Gauge transformations and degrees of freedom

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The general equations \[G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}\] are valid in any coordinate frame, in which the metric obeys eq. 1, so$^*$ we have the freedom to make coordinate transformation \[x^{\mu}\to {x'}^{\mu}=x^{\mu}+\xi^{\mu}(x),\] with four arbitrary functions $\xi^\mu$, which are of the first order by $h_{\mu\nu}$.

$^*$In addition to global Lorentz transformations, which are symmetries of the Minkowski background, or in general the isometries of the background spacetime.

Problem 1: Gauge transformations

Find $h_{\mu\nu}$ in the new (primed) coordinates; show that curvature tensor and its contractions are gauge invariant and do not change their functional form.

In a given frame the metric perturbation $h_{\mu\nu}$ can be decomposed into pieces which transform under spatial rotations as scalars, vectors and tensors (the irreducible representations of the rotation group $SO(3)$) in the following way (spatial components are denoted by Greek indices from the beginning of the alphabet $\alpha,\beta,\gamma\ldots=1,2,3$): \begin{align} h_{00}&=2\Phi;\\ h_{0\alpha}&=-w_{\alpha};\\ h_{\alpha\beta}&=2\big( s_{\alpha\beta} +\Psi\eta_{\alpha\beta}\big), \end{align} where $h_{\alpha\beta}$ is further decomposed in such a way that $s_{ij}$ is traceless and $\Psi$ encodes the trace: \begin{align} h\equiv h_{\alpha}^{\alpha} &=\eta^{\alpha\beta}h_{\alpha\beta} =0+2\Psi \delta^{\alpha}_{\alpha}=6\Psi;\\ \Psi&=\tfrac{1}{6}h;\\ s_{\alpha\beta}&=\tfrac{1}{2}\big(h_{\alpha\beta} -\tfrac{1}{6}h\; \eta_{\alpha\beta}\big). \end{align} Thus the metric takes the form \[ds^{2}=(1+2\Phi)dt^2 -2w_{\alpha}dt\,dx^{\alpha} -\big[(1-2\Psi)\eta_{\alpha\beta} -2s_{\alpha\beta}\big]dx^\alpha dx^\beta\]

Problem 2: Particle's motion, gravo-magnetic and gravo-electric fields

Write down geodesic equations for a particle in the weak field limit in terms of fields $\Phi$, $w_\alpha$, $h_{\alpha\beta}$. What are the first terms of expansion by $v/c$ in the non-relativistic limit?

Problem 3: Dynamical degrees of freedom

Derive the Einstein equations for the scalar $\Phi,\Psi$, vector $w^\alpha$ and tensor $s_{\alpha\beta}$ perturbations. Which of them are dynamical?

Problem 4: Gauge decomposition

Find the gauge transformations for the scalar, vector and tensor perturbations.

Problem 5: Synchronous gauge

This one is equivalent to Gaussian normal coordinates and is fixed by setting \begin{equation} \Phi=0,\qquad w^\alpha=0. \end{equation} Write the explicit coordinate transformations and the metric in this gauge.

Problem 6: Transverse gauge

This is a generalization of the conformal Newtonian or Poisson gauge sometimes used in cosmology, which is fixed by demanding that \begin{equation} \partial_\alpha s^{\alpha\beta}=0,\qquad \partial_\alpha w^\alpha =0. \end{equation} Find the equations for $\xi^\mu$ that fix the transverse gauge.