Some individuals, myself included, like to envision the fundamental geometry of 4D spacetime as including time as an entity on par with the other three spatial directions. In such a framework, a differential position vector \vec{ds} in spacetime between two neighboring events at t,x,y,z and t +dt, x +dx, y +dy, z +dz can be expressed mathematically in component form by:
\vec{ds}=\vec{i_t}cdt+\vec{i_x}dx+\vec{i_y}dy+\vec{i_z}dz
where the i's are unit vectors in the spatial directions.
This is analogous to how we express a vector joining two neighboring points in 3D space. But where is this mysterious 4th time direction that is implied here? Why can't we see into that direction? In any inertial frame of reference, we have complete access to only a specific 3D cut out of 4D hyperspace (and can potentially see infinitely far into any of these three dimensions). The 4th dimension (time direction) is oriented perpendicular to our 3D cut, and, because of our inherent physical limitations as 3D beings, we have no direct access or vision into this 4th spatial dimension.
By analogy, we are like 2 dimensional beings trapped within a flat plane that is immersed in a 3D space. We have no access to the 3rd dimension, except for the 2D cross section that we currently occupy. This 2D cross section may not be stationary in 3D space; it can be moving forward (unbeknownst to us) into the 3rd dimension. If so, as time progresses, we would be sweeping out all of 3D space, and would ultimately be able to sample all of 3D space with our planar cross section. However, at anyone instant of time, we would only have access to a single planar slice out of 3D space.
This is analogous to what we are experiencing in 4D hyperspace. We are 3D beings trapped within a specific 3D slice out of 4D spacetime. This 3D slice is unique to the particular inertial reference frame we currently occupy (i.e., our rest frame). We have no access or vision into our own 4th dimension, except for this 3D cut. The cut is not stationary; it is moving forward (unbeknownst to us) into our 4th spatial dimension (at the speed of light, see below). As time (measured by the synchronized clocks in our 3D reference frame) progresses, we are sweeping out all of 4D spacetime, and will ultimately be able to sample all of 4D spacetime with our moving 3D cut. However, at anyone instant of time, we only have access to a single 3D cut out of 4D spacetime (a 3D panoramic snapshot). Finally, different inertial reference frames in relative motion possesses different 3D cuts, and different time directions perpendicular to the 3D cuts.
If two events occur at the same spatial location within our rest frame of reference, then, according to the equation above, the differential position vector in 4D spacetime between these two events is given by:
\vec{ds}=\vec{i_t}cdt
This implies that, although, as reckoned from our inertial frame of reference, we believe we are at rest, we actually are not at rest relative to 4D spacetime. According to the present interpretation of 4D spacetime geometry, we are actually covering ground in our own time direction with a velocity of \frac{\vec{ds}}{dt}=c\vec{i_t}; clocks in our rest frame are not only clocks, they are also odometers for the distance we cover in spacetime.
Of course, there is a significant geometric difference between 4D spacetime and a 4D Euclidean space. In a 4D Euclidean space, if we dotted the vector \vec{ds} with itself, we would obtain:
(ds)^2=(cdt)^2+(dx)^2+(dy)^2+(dz)^2
In 4D (non-Euclidean) spacetime, we have:
(ds)^2=-(cdt)^2+(dx)^2+(dy)^2+(dz)^2
This implies that in the present geometric interpretation of 4D spacetime, the dot product of the unit vector in the time direction with itself is -1, rather than + 1. This is the key geometric difference between 4D Euclidean space and 4D (non-Euclidean spacetime).
