# If the universe is expanding will gravitational attraction eventually go to zero?

Let's assume that we prove that dark matter exists (after all, only about 4 percent of the entire universal mass is atoms, and 22% dark matter, 74% dark energy (I think I got the numbers right)).

However, if that is the case, than everything would technically pull on everything, right? But if the universe is continuously expanding, but mass cannot be created or destroyed, unless it is from or to energy, than everything pulls on each other, but as time passes it would be weaker and weaker.

1. So my question is, can it ever reach a time when gravitation reaches zero?

2. Or if mass can be created from the expansion of the universe, than does that mean since it is expanding that there is no change in gravitation?

3. Also, how is the Hubble Constant calculated?

• 1) the ratios you presented are for the universe's total energy content, not mass. 2) Objects within the same galactic cluster are gravitationally bound such that they do not expand away from each other. 3) The law of conservation of energy is only valid locally. Energy is not conserved on the scale of the universe. 4) Gravitation never changes, only the force experienced on one body from another can change. – Jim Jul 8 '15 at 16:24
• @JimsBond: Whether energy is or is not conserved on the scale of the universe is unknown and if you look at different theories for vacuum self-energy then you can find a source for the energy that seems to drive the expansion. What the long distance/long time strength of gravity is, is also not known (and can not be known within the limitations of human historical time scales). Be careful not to sell unsettled theory for knowledge. – CuriousOne Jul 8 '15 at 19:13
• @CuriousOne We've been over this. An FRW expanding spacetime in general relativity does not have a time shift symmetry, which means that there is no global conservation of energy law. If the total amount of energy happens to stay the same, there is nothing wrong with that, however that does not make energy a conserved quantity. The stress-energy-momentum tensor is always conserved, but the most general statement is that energy does not need to be conserved globally.... – Jim Jul 9 '15 at 12:22
• We may discover a mechanism that requires global energy to be conserved, but currently no such mechanism is known and so we have no reason to believe energy is conserved. And as for gravity, our best evidence leads us to believe it is constant. If that changes, I will change what I say. Just as I need to be careful not to sell unsettled theories as fact, you must be careful not to undersell widely accepted theories or overinflate the importance of fringe theories. Pop-sci is full of misunderstandings because people too often over-hype the importance of non-mainstream theories or ideas. – Jim Jul 9 '15 at 12:28
• @JimsBond: Yes, we have been over this and you are still assuming that general relativity is the 100% correct theory (even though you seem to be aware that that's a logical mistake), for which there is very little confirmation as of this moment, since, as you admit, nobody has managed to marry it to quantum mechanics. I, on the other hand, am agnostic about this question and I am happy to admit that I don't know things that have not been measured, yet. Experimental physics seems to have an easier time to admit what it doesn't know than theory. – CuriousOne Jul 9 '15 at 15:43

I describe how to calculate the Hubble parameter in How does the Hubble parameter change with the age of the universe?. You should have a quick read through this as it's relevant to the rest of your question.

We know the universe is expanding. We describe its size by a parameter called the scale factor, $a$. The rate of expansion is the rate of change of $a$ with time, which we write as $\dot{a}$.

Our everyday experience of gravity is that it accelerates things i.e. it speeds them up or slows them down. If you drop a ball gravity accelerates it downwards, and if you throw the ball up gravity slows it. In the context of an expanding universe the corresponding quantity is the rate of change of $\dot{a}$ with time, which we write as $\ddot{a}$. If $\ddot{a} \lt 0$ that means the expansion rate is slowing down, while if $\ddot{a} \gt 0$ that means the expansion rate is speeding up. So when you ask if gravitation will ever be zero, I take this to mean that $\ddot{a} = 0$ i.e. the expansion of the universe is neither slowing down nor speeding up.

The equation for $\ddot{a}$ comes from the Friedmann equations. For a flat universe like ours, and assuming the pressure is zero, $\ddot{a}$ is given by:

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\rho + \frac{c^2}{3}\Lambda \tag{1}$$

where $\rho$ is the density of the stuff in the universe and $\Lambda$ is the cosmological constant i.e. the dark energy. Note that the density $\rho$ decreases with time as the universe expands because the same amount of matter occupies a bigger volume of space. However the cosmological constant $\Lambda$ does not change with time.

So is there a time when $\ddot{a} = 0$? If we take equation (1) and set $\ddot{a} = 0$ we get:

$$\frac{4\pi G}{3}\rho = \frac{c^2}{3}\Lambda$$

or rearranging gives us:

$$\rho = \frac{c^2}{4\pi G}\Lambda$$

and this value of $\rho$ happened about 5 billion years ago at the point where the rate of expansion changed from slowing to accelerating. This is the point of inflection in the graph of $a$ versus time:

So there was a moment about 5 billion years ago when the net gravity in the universe was effectively zero. This moment will never come again because dark energy is now accelerating the expansion.

• I think the OP's questions is on the lines of "If $\ddot{a} > 1$, how soon will it be when we stop feeling our weight due to earth's gravitation"? Also, I want to know, if at some point in the past, the net gravity was zero, would the inhabitants of that time felt anything unusual? – Jus12 Aug 24 '16 at 21:40

However, if that is the case, than everything would technically pull on everything, right?

No. A gravitational field is a place where space is "neither homogeneous nor isotropic". You can See Einstein talking about that here. And the FLRW metric "starts with the assumption of homogeneity and isotropy of space". I'm confident that this is correct because the universe didn't contract when it was small and dense. If it did, we wouldn't be here. So a gravitational field is a place where space is inhomogeneous, but in the large scale universe space is homogeneous, so on the large scale, when it comes to the universe, there is no overall gravitational field. So everything doesn't pull on everything else. Instead concentrations of energy in the guise of matter pull on other concentrations of energy in the guise of matter, and these aren't everything.

But if the universe is continuously expanding, but mass cannot be created or destroyed, unless it is from or to energy, than everything pulls on each other, but as time passes it would be weaker and weaker. So my question is, can it ever reach a time when gravitation reaches zero?

Does not parse! The gravitation was zero when the universe was small and dense, and it's zero now. A gravitational field is a place where space is inhomogenous, and as a result of this the motion of light and matter falls down. But it isn't a place where "space falls down". The universe isn't a place that's contracting because of gravity, and never ever was. The expansion of the universe has got a lot to do with general relativity, but not gravity. Why don't you ask another question about the expansion of the universe?

Or if mass can be created from the expansion of the universe, than does that mean since it is expanding that there is no change in gravitation?

Yes. On the large scale space is homogeneous, whilst a gravitational field is a place where space is inhomogeneous. So on the large scale, when it comes to the universe, there is no overall gravitational field. There is no change from zero.