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Of the many ways to write a mass in general relativity (Komar, adm and the like), it seems that none of them are conserved in an expanding universe (or more generally for nonstationary metrics).

There are plausable lines of mathematical reasoning that the energy is transferred to/from the gravitational field, but still the mass of a particular body in and of itself appears to lose/gain energy.

I'm fine with this, until I start reading about determining the redshift of distant celestial bodies. It is assumed that atomic spectra are constant through the cosmological ages..why? It seems like General relativity gives us a roadmap indicating a changing mass, which would clearly effect even a nonrelativistic Schrodinger solution for spectra such as the Rydberg formula for hydrogen:

$$\frac{1}{\lambda}=\frac{m_{e}e^{4}}{8\epsilon_{0}^{2}h^{3}c}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)$$

Where $\lambda$ is the wavelength of radiation emitted/absorbed, $n_{1}$ and $n_{2}$ are any positive integers $n_{1}\neq n_{2}$ ; $n_{1}<n_{2}\;and\;n_{2}<n_{1}$ implying absorption and emission respectively. What is the reasoning for this? I just think it's strange that GR is used to determine redshift, but then not for anything more. I'm probably just missing something.

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  • $\begingroup$ Only the relations between the spectral lines are constant, the wavelengths are depend from the gravitational potential in the location where the photons get emitted. $\endgroup$ – HolgerFiedler Aug 9 '16 at 6:10
  • $\begingroup$ What I meant Holger is wouldn't a changing mass, say decreasing, blueshift the atomic spectra and vice versa. Since GR gives us a reason to expect nonconservation of mass over cosmological times. In other words, why do we assume the emission spectra of hydrogen 6billion years ago was the same as today (as measured locally to the emission) $\endgroup$ – R. Rankin Aug 9 '16 at 6:20
  • $\begingroup$ The constancy of the constants has been checked by Wmap and Planck teams. They published big reports with many indications particularly on the fine structure constant $\endgroup$ – user46925 Aug 11 '16 at 3:15
  • $\begingroup$ It is assumed because there is no evidence that the assumption does not hold. There are sensitive limits to any dependence of the fine coupling constant on redshift that come from quasar absorption spectra. $\endgroup$ – ProfRob Aug 11 '16 at 11:29
  • $\begingroup$ Ps. It only makes sense to look for variations in dimensionless constants. $\endgroup$ – ProfRob Aug 11 '16 at 11:30
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A hydrogen atom 6 billion light years ago (when the universe had expanded less than now) has exactly the same mass as one now. Think mass in its comoving frame of reference which is locally inertial.

It emits the same frequency light as one here on Earth (ignoring our peculiar velocity). The frequency from 6 billion light years away gets redshifted by the expansion equation for redshift. Everything else you brought up is irrelevant. In that sense the mass of a hydrogen atom is conserved. The thing that looses the energy is the photon, and that goes into the expansion if you want to think in terms of energy conservation (that gravitational energy gets created in the expansion), or that the energy is lost if you want to follow strict relativity.

Added in response to Rankins's comments below

Yes to the second, no to the first. Here's why.

The first: why neither the Rydberg constant nor the electron mass change in a gravitational field? The electron mass is a property of the electron. It used to be called the rest mass. The rest mass enters in general relativity (GR) as an invariant, what can change is the energy and momentum. In a local frame, or an inertial frame which in GR is any freely falling (nothing but gravity) that is its mass. In the FLRW universes, for instance with $k=0$ (flat, but if not it's similar relations), $\rho a^{-3}$ is what does not change with the expansion, with a the universe scale factor. The density $\rho$ is mass per unit volume, and so mass stays invariant as the volume goes as $a^3$. The evolution of the universe when matter dominated goes like that. It matches all the cosmological observation and standard model confirmed by the latest CMB observations.

The energy does not change when the universe is matter dominated. But when radiation dominated (e.g. photons), it does loose energy in the redshift. It is simply from the equations for FLRW of the photons. When matter, energy and cosmological constant are all relevant each evolves its own way, mass is conserved, energy is not.

The spectra from stars and galaxies have shown no variation even from quasars and white dwarfs except for that accounted for by the redshift, with measurements of that possible variation to better than 1 part in $10^7$, even at redshift of 2-3 (or time back of 10-12 Gyr).

In special relativity we used to say the mass grows as the velocity gets closer to $c$. Nowadays one says in both SR and GR that the mass of an elementary particle is an invariant, the rest is momentum.

Now for the second, how or why mass energy is not conserved in GR. For energy you already know. For mass, it is for elementary particles, but for composite or macroscopicl bound bodies the mass can change because the binding energy, internal momentum, or simplistically gravitational potential energy, can contribute to the mass, and the energy in this effective view (the total mass or energy, ignoring the external momentum, of the composite body). Thus, when the two black holes merged in 2015 the sum of their masses was 3 solar masses less than the mass of the final black hole. The 3 solar masses were lost to gravitational radiation.

And in general, GR does not have a covariant measure of the energy of gravitational waves, or the total in energy in spacetime. For highly dynamics spacetimes or events it is not generally conserved. For asymptotically flat spacetimes you can get global (but not local) conservation, i.e., from the flux at infinity (or far enough away, like us and the black holes).

If you are not completely sure what is meant to be included in the energy and mass, such as in the simple cases I described above, you have to take the full stress energy tensor as the gravitational source, which includes energy, momentum, angular momentum, and stress.

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  • $\begingroup$ If one says that say the komar mass of the hydrogen atom at one moment is one value and then calculates it after some expansion in the same frame, one obtains a different result. Now I know there is no timelile Killing vector here (rather a conformal one), bringing into question the applicability of the komar equation, but I also understand mass in general is not conserved in an expanding universe. How can this apply to planetary bodies but not their constituent parts? +1 $\endgroup$ – R. Rankin Aug 10 '16 at 7:57
  • $\begingroup$ In another question of mine physics.stackexchange.com/questions/268171/… you seemed to argue that mass energy aren't conserved for a body in an expanding universe (in our addmitedly lenghty comments). I'm just trying to understand why this isn't so for an atom alone. If $\endgroup$ – R. Rankin Aug 10 '16 at 23:36
  • $\begingroup$ @Rankin See my edit to my answer $\endgroup$ – Bob Bee Aug 11 '16 at 2:32
  • $\begingroup$ Would you expect a red shift for light in the photosphere of a black hole with cosmological expansion? I'm guessing not by your arguments. $\endgroup$ – R. Rankin Aug 11 '16 at 3:27
  • $\begingroup$ I don't see why not. Light wherever it is if it travels out through the universe will get redshifted. Now, the photosphere if I remember right is close to the horizon, but can remember how close, so whatever that gravity well is should also affect it. Still, maybe I don't follow what you mean to imply in your question. $\endgroup$ – Bob Bee Aug 11 '16 at 4:01

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