Geometry on Curved Spaces

Differential geometry is the branch of mathematics used by Albert Einstein when he formulated the general theory of relativity, where gravity is the curvature of spacetime. It was originally invented by Carl Friedrich Gauss to study the curvature of hills and valleys in the Kingdom of Hanover.

From what I described, one may guess that differential geometry has something to do with curvature. The geometry we learn in high school only occurs on a flat surface. There we can put coordinates x and y and compute distances, angles, areas, and so on.

To imagine what geometry on curved spaces looks like, imagine a globe. Instead of x and y coordinates, we can use latitude and longitude. One can now see just how different geometry is on this globe. Vertical lines (the lines of constant x) on a flat surface are always the same distance apart. On a globe, the analogues of these vertical lines, the lines of constant longitude, are closer near the poles than they are near the equator.

Other weird things happen on our globe: One can have triangles with angles that sum to more than 180 degrees. Run two perpendicular line segments from the north pole to the equator. They will meet the equator at a right angle and form a triangle with three right angles for a total of 270 degrees. Also on the globe the ratio between the circumference of a circle to its diameter might no longer be equal to the number \pi.

To make things more explicit, we will introduce the concept of a metric (the word “metric” refers to a variety of mathematical concepts related to notion of distance – in this post we use it in the sense of differential geometry to refer to what is also called the metric tensor). The metric is an example of a mathematical object called a tensor, which we will not discuss much of in this post. Instead, we will think of the metric as expressing a kind of “distance formula” for our space, which may be curved. The part of differential geometry that makes use of the metric is called Riemannian geometry, named after the mathematician Bernhard Riemann, a student of Gauss who extended his results on curved spaces to higher dimensions.

We recall from From Pythagoras to Einstein several important versions of the “distance formula”, from the case of 2D space, to the case of 4D spacetime. We will focus on the simple case of 2D space in this post, since it is much easier to visualize; in fact, we have already given an example of a 2D space earlier, the globe, which we shall henceforth technically refer to as the 2-sphere. As we have learned in From Pythagoras to Einstein, a knowledge of the most simple cases can go very far toward the understanding of more complicated ones.

We will make a little change in our notation so as to stay consistent with the literature. Instead of the latitude, we will make use of the colatitude, written using the symbol \theta, and defined as the complementary angle to the latitude, i.e. the colatitude is 90 degrees minus the latitude. We will keep using the longitude, and we write it using the symbol \varphi. Note that even though we colloquially express our angles in degrees, for calculations we will always use radians, as is usual practice in mathematics and physics.

On a flat 2D space, the distance formula is given by

\displaystyle (\Delta x)^{2}+(\Delta y)^{2}=(\Delta s)^{2}.

It will be productive for us to work with extremely small quantities for now; from them we can obtain larger quantities later on using the language of calculus (see An Intuitive Introduction to Calculus). Adopting the notation of this language, we write

\displaystyle (dx)^{2}+(dy)^{2}=(ds)^{2}

We now give the distance formula for a 2-sphere:

\displaystyle R^{2}(d\theta)^{2}+R^{2}\text{sin}(\theta)^{2}(d\varphi)^{2}=(ds)^{2}

where R is the radius of the 2-sphere. This formula agrees with our intuition; the same difference in latitude and longitude result in a bigger distance for a bigger 2-sphere than for a smaller one, and the same difference in longitude results in a bigger distance for points near the equator than for points near the poles.

The idea behind the concept of the metric is that it gives how the distance formula changes depending on the coordinates. It is often written as a matrix (see Matrices) whose entries are the “coefficients” of the distance formula. Hence, for a flat 2D space it is given by

\displaystyle \left(\begin{array}{cc}1&0\\ 0&1\end{array}\right)

while for a 2-sphere it is given by

\displaystyle \left(\begin{array}{cc}R^{2}&0\\ 0&R^{2}\text{sin}(\theta)^{2}\end{array}\right).

We have seen that the metric can express how a space is curved. There are several other quantities related to the metric (and which can be derived from it), such as the Christoffel symbol and the Riemann curvature tensor, which express ideas related to curvature – however, unlike the metric which expresses curvature in terms of the distance formula, the Christoffel symbol and the Riemann curvature tensor express curvature in terms of how vectors (see Vector Fields, Vector Bundles, and Fiber Bundles) change as they move around the space.

The main equations of Einstein’s general theory of relativity, called the Einstein equations, relate the Riemann curvature tensor of 4D spacetime to the distribution of mass (or, more properly, the distribution of energy and momentum), expressed via the so-called energy-momentum tensor (also known as the stress-energy tensor).

