9/20/2023 0 Comments Flat space vs curved space![]() ![]() In the last decade-you may have read this news countless times-cosmologists have found what they say is rather convincing evidence that the universe (meaning 3-D space) is flat, or at least very close to being flat. Incidentally, when cosmologists talk about the expansion of the universe, they mean that space has been expanding, not spacetime. For all we know, space is 3-D, and spacetime is 4-D but if string theory is true, then space turns out to be 9-D, and spacetime 10-D. In this way of looking at things, the nowverse is one of many parallel planes, each of which represent the universe at a particular time of its history. Spacetime, then, would have a more manageable total of three. Because most people (including yours truly) have a hard time visualizing 4-D objects, a common way of thinking of spacetime is to pretend that space had only two dimensions. It is natural to think of the nowverse as a 3-D slice in this 4-D space, just like horizontal planes are 2-D slices in our 3-D world. Time, on the other hand, is indeed an additional dimension, and together with space it forms a larger, four-dimensional entity called spacetime. So, for the time being we may as well just focus on our familiar three dimensions. Call it the nowverse.īut what about all those other dimensions?įanciful theoretical constructs such as string theory postulate that, in fact, there is more to space than we can see, but for now those theories have no experimental evidence to support them. ![]() So here is one natural notion of the universe: all of three-dimensional space at the present time. Thus, we could make the assumption that we can locate anything in the universe using three Cartesian coordinates: at this frozen moment in time that we call the present, every object occupies a certain x, y and z in our three-dimensional continuum. If we take this route, we may first notice that space appears to us to be three-dimensional. Leaving aside the issue of whether "now" can have a universal meaning-and the even subtler ontological question of what it means to exist-it makes sense to think of the totality of space and all of its contents at the present time, and to imagine this totality as a contiguous entity. If we follow this line of thought, the first thing we notice is that the present tense of the verb “to exist” implicitly assumes that we are referring to “everything that exists now.” In colloquial English, the word is often taken to mean “everything that exists.” So this intuitive notion of universe seems like a good place to start. The subtlety is that the word “universe” has different meanings in different contexts. But all of them are true, or at least plausible. Statements like these appear quite frequently in popular science magazines-including Scientific American-and they seem to be in utter contradiction with one another. Or a hall of mirrors, shaped like soccer ball. The universe is 84 billion light-years wide. The universe has nine, or ten or eleven dimensions. That means that parallel transport on the sphere is not the same as parallel transport in the embedding space, since in the former case, we have to project the vector back down to the sphere after every step.The universe is four-dimensional-three for space, one for time. The geometrical difference is that vectors on the sphere must lie tangent to the sphere. You might think the second case should be the same, because the sphere can be embedded into flat space. But this doesn't say anything about the second metric. ![]() You're correct to conclude that the first metric should have zero curvature, since it's related to Cartesian coordinates by a coordinate change and the curvature is a tensor. ![]() The Riemann curvature of the former metric is zero, while the curvature of the latter is not. These are different quantities the spaces involved don't even have the same number of dimensions. Where $r_0$ is a constant, the radius of the sphere. ![]()
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