Is time inflating proportionally to space?

Question

We know space is expanding at a rate of roughly 432 miles/light-year/year. Since Einstein showed that time was intrinsically linked into the 4 dimensional structure of spacetime one would logically wonder if space expansion is linked to the passage of time. Due to the cosmic microwave background we can infer that the universe is 13.5-13.9 billion years old. 1 light-year is ~5.88 trillion miles or $5.8786 × 10^{12}$ miles if we divide $$5.8786 * 10^{12} /432$$ we find that the universe is expanding by a factor of 1/13607870370.4 every year so every year any given piece of space increases in size by about a 13.6 billionth of its size. Looking at the fact that every year the universe also gets a ~13.6 billionth of its age older. Looking at that link made me wonder whether the passage of time, and cosmological expansion are the same cosmological constant type effect but one on space and the other on time. Does space expand proportionally to the age of the universe?

Answer 1

In the FLRW model of the expanding universe, the metric is given in terms of the co-moving coordinates: $$ds^2=dt^2-a(t)^2\bigg[{dr^2\over1-kr^2}+r^2\sin^2(\theta)d\theta^2+r^2d\phi^2\bigg]$$ where the function $a(t)$ is the scale factor of the universe that satisfies the Friedmann equations: $$\bigg({\dot a\over a}\bigg)^2+{kc^2\over a^2}-{\Lambda c^2\over 3}={kc^4\over 3}\rho,$$ and $$2{\ddot a\over a}+\bigg({\dot a\over a}\bigg)^2+{kc^2\over a^2}-\Lambda c^2=-kc^2p.$$ These equations were known some years prior to Hubble's discovery of the expanding universe, at any rate, however, they show the expansion rate of the universe as a function of time, e.g. Hubble's constant is written: $$H_0={\dot a(t)\over a(t)}\bigg|_{t={\text{now}}}.$$ So we can see the explicit dependence of Hubble's constant on matter/energy, including the cosmological constant: $$\bigg(H(t)\bigg)^2=-{kc^2\over a^2}+{\Lambda c^2\over 3}+{kc^4\over 3}\rho.$$

Answer 2

Yes. New time is created (it "expands") as the age of the universe increases. The big bang is at the center. The age of the universe is the radius of the hypersphere. The past universe is a smaller hypersphere inside the present one.