# I Is the Universe a 3-sphere or a 4-sphere?

#### Herbascious J

When discussing the shape of the universe (flatness/curvature), I often hear of three possible examples; spherical, flat and hyperbolic. Presenters will often use a 2-D analogy of how a flat sheet can be curved or shaped, like a saddle, table, or surface of a ball, where triangles can be defined with different sums to their angles. This analogy makes perfect sense to me. In the same discussion they often talk about how the universe is expanding and depending on the shape, rate of expansion and matter density, it may rapidly expand into infinity, just slow down enough to never quite collapse, or turn around and collapse back down to a point. In GR I believe we are talking about space-time, which is a 4-D ‘space’. My question is this, when thinking of the 2-D analogy, do I extend this analog to a 3-D space, or a 4-D space in my imagination? Part of my confusion is whether to include the time-like aspect of space-time into the shape of the universe, or is it handled separately? In the spherical example, I understand the universe to be closed, so does this imply that time also goes in a circle and closes on itself? Can a spherically closed universe just be a 3-sphere and then expand forever without collapse?

#### PeterDonis

Mentor
When discussing the shape of the universe (flatness/curvature), I often hear of three possible examples; spherical, flat and hyperbolic.
These all refer to the spatial geometry (surfaces of constant time according to comoving observers).

In GR I believe we are talking about space-time, which is a 4-D ‘space’.
We are, but unfortunately most discussions of cosmology implicitly split these 4 dimensions up into 3 of space and one of time in a particular way (as noted above, the way comoving observers--observers who always see the universe as homogeneous and isotropic--would naturally do it). So nobody really talks (at least, not in the discussions you refer to) about the overall 4-D geometry of the spacetime of the universe.

In the spherical example, I understand the universe to be closed, so does this imply that time also goes in a circle and closes on itself?
No. Even for the case where a universe that is spatially closed (i.e., spatially a 3-sphere) recollapses to a big crunch, time is not "circular"--the end of the universe does not connect back around to the beginning.

Can a spherically closed universe just be a 3-sphere and then expand forever without collapse?
If there is zero cosmological constant (dark energy), then no; a closed universe will always recollapse. But if there is a positive cosmological constant (dark energy), then it is possible for a spatially closed universe to expand forever (although our best current model says that our actual universe is not an example of this--it's spatially flat, not closed).

#### Herbascious J

Ok, that makes more sense to me. So (disregarding dark energy/lambda) in theory, the time-like aspect is more about the expansion and whether or not it will collapse, correct? So I can disregard 'time' as being being identical to the other dimensions in shape? it's more like the bouncing ball that either gains enough momentum to escape earth's gravity or not and comes crashing down. So then it's not possible for an infinite universe to re-collapse? I get confused on this last note because when I imagine a big bang inflation event, I can't decide it was ever finite with a high density and that somehow the dimensions laid out flat becoming infinite, or if the universe just suddenly exploded from nothing and instantly became an infinite expanding space. It seems strange because if space was ever finite within even 10 to the -35 seconds after it seems like the matter/energy would also be finite and how could that energy evenly fill out an infinite space?

#### PeterDonis

Mentor
the time-like aspect is more about the expansion and whether or not it will collapse, correct?
More or less. But see below.

So I can disregard 'time' as being being identical to the other dimensions in shape?
No. Expansion means there is curvature in the time dimension, even if space is flat (as it is in our best current model).

So then it's not possible for an infinite universe to re-collapse?
It is if there is a negative cosmological constant. But that's not considered physically reasonable so it's not a case that is often discussed.

when I imagine a big bang inflation event, I can't decide it was ever finite with a high density and that somehow the dimensions laid out flat becoming infinite, or if the universe just suddenly exploded from nothing and instantly became an infinite expanding space
Neither of these are correct. A spatially infinite universe was always spatially infinite.

