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From my understanding, dark energy is most likely causing the expansion of the universe and it is accelerating. For this thought experiment assume the rate that the universe is expanding is exactly c.

Two photons move towards each other each at c, however, space itself also pulls them away from each other at c so their total speed should forever be zero.

Since the universe doesn't exactly have a center it pulls all particles in this manner making interaction impossible.

So my question is in this universe where all interaction is impossible what would happen to virtual particles since they can't destroy each other wouldn't the universe create an infinite amount of energy? Even if the total sum of energy is zero and the virtual particles would destroy each other it is now impossible. Does this break the conservation of energy?

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  • $\begingroup$ conservation of energy is not a law in general relativity. We do not have a model yet of this acceleration of expansion so as to see how it fits with conservation of energy in flat spaces. In the end interaction would be impossible anyway, yes , if the expanding "force" in flat space was larger than the attractive forces, nuclear and electromagnetic. $\endgroup$ – anna v Feb 7 '18 at 7:57
  • $\begingroup$ "the rate that the universe is expanding is exactly c" is that assumption true? $\endgroup$ – SRS Feb 7 '18 at 9:05
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For this thought experiment assume the rate that the universe is expanding is exactly $c$.

This isn't how the expansion works. The expansion rate is described by the Hubble parameter, $H$, and the Hubble parameter tells us the (average) recession velocities at a distance $d$. The equation is simply:

$$ v = H d \tag{1} $$

The Hubble parameter is normally given in units of (km/sec)/MPc, where the unit MPc is a megaparsec. So objects at a distance of 1 MPc have an average recession velocity of $H$ km/sec. Objects at a distance of ten MPc have an average recession velocity of $10H$ km/sec, and so on. The current value of $H$ is around $70$ (km/sec)/Mpc, though there is quite a large uncertainty in the value and it could be anywhere in the range $62$ to $82$ (km/sec)/Mpc.

In a universe with no dark energy the value of $H$ falls with time as the mutual gravitational attraction of the matter slows the expansion. In a universe with dark energy the value of $H$ tends to a constant value, and in pathological expansions like the Big Rip the value of $H$ increases with time. For more on this see How does the Hubble parameter change with the age of the universe?

For any value of $H$ there is a distance where the recession velocity is equal to the speed of light. From equation (1) this distance is just:

$$ d_h = \frac{c}{H} $$

And as you suggest in your question light emitted at this distance cannot reach us. This distance is called the particle horizon.

If dark energy behaves like a cosmological constant then in our universe the Hubble parameter will tend to a constant value of about $20$% less than it's current value, which puts the particle horizon at about 16 billion light years.

So in the far future all observers in the universe will have a cosmological event horizon at around 16 billion light years and they will never be able to see farther into the universe than that. However within this distance everything behaves normally.

There is a (wildly speculative) idea that the dark energy density can increase with time eventually driving the Hubble parameter to infinity. This is called the Big Rip. This will in effect tear everything apart and destroy everything, however there is currently no evidence that this will happen.

A couple of final points. Firstly energy is not conserved in the expansion of the universe. This is the case no matter how the universe is expanding and doesn't require dark energy to do anything weird like a Big Rip. See for example:

Secondly the vacuum is't full of virtual particles. The idea of virtual particles being pulled apart is just a toy model originally introduced to give a rough guide to Hawking radiation. Virtual particles are a mathematical device and don't really exist.

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Each type of matter exerts a pressure proportional to its density, viz. $p=w\rho c^2$. For ordinary matter, $w=0$; for radiation, $w=\frac{1}{3}$; for dark energy, $w<-\frac{1}{3}$, although the exact value is unknown. The "Big Rip" scenario you describe occurs if $w\le -1$. It wouldn't prevent virtual particles from annihilating, because they don't obey the same casual energy-momentum relation as real particles.

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