Simplifying Linearized Equations in GR Using Gauge Transformations

In summary, the conversation discusses the linearized field equation in general relativity (GR) and how to use the gauge transformation to simplify it. The linearized equation is expressed in terms of the linearized metric tensor and the gauge transformation involves adding a gauge vector field to the metric tensor. The simplified equation is obtained by using the properties of the metric tensor and solving for the gauge vector field.
  • #1
Psychosmurf
23
2
I need some help with a derivation in GR.

The linearized field equation in GR is:

[tex]G_{ab}^{(1)} = - \frac{1}{2}{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} + {\partial ^c}{\partial _{(b}}{{\bar \gamma }_{a)c}} - \frac{1}{2}{\eta _{ab}}{\partial ^c}{\partial ^d}{{\bar \gamma }_{cd}} = 8\pi {T_{ab}}[/tex]

How would I use the gauge transformation

[tex]{\gamma _{ab}} \to {\gamma _{ab}} + {\partial _b}{\xi _a} + {\partial _a}{\xi _b}[/tex]

to simplify the linearized equation to:

[tex]{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} = - 16\pi {T_{ab}}[/tex]?

EDIT: I should also mention:

[tex]{{\bar \gamma }_{ab}} = {\gamma _{ab}} - \frac{1}{2}{\eta _{ab}}\gamma [/tex]

[tex]\gamma = \gamma _a^a[/tex]

and

[tex]{g_{ab}} = {\eta _{ab}} + {\gamma _{ab}}[/tex]
 
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  • #2
Nvm. I figured it out.
 

1. What are gauge transformations in general relativity (GR)?

Gauge transformations in GR refer to a change in the coordinates used to describe the spacetime geometry in the theory. This change does not affect the physical predictions of the theory, but simply represents a different way of describing the same physical reality.

2. Why are gauge transformations important in GR?

Gauge transformations are important in GR because they allow us to choose the most convenient coordinate system for solving equations and making predictions. They also help us to understand the underlying symmetries of the theory and how different observers may perceive the same physical situation.

3. How do gauge transformations affect the metric tensor in GR?

In GR, gauge transformations do not affect the metric tensor itself, but rather the coordinates used to describe it. This means that the metric tensor, which describes the curvature of spacetime, remains unchanged even when we change the coordinate system used to describe it.

4. Can gauge transformations be used to change the physical properties of a system in GR?

No, gauge transformations do not change the physical properties of a system in GR. They only change the way we mathematically describe the system. Physical properties, such as mass or energy, remain the same regardless of the coordinate system used.

5. How are gauge transformations related to the principle of general covariance in GR?

The principle of general covariance states that the laws of physics must be expressed in a form that is independent of the choice of coordinates. Gauge transformations, which allow us to change coordinates without affecting physical predictions, are a direct consequence of this principle in GR.

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