Gauges for quadratic gravity: the extended transverse gauge and the energy-momentum tensor of the massive spin-2 field

We study the 4D Einstein-Hilbert action extension based on the square of the curvature tensors. Analyses of gauge and perturbation modes are often done considering the Teyssandier gauge condition. Although this approach can be useful for other higher-order extensions of quadratic gravity, a generalized transverse (or Lorentz) gauge is clear and sufficient for the present case, as explained here. We provide a detailed analysis of the generalized transverse gauge condition, its residual gauge symmetry, the physical modes, the induced energy-momentum tensor (EMT) of the massive spin-2 mode (which is gauge-dependent), and a comparison with the Teyssandier gauge. The derivation is valid for any EMT. We also compare the induced EMT in quadratic gravity with the massive spin-2 Fierz-Pauli EMT, which is different from the previous cases. The comparison is further developed by considering a spherical isothermal sphere, which works as an approximation to virialized spherical systems.