The following argument is presented from the Stanford Encyclopedia of Philosophy regarding the existence of perdurantism (temporal parts):
A third argument from STR to perdurantism does not rely on the claim that STR is out-and-out incompatible with endurantism. Instead, the claim is that STR highlights a phenomenon which can better be explained by perdurantists than by endurantists. To understand the phenomenon, imagine a three-dimensional cylindrical chocolate cake which you want to cut up into two-dimensional pieces (you've got a lot of friends). If you cut only at right angles to the length of the cake, then each piece will be circular. But if you cut at a different angle, then each piece will be oval. The cylindrical shape of the cake explains the shapes of the various pieces produced by cutting at various angles.
I understand this part of the argument completely. What I have trouble with is extending this example out into four dimensions:
Now imagine a four-dimensional object, cut into three-dimensional timeslices. In a non-relativistic world, there's only one way to do this, because there's only one way to divide up events into groups of simultaneous events: it's like cutting the cake at right-angles. But, by relativising simultaneity to reference frames, STR gives us lots of different ways to cut the four-dimensional object into three-dimensional slices, just as there are lots of different angles at which to cut the cake. The shape of the three-dimensional slices you get by cutting in different ways, from the perspective of different reference frames, can all be explained by the shape of the four-dimensional object you're cutting up.
My main question is with the general geometry of this 4-D cake. With the "relativised simultaneity" of reference frames, does that mean that each reference frame has a different "now?" Further, would that mean that each "three-dimensional slice" is a slice of the "cake" AND a slice of multiple "nows," thus showing temporal parts alongside spacial parts? Is this the best way to think about these slices? Thanks for your help.