The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance.

To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dx^2 $$

Suppose you want to measure the distance between two spacelike separated points $(t, x_0, y_0, z_0)$ and $(t, x_1, y_1, z_1)$ - the two points are simultaneous in our coordinates so $t_0 = t_1$. To measure the distance you draw a straight line between the points and integrate $ds$ along it. In flat space the integral simply gives:

$$ s^2 = (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2 $$

which you should recognise because it's just Pythagoras' theorem in 3D.

So if you're trying to measure the size of a compact dimension start by setting up your local coordinates then just trace a path parallel to the dimension you want to measure. If the dimension is compact the path will return to your original spatial coordinates, and you measure it's length simply by integrating $ds$ along it.

The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance.

To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just the following:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$

Suppose you want to measure the distance between two spacelike separated points $(t, x_0, y_0, z_0)$ and $(t, x_1, y_1, z_1)$ - the two points are simultaneous in our coordinates so $t_0 = t_1$. To measure the distance you draw a straight line between the points and integrate $ds$ along it. In flat space the integral simply gives:

$$ s^2 = (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2 $$

which you should recognise because it's just Pythagoras' theorem in 3D.

So if you're trying to measure the size of a compact dimension start by setting up your local coordinates then just trace a path parallel to the dimension you want to measure. If the dimension is compact the path will return to your original spatial coordinates, and you measure it's length simply by integrating $ds$ along it.

The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance.

To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dx^2 $$

Suppose you want to measure the distance between two spacelike separated points $(t, x_0, y_0, z_0)$ and $(t, x_1, y_1, z_1)$ - the two points are simultaneous in our coordinates so $t_0 = t_1$. To measure the distance you draw a straight line between the points and integrate $ds$ along it. In flat space the integral simply gives:

$$ s^2 = (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2 $$

which you should recognise because it's just Pythagoras' theorem in 3D.

So if you're trying to measure the size of a compact dimension you just draw a straight line that takes you from your starting point back to your starting point then you integrate $ds$ along it. In flat space a straight line is what you intuitively think it is. If the extra dimensions are curved then the straight line means a geodesic i.e. the path followed by a freely falling object. If you stand on your starting point and throw a stone so it goes round the compact dimension and hits you on the back of the head the path followed by the stone is a geodesic.

The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance.

To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dx^2 $$

Suppose you want to measure the distance between two spacelike separated points $(t, x_0, y_0, z_0)$ and $(t, x_1, y_1, z_1)$ - the two points are simultaneous in our coordinates so $t_0 = t_1$. To measure the distance you draw a straight line between the points and integrate $ds$ along it. In flat space the integral simply gives:

$$ s^2 = (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2 $$

which you should recognise because it's just Pythagoras' theorem in 3D.

So if you're trying to measure the size of a compact dimension start by setting up your local coordinates then just trace a path parallel to the dimension you want to measure. If the dimension is compact the path will return to your original spatial coordinates, and you measure it's length simply by integrating $ds$ along it.