Acceleration

I was thinking last night about the acceleration of massive objects. Objects that move at maximum speed through space move at no speed through time, and stationary objects move at no speed through space and at maximum speed through time. Thus, a space-time graph can be drawn between these extremes... but it struck me that with zeroes and infinities on each end that neither extreme could exist in reality, only the spread in between. Einstein's laws prevent an object from being accelerated to maximum speed (the speed of light) but here I hypothesize that the reverse is also impossible and that no completely stationary object can exist either. An object must always be moving partly in space and partly in time, and if stationary in space or time cannot be moved within that domain.