gravity curved space geometry einstein general relativity

Ultimately, he is persecuted for his teaching, and is arrested and sentenced to prison. From the viewpoint of the Flatlanders, the narrator appears to be insane: again, they have no basis from which to imagine something like a third dimension. The narrator insists on teaching the inhabitants of Flatland what he has learned about Spaceland, and describes a third type of direction called “Upwards, not Northwards”. Think about it: if the creatures of Flatland are only accustomed to life inside of 2 dimensions, they have no basis on which to imagine a 3rd dimension.

In another dream, he is visited by a sphere who teaches him that there is a third dimension, called Spaceland, and describes life in Spaceland.įrom the point of view inside of Flatland, the idea of 3-dimensional space is impossible to imagine. He attempts to teach the inhabitants of Lineland about life in 2-dimensional Flatland, but finds it is impossible to get them to imagine life in more than one dimension. The narrator has a dream about life in a one-dimensional land called Lineland. The narrator, who is a square, describes life in Flatland – the more sides an object has, the bigger its angles are, and the smarter it is. The classic work Flatland, by Edwin Abbott, published in Victorian England in 1884 CE, describes the life of 2-dimensional beings living in 3-dimensional space. Keep in mind that someone on the surface of Rhea can’t detect this curvature – it appears flat from the intrinsic point of view. Here’s a NASA image of the surface of one of Saturn’s moons, Rhea, from the extrinsic point of view, taken by the Cassini-Huygens spacecraft. If you’ve ever flown on a jetliner, you’ve probably seen the curvature in the Earth’s horizon. In mathematical terms, we say that the surface of each of these objects is “locally Euclidean” – from the viewpoint at each point on the surface, the surface appears to be flat, like a table.įrom the external point-of-view – what geometers refer to as the “extrinsic” point of view of a surface – the Earth, the oil pipeline, and the surface of the moon are clearly not flat. From the “intrinsic” point of view of a person standing on the Earth, an ant walking on an oil pipeline, and someone standing on the moon, each of these surfaces appears to be flat. The “intrinsic” view technically means the view from a completely flat object on the surface, not the view of someone standing on the surface, but in practice this distinction is irrelevant if the person is sufficiently small that they can only see a small region around where they’re standing. In geometric terms, we call the view of an object from its surface the “intrinsic” point of view of the surface. Check out this NASA image from the surface of the moon, whose surface also appears flat from the viewpoint of someone standing on it. To a small creature on the surface of either of these objects, the surface appears to be flat. In the same way, an ant walking on a large, round oil pipeline has the impression that it is walking on a flat surface. A glimpse into Differential Geometry “Intrinsic” and “Extrinsic” points of referenceĭespite the fact that the Earth is round, it appears to be flat, because of our relatively small size and position on its surface. In this post we’ll explore the mathematical ideas underlying these key discoveries in 20th-century physics. Physicists in turn used this mathematical formulation to refine our understanding of gravitation. These mathematical tools were in turn generalized to abstract, higher-dimensional surfaces sitting “inside” higher-dimensional spaces – and enabled physicists such as Einstein to develop accurate models of the geometry of space-time. Tools developed by mathematicians working in the field known as “Differential Geometry”, make it possible to determine that the Earth is not flat simply by taking measurements on its surface.

Amazingly, we don’t have to look at the Earth from outer space – or even resort to looking at its shadow on the moon – to determine that it has a curved surface. If we fly into space it’s obvious the Earth is not flat. Yes, we’re fooled by our perspective on the Earth’s surface. But the Earth looks flat from our perspective, right?

Gravity curved space geometry einstein general relativity