To the End of the Universe

I like to remind myself and others how little I know. It’s one of the reasons I like Quora, where I get to occasionally interact with people who know considerably more than me. One such person is Mark John Fernee, a physicist at the University of Queensland. I’ve learned a lot of science from an approach based on scepticism. For example, I was sceptical about relativity theory: that clocks could really slowly down and why did they slow down for one observer but not another, as demonstrated in the famous twin paradox. In fact, it’s nature’s paradoxes that provide the incentive to try and understand it to the extent that one can. 

 

Another example is quantum mechanics. For a long time, I followed David Bohm’s approach, which was really an attempt to bring QM back down to Earth so-to-speak. I believe that both Schrodinger and Einstein also believed in a ‘hidden-variables’ approach.

 

I finally gave this up when I concluded that QM and classical physics obey different rules: superposition and entanglement are not part of classical physics, either experimentally or mathematically. And I found that special relativity only made sense in the context of general relativity (which I discuss in more detail below).

 

And then you have the combination of special relativity with QM, which, from a mathematical perspective, allows anti-particles to exist. As Fernee points out, because an anti-particle can be represented mathematically by a particle going backwards in time, it ensures that charge is conserved by time’s arrow. In other words, you can turn an electron into a positron, or vice versa, by reversing time, which is why it’s never observed.

 

One of the paradoxes I now struggle with is that, according to special relativity, you can have different ‘nows’ in different parts of the universe. This is why most, if not all physicists, argue that the universe is completely deterministic, if someone’s future can be hypothetically observed by someone else’s motion. I confess I’m very sceptical about this. What they're saying is that the ‘now’ in some other part of the Universe is changed by an observer’s motion locally. Fernee quotes Roger Penrose in response to a question: can we theoretically teleport to some other location in the Universe instantaneously, like we see in science-fiction movies? According to Fernee (quoting Penrose), if you could and then teleport back, you might arrive before you left, because a random movement by you could change the ‘now’ in that distant part of the universe into your past. I’m assuming this can be demonstrated mathematically; it’s a consequence of simultaneity changing depending on the observer, according to special relativity. 

 

I’ve discussed this in other posts. I like to point out that, where there’s a causal relationship, the sequence of events can’t be changed, dependent on an observer’s perspective. Which makes me wonder: does a sequence change, dependent on an observer’s perspective, when they’re not causal? Is it possible that there is a sequence of events independent of any observer?

 

And this leads to another paradox that is hardly ever addressed which is that, despite this proliferation of ‘nows’, dependent on observers’ perspectives, we have an ‘age of the Universe’. I actually raised this with Fernee in a dialogue I had with him, and he referenced a paper by Tamara M. Davis and Charles H. Lineweaver at the University of New South Wales, titled, Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe. I’ve lost the link, and I can no longer even find the post on Quora, but I downloaded the paper, which is 24 pages long, not including the references.

 

Of course, it’s an academic paper, yet I found it easier to follow and understand than I might have expected. Which is not to say I have a full grasp of it, but I feel I can relay some of its most pertinent points. The paper is dated 13 November 2013, so it seems apt I’m writing about it on 13 Nov, 2021. Firstly, the cosmological model of the Universe the authors discuss, is referred to as ΛCDM cosmology (Lambda-CDM cosmology), where CDM is an acronym for Cold Dark Matter. Lambda (Λ) is the cosmological constant that gives us ‘dark energy’, so the model includes both dark energy and dark matter.

 

As the title suggests, the authors discuss misconceptions found in the literature concerning the horizon problem, and at the end they provide a list of examples, including one by Richard Feynman (1995), 

 

“It makes no sense to worry about the possibility of galaxies receding from us faster than light, whatever that means, since they would never be observable by hypothesis.” 

 

And this one by Paul Davies (1978): 

 

“. . . galaxies several billion light years away seem to be increasing their separation from us at nearly the speed of light. As we probe still farther into space the redshift grows without limit, and the galaxies seem to fade out and become black. When the speed of recession reaches the speed of light we cannot see them at all, for no light can reach us from the region beyond which the expansion is faster than light itself. This limit is called our horizon in space, and separates the regions of the universe of which we can know from the regions beyond about which no information is available, however powerful the instruments we use.” 

 

What the authors expound upon in the main body of their text is that there are, in effect, a number of horizons, which makes these statements erroneous at best. To be fair to both Feynman and Davies, the ΛCDM model of the Universe wasn’t known at the time. Dark energy wasn’t officially ‘discovered’ until 1998. Davis and Lineweaver provide diagrams to show these various horizons, which I can’t duplicate here, and if I did, I’d have trouble explicating them. But basically, there is a particle horizon, which is the limit of the observable universe, the Hubble sphere, which is the boundary of the expanding universe (where it equals c) and the event horizon. (To quote the authors: Our event horizon is our past light cone at the end of time, t = ∞ in this case.) There is a logical tendency to think they should all be the same thing, but they’re not, as the authors spend a good portion of their 24 pages expounding upon. To quote again:

 

The particle horizon at any particular time is a sphere around us whose radius equals the distance to the most distant object we can see... Our effective particle horizon is the cosmic microwave background (CMB).

 

Whereas:

 

Hubble sphere is defined to be the distance beyond which the recession velocity exceeds the speed of light, DHS = c/H. As we will see, the Hubble sphere is not an horizon. Redshift does not go to infinity for objects on our Hubble sphere (in general) and for many cosmological models we can see beyond it... The ratio of ∼ 3/1 is the ratio between the radius of the observable universe and the age of the universe, 46 Glyr/13.5 Gyr.