#### metastable

Has the scenario of a "collapsing" spacetime co-occurring simultaneously with an expansion of matter within said spacetime ever been seriously considered?

#### PeterDonis

Mentor
the scenario of a "collapsing" spacetime co-occurring simultaneously with an expansion of matter within said spacetime
What would this even mean?

Please review the PF rules on personal speculations.

#### metastable

What would this even mean?
If we take the distance and therefore flight time of light between 2 electrons at rest relative to each other (within a small margin of error), each in a separate intergalactic void, and therefore separated by vast intergalactic distances, I expect due to the effects of observed accelerating expansion, that the distance between these 2 particles will increase as a result of said expansion. But will the flight time of light change linearly as this assumed distance between the electrons changes?

#### PeterDonis

Mentor
If we take the distance and therefore flight time of light between 2 electrons at rest relative to each other (within a small margin of error), each in a separate intergalactic void, and therefore separated by vast intergalactic distances, I expect due to the effects of observed accelerating expansion, that the distance between these 2 particles will increase as a result of said expansion.
If they are sufficiently far apart, yes, this is what will happen.

will the flight time of light change linearly as this assumed distance between the electrons changes?
Linearly with respect to what?

Also, I fail to see what any of this has to do with "a collapsing spacetime co-occurring with expansion of matter", which is what you originally were talking about.

#### metastable

Linearly with respect to what?

Also, I fail to see what any of this has to do with "a collapsing spacetime co-occurring with expansion of matter", which is what you originally were talking about.
Will the flight time of light increase linearly with the increasing distance? I see three possible outcomes:

-the distance increases, and the flight time of light increases linearly with any distance increase
-the distance increases, and the flight time of light stays the same
-the distance increases, and the flight time of light decreases

^I am not saying all of these are possible, I am wondering which is closest to reality, how we know and what it implies about the "expansion," "contraction" or "steady-statedness" of the geometry of spacetime.

#### PeterDonis

Mentor
Will the flight time of light increase linearly with the increasing distance?
Again, linearly with respect to what? Your question can't be answered until you specify that.

#### metastable

Again, linearly with respect to what? Your question can't be answered until you specify that.
At Time = 0 seconds, the electrons are separated by 1 billion light years, and are at rest with respect to each other within a small margin of error in traps on 2 spacecraft with synchronized clocks and each craft detects no acceleration or fictitious forces. One craft emits pulses of light every second that the other craft can eventually detect. Due to accelerating expansion, although neither craft ever detects any acceleration, I expect the distance between both electrons increases. As time passes and the expansion rate increases, does recorded interval between the received pulses as recorded by the second craft increase to greater than one second per pulse, decrease to less than one second per pulse or stay exactly one second per pulse?

#### PeterDonis

Mentor
Due to accelerating expansion, although neither craft ever detects any acceleration, I expect the distance between both electrons increases.
I believe a billion light years of separation is sufficient for that (and I'll assume it is for the rest of this post). If the separation is not large enough, the electrons will start falling towards each other because of the gravity of the matter in the universe between them. The effect of dark energy, which is what you are describing, only overcomes that if the separation is large enough.

As time passes and the expansion rate increases, does recorded interval between the received pulses as recorded by the second craft increase to greater than one second per pulse, decrease to less than one second per pulse or stay exactly one second per pulse?
First, it's important to point out a key factor in the scenario as you've specified it: neither electron is comoving. That is, neither electron sees the universe as homogeneous and isotropic. If they did, they would not start out at rest relative to each other; they would start out moving apart.

Second, since the electrons are a billion light years apart, it will take at least a billion years (actually more, as we'll see in a moment) for a light signal to travel from one to the other. And in that time, the universe will have expanded. Even if the expansion is not accelerating (i.e., even if there were no dark energy), that expansion itself would tend to pull the electrons apart (because the matter in the universe is expanding). So even without dark energy, the electrons would move apart over a billion years and the light travel time between them would increase, which would cause the spacing of the pulses as received by the second electron to increase.

