I think, nobody would reject the fact that Einstein's theory of general relativity is one of the most popular scientific theories nowadays. And probably one of the most hard to understand. But any complex theory in order to survive and spread among the people needs to be presented in the most evident and understandable way. Some popularizers do this by the well-known simple example of so called Flatland and Curvedland.
Let us look through this example, as it is presented in the "Relativity" article of Compton's Encyclopedia (edited by F. Wagner Schlesinger). I will do some comments and conclusions after the excerpts.
"Suppose that somewhere in actual three-dimensional space there is a vast plane inhabited by dotlike creatures who can understand nothing but the two dimensions x and y of their plane. Let us call these creatures dwellers in Flatland, after the title of a noted book once written about them; and let us follow one such creature as he does some experimenting.
Suppose he travels along the straight line between two Flatland cities, A and B. He goes at a steady pace and uses a yardstick to measure the distance. He takes 100 days for the journey; and his yardstick measuring gives the distance as 1,000 miles.
Now he tries another test. He uses a radar outfit to get a reflected signal from A, and finds that the signal passes each way through the intervening space in 1/186th of a second. Since radar signals travel at the speed of light (186,000 miles a second), this means a distance of 1,000 miles. The radar test has confirmed the measure made with the yardstick."
OK.
"Now imagine this creature transferred to the surface of a huge hollow sphere, comparable in size to the Earth. The sphere is in three-dimensional space, and radio signals traverse this space. The transferred Flatlander cannot realize this or other aspects of his actual situation; but he will feel at home because the sphere is huge, and its curved surface seems flat."
Here is the root of mistake. A Flatlander will NOT feel himself "at home", because he is by definition a "dotlike creature who can understand nothing but the two dimensions x and y of his plane". Because of his dotlike nature, however big the curvature of the surface would be, he would find himself "alone", like a dot on the sphere. To use a comparison he would see only himself standing, all his neighborhood would seem to him in a complete fog, which he just can not percept. This would be a very unusual situation for our Flatlander, roughly reminding jumping from hillock to hillock in a foggy swamp.
"Suppose we call his new surroundings Curvedland, and see what will happen if he repeats his distance tests between Curvedland cities C and D."
Alas, but as we see above our Flatlander just can NOT do any distance test in the Curveland. Curveland would all the time have one size for him - one dot, so it wouldn't be any distance inside of one dot. If he would walk ahead, he would apparently find himself as walking on the spot with the only one difference that this spot would change under him as he walks.
Moreover, even if we suppose impossible and let him to measure this distance we still need to remember that out "dotlike creature", must have a "dotlike" yardstick and "dotlike" two-dimensional radar, which of course wouldn't work in Curveland as a surface of the sphere (or even if it will work, it would show the distance exactly the same as in the first "flat" test).
The evidence of mistake is covered with the comparison with the Earth globe. That's why the example itself seems to be true for the fist look. Our planet seems to be flat, though actually it is spherical in a big scale. But the traveler on the surface on the Earth is NOT a dotlike creature who sees only planes but real traveler who can see all "dimensions". Yes, the third dimension (mount-like curvature of Earth's surface) varies very insignificantly while travelling, but it is definitely NOT a complete 0 as in the case with our Flatlander.
If we would see more deeply into this example, we would figure out that Flatlander could not live in Curvedland at all, because he obviously needs a flat plane as his environment.
"This puzzle is only one of many that would arise if the Flatlander continues traveling in Curvedland and making tests. Perhaps strangest of all, if he keeps going long enough, he would be coming to his own back door. This would happen because his supposedly flat space actually is closed and limited. The great circles which he mistakes for straight lines close on themselves and are limited in length."
Alas again. As the Flatlander may not travel in Curvedland (at least in a literal sense of this word) and make any test in it, he would NEVER come back to his door. Again we see a contraband analogy. There is NO home for the Flatlander in Curveland, so there is no back door also. While moving on the spherical surface he would see all the time only himself as a dot on this surface. Nothing more, except the surface have buildings on it. In this case, he would be able to see in some distance cross-sections of the buildings on his "plane of vision".
"Now suppose some "Flatland Einstein," a mathematical genius, studies these matters. He can use as many space dimensions as he likes in working problems even though he can only visualize two. So he devises an explanation, using the shortened yardstick and shortened days. This not only explains the radar signal: it explains the far greater puzzles which would have arisen on a longer journey. His explanation actually amounts to treating all the events as happening in three-dimensional space. But he must state it in mathematical terms, because neither he nor any other Flatlander can think in terms of an actual third dimension, one which they can "see" as existing."
Again misconception connected with the false analogy mentioned above. Of course mathematics is very powerful science and it may be sometimes abstract enough to calculate even such exotic things like four-angled triangle or so. But we need to keep in mind that Flatland Einstein, whatever genius he would be, wouldn't just have the material for his "calculations", because he, as a real dotlike creature, would NOT even "see" as existing anything except his flat plane.
Flatland Einstein is actually a Curvedlander, considering himself as a Flatlander. He tries to think and act in two-dimensions, but, as we saw, fails to do this. He transplants, in a rather naive way, his three-dimensional thinking into two-dimensional environment, continues his analogy further, and requests for an additional dimension to explain 3-D puzzles. But with this additional dimension he actually is not getting into some exotic "four-dimensional" environment, he gets just back to his Curvedland, with the illusion that he is somewhere else. One step back, one step forward brings him to the same position he starts.
