6/20/2023 0 Comments Positively curved spaceThis is analogous to looking at yourself in a fun house mirror or a car mirror that says “objects may be closer than they appear,” both of which have curved surfaces that distort the images they produce. Thus an object will appear to be a different angular size depending on how space is curved. If spacetime is curved, light from an object will follow the curvature of spacetime it will travel along a geodesic. In a hyperbolic space, lines that start off as parallel eventually diverge. In flat space, parallel lines stay parallel. In a spherical space, lines that start off as parallel eventually intersect. Figure 15.14: Geometry in spherical, flat, and hyperbolic spaces. Figure 15.14 illustrates spherical, flat, and hyperbolic spaces. On a globe the string will stretch directly from the beginning to the end of your journey, but the route will look curved when projected onto a flat map of Earth such as the one in the in-flight magazine on your trip. ![]() To map out a great circle route, put a string on a globe at two points (say, your hometown and a distant place you might like to go on vacation-Rome, maybe, or Tahiti). Have a look at an Earth globe and you will see that the shortest distance between the United States and Europe does indeed pass over the Arctic. That is why flights from North America to Europe generally pass over the Arctic. A great circle is the route an airplane flies between two cities because it is the shortest distance between them. An example of a geodesic on a curved surface is a great circle on the surface of Earth-like lines of constant longitude. In a flat space, a geodesic is a familiar straight line. The more general name for the shortest distance between two points is a geodesic. The shortest distance between two points is not a straight line in curved spaces. This type of space is said to have negative curvature. In such a space parallel lines diverge and the angles of a triangle add up to less than 180 degrees. A space with a spherical geometry is said to have a positive curvature.Īnother curved geometry is a hyperbolic geometry, in which space is saddle shaped. In a spherical geometry, such as the surface of Earth, parallel lines converge and the angles of a triangle can add up to more than 180 degrees. Although for short distances a flat geometry works pretty well to describe the relationship between events on Earth (think of a standard city map), for large distances we need to use a curved geometry if we want to accurately describe the relationships. For example, if we consider large enough distances, Earth’s surface is definitely not flat because Earth is a sphere. A flat space in this sense can have three or even more dimensions.Īlthough most situations we are used to involve flat space, or at least a close approximation to it, there are a number of situations in which this is not the case. Flat in this case does not necessarily mean two-dimensional like the surface of a table or flat like a pancake, it just means that normal Euclidean geometry applies and that there is no overall curvature. Space with this type of geometry is often referred to as flat. This is also the geometry in which parallel lines never intersect, angles of a triangle add up to 180 degrees, straight lines are the shortest distance between two points, and standard trigonometry applies. Recall for a moment the subject of Euclidean geometry, the geometry of the Pythagorean theorem. ![]() By comparing models of different curvatures with CMB data, we can determine the geometry of the Universe. The global curvature term k is one of the things that affects the anisotropy pattern in the CMB. The global curvature of the Universe can be described by the Friedmann equation, which includes a term (called k) describing the global curvature of the Universe as well as terms for the expansion and density of the Universe. This idea of curvature applies to not just the space around individual objects but also the Universe as a whole the entire matter and energy content of the Universe will bend the spacetime of the Universe as a whole. ![]() General relativity predicts that matter and energy curve spacetime. There are several possibilities for the geometry of the Universe within the theoretical framework of the Big Bang. What Do You Think: Curvature of the Universe
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