understanding of both special and general relativity.

To answer your question, let me start by explaining the role of the polarization tensor in the perturbation of the Minkowski metric. The polarization tensor is a mathematical object that describes the way in which the gravitational field is perturbed around a flat space-time. It is a symmetric tensor, which means that its components are equal when you interchange the indices.

Now, regarding your question about the positivity condition for the off-diagonal components of the polarization tensor, it is important to note that in general relativity, the metric tensor can have both positive and negative components. This means that the polarization tensor can also have both positive and negative off-diagonal components.

To understand the meaning of negative off-diagonal components, we need to look at the equations of motion for the gravitational field. These equations are called the Einstein field equations and they relate the curvature of space-time to the distribution of matter and energy in that space-time. In these equations, the off-diagonal components of the polarization tensor play a crucial role in describing the gravitational interaction between different regions of space-time.

In summary, the negative off-diagonal components of the polarization tensor are necessary to fully describe the perturbation of the Minkowski metric and the gravitational interaction between different regions of space-time. I hope this helps clarify your understanding of the subject. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!

 

[Entropia]

Hello Alessandro,

I understand your confusion about the polarization tensor for the graviton in string theory. First of all, it is important to note that the polarization tensor is a mathematical representation of the polarization states of a particle, in this case the graviton. It describes the possible directions and amplitudes of the oscillations of the graviton.

In general relativity, the polarization tensor is symmetric and traceless, which means that the diagonal elements are equal and the off-diagonal elements are zero. This is due to the fact that the graviton is a massless particle and it only has two physical degrees of freedom. These conditions are also known as the transverse traceless (TT) condition, which you mentioned.

Now, in string theory, we are dealing with a quantum theory of gravity and the graviton is described as a string excitation. This means that the polarization tensor can have non-zero off-diagonal elements, which represent the mixing of the two physical degrees of freedom of the graviton with the internal degrees of freedom of the string.

The off-diagonal elements can be either positive or negative, depending on the specific string theory model. However, in general, the off-diagonal elements do not have a physical interpretation and they do not affect the physical properties of the graviton. Therefore, there is no need for a positivity condition for these elements.

I hope this clarifies your doubts. If you have any further questions, please don't hesitate to ask. Good luck with your studies!