 

What you have to get your head around is that the universe is dynamic, and given the time it takes for light to reach us from the edge of the Universe, both the edge and the objects (we’re observing) have moved on, quite literally. This means we can observe objects over the horizon so-to-speak. But it’s even more complex than that, because the Hubble sphere, which is expanding, can overtake photons that were emitted beyond the horizon but are travelling towards us. According to the authors, we can observe objects that are ‘now’ travelling at superluminal speeds relative to us. 

 

This is how the authors explain it:

 

Light that superluminally receding objects emit propagates towards us with a local peculiar velocity of c, but since the recession velocity at that distance is greater than c, the total velocity of the light is away from us. However, since the radius of the Hubble sphere increases with time, some photons that were initially in a superluminally receding region later find themselves in a subluminally receding region. They can therefore approach us and eventually reach us. The objects that emitted the photons however, have moved to larger distances and so are still receding superluminally. Thus we can observe objects that are receding faster than the speed of light. 

 

One of the most illuminating aspects of their dissertation, for me, was that one needs to use a general relativistic (GR) derivation of the Doppler redshift and not a special relativistic (SR) derivation, which is usually used. They show graphically that the SR and GR derivations diverge, especially for further distances. On the same graph, they show how a non-relativistic Doppler shift, which would be ‘tired light’ (authors’ term) is actually a horizonal line, so nowhere near. The graph, of course, shows these curves against observations of super novae. As they explain it:

 

The general relativistic interpretation of the expansion interprets cosmological redshifts as an indication of velocity since the proper distance between comoving objects increases. However, the velocity is due to the rate of expansion of space, not movement through space, and therefore cannot be calculated with the special relativistic Doppler shift formula. 

 

What they are saying is that there is a distinction between the movement of the objects in space and the movement of space itself. For me, this ends the debate about whether ‘space’ is an entity or just the distance between objects. As much as I admire and respect Viktor T Toth, I’ve always had a problem with his argument that space ‘doesn’t expand’, but only the objects ‘move’ thus creating more space between them. The Hubble sphere, as I understand it, is where space equals c.

 

Later in their paper, Davis and Lineweaver describe how they derived their equation for the GR redshift.

 

For the observed time dilation of supernovae we have to take into account an extra time dilation factor that occurs because the distance to the emitter (and thus the distance light has to propagate to reach us) is increasing.

 

In other words, in calculating the redshift of a ‘comoving galaxy’, they also have to take into account the constant expansion of space in the photon’s journey to the observer. 

 

....the peculiar velocity of a photon, Rχ ̇, is c. Since the velocity of light through comoving coordinates is not constant (χ ̇ = c/R), to calculate comoving distance we cannot simply multiply the speed of light through comoving space by time. We have to integrate over this changing comoving speed of light for the duration of propagation. Thus, the comoving coordinate of a comoving object that emitted the light we now see at time t is attained by integrating.  (χ ̇is the time dependent expansion of space and R is the radial distance). 

 

Notice that in contrast to special relativity, the redshift does not indicate the velocity, it indicates the distance. That is, the redshift tells us not the velocity of the emitter, but where the emitter sits (at rest locally) in the coordinates of the universe. 

 

In other words, when we integrate χ ̇, we get χ, which is distance. The authors provide another equation for determining the velocity.

 

Now, one of the obvious aspects of this whole exercise is that they are calculating a redshift across space that changes over time, so what does time mean in this context?

 

This is how the authors explain it, just before their conclusion:

 

Throughout this paper we have used proper time, t, as the temporal measure. This is the time that appears in the RW metric and the Friedmann equations. This is a convenient time measure because it is the proper time of comoving observers. Moreover, the homogeneity of the universe is dependent on this choice of time coordinate — if any other time coordinate were chosen (that is not a trivial multiple of t) the density of the universe would be distance dependent. Time can be defined differently, for example to make the SR Doppler shift formula correctly calculate recession velocities from observed redshifts (Page, 1993). However, to do this we would have to sacrifice the homogeneity of the universe and the synchronous proper time of comoving objects.

 

I find it interesting that they adopt a ‘proper time’ for the whole universe. It makes one wonder what ‘now’ really means.

 

Footnote 1: I want to point out that in their acknowledgements, Davis and Lineweaver reference Brian Schmidt, who received a joint Nobel Prize for his work in empirically confirming dark energy, or the cosmological constant (Λ).

Footnote 2: You can download the paper here.

Addendum: This is a video by someone (who knows more than me) and doesn’t give his name. I posted a video by him before, regarding the question: Is gravity a force? His videos on Penrose tiling and the Feigenbaum constant are among the best.

 

In this video, he refutes my claim, arguing that space doesn’t expand. He makes one very compelling point that if space expanded so would atoms and so would we. Victor T Toth makes the exact same point, and I’d have to agree. The size of all atoms is determined by h (Planck's constant), which doesn't change with the expansion of the Universe. I might add that this presenter and Toth disagree on whether gravity is a force or not, so physicists don’t always agree, even in the same field, like cosmology.

 

In the video, he argues that there are 3 types of Doppler shift and contends that they are actually all the same. Most intriguing was the thought experiment that someone in ‘free fall’ wouldn’t see the Doppler shift that another observer would. In other words, it’s observer dependent.

 

But there is a spacetime metric or manifold, which forms the basis of general relativity theory (GR) and this can warp and curve (according to said theory). In fact, there is a phenomenon called ‘frame dragging’, where spacetime is dragged around by a spinning black hole. Light is always c in reference to this spacetime manifold. So when ‘space’ reaches the speed of light at the horizon relative to us, light is still c in that reference frame, even though it is expanding away from us at c or more. Space can travel faster than light, even though massive particles can’t, which is why ‘inflation’, proposed at the birth of the Universe, is possible.

 

Getting back to the Doppler shift the authors cite in their paper, they use a GR Doppler shift, which I believe isn’t covered in the video.