Circumnavigating the Universe in Cosmos (1980)

[bob012345]

TL;DR

In his famous book and TV show Cosmos, Carl Sagan discusses a hypothetical relativistic journey circumnavigating the known universe in about 56 years ship time. I am trying to understand what assumptions is he using.

Sagan describes this on page 207 of his book (1995 reprint hardback edition). In the context of the page he is discussing accelerating at 1g for half the journey and decelerating the other half for trips to Barnard's Star, the center of the Milky Way and the Andromeda galaxy. Then he talks about circumnavigate the known universe. In the context I assume that trip is similar. I wonder what "size" is he using for the known universe in 1980 in order to get that ship time of 56 years? Would circumnavigation imply the whole trip is $2 \pi r$ where r is the radius (half the "size") of the known universe? Any thoughts on how he would have computed this?

 

[Nugatory]

You might start by calculating the distance covered (relative to your starting point, considered to be at rest) in 28 years of 1g proper acceleration. That will give you half the "circumference".

 

[bob012345]

You might start by calculating the distance covered (relative to your starting point, considered to be at rest) in 28 years of 1g proper acceleration. That will give you half the "circumference".

Using $x(\tau)=\frac{c^2}{\alpha}(\cosh(\frac{\alpha}{c}\tau) -1)$ where $\alpha$ is the proper acceleration and $\tau$ is the proper time, also, using units of light years and years where $c=1 ly/y$ and $\alpha$ is very close to $1 ly/y^2$, this gives $x(\tau)\approx\cosh(28)\approx7.23$x$10^{11} ly$ making the size of the known universe estimated at $1.44$x$10^{12} ly$ which seems way too large an estimate to use.

If instead one used the estimate of half being about 1.5x$10^{10}ly$ one gets $\tau_{1/2}\approx24y$ and total proper trip time 48y.

https://imagine.gsfc.nasa.gov/educators/programs/cosmictimes/educators/guide/age_size.html

Maybe he was assuming a certain expansion of the known universe over the trip since it takes a very long time comparable to the age of the universe?

 

[DaveC426913]

It is important to rmember Sagan;s goal and his target audience.

is similar. I wonder what "size" is he using for the known universe in 1980 in order to get that ship time of 56 years? Would circumnavigation imply the whole trip is $2 \pi r$ where r is the radius (half the "size") of the known universe? Any thoughts on how he would have computed this?

If the universe were thought to be 14 billion years old at the time, then it would be 28Gly in diameter. Travelling its entirety and returning to one's starting point might be considered a 56Gly trip. I know that's not technically accurate depending on how you slice it, but IIRC, Sagan wasn't trying to teach people math; he was trying to popularize cosmology. He may have oversimplified the math.

BTW, the innertoobs has some interesting information on how big we thought the universe was at various points in our journey of knowledge:
https://imagine.gsfc.nasa.gov/educators/programs/cosmictimes/educators/guide/age_size.html

1965: Age: 10-25 Billion Years, Size: 25 Billion Light Years
1993: Age: 12-20 Billion Years, Size: 30 Billion Light Years

...which sort of puts the lie to my logic above, since the universe's age wasn't narrowed to ~14Gy until well into the 21st century (according to that page).

 

[bob012345]

It is important to remember Sagan's goal and his target audience.

If the universe were thought to be 14 billion years old at the time, then it would be 28Gly in diameter. Travelling its entirety and returning to one's starting point might be considered a 56Gly trip. I know that's not technically accurate depending on how you slice it, but IIRC, Sagan wasn't trying to teach people math; he was trying to popularize cosmology. He may have oversimplified the math.

Sagan was saying 56 years ship (proper) time.

 

[Ibix]

Using $x(\tau)=\frac{c^2}{\alpha}(\cosh(\frac{\alpha}{c}\tau) -1)$ where $\alpha$ is the proper acceleration and $\tau$ is the proper time, also, using units of light years and years where $c=1 ly/y$ and $\alpha$ is very close to $1 ly/y^2$, this gives $x(\tau)\approx\cosh(28)\approx7.23$x$10^{11} ly$ making the size of the known universe estimated at $1.44$x$10^{12} ly$ which seems way too large an estimate to use.

Well, that's the circumference. That makes the radius about 230Gly. Current estimates of a 14Gy old universe suggest that the observable universe would have a radius of about 45Gly, so four or five times larger than that isn't totally implausible for the upper bounds of the historical estimates of age provided by Dave.

 

[DaveC426913]

Sagan was saying 56 years ship (proper) time.

Sure but IIRC he was flying around in his "Starship of the Mind" which was not limited by physics.
Not sure. Maybe this is a red herring.

 

[PeterDonis]

that's the circumference.

It's possible Sagan actually had in mind a spatially closed universe and was talking about circumnavigating a 3-sphere.

 

[Ibix]

It's possible Sagan actually had in mind a spatially closed universe and was talking about circumnavigating a 3-sphere.

Possibly, but isn't the number a bit low for that? I think we concluded 3 trillion light years round in this post using modern figures, and Dave's historical values seem to suggest we thought the universe was older than we do now. Although obviously age isn't the only factor in size estimates.

