Does an atom of hydrogen today have the same rest mass energy as an atom of hydrogen a billion years in the future?
Standard cosmology seems to tacitly make this assumption.
But surely one can only make this assumption if the FRW metric has a time-like Killing vector?
From V.Moretti's answer to my previous post the FRW metric does not have a time-like Killing vector so that we cannot assert that the energy of an atom of hydrogen is constant over cosmological time.
Is this reasoning correct?
I'm sorry to keep bringing this question up but I think it's an important point that is overlooked in current cosmology.
Answer to responses
I guess I am following the spirit of Sean Carroll's Energy is not conserved Blog post where he says:
When you thought energy was conserved, there was a reason why you thought that, namely time-translation invariance. A fancy way of saying “the background on which particles and forces evolve, as well as the dynamical rules governing their motions, are fixed, not changing with time.” But in general relativity that’s simply no longer true. Einstein tells us that space and time are dynamical, and in particular that they can evolve with time. When the space through which particles move is changing, the total energy of those particles is not conserved.
Energy and momentum evolve in a precisely specified way in response to the behavior of spacetime around them. If that spacetime is standing completely still, the total energy is constant; if it’s evolving, the energy changes in a completely unambiguous way.
According to the FRW metric spacetime is evolving so surely the energy of particles cannot be assumed to be constant? A hydrogen atom today does not necessarily have the same energy as it did a billion years ago or will have a billion years in the future.
Taking the spatially flat FRW metric in Cartesian co-ordinates for simplicity we have: $$ds^2 = -dt^2 + a^2(t)(dx^2+dy^2+dz^2)$$ Now from this representation of the FRW metric one might conclude that it is only the spatial co-ordinates that are evolving. In fact we can transform to conformal co-ordinates: $$ds^2 = a^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ where $d\tau = dt/a(t)$. Now we can clearly see that spacetime is evolving so we cannot assume that the energy of massive particles is constant. However according to V.Moretti massless particles like photons that are described by conformally invariant fields do not have their energy changed by a conformally changing metric as described above. Thus we can rigorously argue that photon energy is constant if we are in fact moving forward in conformal time $\tau$.
So an increasing length scale is associated with a constant energy. Thus a massive particle with a fixed length scale is associated with an energy that increases with the scaling factor $a(\tau)$. In seems that the energies of all massive particles must scale with $a(\tau)$ including the Planck mass. In fact I suppose as all particle masses are ultimately linked to the Planck mass then the the scaling particle masses is ultimately caused by the Planck mass increasing.