Unfortunately, the mistake discussed above has rooted in the works of many astronomers and cosmologists. Let us see, what write Stephen Reucroft and John Swain, professors of physics at Northeastern University in Boston, Mass. in their answer to one of the readers of "Scientific American" (http://www.sciam.com/askexpert/astronomy/astronomy27/astronomy27.html)
"Perhaps the easiest way to see what is meant by an expanding universe is to imagine what life would be like for two-dimensional ants living on the surface of an expanding spherical balloon. They can crawl around, but being unable to fly, or to penetrate the balloon's surface, they live in what is essentially a two-dimensional world. For the ants, provided nothing disturbs them from outside, the universe is the surface of the balloon--that's all there is! Being confined to the surface of the balloon, there is no way for the ants to discover anything at all about what we would term up and down."
As we saw in our Flatlander's case, two dimensional ants just can NOT (at least for those, who respect the logic) live on the surface of an expanding spherical balloon! We, humans, living on the balloon called Earth, are able to discover only because we are spatial beings. If we would be flat beings, we apparently would never understand anything spherical. I think the authors understand it partially, when they add the word "essentially" in the second sentence. So, for the ants, provided nothing disturbs them from outside, the universe is NOT the surface of the balloon--that's NOT all there is!
This two-dimensional universe is finite (the area is finite), but nowhere will the ants find a boundary or an edge. Here you have to ignore the rubber neck of the balloon and the person blowing it up; think of a balloon sealed smoothly into a spherical shape hovering in a tank in which the air pressure could be lowered to make the balloon expand.
I see in the paragraph a very subtle misconception. When we are talking about the "edge"(end) we need to realize very clearly what we are talking about. I would differ between a rational and irrational edges. If I would be asked, if the surface of the balloon has a rational edge, I would answer - yes! Really, the surface of any balloon, evidently, doesn't exist for us as such, until we put a dot (origin) on it. Let's call this origin a pole and remember it. If now we will send the travelers from this pole to any of the possible directions, they will finally collide at the opposite pole, having NO place to go further! This would be a rational edge of that balloon for all travelers. On the other hand, if one traveler would travel for the point called pole strictly in one direction (let us assume he can do it), he would finally come the same point called pole. Of course, he may go further, but what is the rational reason for him to do it? He knows everything from his first travel (assuming nothing has been changed). Again, this will be a rational edge of the surface of the balloon. It would be useful also to remember that our Earth also has the equator and the main meridian - rational edges of the Earth surface.
As regards the irrational edge, of course, we can imagine some crazy traveler, without memory and any intention, who would wander about the surface. Surely, for such a traveler it would NOT be any edge or end. But such a traveler scarcely may be a witness for a truly scientific research.
Now as the balloon expands, the ants see each other getting farther and farther apart. Each sees the same thing: all its neighbors are moving away. The ants live in, what for them, is an expanding universe, with no physical edge. If an ant walks quickly enough, she could conceivably get all the way around the balloon and return to her starting point without encountering an edge.
See the above.
You may object at this point and claim that we see the balloon expanding into the surrounding space. But we have access to an extra dimension in which to move--the one that would correspond to up and down for the ants, were they able to move in those directions. As far as the ants are concerned, they can learn everything they want to know about their world by making measurements on the balloon's surface, with no reference to the surrounding space. The surrounding space is inaccessible to them.
Experience in physics has taught us that when we find a concept that is spurious--in the sense that it leads to no predictable effects-- we do better just to assume it's not there. In other words, the ants would do well not to talk about their space expanding into something that they can't measure. Nothing is lost and there is a substantial gain in simplicity.
Again and again. The "two-dimensional" ants are NOT able to be concerned, they can NOT learn anything about their world, can not measure it, can NOT talk about their "space" (surface) expanding, just because the spherical surface doesn't exists for them as such. All the y would see all the time would be a very simple "one-dimensional" dot, at which their "horizon" would touch the surface of the balloon.
To make an analogy between the ant's situation and our own, you have to imagine space expanding in all directions. Everybody in the universe sees everything rushing away from everything else; but the universe need have no physical edges, and there is no need to describe it as expanding into anything; it can just expand.
I consider it could be a wrong analogy between the ant's situation and our own. If we ARE aware of any expansion, we evidently MUST be able to describe it in plain terms. If we are not able to do it so far, we need to acknowledge this and try another way without a fainthearted attempt to avoid the question.
To finish aforesaid, somebody may say, that my criticism is based on the Euclidean" geometry, though there are many "non-Euclidean" geometries appeared recently, that I have an "Euclidean" mind, etc. Not speaking about the value of such "alternative" geometries now, I would say that the cited examples itself are based upon the classical geometry, so it's absolutely correct to criticize them in the same way.
This complex mistake in the simple example, which stands in the very bottom of somebody's attempt to imagine one more additional dimension leads us to the major problems for the concept of space-time, which in it's turn is one of the basic concepts of the whole general relativity theory.
The time will show, if his followers would be able to fix it after a short space.