While looking at the metrics of different spacetimes, i came across the "Ellis wormhole", with the following metric:
I note that the temporal term has a constant coefficient. The Wikipedia article mentions:
There being no gravity in force, an inertial observer (test particle) can sit forever at rest at any point in space, but if set in motion by some disturbance will follow a geodesic of an equatorial cross section at constant speed, as would also a photon. This phenomenon shows that in space-time the curvature of space has nothing to do with gravity (the 'curvature of time’, one could say).
So this metric would not result in any "gravitational effects".
Looking at the Schwarzschild metric:
Here we have a non-constant coeffcient for the first component. And this metric clearly has an attractive effect on particles, e.g. it's geodesics have the tendency towards $r\rightarrow0$.
Does that mean the gravitational effect comes primarily from a "curvature of time" and not from spatial curvature? I assume part of the answer has to do with the motion through time being dominant for all but the fastest particles?
Is the spatial curvature the primary cause of the visual distortion, e.g. the bending of light paths, in these metrics?
I'm getting the picture that temporal curvature primarily affects objects moving fast through time (static and slow objects), and spatial curvature primarily affects objects moving fast through space (photons). Is this a good picture or completely wrong?
If the spacetime around an "Ellis wormhole" is purely spatial, does that mean the faster i move (through space), the more i would feel the attraction and also second order effects like tidal forces?
Are there physical metrics, e.g. valid solutions for the EFE which only have temporal curvature but no spatial curvature? Would such an object behave like a source of gravity, without the gravitational lensing?
If such objects would be valid, would that mean you could pass them unharmed or even unnoticed at high speeds (moving fast through space), but would be ripped to pieces if you are moving slowly (moving fast through time)?