# Is the Universe self-similar?

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• Jebediah Kerman
The other possibility is that the universe is a 4-dimensional hypersphere, in which case the milky way would be visible multiple times. However, even in models like this the universe appears to be finite in size, so it's not clear how you could see it more than once.f

#### Jebediah Kerman

It’s confusing how a universe can have finite mass but infinite size. Could the universe be like a tesselation, where you always end up where you started? Not like a closed universe, but where space is just copied over and over?

It’s confusing how a universe can have finite mass
It doesn't. The total mass of the universe isn't a well-defined quantity.

Edit: it's also worth noting that our current best-fit model of the universe is spatially flat, not closed.

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It’s confusing how a universe can have finite mass but infinite size. Could the universe be like a tesselation, where you always end up where you started? Not like a closed universe, but where space is just copied over and over?
I don't think your first sentence is worded very well. If it's reworded as follows, it becomes (possibly) accurate:

"The observable universe is finite, but the universe as a whole is infinite."

Note: we don't know if the universe is infinite or not. The simplest models are infinite, but that doesn't necessarily mean they're correct. The universe may be finite.

That said, in the simplest infinite model, this is exactly what happens, but it isn't any sort of regular copying: it stems from the fact that within a finite observable universe, there are only a finite number of degrees of freedom. That means that if you were able to travel far beyond the cosmic horizon somehow, you'd eventually reach a different observable universe with an identical configuration. So there'd be another Sun, another Earth, and another you sitting there. But you'd have to travel really really far. You'd probably have to cross something like ##10^{10^{100}}## observable regions before seeing an exact copy, or something ridiculous like that. But since the universe is infinite, there has to eventually be a copy somewhere.

The problem with spatial infinity is it implies temporal infinity and consensus among cosmologist is the universe is of finite age, This concession puts a strain on the concept of spatially infinite or that it eventually repeats ad nauseum over sufficiently vast distances. This leads to curious scenarios like the pac-man universe which is finite, but, unbounded [spacetime boundaries present special problems that have no easy cosmological, or philosophical, solutions]. Since the risk of being proven wrong is low, most cosmologists are content with the easy way out offered by spatial nfinity, and reply 'What can of worms?', when pressed on the issue.

The problem with spatial infinity is it implies temporal infinity and consensus among cosmologist is the universe is of finite age, This concession puts a strain on the concept of spatially infinite or that it eventually repeats ad nauseum over sufficiently vast distances. This leads to curious scenarios like the pac-man universe which is finite, but, unbounded [spacetime boundaries present special problems that have no easy cosmological, or philosophical, solutions]. Since the risk of being proven wrong is low, most cosmologists are content with the easy way out offered by spatial nfinity, and reply 'What can of worms?', when pressed on the issue.
I don't know if I buy the logic that spatial infinity implies temporal infinity. But it is definitely the case that models which are spatially-infinite are difficult to make sense of (not due to the objections in the original post, but just because it's difficult to come up with a physical process that would generate such a universe).

That said, you're absolutely correct that most physicists ignore this sort of thing. Still, it is the expected result from eternal inflation models (though the "copies" are separated in both time and space in these models).

Is it true to say that space and time do not exist in same way that ponderable matter exists and therefore trying to estimate their total size, their beginning and their end is meaningless? We do seem to make relative spatial and temporal measurements on matter.

I have also wondered if Cosmologists consider the possibility that the Universe is circular? (like the surface of a balloon), so that we can perhaps see the milky way one or even more times, as its light travels the universe one or multiple times. I remember playing a game called asteroids like that :)

I have also wondered if Cosmologists consider the possibility that the Universe is circular? (like the surface of a balloon), so that we can perhaps see the milky way one or even more times, as its light travels the universe one or multiple times. I remember playing a game called asteroids like that :)

You seem to be alluding to at least two different possibilities here. One is that the universe is spatially a 3-sphere instead of flat; in principle that would allow light to circumnavigate the universe and come back to its starting point. However, in all such models it turns out to be the case that, either the universe recollapses to a big crunch before light has had time to circumnavigate the universe once, or, in the presence of a positive cosmological constant, the universe expands too fast for light to make it around (it gets larger faster than the light travels).

The second possibility is that the universe's spatial topology is not simply connected; for example, it might be a flat 3-torus, which is the 3-dimensional version of the asteroids game (I played it a lot too ). Unlike the 3-sphere case, the universe in a model like this can actually be spatially flat in terms of local geometry, so measurements of the universe's spatial curvature, by themselves, can't tell us that the universe isn't like this (whereas they can tell us that the universe is not spatially a 3-sphere, since that requires spatial curvature). As far as I know, in a model like this it is possible for light to return to its starting point, but we have no evidence of this actually happening, so we have no evidence that a model like this describes our actual universe.

