Where does the line element of Minkowiski space come from?

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Discussion Overview

The discussion revolves around the derivation of the line element of Minkowski space, specifically the expression ds² = -cdt² + dx² + dy² + dz². Participants explore its origins, implications, and connections to other mathematical concepts, including the Pythagorean Theorem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants inquire about the derivation of the Minkowski line element, questioning how it is formulated.
  • One participant suggests that if the line element were expressed as ds² = +cdt² + dx² + dy² + dz², it would resemble the Pythagorean Theorem, indicating a contrast between Minkowski and Euclidean spaces.
  • Another participant asserts that the Minkowski line element is not derived in a traditional sense but is instead arrived at through induction, possibly as an approximation to more general spacetimes.
  • Conversely, some argue that the Minkowski metric can be derived from the assumption of Lorentz transformations, positing that it is the only metric invariant under such transformations.
  • Additionally, it is mentioned that assuming the Minkowski metric allows for the derivation of the Lorentz transform.

Areas of Agreement / Disagreement

Participants express differing views on whether the Minkowski line element can be derived or is merely an inductive result. There is no consensus on the nature of its derivation, with multiple competing perspectives presented.

Contextual Notes

The discussion highlights the dependence on assumptions regarding the Lorentz transform and the nature of spacetime metrics. The implications of these assumptions on the derivation process remain unresolved.

Ahmed Atef
How is it derived?
ds^2 =-cdt^2+dx^2+dy^2+dz^2
 
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Had this been ds^2 =+cdt^2+dx^2+dy^2+dz^2 it would've been the Pythagorean Theorem.
 
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puzzled fish said:
Had this been ds^2 =+cdt^2+dx^2+dy^2+dz^2 it would've been the Pythagorean Theorem.
If it was ecludian space
 
Ahmed Atef said:
How is it derived?
ds^2 =-cdt^2+dx^2+dy^2+dz^2

I would say that it is not derived (except maybe as an approximation to more general spacetimes), i.e., it is arrived at by induction, not by deduction.
 
Well, it can derived depending on what your assumptions are. If you assume the Lorentz transform, then the Minkowski line element follows as the only possible metric that is invariant under Lorentz transforms. Similarly, if you assume the Minkowski metric, you can derive the Lorentz transform.
 

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