Trip Around Universe: A Journey in an Inertial Frame

[Snip3r]

if i were to leave Earth travel in as straight a line as possible with constant velocity eventually i would return to earth.
is mine a valid inertial frame?

 

[HallsofIvy]

if i were to leave Earth travel in as straight a line as possible with constant velocity eventually i would return to earth.

What evidence do you have for this assertion?

is mine a valid inertial frame?

In special or general relativity?

 

[morfin56]

if i were to leave Earth travel in as straight a line as possible with constant velocity eventually i would return to earth.

Of coarse not you would eventually wander into another galaxy/

 

[elfmotat]

WMAP has confirmed that the universe is flat to within about ±0.5%. If the universe is flat then you can't travel in a "straight" line and return to your original location.

 

[Nugatory]

if i were to leave Earth travel in as straight a line as possible with constant velocity eventually i would return to earth.
is mine a valid inertial frame?

What is this "constant velocity" of which you speak? Constant relative to what? And that phrase "in a straight line" has the same problem...

OK, let me try the question that (I think) you're really asking.

Q: If you travel in a locally straight straight line, free fall, no acceleration, no deliberate turning and twisting and course changes, through a curved universe, do the curvature effects mean that you aren't in a valid inertial frame?
A: You are always in a locally inertial frame. There is a region of space-time around you in which the curvature effects are too small to measure, and as long as you only do experiments within that region, you'll get the results predicted by special relativity, which works within inertial frames. If you do experiments on a scale large enough for the curvature to matter, you need general relativity. The stronger the curvature, the smaller the locally flat region - but (except at a singularity) you can always find a region small enough to be locally flat, and within that region you're always in a valid inertial frame.

Consider that when you're laying out the foundations of a house, you don't worry about the curvature of the Earth's surface; the Earth is locally flat. If you're laying out a flight path between London and Tokyo, you do consider the curvature of the earth.

(Whether you would eventually return to Earth or not depends on how and how much the universe is curved. Others have already commented on that).

 

[phinds]

WMAP has confirmed that the universe is flat to within about ±0.5%. If the universe is flat then you can't travel in a "straight" line and return to your original location.

Only if it is unbounded. My understanding is that flat and bounded are possible, in which case you most certainly would, in theory, return.

 

[phinds]

Of coarse not you would eventually wander into another galaxy/

You either don't understand the question or you don't understand cosmology.

 

[DrGreg]

Actually, even if the Universe is bounded, you'd never make the journey all the way round it because it's expanding too fast. There are distant parts of the Universe we cannot see because the light from there is traveling slower than the rate of expansion.

 

[Matterwave]

Only if it is unbounded. My understanding is that flat and bounded are possible, in which case you most certainly would, in theory, return.

Do you have a reference to "flat and bounded"? As far as I know, a flat metric would have a 3-volume which would integrate to infinity:
$$a(t_0)\int_{-\infty}^{\infty} dxdydz\rightarrow\infty$$

 

[phinds]

Actually, even if the Universe is bounded, you'd never make the journey all the way round it because it's expanding too fast. There are distant parts of the Universe we cannot see because the light from there is traveling slower than the rate of expansion.

Yes, I agree completely. That's what I intended to imply by my "in theory" --- I meant to imply that an FTL beam would hit you on the back of the head because of topology, not that it could ever actually happen.

 

[phinds]

Do you have a reference to "flat and bounded"? As far as I know, a flat metric would have a 3-volume which would integrate to infinity:
$$a(t_0)\int_{-\infty}^{\infty} dxdydz\rightarrow\infty$$

Nope, I'm just repeating what I have read here, or maybe what I THINK I have read here.

