Is there an analog of the Komar mass for conformal timelike Killing vector Fields?

For a stationary spacetime M, the Komar mass may be given as:

$$M=\oint_{\partial M}\sqrt{g_{00}}(2T_{\mu\nu}-g_{\mu\nu}T)e^{\mu}\xi^{\nu}$$

Where $e^{\mu},\xi^{\nu}$ are respectively a unit timelike vector and a Killing vector (the latter normalized at a great distance from the massive body). I get that this only holds for stationary metrics; however, there are many spacetimes that, while lacking timelike Killing vectors, still possess conformal symmetry (and thus conformal Killing vectors).

a standard Killing vector, X, is defined as:


While a conformal one is of the form:

$$L_{x}g=\lambda g$$

I feel like there ought to still be a conserved quantity for the Komar mass if properly set up for conformal symmetry. anyone know of anything along these lines? Worth noting is that I'm working on a closed FLRW universe, and while Komar mass is meant for asymptotically flat spacetimes, it seems to work just fine on a three-sphere when properly normalized.


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