QuantumQuest and Tanelorn
Thanks Peter. The best thing about these Forums is when Pros such as yourself are able to guide part time amateurs such as myself.
I suspected that these concepts were considered but I just didn't know that they were called "3 sphere" and "flat 3 torus". I will try to find more information about these.
I guess it would be very difficult to prove that an extremely distant galaxy is the milky way of the past.
Thanks again.

(whereas they can tell us that the universe is not spatially a 3-sphere, since that requires spatial curvature)
I wouldn't go that far. Two things muck this up:
1) The 3-sphere topology only specifies the average curvature. It is certainly possible for localized regions to have curvature which differs from this value. If those local variations are large enough compared to the size of the 3-sphere, then the curvature within the observable universe might still end up with the wrong sign despite the overall topology.
2) Even if the local variations in curvature were extremely tiny, the 3-sphere could simply be large enough that its curvature is too small to be measurable.

Unfortunately, there's little chance we'll ever be able to definitively measure the overall topology of the universe.

Two things muck this up

These are both valid points, yes. In principle it is still possible to measure over a large enough scale to overcome these obstacles, but I agree that in practice in our actual universe there is very little chance we'll be able to get a definite answer.

Is it possible that the mathematics currently used to speculate about the universe isn't up to the task? For example, pretend there's a civilization whose mathematical systems are all based on counting. They've developed extremely sophisticated ways to count, to describe things using counting, even to talk about counting. Sure they have some anomalies, infinities, etc. that they just shrug off, but all in all they've been able to develop sophisticated means of description. But for whatever reason they never discovered geometry. So their descriptions of the universe would be necessarily limited, though they wouldn't know it. How do we know that our cosmologists aren't in the same boat? All these anomalies and infinities and breakdowns are just because they're using the wrong type of math.

Is it possible that the mathematics currently used to speculate about the universe isn't up to the task? For example, pretend there's a civilization whose mathematical systems are all based on counting. They've developed extremely sophisticated ways to count, to describe things using counting, even to talk about counting. Sure they have some anomalies, infinities, etc. that they just shrug off, but all in all they've been able to develop sophisticated means of description. But for whatever reason they never discovered geometry. So their descriptions of the universe would be necessarily limited, though they wouldn't know it. How do we know that our cosmologists aren't in the same boat? All these anomalies and infinities and breakdowns are just because they're using the wrong type of math.
The mathematical models used are intrinsically tied into the physical theories used. You can't separate the two.

Is it possible that our physical theories are mistaken? Absolutely! In fact, we know they have to be wrong on some level.

But the infinities that exist in this case are "real" in the sense that they aren't simply artifacts of the way things are calculated. Getting around them is fundamentally impossible without a different physical model.

This is as opposed to quantum field theory, where many of the infinities which exist are a result of the approximations used to perform the calculations, rather than being intrinsic to the calculations themselves.

The mathematical models used are intrinsically tied into the physical theories used. You can't separate the two.
Can you explain this part a little bit more. Couldn't the same be said in the context of my example of the counting based physics? Doesn't the math preceed the physical theory?

Can you explain this part a little bit more. Couldn't the same be said in the context of my example of the counting based physics? Doesn't the math preceed the physical theory?
I think the point being made is that Newtonian physics isn't usually described in terms of geometry, but it can be - Cartan did it. Likewise, it would not surprise me if it's possible to develop GR without using geometry, so it's not at all clear that your hypothetical system would be different from GR. In fact, if it made the same predictions then it would not be distinct. If it made different predictions then it would be a different physical theory, one that may or may not be better than GR.

I think the point being made is that Newtonian physics isn't usually described in terms of geometry, but it can be - Cartan did it. Likewise, it would not surprise me if it's possible to develop GR without using geometry, so it's not at all clear that your hypothetical system would be different from GR. In fact, if it made the same predictions then it would not be distinct. If it made different predictions then it would be a different physical theory, one that may or may not be better than GR.
The reference to geometry was unimportant, it was just part of the analogy. Pretend I used the word "phlemomotry". I guess in terms of the second part of your response, my question is, does the math determine the predictions? Point being, if the math predicts something weird like infinities, or some of the aspects of the big bang theory (the "right way to think about it" as described in the Sci Am article), maybe it's not that the universe has those odd properties, but that the math used to get to the predictions is the wrong type of math. (I know this is straying into philosophy of physics. Pardon my wandering.) To put it extremely crudely, if all you have is a hammer (current math) then everything looks like a nail.

