Motivation for the usage of 4-vectors in special relativity

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Discussion Overview

The discussion centers on the motivation for using 4-vectors in special relativity, exploring their theoretical foundations, implications for spacetime, and potential applications in both relativistic and non-relativistic physics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants explain that 4-vectors arise from the need to maintain Lorentz invariance in physical equations, as time and space are interrelated in special relativity.
  • Others argue that the spacetime interval is the invariant quantity in special relativity, contrasting it with the traditional Pythagorean line element in classical mechanics.
  • A participant suggests that the symmetry analysis leading to Lorentz transformations can be formalized using Minkowski's pseudo-scalar product, highlighting the mathematical elegance of this approach.
  • Some contributions point out that the 4-D perspective can also be useful in non-relativistic physics, providing examples such as the formulation of charge density and current density as a 4-vector.
  • A later reply questions whether one assumes a 4D spacetime from the outset when discussing the pseudo-scalar product and its invariance under Lorentz transformations.
  • Participants discuss the nature of time in Newtonian mechanics, with some suggesting it is treated as a parameter, while others argue it can also be viewed as a coordinate that remains unchanged under Galilean transformations.

Areas of Agreement / Disagreement

While some participants express agreement with the initial explanation of 4-vectors, there remains a lack of consensus on the interpretation of time in Newtonian mechanics and the foundational assumptions regarding spacetime in the context of Lorentz transformations.

Contextual Notes

There are unresolved questions regarding the assumptions made about spacetime and the implications of different transformation laws in Newtonian versus relativistic frameworks.

  • #31
PeterDonis said:
In Newtonian physics, yes.

Would it be correct to say that in Newtonian physics, since time (and simultaneity) is (are) absolute the approach of considering a 3 dimensional space at each instant in time is well-defined, since it is observer independent, hence we can consider 3 dimensional space parametrised by time, which governs how the 3D space evolves. In special relativity, both time and simultaneity are relative and so there is no well-defined way of considering 3-dimensional space at each instant in time, since this can not be achieved in an observer independent manner. Since there is no observer independent way to separate time and space coordinates we must consider them as coordinates of a single 4-dimensional spacetime.
 
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  • #32
Frank Castle said:
Would it be correct to say...

All of this looks ok to me.
 
  • #33
Frank Castle said:
Would it be correct to say that in Newtonian physics, since time (and simultaneity) is (are) absolute the approach of considering a 3 dimensional space at each instant in time is well-defined, since it is observer independent, hence we can consider 3 dimensional space parametrised by time, which governs how the 3D space evolves. In special relativity, both time and simultaneity are relative and so there is no well-defined way of considering 3-dimensional space at each instant in time, since this can not be achieved in an observer independent manner. Since there is no observer independent way to separate time and space coordinates we must consider them as coordinates of a single 4-dimensional spacetime.

Hi Frank. 3D space being relative is what Einstein referred to in his quotes I posted in that other thread.
 

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