The only assumption that isn't true is the one asserting that there should be no dark energy. One you accept isotropy, then homogeneity can only fail if you accept that our location in space is somehow special, an idea that physicists reject. Based on current observations, this average really is the same everywhere, so the Universe is indeed likely to be isotropic. However, this assumption refers to the average look over large volumes of space. This idea may seem strange at first, since the world definitely looks different in different directions. Another assumption is that the Universe is isotropic, that is, as we look in the various directions in the sky, no direction stands out over an other, they are all the same, on average. The theory has been tested many times and has always stood up to the tests admirably, so physicists on the whole take it to be true (though there are dissenters, see here). This would imply eternal expansion, at least if the assumptions above are true.īut are they? The first is that general relativity is true. Left-over radiation from the Big Bang) suggest that the Universe is indeed flat, or Observations of the cosmic microwave background (the May be that matter can slow the rate of expansion, but it'll never be Angles of triangles add up to exactlyġ80 degrees and the Universe is infinite. If the Universe has zero curvature, then its geometry is the ordinary 3D space we learn about at school.įinally, it could be that there's just enough matter for the The angles of a triangle drawn on this plane add up to exactly 180 degrees.
Unlike in flat 2D space, its angles add up to more than 180Ī piece of the flat plane. Whose sides are made from pieces of great circles, you will find that, Great circles are theĪnalogues of straight lines on the flat plane. Through two opposite points on the sphere. Should travel along a great circle: a circle that passes One thing that allows you to see whether a given space isīetween two points on a sphere along the shortest route, then you Mass will eventually stop it from expanding (which it is currentlyĭoing) and cause it to contract. What is more the gravitational pull exerted by all the Positively curved, then it is finite, just like the surface of a (Curvature is a precisely defined mathematicalĬoncept, see here to find out more.) If the Universe is indeed Is hard to imagine, but it's easily described mathematically and comes The first possibility is that the density of matter (the averageĪmount of matter per unit volume) is so high, it curves the UniverseĪround on itself to form the 3D analogue of a sphere. The shape of the Universe, depending on how much matter there is If you assume that the Universe is isotropic (looks the same, on average, in every direction in the sky) and homogeneous (looks the same, on average, at every point), and that there is none of that mysterious substance called dark energy, then general relativity tells you that there are only three possibilities for If the Universe has positive curvature, then its geometry is the 3D analogue of a sphere.Īccording to Einstein's general theory of relativity massive The angles of a spherical triangle add up to more than 180 degrees. Work with generalized metric spaces and study some of their properties.The surface of a sphere is finite, but it doesn't have an edge. Since the category of metric spaces is not cocomplete, we are lead to
That our conditions on precubical sets actually coincide with those for metric Study the geometric realization of precubical sets in metric spaces, to show Homotopy coincide for directed paths in these precubical sets. Rewriting techniques, we are then able to show that directed and non-directed Programs have non-positive curvature, a notion that we introduce and study hereįor precubical sets, and can be thought of as an algebraic analogue of the The particular case of programs using only mutexes, which are the most widely "geometry" of the space of possible executions of the program. Order to study (higher) commutations between the actions, thus encoding the
POSITIVELY CURVED SPACE PDF
Download a PDF of the paper titled Directed Homotopy in Non-Positively Curved Spaces, by Eric Goubault and 1 other authors Download PDF Abstract: A semantics of concurrent programs can be given using precubical sets, in
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