If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve). As 3D beings, we experience just the "surface" of the 4D ball, just as Flatlanders experience only two dimensions of their space that you can curve into three dimensions.Luboš's answer is of course perfectly correct. Scientists believe, however, that our space is smoothly curved - more like a ball than a wad of paper - and thus the effects are only noticeable on cosmological scales. You might notice some unusual phenomena, like walking to your mailbox and ending up in Mongolia.
If a 4D being decides to "crumple" our 3D space, like the Flatlanders, you wouldn't notice anything right away, either. They might experience some interesting phenomena, though - triangular residents might find that their interior angles no longer add up to 180 degrees, and parallel lines everywhere might begin to intersect. Doing so would not be immediately apparent to Flatlanders, because light still travels in straight lines within their 2D space. You can curve Flatland in your three-dimensional reality by rolling the piece of paper into a tube, folding it into a paper airplane, or just crumpling it into a wad. In other words, an additional dimension must exist into which the other dimensions can be curved. Regardless of the number of dimensions that describe a space, the curvature of that space, by definition, implies the existence of at least one more. And while Flatlanders cannot see beyond the boundaries of their world, to you they are nothing more than hollow geometrical shapes. Say "Hello" to Flatland - your voice is also omnipresent. Place your fingertips on Flatland - you appear to them in many places at once. So as a three-dimensional being, your actions would violate the Flatlanders' laws of physics. The laws of 2D space as understood by 2D beings would restrict light (and everything else) from moving up and down in fact, there would be no up or down. If it helps, draw Flatland and its residents on a sheet of paper and place it flat on a table in front of you.
#CURVED SPACE COORDINATE RADIUS FULL#
To assist you in imagining space curved in four dimensions, pay a visit to Flatland, a two-dimensional world full of square, triangular, and circular beings. Some theoretical equations incorporate many other "dimensions" - velocity, temperature, or density, for example. In this way, time merely serves as a yardstick for our personal experiences and our laws of physics. Unlike width, length, and height, we only know time to move in one direction: past to present to future. In Einstein's Theories of Relativity, time helps us understand the three-dimensional experience of gravity.īut time is not a true spatial dimension like the three we're familiar with. This is the stage on which planets, stars, galaxies - all matter in the universe - act their gravitational roles.
Einstein used time as a fourth dimension to describe a coordinate system called space-time. Many people think the fourth dimension is simply time, and for some astronomical equations, it is. Space is indeed curved - in four dimensions.
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