I understand that if an object is moving at relativistic speeds in space, it will noticeably contract in the direction that it was traveling, but I'm not sure "towards" which point the contraction occurs (i.e. which point is fixed). e.g. Suppose you had the following object:

<------------------------->

and you observed it to accelerate to relativistic speeds. Then, in comparison to its position if length contraction did not occur (but say stuff like time dilation and whatnot did, not sure what this would affect, let me know if this assumption is wherein lies the issue), then these all seem to be possible candidates for the contracted object:

<---------->

             <---------->

       <---------->

My issue with this actually goes a bit further: whatever the answer to this question is, I claim something annoying happens when you split the object into components and then measure contraction on each part: you'll get disjoint parts, which are seemingly connected by... nothing? For instance,

<----------><----------><---------->

becomes (for example):

  <--->        <--->        <--->

One argument I could see being made is that the space within also contracts. But, if you had a Lorenz factor of say 1/2, the space between can only be contracted to 1/2 the amount; the components will still remain separated by a non-zero amount of space. But this leads to two (in fact, infinite) contradictory views of what a length contracted object would become: either one contiguous segment or multiple disjoint segments.