# How the mass is conserved during continuous expansion?

If universe is continuously expanding doesn't it contradict the fact that mass is conserved? Because if mass is conserved so how is universe expanding?

• The mass density is constantly falling. Galaxies are moving away from each other. If the acceleration of the expansion keeps continuing, then in a few ten billion years the only thing visible in the night sky would be the galaxy that is left over from the merger of the Milky Way and Andromeda. The last inhabitants of this universe may even live on a planet circling a lonely star surrounded by nothing but blackness without even a trace of cosmic microwave radiation. They will be thinking that their creator made nothing but them and they will have every right to think so in their loneliness. :-) – CuriousOne Jun 19 '15 at 5:55
• Doesn't this mass density effects the gravitational field of masses? – aish Jun 19 '15 at 6:04
• Yes, it does. Galaxy mergers will become rare and then seize and star formation will seize, too. – CuriousOne Jun 19 '15 at 6:56

There are three types of matter/energy we consider when calculating how the universe expands:

1. Matter - both normal matter and dark matter
3. Dark energy

We measure the expansion of the universe using a scale factor that we normally denote by $a$. The scale factor increases with time as the universe expands, and if we look backwards in time we see $a$ decrease as the universe contracts. We take the value of $a$ to be $1$ right now, so if the value of $a$ is $2$ that mean the universe has expanded by a factor of $2$ and if the value of $a$ is $0.5$ that means the universe has contracted to half its original size.

If we represent the density of matter as $\rho_M$, radiation as $\rho_R$ and dark energy as $\rho_\Lambda$, then the densities are related to $a$ by:

$$\rho_M = \frac{\rho_{M_0}}{a^3} \tag{1}$$

$$\rho_R = \frac{\rho_{R_0}}{a^4} \tag{2}$$

$$\rho_\Lambda = \rho_{\Lambda_0} \tag{3}$$

where the suffix $0$ means the present time i.e. $\rho_{M_0}$ means the density of the matter right now.

If you look at equation (1) this tells you that the density of matter falls as the universe expands in exactly the way you would expect. If the universe expands by a factor of two, i.e. $a = 2$, then the volume increases by a factor of $8$ and the density of matter falls by a factor of $a^3 = 8$. The density of matter falls in the way you would expect if matter is conserved.

However note that $\rho_M$ is the average density of the matter over large scales. The expansion of the universe doesn't change the density of the Earth or the Sun because the gravity of bodies like the Earth and the Sun is large enough to conteract the expansion and keep the volume of the Earth constant.

It's more interesting to look at equations (2) and (3). Starting with equation (2) we see that the desnity of radiation falls as $a^4$ not $a^3$. This is because as the universe expands the radiation is red shifted as well as diluted, and the end result is that its density falls faster than matter.

Equation (3) is the really odd one, because it shows that the density of dark energy is constant i.e. as the universe expands the density of the dark energy dosn't change. This looks as if it violates conservation of energy, and well it does. An expanding universe violates time shift symmetry and that means the law of conservation of energy doesn't apply.

I should add that the claim an expanding universe doesn't conserve energy is a bit controversial because it depends on how you calculate total energy and what you include.

• One should note that we don't really know what equation (3) should be. That's just the current model. Whether conservation of energy applies or not is unknown, since we don't know if there is an energy source that feeds dark energy. We could simply be looking at an incomplete description of the system we call "universe". If I had to bet a case of wine, that's what I would bet on. – CuriousOne Jun 19 '15 at 7:01

The expansion of space is like stretching a rubber sheet. (Don't take this analogy too seriously. It works for this explanation but fails elsewhere.) The mass of the rubber sheet stays the same as it gets bigger. Space expands, but mass does not increase with it.