- #1
jonmtkisco
- 532
- 1
Hi SpaceTiger, Pervect & Hellfire,
Here are some follow-up thoughts about the Friedmann equation for expansion. Pervect, thank you for using Noether's Theorem to demonstrate that normal momentum (of movement) is conserved. The next challenge is to demonstrate that the "momentum-like" continuation of the original expansion of space is conserved.
1. By playing around with a spreadsheet, I have determined that if mass is held constant, the "momentum-like" continuation of expansion is conserved, by the formula:
PP = [tex]\Delta[/tex]volume[tex]^{2}[/tex] /volume
where PP is the "momentum-like" continuation of expansion. So my earlier suggestion that [tex]\Delta[/tex]volume/ [tex]\Delta[/tex]time might be the metric turns out to be wrong. It's a relief to find that this "momentum-like" quantity remains constant in the Friedmann equation when mass/energy is constant.
2. However, that finding merely leads to the next question, which I find to be of great concern. That is, that the Friedmann equation calculates that when the mass/energy of the universe increases (e.g., due to the cosmological constant), the expansion rate increases. And most relevant, when mass/energy decreases, the expansion rate decreases. This is an important scenario, because under the traditional [tex]\Lambda[/tex]CDM model, the total mass/energy of the universe is held to have decreased dramatically during the radiation dominated era, as radiation gave up energy to expansionary redshift. Sure enough, the Friedmann equation dutifully calculates that the "momentum-like" continuation of expansion declines throughout the radiation-dominated era. Yet if mass/energy is held constant at its pre-decline value, Friedmann calculates that the expansion rate remains higher (and higher than we observe). You can convince yourself of this just by noting that mass is in the top line of the Friedmann equation, and R is not.
Now I really need a clear explanation as to how, in a GR-based model, a large decline in mass/energy can cause a large decline in the expansion rate. That certainly violates the expected behavior of a "momentum-like" quantity. Less gravity ought to cause faster expansion than more gravity. That's pretty basic.
The Friedmann equation was created in the 1920's when there was no observational evidence that the universe was expanding, and therefore the idea that the universe might not be purely adabiatic, because its mass/energy might actually change over time, was not incorporated in the formula. If anyone is aware of this specific question having been addressed subsequently by mathematicians, I would very much appreciate a reference.
Jon
Here are some follow-up thoughts about the Friedmann equation for expansion. Pervect, thank you for using Noether's Theorem to demonstrate that normal momentum (of movement) is conserved. The next challenge is to demonstrate that the "momentum-like" continuation of the original expansion of space is conserved.
1. By playing around with a spreadsheet, I have determined that if mass is held constant, the "momentum-like" continuation of expansion is conserved, by the formula:
PP = [tex]\Delta[/tex]volume[tex]^{2}[/tex] /volume
where PP is the "momentum-like" continuation of expansion. So my earlier suggestion that [tex]\Delta[/tex]volume/ [tex]\Delta[/tex]time might be the metric turns out to be wrong. It's a relief to find that this "momentum-like" quantity remains constant in the Friedmann equation when mass/energy is constant.
2. However, that finding merely leads to the next question, which I find to be of great concern. That is, that the Friedmann equation calculates that when the mass/energy of the universe increases (e.g., due to the cosmological constant), the expansion rate increases. And most relevant, when mass/energy decreases, the expansion rate decreases. This is an important scenario, because under the traditional [tex]\Lambda[/tex]CDM model, the total mass/energy of the universe is held to have decreased dramatically during the radiation dominated era, as radiation gave up energy to expansionary redshift. Sure enough, the Friedmann equation dutifully calculates that the "momentum-like" continuation of expansion declines throughout the radiation-dominated era. Yet if mass/energy is held constant at its pre-decline value, Friedmann calculates that the expansion rate remains higher (and higher than we observe). You can convince yourself of this just by noting that mass is in the top line of the Friedmann equation, and R is not.
Now I really need a clear explanation as to how, in a GR-based model, a large decline in mass/energy can cause a large decline in the expansion rate. That certainly violates the expected behavior of a "momentum-like" quantity. Less gravity ought to cause faster expansion than more gravity. That's pretty basic.
The Friedmann equation was created in the 1920's when there was no observational evidence that the universe was expanding, and therefore the idea that the universe might not be purely adabiatic, because its mass/energy might actually change over time, was not incorporated in the formula. If anyone is aware of this specific question having been addressed subsequently by mathematicians, I would very much appreciate a reference.
Jon