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

#### PAllen

I thought of a more specific reason to expect that the second derivative must be involved:

The Milne model has positive Hubble Parameter but zero deceleration parameter, and consistent with this, initially mutually at rest test bodies remain that way and remain at constant distance (e.g. as measured by light round trip).

The paper I had seen before was different from the one that @Bandersnatch posted, but obviously consistent with it. What I remember as initially surprising to me was that you could have converging test bodies in an expanding universe and diverging test bodies in a contracting universe, depending on the particular a(t).

#### Bandersnatch

The Milne model has positive Hubble Parameter but zero deceleration parameter, and consistent with this, initially mutually at rest test bodies remain that way and remain at constant distance (e.g. as measured by light round trip).
That's, likewise, what I always start off with when attempting to imagine what's happening, intuitively. Still, even when considering the Milne universe, the idea that there should be a 'drag' associated with unaccelerated expansion always creeps in, and it takes pains to extirpate it.

I'm not sure any of this is helping @Herbascious J or @metastable .

#### metastable

Suppose I mathematically describe an "imaginary" 2-D "spacetime" with a singularity in the middle, and in this 2D universe the trajectory of any particle moving "through" any of the observable universes A-F, relative to the non-accelerating stationary reference lines in the middle circle, tends to be curved proportionally to its distance from the singularity. I am in the observable universe labeled F, so I can't see the singularity, and no particle that crosses my entire observable universe appears to "loop back around on itself." In this imaginary universe, even though I can't see the singularity from my position F, the curvature of the spacetime in my observable universe "F" is still being affected by the singularity.

In the actual universe, is the total universe thought to be much larger than our currently observable universe? What are the most promising methods to "rule out" the possibility of a spacetime singularity located outside the observable universe, but still located well within the "total" universe?

#### Bandersnatch

There seems to be some confusion here. The cosmological singularity is not in space, but in time.
Unless you mean black hole singularities, which should be aplenty without as well as within the observable universe.

But yes, the observable universe is a subset of some larger universe.

#### metastable

But yes, the observable universe is a subset of some larger universe.
Thank you.

Does light "climbing" "out" of a gravitational potential experience redshift?

Suppose our observable universe is at position F in the above chart (oversimplified), and outside the "observable" part F of the universe is an "immense" black hole represented in the middle, and the matter in our small portion F of the "total" universe predominantly/statistically has orbital velocity "away" from the immense gravitational singularity in the middle, which is outside of our "observable" portion. Could such a setup lead to observations in our "universe portion" of "increasing" measured redshift over time observed in all directions statistically correlated with distance from our position & according to known laws?

#### Nugatory

Mentor
Does light "climbing" "out" of a gravitational potential experience redshift?
It depends on the observer. If two observers are moving relative to one another as they pass through the point where the redshift is being measured, they will in general measure different redshifts, or even blue shifts.

The gravitational redshift you’ll read about in popular treatments can be said to be caused by light “climbing out” of a potential well, but this is a special case: limited to spacetimes in which the notion of potential is meaningful; and then comparing the redshift measured by observers at different heights in that well and using a particular definition of “at rest” relative to one another. The intuition you get from considering this special case is of very little value in understanding cosmological expansion.

#### metastable

I'd expect to see redshift from the pre-agreed transmission frequency if the nearby sender was moving directly away from me at the transmission time, assuming I've also detected no accelerations or fictitious forces since the transmission time.

What if I detect redshift from the pre-agreed transmission frequency and both myself and the sender have escape velocity along the same vector directly away from the center of nearby supermassive black hole, and I am "above" the sender with respect to the black hole, neither of us detect accelerations or fictitious forces, can I be certain my distance from the sender has increased since transmission time?

#### PeterDonis

Mentor
Could such a setup lead to observations in our "universe portion" of "increasing" measured redshift over time observed in all directions statistically correlated with distance from our position & according to known laws?
No. We would not observe redshifts to be isotropic (the same in all directions, on average) in the setup you describe.