The application of differential geometry is not limited to general relativity of course, and its objects of study are not limited to the metric. For example, in particle physics, gauge theories such as electrodynamics (see Some Basics of (Quantum) Electrodynamics) use the language of differential geometry to express forces like the electromagnetic force as a kind of “curvature”, even though a metric is not used to express this more “abstract” kind of curvature. Instead, a generalization of the concept of “parallel transport” is used. Parallel transport is the idea behind objects like the Christoffel symbol and the Riemann curvature tensor – it studies how vectors change as they move around the space. To generalize this, we replace vector bundles by more general fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles).

To give a rough idea of parallel transport, we give a simple example again in 2D space – this 2D space will be the surface of our planet. Now space itself is 3D (with time it forms a 4D spacetime). But we will ignore the up/down dimension for now and focus only on the north/south and east/west dimensions. In other words, we will imagine ourselves as 2D beings, like the characters in the novel Flatland by Edwin Abbott. The discussion below will not make references to the third up/down dimension.

Imagine that you are somewhere at the Equator, holding a spear straight in front of you, facing north. Now imagine you take a step forward with this spear. The spear will therefore remain parallel to its previous direction. You take another step, and another, walking forward (ignoring obstacles and bodies of water) until you reach the North Pole. Now at the North Pole, without turning, you take a step to the right. The spear is still parallel to its previous direction, because you did not turn. You just keep stepping to the right until you reach the Equator again. You are not at your previous location of course. To go back you need to walk backwards, which once again keeps the spear parallel to its previous direction.

When you finally come back to your starting location, you will find that you are not facing the same direction as when you first started. In fact, you (and the spear) will be facing the east, which is offset by 90 degrees clockwise from the direction you were facing at the beginning, despite the fact that you were keeping the spear parallel all the time.

This would not have happened on a flat space; this “turning” is an indicator that the space (the surface of our planet) is curved. The amount of turning depends, among other things, on the curvature of the space. Hence the idea of parallel transport gives us a way to actually measure this curvature. It is this idea, generalized to mathematical objects other than vectors, which leads to the abstract notion of curvature – it is a measure of the changes that occur in certain mathematical objects when you move around a space in a certain way, which would not have happened if you were on a flat space.

In closing, I would like to note that although differential geometry is probably most famous for its applications in physics (another interesting application in physics, by the way, is the so-called Berry’s phase in quantum mechanics), it is by no means limited to these applications alone, as already reflected in its historical origins, which barely have anything to do with physics. It has even found applications in number theory, via Arakelov theory. Still, it has an especially important role in physics, with much of modern physics written in its language, and many prospects for future theories depending on it. Whether in pure mathematics or theoretical physics, it is one of the most fruitful and active fields of research in modern times.

Bonus:

Since we have restricted ourselves to 2D spaces in this post, here is an example of a metric in 4D spacetime – this is the Schwarzschild metric, which describes the curved spacetime around objects like stars or black holes (it makes use of spherical polar coordinates):

\displaystyle \left(\begin{array}{cccc}-(1-\frac{2GM}{rc^{2}})&0&0&0\\0&(1-\frac{2GM}{rc^{2}})^{-1}&0&0\\0&0&r^{2}&0\\ 0&0&0&r^{2}\text{sin}(\theta)^{2}\end{array}\right)

In other words, the “infinitesimal distance formula” for this curved spacetime is given by

\displaystyle -(1-\frac{2GM}{rc^{2}})(d(ct))^{2}+(1-\frac{2GM}{rc^{2}})^{-1}(dr)^{2}+r^{2}(d\theta)^{2}+r^{2}\text{sin}(\theta)^{2}(d\varphi)^{2}=(ds)^{2}

where G is the gravitational constant and M is the mass. Note also that as a matter of convention the time coordinate is “scaled” by the constant c (the speed of light in a vacuum).

References:

Differential Geometry on Wikipedia

Riemannian Geometry on Wikipedia

Metric Tensor on Wikipedia

Parallel Transport on Wikipedia

Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo

Geometry, Topology, and Physics by Mikio Nakahara

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4 thoughts on “Geometry on Curved Spaces

  1. Pingback: Connection and Curvature in Riemannian Geometry | Theories and Theorems

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  3. Pingback: Rotations in Three Dimensions | Theories and Theorems

  4. Pingback: Differentiable Manifolds Revisited | Theories and Theorems

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