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HSS - "Hyper Surface of Simultaneity"

Listening to different sources online I understood that HSS for a observer represents the points that are at same moment of time.

Consider a 1d world. Below X axis represents 1d space and y axis is time. Now observer is at rest. The instants which are visible to the observer at that moment is given by past light cone drawn.

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Hence the points on light cone are at same moment according to observer hence it should represent the HSS for the observer.

But we say a horizontal line represents the HSS. So what is wrong in my understanding of definition of HSS.

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No, the further back you go along the slope of the light cone, the further back you are going in time. If you take a horizontal slice through the cone, you end up with a circle of events each of which is the same age from your perspective (ie the same distance away in time). So, for example, suppose you go an hour back in time and slice through the cone horizontally- all the points on that slice happened an hour ago, so they were all simultaneous. If you go back further by the same amount and take another horizontal slice, all the points on that slice happened two hours ago and thus were simultaneous with each other, but they were not simultaneous with the events that occurred an hour ago. So the point is that horizontal slices give you sets of events that are simultaneous with each other in your reference frame.

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