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I'm confused about light cones and I've driven a crazy scenario to try to represent it.

Let's say I can move through my past light cone to the exact time of my birth. If I kept on moving back could I move to a time before I was born? Therefore my question is, when they show light cones in space-time diagrams do they apply to our own past our the past of the time throughout the universe. Does this mean that in the -t direction, time for us only goes as far back as our birth. Sorry if this sounds too confusing.

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Past and future light-cones really only apply to a single event (location in space and time, call it $P$): the future light cone is the set of all events that can be causally affected by $P$; the past light cone is the set of all events that could have causally affected $P$. If you examine the event of your birth (call it $B$), it will have its own past and future light cones (the future light cone of $B$ overlaps with the past light cone of $P$, and all of $P$'s future light cone is contained in the future light cone of $B$).

While you as a organism with 23 pairs of chromosomes only existed since the event of conception, that event also has an expansive past light cone involving all the molecules and atoms that made up that cell, so no, past light-cones go "all the way" back to the origin of the universe.

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