Does a space varying magnetic field produce an electric field

We have recently finished the chapter on Electromagnetic Induction and it was only in the last week that I finally got the hang of the part about induced electric field. My book in its intensely concise and simple language states:

A time varying magnetic field produces an induced electric field.

Now that we have started upon the chapter not-so-modern Physics and I am doing a bit of side reading I found out that space and time are basically one and the same thing/two sides of the same coin/(Insert any other metaphor you prefer)

So since time and space are sort of one thing shouldn't a space varying magnetic field also produce an electric field.

• "I found out that space and time are basically one and the same thing" - but this isn't true just as rulers and clocks are not basically one and the same thing. – Alfred Centauri Nov 6 '17 at 14:16
• A similar question would be: If the magnetic field source is moving and there is a stationary electric charge nearby, will the charge be accelerated? – Andrei Geanta Nov 6 '17 at 14:17
• space and time are not the same thing, space itself is a euclidean space, but when time is included it becomes a pseudoeuclidean four dimensional space en.wikipedia.org/wiki/Pseudo-Euclidean_space – anna v Nov 6 '17 at 14:18

What matters is the rate of change of the magnetic flux through a closed loop (or an open surface): $${\cal E}=-\frac{d}{dt}\int_C \vec B\cdot d\vec S$$ It doesn't matter how this change in flux is produced.
You can start, for instance, with a static magnetic field produced by a wire carrying a constant current, and drag a square frame at constant speed away from this wire, and the result will be a change in flux, and thus an EMF, and thus an associated electric field. In this case, the change in flux is produced by the change in the strength of $\vec B$ over the frame when the frame is moved, even if $\vec B$ is not time-varying.
Alternatively, you can have a fixed frame but a time-dependent $\vec B$ so that $\vec B\cdot d\vec S$ changes in time. You can also deform the frame in a constant magnetic field so the change in flux is not contained in the changing $d\vec S$. You can also change the angle between $\vec B$ and $d\vec S$ at some rate so that $\vec B\cdot d\vec S$ changes in time .