Curved space YouTube

Metrics in general

The most general metric we could write down for a given set of coordinates $(x, y)$ in two-dimensional space has the form $$g_{i j} \rightarrow \begin{bmatrix} g_{1 1}(x, y) & g_{1 2}(x, y) \\ g_{2 1}(x, y) & g_{2 2}(x, y) \end{bmatrix}$$

The metric completely determines the distances between neighbouring pairs of points. If our choice of coordinates allows us to be able to mark off units as constant lengths, then the components of the corresponding metric will be constant.

Nb. Futhermore, if we can find one set of coordinates in which the metric has constant components, then the space is flat.

Figure 8.4 - Arbitrary curvilinear coordinates

Figure 8.4 - Arbitrary curvilinear coordinates

If, however, the coordinate mesh is not uniform - as is often the case in the general theory of relativity - then the components of the metric will be functions of the coordinates.

We refer to such coordinates as curvilinear.

The proper time of general relativity is thus given by the formula, $$\mathrm{d}\tau^2 = g_{\mu \nu} \mathrm{d}X^\mu \mathrm{d}X^\nu$$ where $X^\mu$ can be any curvilinear coordinates and the components of the metric are functions of those coordinates, $$g_{\mu \nu} = g_{\mu \nu}(X^\mu)$$

Nb. It is interesting that (we won't prove it) if the components of a particular metric form a diagonal matrix, then the mesh-lines of the components will be perpendicular to each other everywhere.

Intrinsic and extrinsic curvature

The straightforward, mathematical way to define curved space is to say that it's not flat.

That is, for every set of coordinates that can be defined upon a curved space, the components of the corresponding metric are not all constant.

A common difficulty with the notion of curvature is to do with the way we (humans) visualise objects and spaces. We have to embed them in three-dimensional space, even if they do not have three dimensions themselves.

Consider an idealised flat piece of paper lying on a table. We can bend this surface into many different shapes, without stretching it, or squeezing it (or making a fold in it) and it will remain a flat surface.

For example, consider rolling up the paper into a cylinder. The curvature we see in the cylinder is extrinsic curvature in the three dimensions of the space we live in.

But there is no intrinsic curvature in the two-dimensional space itself, since so stretching or squeezing has occurred. Distances between points on the surface of the cylinder are the same as the original piece of paper.

Spherical surfaces

Spherical surfaces do have intrinsic curvature. A simple way to see this is to note that we can't flatten out a hemisphere at all without stretching and squeezing.

Now, mathematically, a sphere always refers to a surface, and the volume that it contains is called a ball. In particular, a two-dimensional sphere, or $2$-sphere, is the surface of a three-dimensional ball, or $3$-ball.

Going down one dimension, a circle, or $1$-sphere, is the surface of a $2$-ball, or disc and we can still visualise both the surface and the interior. But in going up a dimension, we encounter a difficulty - we cannot visualise a $3$-sphere surface or the $4$-ball that it contains.

However, the mathematics of spheres and balls generalise to any dimension, so we can concentrate on $2$-spheres.

Figure 8.5 - Distance on a sphere

Figure 8.5 - Distance on a sphere

A standard way to coordinatise a sphere (of radius $R$, say) is to use latitude, $\phi$, and longitude, $\theta$. For simplicity, the latitude is zero at the North Pole and increases to $180^{\circ}$ at the South Pole.

We want to calculate the distance from $P (\phi, \theta)$ to $Q (\phi + \mathrm{d}\phi, \theta + \mathrm{d}\theta)$.

Now, lines of longitude are great circles, which have radius $R$, hence the length of the green arc in figure 8.5 is simply $R \mathrm{d}\phi$.

But lines of latitude are not great circles and depend on upon the height above (or below) the equator. It it easy to show that the radius of the red arc is $R \sin \phi$ hence its length is $R \sin \phi \mathrm{d}\theta$.

Thus, the distance (from $P$ to $Q$) is $$\mathrm{d}s^2 = R^2 \mathrm{d}\phi^2 + R^2 \sin^2 \phi \mathrm{d}\theta^2$$ and so the metric for a $2$-sphere is given by $$g_{i j} \rightarrow \begin{bmatrix}R^2 & 0\\0 & R^2 \sin^2 \phi\end{bmatrix}$$

This metric is comparable to the metric for polar coordinates. The difference is that the space in which we can define polar coordinates is flat, since we can just use Cartesians instead, but the space in which we can define spherical coordinates is curved meaning that there are no sets of coordinates in which the components of the metric are constant.

Positive and negative curvature

We have seen that a space is flat if there exists at least one set of coordinates in which the metric has constant components and a space is curved if no set of coordinates exist in which the metric has constant components.

We could say that a flat space has zero curvature and a curved space has non-zero curvature. We can go further and distinguish between positive and negative curvature.

A simple way to visualise this for two-dimensional spaces (which we won't prove) is to draw a triangle on the surface and measure the three interior angles. Then,

  • If the angles add up to $180^{\circ}$, then as expected, the space is flat, with zero curvature.
  • If the angles add up to more than $180^{\circ}$ then the surface has positive curvature.
  • If the angles add up to less than $180^{\circ}$ then the surface has negative curvature.
Figure 8.6 - Positive curvature

Figure 8.6 - Positive curvature

In figure 8.6, we show that a sphere has positive curvature.

We mark out a triangle in the following way; start at the north pole and head south towards the equator. Then travel eastwards one-quarter way around the equator before heading north, back up to the north pole.

All three angles of the triangle are right-angles, hence the sum adds up to $270^{\circ}$, much greater than $180^{\circ}$.

Global curvature versus local flatness

Note that the amount by which the sum of the angles of a triangle differs from $180^{\circ}$ is a measure of the curvature.

If we had chosen to draw a much smaller triangle in figure 8.6 above, then the difference in the sum of the angles from $180^{\circ}$, and hence the curvature, would have been much smaller too.

For example, if we were to draw a triangle on the surface of the earth, then it would have to be a pretty big triangle before we would measure any difference from $180^{\circ}$. So we can say that, although a sphere is curved globally, it is locally flat.

Figure 8.7 - Global curvature versus local flatness

Figure 8.7 - Global curvature versus local flatness

In figure 8.7, we have drawn several circles of increasing size.

Notice that, at this scale, as the size of the circle increases, the curvature decreases. That is, the arcs become flatter as their radii increase in length.

The dotted lines indicate (roughly) the size of arcs considered to be equally flat on each of the circles. Again, these arcs get longer and longer as their radii increase in length.

For an arbitrary curve, we could approximate the curve at a given point with an arcs of a circle with a certain radius, which we could call the radius of curvature of the curve at that point. Note that the larger the radius of curvature, the smaller the curvature itself.

We can use local curvature to set limits on the degree of global curvature of a given space.

That is, the larger a region of space is measured to be locally flat, the smaller the global curvature has to be, in comparison.

Saddle-like surfaces

Figure 8.8 - A surface with negative curvature

Figure 8.8 - A surface with negative curvature

Figure 8.8, courtesy of Wolfram|Alpha, is called a saddle-like surface and is an example of a two-dimensional surface with negative curvature.

To confirm that the curvature is negative, we could draw a triangle on the surface and then notice that the sum of the angles is less than $180^{\circ}$.

The remark about global versus local curvature also applies to negatively curved surfaces, in that they can appear locally flat.