[Understanding Riemannian Geometry] 2. From the Pythagorean Theorem to the Riemannian Metric

Riemannian Metric

The word "Geometry" originally means "earth measurement." Since it involves measurement, there must be a reference object and a method to calculate distances.

With a reference object, we can establish a coordinate system and record the coordinates of every point. As for calculating distances, we have the great Pythagorean theorem: ds^2 = dx^2 + dy^2 \tag{1} However, we have overlooked two issues here.

The first issue is that we do not necessarily use a Cartesian coordinate system. If we use polar coordinates, the formula should be: ds^2 = dr^2 + r^2 d\theta^2 \tag{2} Therefore, one can imagine that the most general form should be: ds^2 = E(x^1, x^2)(dx^1)^2 + 2F(x^1, x^2)dx^1 dx^2 + G(x^1, x^2)(dx^2)^2 \tag{3} Here, x^1, x^2 are generalized coordinates. We use superscripts instead of subscripts to label indices to remain consistent with traditional textbook notation. What does this formula mean? It is actually quite simple. Just as there is no reason to require the whole world to use the same currency, there is no need to require every location to use the same coordinate system. A more reasonable approach is for each location to use its own coordinate system (a local coordinate system) and then provide a method for calculating distances locally. Thus, the above formula states that at position (x^1, x^2), the formula for calculating the length of the vector (dx^1, dx^2) (the local Pythagorean theorem) is ds^2 = E(x^1, x^2)(dx^1)^2 + 2F(x^1, x^2)dx^1 dx^2 + G(x^1, x^2)(dx^2)^2.

The second issue is that we are not only studying 2D planes; we also need to study n-dimensional spaces. Therefore, the most general formula is: ds^2 = g_{\mu\nu}(\mathbf{x}) dx^{\mu} dx^{\nu} \tag{4} Here, \mathbf{x}=(x^1, x^2, \dots, x^n), and the Einstein summation convention is used, meaning that identical upper and lower indices in a monomial imply summation. g_{\mu\nu} is what we call the Riemannian metric. We can choose a symmetric metric, i.e., g_{\mu\nu}=g_{\nu\mu}, without changing the form of ds^2. Based on our previous discussion, the entire ds^2 is simply the different ways of calculating distances used at different positions in a high-dimensional space. Here, we restore the measurement-based meaning of geometry.

On the other hand, the Riemannian metric can also be viewed as a standard of measurement. Just as different countries have different currencies, if there were a formula that could convert any country’s currency into an equivalent amount of gold, the problem of comparing different amounts of currency would be solved. The Riemannian metric serves a similar purpose; rather than saying it provides different ways of calculating distances at different locations, it is better to say it unifies the ways distances are calculated across all locations.

It should be noted that from the perspective of merely defining distance, we do not strictly have to use a quadratic form; for instance, cubic forms, quartic forms, or even more complex forms could be used. However, in terms of practical value, we only study quadratic forms. Even so, there are already more problems than we can ever finish studying.

Some Examples

Under what circumstances does a non-constant Riemannian metric arise (or become necessary)? As seen earlier, transforming Cartesian coordinates into polar coordinates results in a non-constant Riemannian metric. That is to say, even in a flat space, as long as a curvilinear coordinate system is used, a non-constant Riemannian metric will appear.

Furthermore, the Riemannian metric of a curved space must be non-constant. Are there any specific examples? The most classic should be the two-dimensional sphere (it is a 2D sphere, not 3D spherical coordinates; readers should not confuse the two). Setting the radius to 1, the spherical coordinates are: \left\{\begin{aligned} &x=\sin\theta\cos\varphi\\ &y=\sin\theta\sin\varphi\\ &z=\cos\theta \end{aligned}\right. \tag{5} Then its Riemannian metric is: ds^2=dx^2+dy^2+dz^2=d\theta^2+\sin^2\theta d\varphi^2 \tag{6}

Another very vivid example is learned from the second volume of The Feynman Lectures on Physics. Suppose we are in a flat space, but the temperature varies everywhere. We use a ruler with a very large coefficient of thermal expansion as our measurement tool. What will happen? In high-temperature areas, the ruler expands, so the measurement result will be smaller; conversely, in low-temperature areas, the ruler shrinks, and the measurement result will be larger. For the same distance, the measurement might be 50 cm in a high-temperature area and 100 cm in a low-temperature area. Therefore, a non-constant metric is needed to unify them—either by multiplying 50 by 2, or dividing 100 by 2, or multiplying 50 by 4 while multiplying 100 by 2, and so on. It is not hard to imagine that the Riemannian metric in this case should take the form (considering 2D or 3D): ds^2 = f(x,y,z)(dx^2+dy^2+dz^2) \tag{7}

This leads to the emergence of a curved space—in this case, it is not the space that is "curved," but the ruler that is "curved." This example also appears in The Feynman Lectures on Gravitation; according to Feynman, it was invented by a student of Robertson. Due to its clear physical meaning, it is also known as "isothermal parameters" or an "isothermal coordinate system." In fact, this is somewhat similar to the idea that "motion is relative." The appearance of a curved space may be due to the curvature of the space itself (like a sphere) or the "curvature" of the ruler (like a thermally expanding ruler), but their mathematical results are the same.

