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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.

  • There is no distinction between temporal and spatial parts in relativity, time and space separately have no physical meaning, only spacetime has. Spacelike slices can serve as pretend "nows" in a reference frame, but again, there is nothing physical (Lorentz-invariant) to them, it is merely a bookkeeping device for that reference frame's user. It means not that each reference frame has a different "now", but rather that none of them has any. There is only here-and-now in relativity. – Conifold Dec 3 at 1:16
  • @conifold So how would that argument support temporal parts? Is it essentially saying the objects have spacial parts, so events must have temporal parts? – N. Bar Dec 3 at 2:29
  • It rather supports that we should think in terms of spatiotemporal parts, if any, as they say at the end. But if one insists on "explaining" correlated spacelike slices one can adopt a Lorentz-like interpretation of relativity that specifies a "hidden" privileged frame with "canonical nows". – Conifold Dec 3 at 3:45
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The key to understanding the point being made here is the difference between spacetime in relativity theory and the sort of spacetime you might use for non-relativistic physics (sometimes called "Newton-Cartan spacetime").

In NC spacetime you have three space axes and one time. What all our clocks measure is progress along the time axis. A slice of the spacetime made perpendicular to the time axis represents the state of the universe at that time for everyone. A wordline is the path an observer (or any object) might make through this structure, wandering around in space as they go to work, walk the dog etc, but always inexorably proceeding along the time axis at the same rate as everyone and everything else.

Note that in this picture the length of a wordline has no particular physical meaning (only the projection of its length on the time or the space axes but independently). Note also that slices that aren't made perpendicular to the time axis (or perhaps parallel to it) have no obvious interpretation.

For these reasons this way of looking at spacetime was never used much in physics but it is the sort of mental picture someone like Augustine probably had when he described time from the point of view of God. Its main use today is to contrast it with the much more interesting, physically relevant but counter-intuitive "Minkowski" spacetime of relativistic physics.

In Minkowski spacetime we have three space and one time axis as before. But the time you measure on a clock is the length of your worldline, not your distance along the time axis. Moreover length is not measured the way you might expect. The length we're talking about here is given by the square root of the sums of the squares of the three spatial displacements minus the square of the temporal displacement. This way of measuring length is called the "Minkowski metric".

In this picture simultaneity is no longer as simple as slices perpendicular to the time axis, and is no longer absolute. The events that a particular observer will experience as simultaneous with a point on her worldline are those that can be reached by lines orthogonal to her worldline, where orthogonality is measured using the Minkowski metric. No space for a full discussion here but I will just mention that this way of measuring orthogonality means that as a wordline rotates away from the time axis, in other words as the observer's velocity increases, the plane of events orthogonal to that worldline do not remain at right angles to it but the angle gets tighter going to zero at the speed of light, at which point your now would be simultaneous with your own past and future.

The point your writer is getting at here I think is that slices through Minkowski spacetime don't have to be parallel to the time axis, but can tilt away from it up to the speed of light and will still represent the sets of simultaneous events perceived by observers at different velocities relative to each other.

A very good book I highly recommend about all this is Relativity and Geometry by Roberto Torretti.

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