#### PeterDonis

Mentor
What if I detect redshift from the pre-agreed transmission frequency and both myself and the sender have escape velocity along the same vector directly away from the center of nearby supermassive black hole, and I am "above" the sender with respect to the black hole, neither of us detect accelerations or fictitious forces, can I be certain my distance from the sender has increased since transmission time?
No. In fact, the way you have specified the scenario--you and the sender both have escape velocity, but you are above the sender--you can be certain of the opposite, that your distance from the sender will decrease with time, without even looking at the frequency of the signals you're receiving.

#### metastable

*sorry I meant "at least" escape velocity... both have at least escape velocity.

#### metastable

Ok suppose the black hole has an event horizon radius equal to the present observable universe radius. The receiver has at least escape velocity directly away from the center of the singularity at a distance from the center of 2 event horizon radii. The sender is 1 billion light years directly “above” the receiver with respect to the singularity. Assume in the scenario “spatial expansion” = 0 and at transmission time the sender is approaching the receiver. At the same time as the pulse is received, the sender is also approaching the receiver. There are no other gravitational bodies than the singularity and the 2 craft, and neither craft detects acceleration or fictitious forces. Can the receiver detect redshift from a pre-agreed frequency?

#### PeterDonis

Mentor
Can the receiver detect redshift from a pre-agreed frequency?
It depends on how fast the sender is approaching the receiver when the pulse is sent, as compared to the difference in height between them.

#### metastable

It depends on how fast the sender is approaching the receiver when the pulse is sent, as compared to the difference in height between them.
So if it’s “possible” then to detect redshift from a distant approaching object when both the observer and object have at least escape velocity from a large singularity which is outside their observable universe, and the equations describing this redshift are already known, why is “spatial expansion” via an “unknown” mechanism considered a “more likely” explanation for the observation of red shifts which vary in proportion to distance from an observer?

#### PeterDonis

Mentor
why is “spatial expansion” via an “unknown” mechanism considered a “more likely” explanation
First, the mechanism is not unknown; it's just inertia from the Big Bang.

Second, as has already been pointed out, redshifts in your scenario would not be isotropic. But we observe them to be isotropic.

#### Ibix

why is “spatial expansion” via an “unknown” mechanism considered a “more likely” explanation for the observation of red shifts which vary in proportion to distance from an observer?
All versions of your scenario imply an overall "upwards" and a "downwards" direction to the observable universe, with different redshift-versus-distance profiles if you look "up" or "down". We don't see that - once you correct for our velocity compared to a comoving observer the redshift profiles are the same in all directions - isotropic, as PeterDonis says.

#### metastable

Second, as has already been pointed out, redshifts in your scenario would not be isotropic. But we observe them to be isotropic.

^If a formation of transmitter spaceships all at rest with respect to the receiver (also comoving, assume spatial expansion = 0) at transmission time were all traveling close enough to light speed directly away from the singularity in the scenario (each separated by either 1b or 2b ly from receiver at transmission time - some higher, some lower, some same height above singularity as receiver), I thought that most light emitted by these ships, even light emitted in a very “downward” direction from their rest frame, would actually end up traveling in a very “upward” direction with respect to the singularity, and become redshifted due to the relativistic aberration... Is this correct? I thought with increasing separation between craft, the light must climb higher out of the singularity’s gravity well in transit between craft and thus the pulses become more redshifted with increasing separation distance . In other words, with enough velocity away from the singularity can relativistic aberration redirect nearly all light emitted by the craft away from the singularity and thus produce redshift from all directions in the receiver’s relativistic rest frame with respect to the singularity?

#### PeterDonis

Mentor
with enough velocity away from the singularity can relativistic aberration redirect nearly all light emitted by the craft away from the singularity and thus produce redshift from all directions in the receiver’s relativistic rest frame with respect to the singularity?
No.

#### PeterDonis

Mentor
The OP question has been answered, and the current subthread is verging on personal speculation. Thread closed.

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