All that dark energy really does is make the spacing between the pulses increase faster than it otherwise would.

#### Herbascious J

If there is no dark energy, shouldn't the electrons signals remain at a constant interval? I assume this because even though they would be moving apart, their apparent 'velocity' between them would be constant and any Doppler like shift would be constant. It would just take the impulses longer to reach them (hence the constant doppler shift)?

#### PeterDonis

Mentor
even though they would be moving apart, their apparent 'velocity' between them would be constant
No, it wouldn't. Remember that you specified that the electrons start out at rest relative to each other. The rest of the matter of the universe, as it expands, will pull the electrons apart given that initial condition. That means the electrons will accelerate away from each other even without dark energy. Dark energy, as I said, simply increases the acceleration.

#### metastable

I believe a billion light years of separation is sufficient for that (and I'll assume it is for the rest of this post). If the separation is not large enough, the electrons will start falling towards each other because of the gravity of the matter in the universe between them. The effect of dark energy, which is what you are describing, only overcomes that if the separation is large enough.

First, it's important to point out a key factor in the scenario as you've specified it: neither electron is comoving. That is, neither electron sees the universe as homogeneous and isotropic. If they did, they would not start out at rest relative to each other; they would start out moving apart.

Second, since the electrons are a billion light years apart, it will take at least a billion years (actually more, as we'll see in a moment) for a light signal to travel from one to the other. And in that time, the universe will have expanded. Even if the expansion is not accelerating (i.e., even if there were no dark energy), that expansion itself would tend to pull the electrons apart (because the matter in the universe is expanding). So even without dark energy, the electrons would move apart over a billion years and the light travel time between them would increase, which would cause the spacing of the pulses as received by the second electron to increase.

All that dark energy really does is make the spacing between the pulses increase faster than it otherwise would.
So in your view dark energy causes the durations between received pulses at the second craft to increase over time, by a greater amount than can be explained through gravity alone.

I'd like to clarify a few points of the thought experiment. I define Time = 0 as the same time a light pulse is received by the second craft which is measured by the second craft to be the same wavelength as a pre-agreed transmission wavelength. We've also pre-agreed the first craft will cease transmissions if it detects fictitious forces or acceleration and the second craft will cease observations if it detects acceleration or fictitious forces.

Suppose I'm on the second spacecraft and I don't have enough information about the universe to know in advance or deduce what the outcome of the measurements on red shift and interval between pulses will be. In my craft I consider what possible outcomes might be measured. Since Time = 0 is the time that I first measure a pulse which from my spacecraft appears not to be red or blue shifted from the pre-agreed transmission frequency and I know the first spacecraft is sending pulses every second and I know the wavelength, over time I can observe both whether these pulses appear red or blue shifted and whether the interval between arrival times changes.

Since I've stated I don't have enough information on the second craft to know what the results will be in advance, I consider the possible measurements and what implications they have.

I consider a subset of possibilities-- I am now only considering the range of possibilities in which measurements show that over time the pulses become more and more redshifted from the pre-agreed frequency.

From this subset of red-shifting possibilities, I consider the implications of what it means if the following occurs:

-Pulses become more and more red shifted and interval between received pulses increases to more than one second
-Pulses become more and more red shifted and interval between received pulses remains exactly one second
-Pulses become more and more red shifted and interval between received pulses decreases to less than one second

Now from this list of possibilities I consider the implications of only one:

-Pulses become more and more red shifted and interval between received pulses increases to more than one second

I can list possible meanings of this particular result without being entirely certain if any of the meanings I've listed are true:

Possibility A: Subtracting the effects of gravity, since light is taking a longer and longer time to reach me from the first craft and the light is redshifted, the second craft must be increasing its distance, because light always takes the same amount of time to cover a given distance between 2 co-moving points in spacetime

Possibility B: Subtracting the effects of gravity, Since neither ship measured acceleration or rotation, the two ships are in fact the same distance according to some metric, but the geometry of spacetime is contracting, causing light to take a greater and greater time to cover the same distance

From the results of the experiment plus any other information I could gather from the second ship, how do I correctly interpret the meaning of the results of the described a measurement showing the pulses become more and more red shifted and interval between received pulses increases to more than one second as measured on the second craft?

