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Four-dimensionalism holds that objects are four-dimensional entities extended in time as well as space, with temporal parts analogous to spatial parts. A person at age 10 and at age 20 are temporal parts of a single 4D object; this view dissolves worries about how objects change and maintain identity through time by denying that any part of the 4D object persists unchanged.
From your prerequisites on the philosophy of time and persistence, you know the central puzzle of change: how can an object remain the same thing through time while undergoing genuine alteration? A leaf is green in summer and brown in autumn — if it is one and the same leaf, how can it have contradictory properties? From your study of A-theory versus B-theory, you know that the B-theorist treats time as a dimension analogous to space, with all moments equally real. Four-dimensionalism is the metaphysics of persistence that fits naturally with the B-theoretic picture.
The core move is the introduction of temporal parts. Just as a physical object can have spatial parts — the left half and the right half of a road are distinct spatial parts of one road — four-dimensionalism claims that objects have temporal parts: distinct stages that exist at different times. You have a Monday-stage, a Tuesday-stage, a ten-year-old-stage, a thirty-year-old-stage. These stages are not identical to one another — they are different parts of the whole four-dimensional object that is you. The complete you is the aggregate of all your temporal stages, stretched across your entire lifetime like a spacetime worm.
This dissolves the problem of change elegantly. The leaf is green in summer and brown in autumn — but there is no contradiction, because the green-property is had by the summer-stage and the brown-property is had by the autumn-stage. Different parts of the leaf have different properties, just as the left half of a road can be smooth and the right half bumpy. On the rival three-dimensionalist view (endurantism), the same whole leaf exists at both times and somehow has incompatible properties; the four-dimensionalist avoids this by saying only a temporal part of the leaf exists at any given time.
Four-dimensionalism also handles personal identity in a distinctive way. The question "what makes you at 30 the same person as you at 10?" gets the answer: they are temporal parts of a single four-dimensional person-worm, connected by psychological and physical continuity chains. There is no separate fact about identity over and above this overlapping chain of connected stages. This dissolves certain puzzles about fission (if a person's brain is split, which resulting person is the original?) by allowing that the original person-worm simply branches into two future-directed continuers.
The main objection to four-dimensionalism is that temporal parts feel metaphysically extravagant and phenomenologically odd — you don't experience yourself as a spacetime worm but as a persisting whole. The three-dimensionalist insists that objects are wholly present at each moment they exist, not spread across time like spatial objects. The debate turns on which metaphysical picture better fits our intuitions about identity, change, and what we mean when we say the same object exists before and after a change — a question your study of persistence and the A-B theory distinction has prepared you to engage directly.
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