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物理代写|广义相对论代写General Relativity代考|CURVED SURFACES

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物理代写|广义相对论代写General Relativity代考|CURVED SURFACES

Gauss’ brilliant realisation is that there are two distinct ways in which a surface can be curved: it can have ‘extrinsic’ or ‘intrinsic’ curvature. Understanding this distinction is the basis for understanding general relativity.

Extrinsic curvature is simple: we say that a $2 \mathrm{~d}$ surface immersed in $3 \mathrm{~d}$ Euclidean space has extrinsic curvature if it is not (a portion of) a $2 \mathrm{~d}$ plane. The definition of intrinsic curvature, on the other hand, is Gauss’ stroke of genius.

Intrinsically flat surfaces
Imagine a flat sheet of paper, which can bend but not stretch, with some geometrical figures drawn on it (see Figure 3.1, first panel). These obey two-dimensional Euclidean geometry. Imagine bending the paper (Figure 3.1, second panel). When we bend the paper, a straight segment drawn on the paper becomes bent, but it is still the shortest among all the lines between its extremes that can be drawn on the surface. Call the shortest line on the surface between two given points ‘(intrinsically) straight’, and call its length the ‘(intrinsic) distance’ between its extremes. Obviously these ‘(intrinsically) straight lines’ and the ‘(intrinsic) distances’ between the points on the bent sheet satisfy the same properties as the straight lines and the distances on a 2 d plane.

For instance, imagine a triangle drawn on the paper. Neither the length of its sides nor the amplitude of its angles changes when bending the paper. Therefore, the triangle drawn on the bent paper satisfies the standard properties of $2 \mathrm{~d}$ Euclidean triangles: if one angle is straight, the lengths $a, b, c$ of its sides satisfy Pythagoras’ theorem $a^2+b^2=c^2$; the sum of the three angles is $\pi(\mathrm{rad})$. Similarly, a circle drawn on the paper (the set of points at equal intrinsic distance $r$ from a central point) has a perimeter $p$ that still satisfies $p=2 \pi r$ when drawn on the paper. And so on: standard Euclidean geometry holds.

Let’s put it visually: if you were a small ant moving on the bent paper and capable of measuring angles and lengths of lines on the surface, but incapable of looking ‘outside’ the paper, you would not be able to figure out that you are on a surface that is not a plane. The two-dimensional geometry defined by the length of the intrinsically straight lines is the same geometry as the geometry of a plane. This geometry is called ‘intrinsic geometry’. Hence we say that ‘the intrinsic geometry of the bent sheet of paper is flat’, even if the paper itself is actually curved.

物理代写|广义相对论代写General Relativity代考|Intrinsically curved surfaces

What is described above, however, is not true for generic curved surfaces. Consider, for instance, a sphere of unit radius. The intrinsic geometry defined by the length of the lines drawn on the sphere itself is not the geometry of a plane.

Given two points on the sphere, the shortest line on the sphere connecting them is a portion of a maximal circle. These are the ‘intrinsically straight’ segments on the sphere. Take the North Pole of the sphere and two points on the equator at one quarter of the equator length from one another. These define a triangle, formed by a portion of the equator and two meridians. It is evident immediately that the sum of the angles of this triangle is not $\pi$ but $\frac{3}{2} \pi$ ! Similarly, the equator is a circle or length $p=2 \pi$ at intrinsic distance $r=\pi / 2$ from the North Pole, hence it does not satisfy $p=2 \pi r$, but rather $p=4 r$

Thus, straight lines on the sphere define an intrinsic geometry which is different from the geometry of the $2 d$ plane. If you were a small ant moving on the sphere, capable of measuring lengths of lines on the surface but incapable of looking ‘outside’ the surface, you would be able to figure out that you are not on a plane: it would suffice to measure the length $p$ of the line formed by points at a distance $r$ from a centre: if $p \neq 2 \pi r$, your intrinsic geometry is not flat. When the intrinsic geometry is not flat, we say that the surface has ‘intrinsic curvature’.

These examples illustrate the difference between extrinsic and intrinsic curvature.

The ‘intrinsic geometry’ of a surface is the geometry defined by the length of the lines lying on the surface. If this geometry is the same as that of the lines on a plane, we say that the surface is ‘intrinsically flat’, or that it has no ‘intrinsic curvature’. If instead, the geometry defined by these lines is different from the geometry of the lines on a plane, we say that the geometry is ‘intrinsically curved’, or that there is ‘intrinsic curvature’.

The importance of this intuition by Gauss is that in this manner we can talk about the curvature of a surface using only the geometry of the distances on the surface itself, with no need of looking at how the surface is embedded in a larger space. This is exactly what we shall do in general relativity.

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