 

[Nugatory]

This "what DID Sagan mean?" ambiguity is an example of why popularizations can be so unsatisfying - it is possible that we collectively have put as much effort into understanding it as Sagan put into writing it.

 

[bob012345]

Possibly, but isn't the number a bit low for that? I think we concluded 3 trillion light years round in this post using modern figures, and Dave's historical values seem to suggest we thought the universe was older than we do now. Although obviously age isn't the only factor in size estimates.

I think he must have assumed that since the journey would take longer than the age of the universe at that time wrt the earth frame that the distance travelled would be much larger than the size of the universe at that time. Using a naive 3 sphere universe and a Hubble expansion rate of about 7% per billion years I think we get into the trillion light year regime easily.

 

[bob012345]

This "what DID Sagan mean?" ambiguity is an example of why popularizations can be so unsatisfying - it is possible that we collectively have put as much effort into understanding it as Sagan put into writing it.

Right. I’m only asking out of curiousity and I hope people don’t waste valuable time on this. I hope responses are only because people are curious and have fun with the question.

 

[Nugatory]

I hope responses are only because people are curious and have fun with the question.

Seems to be turning out that way

 

[PeterDonis]

isn't the number a bit low for that?

With what we now know, yes. But we didn't have that data in 1980; what numbers we had then made it possible that the universe was closed with a 3-sphere size significantly smaller than would be within the error bars today.

 

[cianfa72]

With what we now know, yes. But we didn't have that data in 1980; what numbers we had then made it possible that the universe was closed with a 3-sphere size significantly smaller than would be within the error bars today.

When we make statements like an Universe as closed 3-sphere, I think we're actually talking of the geometry/topology of spacetime's spacelike hypersurfaces at a given cosmological time.

The latter should be the proper time as measured by expanding galaxies since the Big Bang.

 

[PeterDonis]

When we make statements like an Universe as closed 3-sphere, I think we're actually talking of the geometry/topology of spacetime's spacelike hypersurfaces at a given cosmological time.

Yes, that's the standard usage in cosmology.

The latter should be the proper time as measured by expanding galaxies since the Big Bang.

More precisely, the proper time of comoving observers, which is also coordinate time in standard FRW coordinates.

 

[cianfa72]

More precisely, the proper time of comoving observers, which is also coordinate time in standard FRW coordinates.

Ok, so comoving observers are "at rest" in standard FRW coordinates (i.e. their timelike worldlines are described by fixed spacelike coordinates and varying coordinate time $t$ in that chart/coordinates).

Now "Universe is expanding" actually means that proper distance between galaxies evaluated on any spacelike hypersurface of constant cosmological time (i.e. the length of spacelike geodesics connecting them on any constant cosmological time's spacelike hypersurface) increases in cosmological time $t$.

 

[PeterDonis]

comoving observers are "at rest" in standard FRW coordinates

Yes. We have been over this in multiple previous threads. It is something that can be assumed to be common knowledge in an "I" level thread and does not need to be stated again every time.

"Universe is expanding" actually means that proper distance between galaxies evaluated on any spacelike hypersurface of constant cosmological time (i.e. the length of spacelike geodesics connecting them on any constant cosmological time's spacelike hypersurface) increases in cosmological time $t$.

This is one way of putting it, but it is coordinate-dependent and so does not make it obvious that there is an invariant corresponding to "the universe is expanding". That invariant is the expansion scalar of the congruence of comoving worldlines, which is positive.

 

[cianfa72]

This is one way of putting it, but it is coordinate-dependent and so does not make it obvious that there is an invariant corresponding to "the universe is expanding". That invariant is the expansion scalar of the congruence of comoving worldlines, which is positive.

You mean coordinate-dependent since one has to pick the family/foliation of spacelike hypersurfaces (even though on any of them the notion of geodesics connecting the comoving observers is actually invariant).

Therefore, as you pointed out, is better to refer to the expansion scalar of that comoving congruence.

 

[PeterDonis]

You mean coordinate-dependent since one has to pick the family/foliation of spacelike hypersurfaces

That family of hypersurfaces is picked out by the invariant geometric properties of homogeneity and isotropy. It is not coordinate-dependent. FRW coordinates are the ones best adapted to those geometric properties, but the properties themselves, and the hypersurfaces that satisfy them, are invariant.

 

[cianfa72]

That family of hypersurfaces is picked out by the invariant geometric properties of homogeneity and isotropy. It is not coordinate-dependent.

You mean spatial homogeneity and isotropy (on any of those spacelike hypersurfaces).

FRW coordinates are the ones best adapted to those geometric properties, but the properties themselves, and the hypersurfaces that satisfy them, are invariant.

Sorry, what is coordinate-dependent then as implied in your post #18 ?

 

[PeterDonis]

You mean spatial homogeneity and isotropy (on any of those spacelike hypersurfaces).

Of course. That's obvious since the hypersurfaces are spacelike. You don't need to keep pointing out the obvious.

what is coordinate-dependent then as implied in your post #18 ?

The fact that you used the term "cosmological time", by which you meant FRW coordinate time.