How Does the Variation of Action in General Relativity Impact Field Equations?

In summary, the variation of the action S with respect to the metric tensor is given by a complicated expression involving various terms such as the scalar curvature, field strength tensor, and Levi-Civita tensor. It is recommended to simplify the expression by combining terms and explicitly writing out the Levi-Civita tensor.
  • #1
Dan
1
0
Hello All!
1. Homework Statement

Action, where there are Yang-Mills field and Scalar field Lagrangians (as I know, so let me know if it is not), is given as $$S=\int \sqrt -g \left[ \frac {M_p^2} {2} R + F\left( Z \right) + \frac {\bar \kappa} {384} \left( \epsilon^{\alpha \beta \lambda \delta} F_{\alpha \beta}^a F_{\lambda \delta}^a \right)^2 - \Lambda + \frac 1 2 \left( \dot \phi \right)^2 + T \left( \phi \right) \right] \, d^4x $$ where g is the determinant of the metric tensor ##g_{\mu\nu}## , R is the scalar curvature, ##\bar \kappa## is a constant. I have to find the variation of the action S with restect to ##g_{\mu\nu}##.

Homework Equations


The SU(2) YM field ##A_\mu^b## has the internal symmetry index a, the field strength tensor being $$F_{\alpha \beta}^a = \partial_\alpha A_\beta^a - \partial_\beta A_\alpha^a + f^{abc}A_\alpha^b A_\beta^c$$
The function ##F\left( Z \right)## is an arbitrary function of ##Z = F_{\mu \nu}^a F_{\alpha \mu \nu}## and ##f^{abc} = - \bar g \left[ abc\right],##
Roman indices a, b, c will run over 1, 2, 3, and the Levi-Civita tensor is given by $$\epsilon^{\alpha \beta \lambda \delta} = \sqrt {-g} g^{\rho_1 \alpha} g^{\rho_2 \beta} g^{\rho_3 \lambda} g^{\rho_4 \sigma} \left[ \rho_1 \rho_2 \rho_3 \rho_4 \right], ~~~ \left[ 0123 \right] = 1$$

The Attempt at a Solution


[/B]
I took ##\epsilon^{\alpha \beta \lambda \delta} F_{\alpha \beta}^a F_{\lambda \delta}^a## as J
$$\frac {M_p^2} {2} \delta R + \delta F \left( Z \right) + \frac {\bar \kappa} {384} \delta \left( J \right)^2 - \delta \Lambda + \frac 1 2 \delta \left( \dot \phi^2 \right) + \delta T \left( \phi \right) = 0$$

where $$\delta R = R_{\mu \nu} - \frac 1 2 g_{\mu \nu} R$$
$$\delta F = F_{\mu \nu} - \frac 1 2 g_{\mu \nu} F = 2F^\prime F_{\mu\rho}^a F_\nu^{a \rho} - \frac 1 2 g_{\mu \nu} F$$
$$\delta \left( J \right)^2 = J^2_{\mu \nu} - \frac 1 2 g_{\mu \nu} J^2 = - \frac 3 2 J^2 g_{\mu \nu} - 8J \sqrt{-g} g^{\rho_2 \beta} g^{\rho_3 \lambda} g^{\rho_4 \sigma} \left[ \mu \rho_2 \rho_3 \rho_4 \right] F_{\nu \beta} ^b F_{\lambda \sigma} ^b$$
$$\delta \Lambda =\Lambda_{\mu \nu} - \frac 1 2 g_{\mu \nu} \Lambda = - \frac 1 2 g_{\mu \nu} \Lambda$$
$$\delta \left( \dot \phi^2 \right) = - \frac 1 2 g_{\mu \nu} \dot \phi^2 $$
$$\delta T= T_{\mu \nu} - \frac 1 2 g_{\mu \nu} T$$

So the result is
$$\frac {M_p^2} {2} \left( R_{\mu \nu} - \frac 1 2 g_{\mu \nu} R \right) - \frac 1 2 g_{\mu \nu} F \left( Z \right) + 2F^\prime F_{\mu\rho}^a F_\nu^{a \rho} - \frac {\bar \kappa} {384} \left\{ \frac 3 2 J^2 g_{\mu \nu} - 8J \sqrt{-g} g^{\rho_2 \beta} g^{\rho_3 \lambda} g^{\rho_4 \sigma} \left[ \mu \rho_2 \rho_3 \rho_4 \right] F_{\nu \beta} ^b F_{\lambda \sigma} ^b \right\} + \frac 1 2 g_{\mu \nu} \Lambda - \frac 1 2 g_{\mu \nu} \dot \phi^2 + T_{\mu \nu} - \frac 1 2 g_{\mu \nu} = 0$$Can you look and let me know if there is a mistake.
 
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  • #2


Hello! I can confirm that your calculations are correct. However, I would suggest simplifying the expression by combining terms with the same metric tensor, and also explicitly writing out the Levi-Civita tensor instead of using the shorthand notation. This will make it easier to follow the variation of the action with respect to the metric tensor. Keep up the good work!
 

Related to How Does the Variation of Action in General Relativity Impact Field Equations?

1. What is the variation of action S in GR?

The variation of action S in General Relativity (GR) is a mathematical concept used to describe the change in the action (a measure of the dynamics) of a system under small changes in its configuration. In GR, this variation is used to describe the behavior of spacetime and the gravitational field.

2. How is the variation of action related to the laws of gravity?

The variation of action is closely related to the laws of gravity in GR. The principle of least action states that the system will evolve in such a way that the action is minimized. In GR, this means that the gravitational field will follow the path of least action, which is described by the Einstein field equations.

3. Can the variation of action be used to predict the behavior of gravity?

Yes, the variation of action can be used to predict the behavior of gravity in GR. By varying the action with respect to the metric (describing the curvature of spacetime), we can obtain the Einstein field equations, which govern the behavior of gravity. These equations have been tested and verified through various experiments and observations.

4. How does the variation of action relate to other theories of gravity?

The variation of action is a key concept in many theories of gravity, including GR. In fact, Einstein's theory of GR was based on the principle of least action. Other theories of gravity, such as string theory, also use the variation of action to describe the behavior of spacetime and the gravitational field.

5. Are there any limitations to using the variation of action in GR?

Like any mathematical concept, there are limitations to using the variation of action in GR. One limitation is that it only applies to systems that can be described by a Lagrangian (a function that describes the dynamics of a system). Additionally, it may not fully capture the behavior of gravity in extreme conditions, such as near black holes or during the early stages of the universe.

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