If we want to cover this sphere in fabric, for instance, we will need a piece 4 × π (about 12.566) times as big as a square with edges as long as our piece of rope, cut up into little patches, and sewn together just right. The places where the free end of the rope reach define a sphere. To use this formula, we make a sphere just as we made a circle, but we move everywhere the rope will let us. $$Area = 4 \times \pi \times Radius \times Radius = 4 \times \pi \times (Radius)^2.$$ Now, to think about curvature in three dimensions, we recall another high-school formula that gives the surface area of a sphere in terms of its radius: The classic formula doesn't apply in these curved space examples. If you now measure around the circumference of the circle you drew, you get a number that is less than 2 × π times the length of the rope! If we tried to make a circle centered on a mountain pass (or in the middle of a horse's saddle), we would get a circumference that is greater than 2 × π times the length of the rope. Draw the circle on the lower parts of the hill. Choose the center point of the circle to be the top of a hill and pretend that you are trapped on the hill you and your rope can't rise up off of it, or dig down inside of it. Now imagine that you try to use this formula in a surface that is not flat. If we measure around the circle - its circumference - we should find that it is 2 × π times the length of our rope. Stretch the rope taut, and rotate around the center point, tracing out the position of the free end of the rope. Choose any point, and tie one end of a length of rope to that point. To use this formula, we make a circle in a particular way. $$Circumference = 2 \times \pi \times Radius$$ We remember the number π - pi, roughly 3.1416 - showing up in the formula One familiar fact from this exploration is the formula for the circumference of a circle in terms of its radius. In high-school geometry class, we typically explore two-dimensional space, which is almost always assumed to be flat. Leave the page down, and you see a flat surface. Turn up half the page in a book, and you will see that it is curved. We also recognize a curved surface when we see it. Movie screens, pages in a book, and the surface of an apple are all examples. We are all familiar with two-dimensional surfaces. To understand this warping, it helps to think about warping in two-dimensional space. One-dimensional straight lines became one-dimensional curvy lines two-dimensional flat sheets became two-dimensional curvy sheets three-dimensional flat spaces became three-dimensional curvy spaces. For his second - and greater - revolution, Einstein allowed the slices to be curved and warped. In a similar way, the three-dimensional slices Einstein took were also flat, in a way we will explore below. Each of the slices was flat: a one-dimensional slice was just a straight line a two-dimensional slice was a flat sheet. From Lord Byron's Childe Harold's Pilgrimageįor his first revolution, Einstein unified space and time, and showed that a given observer just took a slice out of this spacetime.
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