My point was that two systems of maths that give the same predictions aren't really different - Newton and Cartan being an example. So the question is, does your hypothetical system give different predictions from GR? If it doesn't then it's the same theory as GR, just expressed in a different language. If it does give different predictions then it's a different theory.

As @kimbyd says, we are fully aware that GR is very likely not completely accurate. The singularities it predicts almost certainly do not exist in reality. We continue to look for successor theories, which may or may not be expressed (or expressible) in terms of geometry. The only real constraint on the form of such a theory is that its predictions must match GR in cases we have tested to at least the precision of our tests.

Carpe Physicum
As @kimbyd says, we are fully aware that GR is very likely not completely accurate. The singularities it predicts almost certainly do not exist in reality. We continue to look for successor theories, which may or may not be expressed (or expressible) in terms of geometry. The only real constraint on the form of such a theory is that its predictions must match GR in cases we have tested to at least the precision of our tests.
Ah ok that makes a lot of sense. I was not aware that cosmologists think the singularities probably don't exist in reality. I thought that was the forgone conclusion which then to us laymen indicated just how strange the universe is. (I guess you can't make big budget sci fi thrillers on the reality of current theory. ;) ) Thanks for being patient and giving honest, not condescending replies.

Can you explain this part a little bit more. Couldn't the same be said in the context of my example of the counting based physics? Doesn't the math preceed the physical theory?
Not necessarily. Experimental results have often provided hints as to new possible mathematical models that might be used. The mathematics involved in quantum mechanics, for example, would probably never have developed as far as it has if there was no demand for that kind of theory in physics. Physicists generally don't just pick pre-existing mathematical models off the shelf to make use of them. Some theorists explore new regimes of mathematics that hadn't yet been explored, or extend regimes that had only been examined briefly by mathematicians. Furthermore, many mathematicians have a great deal of interest in math as it is applied to physics, such that they pursue investigations with the specific purpose of understanding physics better.

There have also been situations where physicists have come up with their own physical models without understanding the full mathematical context, and a mathematician will come along and show that this "new" thing the physicist came up with is actually this other concept that mathematicians have studied for a long time, and the mathematicians help to clarify and expand upon the physical theory.

Right now, the overwhelming challenge in physics lies not with better math, but rather with unexplored experimental regimes.

But since the universe is infinite, there has to eventually be a copy somewhere.
Hi @kimbyd:

I have found this thread quite interesting, and in particular the very clear explanations you made regarding the OP. However, the above quote does not quite make sense to me. There seems to me to be two problems with this assumption.

I assume the universe we are discussing to be the current best cosmological model for out actual universe.

(1) Each observable universe (OU) has its own history based on it's "initial conditions" at some arbitrarily chosen past time T using comoving coordinates. If we assume that there is some other very distant OU to ours which is identical to ours now, it must have been identical to ours at the earlier time T. My thought is that the influences of interactions among the various components of the two OUs dependent on QM effects, would necessarily introduce small changes between the two OUs over the time between T and now, and each small change would grow to become a significant change. Thus the two OUs would not be identical now even they were identical at time T.

(2) For the purpose of presenting this problem I ignore the QM influence in (1). I think you are confusing two different kinds of infinity: ℵ0 and ℵ1.
The number of OUs is ℵ0. The number of possible configurations of time T initial conditions is ℵ1.
Therefore the likelihood that there is another OU with initial conditions identical to ours in infinitesimal. If the initial conditions are different, even when the difference is small, they will produce current configurations which will be greatly different for chaos theory reasons.

If you disagree with either the (1) or (2) conclusion, I would much appreciate your explanation.

Regard,
Buzz

The number of OUs is ℵ0.

Why?

Hi @PeterDonis:

Imagine a line of infinite length, from zero to infinity. How many meters are in this line? You can set up a map of the line by naming each meter with it's lower boundary. Thus the first meter is "0", the next is "1", the next is "2", etc. Therefore the number of meters is the same as the number of non-negative integers.

Is this sufficient, or do you also want an explanation about why this also applies to the number of cubic meters in the infinite universe? Do you also want an explanation about why the number of initial conditions is ℵ1?

Regards,
Buzz

Why?
I think the relevant point is slightly weaker than the one that @Buzz Bloom is trying to make.

One can nail down the initial conditions for the entire universe in terms of the initial conditions for a countable collection of observable universes. The uncountably many other observable universes will overlap with this set and will have their initial conditions determined thereby.

Buzz Bloom
One can nail down the initial conditions for the entire universe in terms of the initial conditions for a countable collection of observable universes. The uncountably many other observable universes will overlap with this set and will have their initial conditions determined thereby.

Ah, ok.

