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The potential encoded information in a photon that is at the edge of the observable universe would seem to be lost as the universe expands. Does that loss of information contribute to the overall entropy of the universe or is the information fully encoded in our measurement apparatus just before the photon is lost?

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    $\begingroup$ By definition the entropy of the universe keeps increasing, but that is only so for well defined volumes, which the entire expanding universe is not. I don't know how you want to measure a photon at the edge of the visible universe, though. To measure a photon it has to be where the measurement is made and by definition we are never at the edge of the visible universe. We are always "here" and a photon at the edge will not make it "here" until a long time from now, if ever. In that sense it's not clear what entropy should even mean, since it's a completely open system. $\endgroup$ – CuriousOne Jul 1 '15 at 22:06
  • $\begingroup$ @CuriousOne thanks for that, was thinking about it, didn't realise a definition could help sort it out, $\endgroup$ – user81619 Jul 1 '15 at 23:12
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    $\begingroup$ Obviously the photon is measured 'here'; it was produced 'there' some 13 Billion yrs ago. The information in the sum is the same. What happens when the source of that information becomes inaccessible to measurement as it expands beyond reach? (or another way, when we can no longer detect that photon, does it count in the measure of entropy of the system, the observable universe) $\endgroup$ – A Monroe Jul 1 '15 at 23:13
  • $\begingroup$ @AlanMonroe: The problem with that is that the information content "there" is very, very low: it's the total of the structure in the CMB, so if you pick that definition you end up with the universe having the "information content" of a few thousands bits, or so. Having said that, this is NOT an information problem but an open system thermodynamics problem and one should treat it for what it is, rather than for something that it isn't. For one thing we don't know what the relevant temperature scale is, it could be extremely high but almost completely decoupled from us. $\endgroup$ – CuriousOne Jul 1 '15 at 23:36
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    $\begingroup$ I find this problem really interesting. Somewhere I am sure I encountered Susskind talking about this, and the general principle of treating event horizons as holographic surfaces which encode the information 'beyond' them. But am not readily finding anything about this on cosmic horizons. Looking at en.m.wikipedia.org/wiki/Black_hole_information_paradox it seems the holographic principle is by no means the only game in town. The Big Rip now expected at the end of the universe, is another issue to reconcile in terms of information conservation $\endgroup$ – CriglCragl Oct 4 '18 at 15:21
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If the whole universe is homogeneous, it will be as likely to loose a photon than to gain one that come from the other side, so on average the entry does not change. However, we are actually in an accelerating universe with accelerated rate of expansion. In this case the observable universe actually diminishes in size, and so the entropy. The total entropy of the observable universe only grows if it is a closed system. The visible universe of a particular observer, does not define any physical border, just a causal/observable one. The entropy inside the observable universe (and its radius) does not grow, and actually, by the holographic principle, the entropy gets actually reduced. This is because of the reduction of mass and energy and of available microstates as the radius diminishes. The extreme case will be in the future if the big rip happens, every fundamental particle and every black hole will become isolated from the rest of the universe, so each disconnected part of the universe will in fact have very low entropy (although the entropy of the whole universe will grow)

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The Second Law of Thermodynamics states that the state of entropy of the entire universe, as an isolated system, will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative. Put another way as the universe increases in volume there are more ways to arrange the mass of the universe in more configurations. Therefore as the size and volume increase at an accelerated rate the potential for even more configuration of mass increases. This is true regardless of the speed of light.

The speed of light is relevant when we examine the observable cosmic universe or hubble horizon -

One can define a so-called "Hubble Horizon" which shows roughly how far light would travel if space were not expanding. This size is

$\chi = c t$

where t is the lookback time since the Big Bang (otherwise known as the age of the universe) which, according to the Friedmann Equations, is:

$t = \int^{a}_{0}{\frac{da}{H_0 \sqrt{\Omega_R a^{-2} + \Omega_m a^{-1} + \Omega_k +\Omega_\Lambda a^2}}}$

where $H_0$ is the Hubble Constant and the $\Omega$ density parameters are, in order, the density of radiation, matter, curvature, and dark energy scaled to the critical density of the universe.

Today, roughly:

$\chi_0 = \frac{c}{H_0}$

yielding a Hubble horizon of some 4.2 Gpc. This horizon is not really a physical size, but it is often used as useful length scale as most physical sizes in cosmology can be written in terms of those factors.

You ask -

The potential encoded information in a photon that is at the edge of the observable universe would seem to be lost as the universe expands. Does that loss of information contribute to the overall entropy of the universe or is the information fully encoded in our measurement apparatus just before the photon is lost?

Very much doubt that it would be encoded in the measuring apparatus. And yes since it is no longer observed in the local Hubble Horizon or sphere the data would appear to be lost locally and a net local increase in entropy could said to take place. All the while as the entropy of the universe is increasing at an increasing rate along with the exponentially increasing volume of the universe. Basically just a subset of the whole.

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