Local Cartesian Coordinate Systems

Now we attempt to describe the Riemannian metric in matrix form. Let \mathbf{g}=g_{\mu\nu}, \mathbf{x}=x^{\alpha}, and d\mathbf{x}=dx^\alpha. Here, vectors are column vectors, and we do not distinguish between the components of a vector and the vector itself. Therefore, the Riemannian metric can be written as: ds^2 = d\mathbf{x}^T \mathbf{g} d\mathbf{x} \tag{8}

Note that the matrix \mathbf{g} is symmetric, so in general, it can be decomposed as \mathbf{h}^T \mathbf{h}, where \mathbf{h} is a matrix of the same order as \mathbf{g}. In this case: ds^2 = d\mathbf{x}^T \mathbf{h}^T \mathbf{h} d\mathbf{x} = (\mathbf{h}d\mathbf{x})^T (\mathbf{h}d\mathbf{x}) = |\mathbf{h}d\mathbf{x}|^2 \tag{9} That is to say, it is ultimately converted into the norm of \mathbf{h}d\mathbf{x}, and this norm is consistent with the Pythagorean theorem in flat space. We can consider that the matrix \mathbf{h} exactly describes the local coordinate system. The vector d\mathbf{x} in the coordinate system \mathbf{h} is exactly equivalent to the vector \mathbf{h}d\mathbf{x} in the local Cartesian coordinate system. Or rather, \mathbf{h} is the transformation matrix (Jacobian matrix) from the local coordinate system to the local Cartesian coordinate system.

With the transformation to a Cartesian coordinate system, we can define many geometric quantities, all of which are extended from flat space. For example, given a vector \mathbf{A}=A^{\mu}, the square of its norm is: |\mathbf{h}\mathbf{A}|^2 = \mathbf{A}^T \mathbf{h}^T \mathbf{h} \mathbf{A} = \mathbf{A}^T \mathbf{g} \mathbf{A} = g_{\mu\nu}A^{\mu}A^{\nu} \tag{10}

Given two vectors \mathbf{A} and \mathbf{B}, their inner product is: (\mathbf{h}\mathbf{A})^T (\mathbf{h}\mathbf{B}) = \mathbf{A}^T \mathbf{h}^T \mathbf{h} \mathbf{B} = \mathbf{A}^T \mathbf{g} \mathbf{B} = g_{\mu\nu}A^{\mu}B^{\nu} \tag{11} If you wish, you can also define the angle \theta between them as: \theta = \arccos \frac{g_{\mu\nu}A^{\mu}B^{\nu}}{\sqrt{g_{\mu\nu}A^{\mu}A^{\nu}}\sqrt{g_{\mu\nu}B^{\mu}B^{\nu}}} \tag{12}

At the same time, one can calculate the area of the parallelogram spanned by two vectors \mathbf{A} and \mathbf{B}: |\mathbf{h}\mathbf{A}| \times |\mathbf{h}\mathbf{B}| \times \sin\theta = \sqrt{(g_{\mu\nu}A^{\mu}A^{\nu})(g_{\mu\nu}B^{\mu}B^{\nu})-(g_{\mu\nu}A^{\mu}B^{\nu})^2} \tag{13}

If one wants to calculate a (hyper)volume integral, the volume element is: \det(\mathbf{h})\prod_{\alpha} dx^{\alpha} = \sqrt{\det(\mathbf{g})} \prod_{\alpha} dx^{\alpha} = \sqrt{g}d\Omega \tag{14} Here g is shorthand for \det(\mathbf{g}), and d\Omega is shorthand for \prod_{\alpha} dx^{\alpha}. Note that we have \det(\mathbf{g}) = \det(\mathbf{h}^T \mathbf{h}) = (\det(\mathbf{h}))^2. \sqrt{g} is actually a volume scaling factor. With this result, we can write: given n vectors \mathbf{A}^1, \mathbf{A}^2, \dots, \mathbf{A}^n, written as column vectors, the hypervolume of the n-dimensional parallelotope they span is: \sqrt{g}\det(\mathbf{A}^1, \mathbf{A}^2, \dots, \mathbf{A}^n) \tag{15} Here (\mathbf{A}^1, \mathbf{A}^2, \dots, \mathbf{A}^n) refers to the n \times n matrix formed by arranging these n column vectors.

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