Hi @kimbyd:

I have found this thread quite interesting, and in particular the very clear explanations you made regarding the OP. However, the above quote does not quite make sense to me. There seems to me to be two problems with this assumption.

I assume the universe we are discussing to be the current best cosmological model for out actual universe.

(1) Each observable universe (OU) has its own history based on it's "initial conditions" at some arbitrarily chosen past time T using comoving coordinates. If we assume that there is some other very distant OU to ours which is identical to ours now, it must have been identical to ours at the earlier time T. My thought is that the influences of interactions among the various components of the two OUs dependent on QM effects, would necessarily introduce small changes between the two OUs over the time between T and now, and each small change would grow to become a significant change. Thus the two OUs would not be identical now even they were identical at time T.

(2) For the purpose of presenting this problem I ignore the QM influence in (1). I think you are confusing two different kinds of infinity: ℵ0 and ℵ1.
The number of OUs is ℵ0. The number of possible configurations of time T initial conditions is ℵ1.
Therefore the likelihood that there is another OU with initial conditions identical to ours in infinitesimal. If the initial conditions are different, even when the difference is small, they will produce current configurations which will be greatly different for chaos theory reasons.

If you disagree with either the (1) or (2) conclusion, I would much appreciate your explanation.

Regard,
Buzz
In quantum mechanics, any finite region has a finite number of degrees of freedom. In our case, as long as the cosmological constant is positive, the total number of configurations within a given cosmological horizon can be counted precisely based upon the entropy of the horizon. The finite number of degrees of freedom guarantees infinite repetitions if the universe is genuinely spatially-infinite (with the same cosmological constant everywhere).

The far-away identical regions would include not only identical current states, but also identical initial conditions and histories.

Also, let me just point out a part of the argument above that I ignored: in principle it's possible to conceive of an infinite universe with some initial conditions only being replicated a finite number of times. For a very simple example, consider the decimal form of 1/6: 0.1666666...

This is an infinite sequence, with the digit 6 repeating an infinite number of times, while the digit 1 only appears a single time. Thus to close the loop and complete the argument you also have to guarantee that the physics which set up the initial conditions explore all of the possible configurations an infinite number of times. In this simplest models of cosmic inflation, this is trivially true: the zero-point fluctuations which set the initial conditions are purely random.

I imagine it would be conceivable to set up a model where some finite number of initial configurations are genuinely unique, but that would be really hard to achieve in practice, and regardless of the model you'd expect it to be basically impossible to find yourself in one of those states.

In quantum mechanics, any finite region has a finite number of degrees of freedom. In our case, as long as the cosmological constant is positive, the total number of configurations within a given cosmological horizon can be counted precisely based upon the entropy of the horizon.
Hmm, I'm having difficulty following this argument. When you say "any finite region", do you mean a finite 3D spacelike slice, or a general 4D spacetime volume?

Hmm, I'm having difficulty following this argument. When you say "any finite region", do you mean a finite 3D spacelike slice, or a general 4D spacetime volume?
In QM, the 4D space-time is degenerate, as you can time-evolve the state of the system at anyone time to obtain the state of the system at another. Thus the full configuration of the system is described by a single time-slicing (and it can be evolved forward and back to get the full 4D state).

Pedantic note:
This requires an interpretation of QM that doesn't collapse, such as the Many Worlds interpretation, and knowing the full wavefunction. More precisely, it requires unitary evolution.

In QM, the 4D space-time is degenerate, as you can time-evolve the state of the system at anyone time to obtain the state of the system at another. Thus the full configuration of the system is described by a single time-slicing (and it can be evolved forward and back to get the full 4D state).
Yes, so,... what precisely did you mean earlier by "a given cosmological horizon"? Do you mean the horizon of our distant (but finite) past, or some other horizon at a finite spacelike distance from us?

I'm familiar with horizons, their thermal radiation, entropy, etc, but I'm still having trouble meshing this with what you said. Could you elaborate in more detail, pls?

Yes, so,... what precisely did you mean earlier by "a given cosmological horizon"? Do you mean the horizon of our distant (but finite) past, or some other horizon at a finite spacelike distance from us?

I'm familiar with horizons, their thermal radiation, entropy, etc, but I'm still having trouble meshing this with what you said. Could you elaborate in more detail, pls?
I believe the horizon in question is the cosmological event horizon, beyond which it is impossible for signals to reach a given observer. That's the horizon that has a temperature, as I understand it.

I believe the horizon in question is the cosmological event horizon, beyond which it is impossible for signals to reach a given observer. That's the horizon that has a temperature, as I understand it.
Oh, I see now -- like (for example) in de Sitter spacetime which has a Killing horizon at ##r = \sqrt{3/\Lambda}## .