#### PAllen

No, it wouldn't. Remember that you specified that the electrons start out at rest relative to each other. The rest of the matter of the universe, as it expands, will pull the electrons apart given that initial condition. That means the electrons will accelerate away from each other even without dark energy. Dark energy, as I said, simply increases the acceleration.
Actually, it depends on the form of a(t) on the FLRW metric. While still having expansion, you can have the particles converge, diverge, or stay same distance, per some reasonable definition. It depends on a derivative of a(t) - I can’t remember right now whether it is the first or second derivative that determines this behavior.

#### PeterDonis

Mentor
So in your view dark energy causes the durations between received pulses at the second craft to increase over time, by a greater amount than can be explained through gravity alone.
More precisely, than can be explained by expansion of the universe without dark energy. Calling the effect of the expansion of the universe without dark energy "gravity", while not technically incorrect, is probably not a good choice of terminology, since the effect is very different from the gravity of an ordinary isolated gravitating mass like a planet or star.

I define Time = 0 as the same time a light pulse is received by the second craft which is measured by the second craft to be the same wavelength as a pre-agreed transmission wavelength.
There will be no such pulse for the scenario as you have given it, unless I am misunderstanding the scenario. In the scenario as you have given it, the first craft starts emitting pulses when it and the second craft are at rest relative to each other. If that is the case, then the very first pulse the second craft receives will be redshifted.

-Pulses become more and more red shifted and interval between received pulses increases to more than one second
-Pulses become more and more red shifted and interval between received pulses remains exactly one second
-Pulses become more and more red shifted and interval between received pulses decreases to less than one second
Only the first of these is possible. The redshift factor is also the factor by which the time between pulses will increase. You can see why this must be the case if you consider the behavior of wave crests and realize that it will be the same as the behavior of pulses.

Now from this list of possibilities I consider the implications of only one
Which is a good choice since it's the only one that's actually possible. See above.

Possibility A: Subtracting the effects of gravity
Possibility B: Subtracting the effects of gravity
You can't subtract the effects of gravity; the effects of gravity are the effects of spacetime geometry, and if you change the spacetime geometry, you change the whole scenario, including the predicted observations. If there were no gravity in the scenario, you would be in flat Minkowski spacetime, and if the two craft started off at rest relative to each other, and neither one ever fired its rocket engines, the redshift of the light pulses would start at zero and would stay at zero forever.

how do I correctly interpret the meaning of the results of the described a measurement showing the pulses become more and more red shifted and interval between received pulses increases to more than one second as measured on the second craft?
As showing you that the two craft are in an expanding FRW spacetime geometry. The exact behavior of the redshift/pulse interval as a function of time tells you exactly which particular expanding FRW spacetime geometry the two craft are in (i.e., what kind of "stuff" is present--ordinary matter, or ordinary matter + dark energy).

#### PeterDonis

Mentor
it depends on the form of a(t) on the FLRW metric. While still having expansion, you can have the particles converge, diverge, or stay same distance, per some reasonable definition
I was assuming that the expansion scalar is positive, i.e., $\dot{a} / a$ is positive.

