Calibration of Minkowski Diagram: Explaining Invariance of S^2

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SUMMARY

The discussion focuses on the invariance of the spacetime interval, denoted as ##s^2##, in the context of Minkowski diagrams. It establishes that if ##s^2## is zero in one inertial frame, it remains zero in all frames, indicating that events are light-like separated. A positive ##s^2## indicates that events are time-like separated, meaning no signal can travel between them, while a negative ##s^2## indicates space-like separation, ensuring a consistent time order across all frames. The discussion also touches on determining the relative velocity of frames using displacement-time graphs.

PREREQUISITES
  • Understanding of Minkowski diagrams and spacetime concepts
  • Familiarity with the spacetime interval and its mathematical representation
  • Knowledge of the -+++ sign convention in special relativity
  • Basic principles of velocity determination from displacement-time graphs
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  • Study the mathematical formulation of the spacetime interval in special relativity
  • Learn about the implications of light-like, time-like, and space-like separations
  • Explore the concept of simultaneity in different inertial frames
  • Investigate the relationship between velocity and spacetime diagrams in special relativity
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Students and educators in physics, particularly those studying special relativity, as well as researchers interested in the geometric interpretation of spacetime events.

Pushoam
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min dia1.png
min dia 2.png

Can anyone please illustrate highlighted part?
Can anyone please explain me how invariance of s^2 is useful in characterization of events?
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res 2.png

I didn't understand the highlighted part i.e. how to determine velocity of S' relative to S?
 
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##s^2## is useful for characterising the separation between events. If it's zero in one frame it's zero in all, and the distance between the events must be equal to the distance light can travel in the time between them in all frames.

If it's positive in one frame (using the -+++ sign convention your book has, which is not universal) then it's positive in all, so not even light can get from one event to another. Note that the x difference cannot change sign in this case - because it cannot do so without passing through zero which must yield a non-positive ##s^2##.

Likewise a negative value is negative in all frames; this time it's the t difference between the events that cannot change sign. This means that all frames must agree on the time order of the events - which is good because influences traveling below lightspeed can get from one event to the other, and we don't like theories that let effect precede cause.

The velocity of the frame is easy. How do you get the velocity of an object from a displacement-time graph? What's the relationship between the time axis of S' and the graph of an object at rest in S'?
 
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