If the universe is finite it cannot be flat. If it is not flat, then it does not have parallel geodesic lines. If the geodesics diverge, then a volume gets expanded on the transport along the geodesics. A volume expansion would explain the observed red shift.
You should side with the evidence. It seems that our most informed perspective on this is that the observable universe is just about flat. The Lambda CDM model allows for parameters that fit the observations very well.
Now, outside the observable universe we just don't know. That could be infinite in space and be eternal, or maybe not. It could have some non-trivial topology. The problem is none of this is observable.
Flat means locally flat. What if "the local view" is the problem? The concept of Riemannian curvature is proven on the base of small closed paths. If the universe is finite, then the term "outside of the observable" is not defined.
Why can't the entire universe be finite? We should be careful if we are discussing just finite in space or finite in space-time.
If you want to discuss the global shape of a space-time, then you are really thinking of topology. General relativity say almost nothing about topology.
Indeed, time can't be finite. Therefore "finite" only concerns space. And the discussion is not about general relativity. We only consider consequences of an isotropic space, which is finite/closed in the sense, that if you follow any geodesic you finally will arrive at your starting point.
WK: What's more speculative, an expanding universe or a curved closed universe with diverging geodesic lines?
That is not a dichotomy, an expanding, closed universe with positive curvature is one of the valid descriptions of the current "Big Bang" model. Those who work with QM like that model because it removes some difficulties with not having any "boundary conditions" in an infinite universe.
WK: If the universe is finite it cannot be flat.
Yes it can, it can be a flat 3-torus.
Why do you keep repeating false statements? I don't expect you to take my word for it but having been given a hint, why don't you research what I said and confirm it from reliable sources, I've pointed this out several times now?
WK: If it is not flat, then it does not have parallel geodesic lines.
True.
WK: If the geodesics diverge, then a volume gets expanded on the transport along the geodesics.
OK.
WK: A volume expansion would explain the observed red shift.
No it wouldn't. Spatial curvature only results in angular size distance differing from proper distance.
WK: We only consider consequences of an isotropic space, which is finite/closed in the sense, that if you follow any geodesic you finally will arrive at your starting point.
We could call the length of that space-like path the circumference (which is an accurate term if the geometry is a 3-sphere). If that length is unchanging with time as in a "static universe" then there will be no redshift because the time taken to circumnavigate the 3-sphere is also unchanging in time hence the period of any repetitive event as seen by an observer must be equal to the period of the source. No change of period means no redshift.
GD: No it wouldn't. Spatial curvature only results in angular size distance differing from proper distance.
This is again an argument which considers local disturbances. It is not applicable to effects caused by a homogeous and isotropic curved space geometry.
It is applicable, read my final paragraph which explains why it is valid for signals that circumnavigate the entire space.
Hi George,
sorry, but the "length reasoning" in your final paragraph is not correct. Instead you should consider the volume transport along geodesics around the complete space.
The length determines the time it takes a signal to traverse the path at the speed of light regardless of volume effects. If that is constant as it would be in a static universe, the period received equals the period transmitted, hence no red shift. The argument is completely valid as it stands.
Distributing a quantity of light, say the flash of a supernova, over an expanding volume means it grows dimmer (for a given area of intercepted light) but has no effect on the colour of the light.
Hi George,
If you receive light which fills an expanded volume, you will experience a time dilatation according to the length expansion. Please check if your arguments would disprove gravitational red shift.
If you think of a flash from a brief light source, as the light moves away, it spreads over a larger surface area but the thickness of the shell of light (due to its finite duration) doesn't alter, the expansion is only in the transverse direction which is what creates the classical inverse square law. Length doesn't expand unless the background metric is expanding and that is the usual FLRW model.
WK: Please check if your arguments would disprove gravitational red shift.
It would not. There would be a growing increase in the period of event for a source and observer in free fall which is equivalent to the cosmological situation.
I've answered your questions, now how about answering mine:
In your closed universe with positive curvature, light from a supernova would go round the universe and return to the location of the source producing a flash across the whole sky billions of years later. Do you agree that if the universe is not expanding, two such supernovae happening one year apart in a galaxy would produce two flashes one year apart when the light returns to the origin?
Imagine that travelling around the entire universe is comparable to travel away from a heavy mass. The heavy mass produces two flashing events with in one year according to a clock on the surface of that mass. We will receive those flashes red shifted and within an increased time delay of more than one year according to the red shift.
Does this answer your question?
In this analogy I only assume, that crossing a region with diverging geodesics is the same as crossing a gravitational potential difference.
WK: Does this answer your question?
You have at least answered, thank you for that. However there are two flaws in what you say though they are closely related.
WK: In this analogy I only assume, that crossing a region with diverging geodesics is the same as crossing a gravitational potential difference.
The first problem is that you assume a homogeneous and isotropic universe in which case there can be no change in gravitational potential moving from one region to another (ignoring local disturbances).
WK: Imagine that travelling around the entire universe is comparable to travel away from a heavy mass.