 

[Matterwave]

I think there can be confusion in using the word "bounded". I take it to mean "finite volume", but it can sometimes be interpreted to mean "will collapse back in on itself in a finite time". These two questions are distinct from each other. One can see:http://www.lbl.gov/Science-Articles/Archive/images1/omegamomegal3.gif

 

[George Jones]

A flat FRW universe satisfies the cosmological principle, i.e., is spatially homogeneous and isotropic, and has space topologically equivalent to R^3. Either of these principles can be relaxed. A flat, homogenous, non-isotropic universe can have space topologically equivalent to the 3-torus T^3. See chapter of Gron and Hervik,



Einstein's Einstein's equation doesn't determine the topology of spacetime.

 

[phinds]

A flat FRW universe satisfies the cosmological principle, i.e., is spatially homogeneous and isotropic, and has space topologically equivalent to R^3. Either of these principles can be relaxed. A flat, homogenous, non-isotropic universe can have space topologically equivalent to the 3-torus T^3. See chapter of Gron and Hervik,



Einstein's Einstein's equation doesn't determine the topology of spacetime.

George, I appreciate your erudition, but I don't have a clue what you just said. Is it possible for there to be a flat but bounded universe, or is that not a meaningful question?

 

[Matterwave]

He means that in a standard FLRW universe, where homogeneity and isotropy of space are strictly enforced, then a flat universe will be isomorphic to R^3 and therefore cannot close in on itself like a sphere or torus.

But apparently, if you start to mess around with the homogeneity and isotropy conditions, you can get a flat universe which is "equivalent to the 3-torus (SxSxS)" (i.e. "bounded"). Non-isotropic universes are not nearly as well studied as the FLRW solutions (which most modern cosmology uses).

I think the point is, one should note that the Einstein equations are local equations and objects such as the Riemann curvature tensor can only "see" intrinsic curvature. This means that global, extrinsic curvature (such as one may have for a toroidal configuration) may not be able to be "seen" by objects like the Riemann tensor since the geodesics do not converge or diverge.

I haven't studied non-isotropic universes, so, correct me if I'm wrong here.

 

[yenchin]

To sum up:

1. We know the universe is almost [maybe really] flat, in terms of spatial curvature.

2. The cosmological principle which demands that the universe is homogeneous and isotropy can be relaxed [since they are approximate anyway] into "locally isotropy" or "locally homogeneous". For discussions on geometry in this sense from mathematically rigorous standpoint, see e.g. Wolf's "Spaces of Constant Curvature" text.

3. If homogeneity and isotropy is relaxed, one can allow for nontrivial topology such as the three-dimensional flat torus as mentioned by other posters earlier. In fact many people are trying to look for patterns in CMB that might indicate such nontrivial topology, though none has been found conclusively. See e.g. http://arxiv.org/abs/1104.0015v2.

4. As DrGreg pointed out, due to the current accelerating expansion, we can no longer circumnavigate the universe without going faster than light, even if the universe has non-trivial topology.

Here is a nice article about nontrivial topology in cosmology.

 

[yenchin]

If you look at Table 1 and Fig 1 on page 3 of this paper: http://arxiv.org/abs/1103.1466, you can see at a glance, which of the 18 flat space geometry is orientable, conpact and homogeneous.

 

[bahamagreen]

In the spirit of the question, and taking some liberties with the thought experiment...

Assume that you choose a direction, and instead of traveling that way or looking to receive light from that direction, just extend your conceptual coordinate in that direction to the distant point which corresponds to a full lap "around" the universe back to your present location... assuming that is the geometry, one might draw a few conclusions about related things...?

If you continue the extension (make a second lap), you would find your location again. Each lap "around" the universe, you would again find a subsequent further version of your present point.

Since any direction you choose makes this happen, each one of these distant copies of your present location takes the form of a sphere.

Because of expansion and Hubble, the radial length contraction makes each successive distance to the next sphere less than the previous... they approach a limit where recession approaches c, plus expansion.

So the universe has you in the middle, then a copy of your location spread out as a distant sphere surrounding you, then a succession of spheres each less further than the previous, all approaching a limit as the concetual laps goes to infinity.