Linearized Gravity and the Transverse-Traceless Gauge Conditions

In summary, the conversation discusses the justification for imposing the conditions (1.1) and (1.2) in the Lorenz gauge in order to find the two physical polarization states of a gravitational wave. The conversation also mentions the relevant equations, including the vacuum Einstein field equation and the vacuum wave equation, as well as the fact that the wave amplitude is orthogonal to the wave vector. The attempt at a solution involves confusion and a discussion of Schutz's book "A First Course in General Relativity," where it is shown that infinitesimal coordinate transformations can lead to equations that allow for the imposition of conditions (1.1) and (1.2). However, it is not clear how this implies that the conditions can be imposed.
  • #1
Alexrey
35
0

Homework Statement


I'm working on some things to do with linearized gravitational radiation and I'm trying to justify the claim that in the Lorenz gauge, where [tex]\partial_{\nu}\bar{h}^{\mu\nu}=0 (1.1),[/tex] we are able to impose the additional conditions [tex]A_{\alpha}^{\alpha}=0 (1.2)[/tex] and [tex]A_{\alpha\beta}u^{\beta}=0 (1.3)[/tex] in order to find the two physical polarization states of a gravitational wave. All of the books that I have looked at so far have just stated that we are able to impose (1.1) and (1.2) without any workings of how they achieved this claim.


Homework Equations


Equations (1.1), (1.2), (1.3) as well as the vacuum Einstein field equation [tex]\square\overline{h}_{\mu\nu}=0[/tex] (where the bar denotes the use of the trace reverse metric perturbation) which leads to the vacuum wave equation [tex]\overline{h}_{\mu\nu}=\Re(A_{\mu\nu}e^{ik_{\sigma}x^{\sigma}}).[/tex] In addition to this, it might be helpful to know that the wave amplitude [tex]A_{\mu\nu}[/tex] is orthogonal to the wave vector [tex]k_{\nu},[/tex] that is, [tex]k_{\nu}A^{\mu\nu}=0.[/tex] which removes 4 degrees of freedom from the metric perturbation.

The Attempt at a Solution


As it stands I am quite confused and do not know really know where to start with proving that equations (1.1) and (1.2) are possible. In Schutz book "A First Course in General Relativity" after some calculations (on page 205 if you have the book) he does show that under an infinitesimal coordinate transformation we get [tex]A_{\alpha\beta}^{'}=A_{\alpha\beta}-ik_{\beta}B_{\alpha}-ik_{\alpha}B_{\beta}+i\eta_{\alpha\beta}k_{\mu}B^{\mu}.[/tex] where we can choose the [tex]B_{\alpha}[/tex] to impose (1.1) and (1.2).
 
Physics news on Phys.org
  • #2
However, I do not really understand how this implies that we can impose conditions (1.1) and (1.2) as it doesn't seem to be explicitly stated. Any help would be much appreciated!
 

What is linearized gravity?

Linearized gravity is a mathematical approximation of general relativity that simplifies the equations by assuming that the gravitational field is weak. This approximation is useful for studying small perturbations in the curvature of spacetime.

What are the transverse-traceless gauge conditions?

The transverse-traceless gauge conditions are a set of constraints that are used to simplify the equations of linearized gravity. They require that the perturbations in the metric tensor be transverse to the direction of propagation and have no trace, meaning that the sum of the diagonal elements of the tensor is zero.

What is the physical significance of the transverse-traceless gauge conditions?

The transverse-traceless gauge conditions have physical significance because they correspond to the physical degrees of freedom of gravitational waves. This means that in this gauge, the equations of linearized gravity only describe the behavior of gravitational waves, making it easier to interpret the results.

What are the advantages of using the transverse-traceless gauge conditions?

There are several advantages to using the transverse-traceless gauge conditions in the study of linearized gravity. They simplify the equations, making them easier to solve, and they correspond to the physical degrees of freedom of gravitational waves. Additionally, these conditions are invariant under coordinate transformations, making them a useful tool for comparing results from different coordinate systems.

Are there any limitations to using linearized gravity and the transverse-traceless gauge conditions?

While linearized gravity and the transverse-traceless gauge conditions are useful for studying weak gravitational fields, they are not suitable for describing strong fields, such as those near black holes. This is because the assumptions made in the linearization process break down in these extreme scenarios. Additionally, the transverse-traceless gauge conditions only apply to gravitational waves and cannot be used to describe other phenomena, such as the effects of matter on spacetime.

Similar threads

  • Advanced Physics Homework Help
Replies
18
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
947
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
3K
  • Advanced Physics Homework Help
Replies
8
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Special and General Relativity
Replies
2
Views
1K
Back
Top