#### metastable

From this subset of red-shifting possibilities, I consider the implications of what it means if the following occurs:

-Pulses become more and more red shifted and interval between received pulses increases to more than one second
-Pulses become more and more red shifted and interval between received pulses remains exactly one second
-Pulses become more and more red shifted and interval between received pulses decreases to less than one second
Only the first of these is possible. The redshift factor is also the factor by which the time between pulses will increase. You can see why this must be the case if you consider the behavior of wave crests and realize that it will be the same as the behavior of pulses.
You can't subtract the effects of gravity; the effects of gravity are the effects of spacetime geometry, and if you change the spacetime geometry, you change the whole scenario, including the predicted observations.
If I understand what you are saying correctly:

-I will certainly detect the redshift caused by a "recessional" velocity by the time I receive the 1st pulse if the 2 ships were both "at rest" with respect to each other when the first pulse was sent, and this redshift effect increases with increasing initial distance.

-I can be certain in advance given the setup scenario that the interval between subsequent received pulses increases to more than one second

-None of this requires invoking a faster than light interaction between the 2 craft

-Effects of gravity travel at no more than the speed of light

#### PAllen

I was assuming that the expansion scalar is positive, i.e., $\dot{a} / a$ is positive.
But That has more to do with the behavior of comoving observers, not observers starting with no mutual spectral shift. I am having trouble finding a paper I saw on this, but I’m thinking the criteria involved the second derivative of a(t).

#### PeterDonis

Mentor
-I will certainly detect the redshift caused by a "recessional" velocity by the time I receive the 1st pulse if the 2 ships were both "at rest" with respect to each other when the first pulse was sent, and this redshift effect increases with increasing initial distance.
Yes, because the separation between the craft is large enough that the expansion of the universe cannot be ignored. If the craft were only separated by one light-year, the expansion of the universe wouldn't have enough of an effect to be detectable (at least I don't think so given our current level of technology at detecting frequency shifts).

-I can be certain in advance given the setup scenario that the interval between subsequent received pulses increases to more than one second
Yes.

-None of this requires invoking a faster than light interaction between the 2 craft
Yes.

-Effects of gravity travel at no more than the speed of light
This is true, but irrelevant to the scenario since no effects of gravity need to propagate; the effects of the expansion of the matter in the universe are felt locally by the light pulses as they travel, they don't have to propagate anywhere.

#### PeterDonis

Mentor
That has more to do with the behavior of comoving observers
The expansion scalar describes the congruence of comoving observers, yes. But it also constrains the overall spacetime geometry, which affects everything. You're right, though, that it would be good to actually look at the math for that spacetime geometry as it applies to objects that start off at relative rest instead of being comoving.

#### metastable

I'm confused how to interpret the meaning if I observe a null spectral shift at distance of a billion light years...

As we've already discussed, If the 2 spaceships are both "at rest" at time = 0 seconds at distance = 1 billion light years, the second spaceship sees redshift.

So if the second ship sees null spectral shift, does the recessional velocity match the approach velocity? do we say the distance is changing?

#### Bandersnatch

But That has more to do with the behavior of comoving observers, not observers starting with no mutual spectral shift. I am having trouble finding a paper I saw on this, but I’m thinking the criteria involved the second derivative of a(t).
The expansion scalar describes the congruence of comoving observers, yes. But it also constrains the overall spacetime geometry, which affects everything. You're right, though, that it would be good to actually look at the math for that spacetime geometry as it applies to objects that start off at relative rest instead of being comoving.
@PAllen is right, the behaviour of initially stationary observers depends on the sign of the deceleration parameter. In the decelerating scenario, as in the example discussed above, the observers approach each other.
Paper here:
Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects; Tamara M. Davis, Charles H. Lineweaver, John K. Webb

#### PAllen

@PAllen is right, the behaviour of initially stationary observers depends on the sign of the deceleration parameter. In the decelerating scenario, as in the example discussed above, the observers approach each other.
Paper here:
Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects; Tamara M. Davis, Charles H. Lineweaver, John K. Webb
Thanks!

I thought of a heuristic argument why you would expect this to be the case. The tendency of initially at mutual rest test bodies to diverge or converge is a tidal gravity effect. Tidal gravity is expected to be determined by second derivatives of the metric, not first derivatives.

"Is the Universe a 3-sphere or a 4-sphere?"

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