Extra Dimensions

Background of Terms

Definitions

A 'dimension' is a direction or axis in which we measure components of length or interval in that direction.

A 'space' is a manifold which is described by some number of dimensions. That number is called the 'dimensionality'. Our daily physical space has a dimensionality of 3.

General covariance

Einstein points out that, though we may use coordinates for each dimension, our theories must never depend on exactly which directions we choose for each dimensional coordinate axis.

In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. While this concept is exhibited by general relativity, which describes the dynamics of spacetime, one should not expect it to hold in less fundamental theories. (Wikipedia)

Extra dimensions vs Extra spaces

We have here to distinguish theories which give an extra dimension to physical space, from those which give a separate extra space with its own set of dimensions and dimensionality.

You can move to 'extra dimensions' just by rotating where you are, or freely choose different coordinate axes for the local dimensionality.
You can not move to extra spaces by traveling continuously.

Einstein's Special Relativity describes how to have 4 dimensions for 3-space and 1-time together in 'spacetime'.
Rotating in the same place towards the time dimension then corresponds to changing your travel velocity.