The second problem is that my question asked about seeing the flashes in the same galaxy in which the supernovae originated. Your answer should reflect that thus you might say "Imagine that travelling around the entire universe is comparable to travel away from a heavy mass followed by return back to the starting point." If you saw redshift on the outward half, you would see an equal blueshift on the second half, back to the start.
Even that isn't correct due to the first point To look at it another way, if travel away from the starting point is similar to moving away from a mass, your starting location in the universe would be a minimum of potential and the antipodal location would have to be a maximum, but the Copernican Principle means every location could be considered a possible starting point hence all locations must be both a maximum and a minimum which is only possible if all points are at the same potential.
Imagine plotting potential over the surface of a sphere, then a 3-sphere (glome) by extension from the linked image.
https://en.wikipedia.org/wiki/Antipodal_point
George, the gravitational potential difference is only an analogon to the impact of diverging geodesics over cosmic distances. The geodesics diverge in all directions, which means that in the opposite direction there is again a red shift and not a blue shift as a real gravitational potential would cause.
The impact of the isotropic and homogenous curved geometry is like an also isotropic an homogenous gravitational potential difference which is only noticeable over cosmic distances. This means it is exactly the same for any pair of two points with the same (cosmic) distance and the algebraic sign of the potential difference depends on the displacement direction.
Because the argumentation is purely geometrical, it does not contain any flaws.
WK: The geodesics diverge in all directions,
For positive curvature, initially parallel lines converge.
If you start with diverging lines (radial from a source), they would be parallel and maximally separated after traversing quarter of the circumference, converge to the antipodal point then separate, be parallel after three quarters of the circumference and then converge again on the source.
WK: The impact of the isotropic and homogenous curved geometry is like an also isotropic an homogenous gravitational potential difference ..
There is no gravitational potential difference in an isotropic and homogeneous universe as I explained in the previous post.
WK: potential difference depends on the displacement direction.
Potential by definition has a single value at any location. Returning to the origin restores the original potential hence there can be no global gravitational redshift in your model. only local effects.
You said: "We will receive those flashes red shifted and within an increased time delay of more than one year according to the red shift." which is a consistent statement but since both flashes of light had to travel at the same speed, an increased delay means a greater distance travelled and increased circumference is commonly described as an "expanding universe".
WK: Because the argumentation is purely geometrical, it does not contain any flaws.
The geometry of a glome is consistent but your statements about that geometry are flawed.
George, now you are ignoring arguments.
The geometrical model is homogeneous and isotropic and does not include a gravitational potential. It is only that the impact of diverging geodesics is the same as caused by a gravitational potential difference. The "virtual gravitational potential" I am talking about, does not generate a force, it only influences a displacement.
GD: ..since both flashes of light had to travel at the same speed, an increased delay means a greater distance travelled..
Thats wrong because the theory of General Relativity implies: Travelling across a potential difference causes a time dilatation. If you put your clock on Jupiters surface and observe it from a place far away from Jupiter you will see a (slightly) slower time progression.
In all your arguments you completely ignore any possible effect of diverging geodesics. This is why you think, none of your arguments could be flawed.
You dont't make any difference between a distance and a transport over a distance. But in a non Euclidian geometry there is a difference!
WK: George, now you are ignoring arguments.
Hardly.
WK: The geometrical model is homogeneous and isotropic and does not include a gravitational potential.
OK, I have no problem with that statement.
WK: It is only that the impact of diverging geodesics is the same as caused by a gravitational potential difference.
That is where you are ignoring my comments, not vice versa. I have repeatedly pointed out that spatial curvature is NOT equivalent to a gravitational potential, it is equivalent to a lens that gives magnification but NO frequency change.
WK: The "virtual gravitational potential" I am talking about, does not generate a force, it only influences a displacement.
The "force" of gravity is a pseudo-force which appears because test masses moving inertially follow geodesics. If the geodesics diverge, test masses initially on parallel paths move apart as if there were a differential force acting. The two go hand in hand, you cannot separate them.
GD: ..since both flashes of light had to travel at the same speed, an increased delay means a greater distance travelled..
WK: Thats wrong because the theory of General Relativity implies: Travelling across a potential difference causes a time dilatation.
Correct, but in a homogeneous and isotropic universe, there is no variation of potential from location to location as you agreed in what I quoted at the top.
WK: If you put your clock on Jupiters surface and observe it from a place far away from Jupiter you will see a (slightly) slower time progression.
Correct, and if the clocks are inertial they will move apart. That indicates curvature involving space and time. A static, closed universe has curvature in space but not in time (other than locally) so your analogy is inapplicable.
WK: In all your arguments you completely ignore any possible effect of diverging geodesics. This is why you think, none of your arguments could be flawed.
That is completely untrue! Over and over again, right from the beginning, I have told you that the effect of spatial curvature is well known and results in an effect on angular size distance but not frequency. I even gave you a link to Prof. New Wright's tutorial where he discusses that and gives the equations for it. You'll find it in most undergraduate textbooks on cosmology. It is you who have consistently ignored those replies.
WK: You dont't make any difference between a distance and a transport over a distance. But in a non Euclidian geometry there is a difference!
You might claim that for a single light flash in the question I asked, but when you say that the second flash takes longer it must have travelled farther. That is independent of what happens to each flash as a result of their transport such as an apparent change of size (which should be the same effect on both flashes in your static model).
GD: ..spatial curvature is NOT equivalent to a gravitational potential, it is equivalent to a lens that gives magnification but NO frequency change.
WK: A transport in a curved space across a region with diverging geodesics is equvalent to a transport across a gravitational potential difference.
Would you agree, that this two statements are somehow the root of our dissention?
I think the root problems are that (a) you are not differentiating between space-space curvature and space-time curvature and (b) you are not grasping that spatial geodesic deviation is only transverse to the direction of the paths and (c) positive curvature causes parallel lines to converge, not diverge.
Wolfgang, we could argue forever if we only express our own opinions. I've added some links that show my understanding as I think some third-party sources are the only way to resolve the question.
The first shows a spacetime diagram, you can see the curvature responsible for gravitational time dilation. The others talk about spatial curvature only and explain why positive curvature causes geodesics to converge.
http://visualrelativity.com/LIGHTCONE/pics/bh3.gif
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_variable/index.html
https://image.slidesharecdn.com/presentation-friday-new-160218172226/95/einsteins-gravitational-lines-17-638.jpg?cb=1455816259
Hi George,
Though we had not been able resolving our dissention I thank you for your patience and your insistence on your point of view, which shows your deep factual interest.
Please understand that I am still convinced that diverging geodesics have an impact on transport. (I never mentioned a positive curvature, that was only your assumption)
WK: Please understand that I am still convinced that diverging geodesics have an impact on transport.
They do but (a) in positive curvature geodesics converge and (b) the effect is called "gravitational lensing" is is well known, it is a powerful astronomical tool.
I thought that earlier in the discussions in other threads you talked of positive curvature but I may be mistaken, I may have inferred that from your statements. Negative curvature gives an open universe and you have consistently said your model is a closed universe which requires that it have positive curvature (or be a flat 3-torus but that has no curvature).
It is a somewhat moot point though, spatial curvature doesn't affect frequencies either way.
In fact I am analysing a closed homogenous and isotropic geometry characterized by a homology between displacement and rotation in three dimensions. The evaluation of this geometry leads to a zero curvature and to a (long distance) divergence of geodesics. (Locally the geodesics are parallel)
Apart from the mathematical implications, this geometry raises the question about the possible impact of diverging geodesics on the transport.
WK: zero curvature and to a (long distance) divergence of geodesics
My understanding is that curvature is derived from the global behaviour of geodesics that are locally parallel as indicated in the diagram but obviously that is only an illustration of the maths so perhaps what you suggest might be possible.
WK: this geometry raises the question about the possible impact of diverging geodesics on the transport.
Sure. The divergence would certainly lead to a variation in the angle subtended at the observer by a distant object of known width hence the "magnification" effect I discussed previously. I don't think that is contentious.
However, the time taken for a light signal to propagate along a (spatial) geodesic depends only on the speed of light and the length of the geodesic. If that length is unchanging, the time taken will be unchanging, the "two supernovae" thought experiment I suggested says that consequently the time between arrivals will be the same as the time between the supernovae at the source because the light from each travels along the same geodesic. That means no frequency change and no redshift.
https://image.slidesharecdn.com/presentation-friday-new-160218172226/95/einsteins-gravitational-lines-17-638.jpg?cb=1455816259
It is a fact that the transport away from a heavy mass enconters diverging geodesics. Therefore I expect a similar impact including time dilatation and a corresponding red shift.
WK: It is a fact that the transport away from a heavy mass enconters diverging geodesics. Therefore I expect a similar impact including time dilatation and a corresponding red shift.
No, the curvatures are different as I've explained before. In a static universe, objects stay at fixed locations so there is no diverge in time which is what causes redshift. You might have divergence in space but that causes only geometric effects, not frequency changes. Unless the length of the geodesic changes, two independent light flashes will take the same time to travel, arrive with the same time separation and consequently show no red or blue shift. That is your fundamental error.
As I said before:
GD: I think the root problems are that (a) you are not differentiating between space-space curvature and space-time curvature
I also think you are not being consistent in your statements. You question above states "If the universe is finite it cannot be flat" but now you are saying "The evaluation of this geometry leads to a zero curvature" in which case by your own claim it cannot be finite.
GD: No, the curvatures are different as I've explained before.
All your explanations consider gravitational potentials as an entiere phenomenon including taking into account inside and outside. I am only considering it partially on a path from inside to outside.
As the physical part of my analogy I only use: "time is slowed down inside a strong gravitational potential". Therefore a time dilatation occurs on a path from inside to outside.
The inconsistence of "zero curvature" and "finite" has to do with the mathematical tradition to consider everything locally. The local view does not exclude the possibility that non linear effects incidentally are cancelled out locally. But this touches the mathematical implications, which are not subject of that thread.