Principle of Relativity, from a mathematical perspective

Click For Summary

Discussion Overview

The discussion revolves around the mathematical necessity of invariance in physical laws with respect to transformations, specifically in the context of Newton's Laws and Maxwell's equations. Participants explore whether laws of physics must exhibit invariance under some transformation and the implications of adding or removing terms from these equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that while invariance under transformations is not mathematically necessary, it is physically advantageous.
  • One participant mentions that laws of physics do not have to possess non-trivial symmetries, referencing a hypothetical Aristotelian world.
  • Another participant emphasizes the importance of seeking out symmetries in physical laws, suggesting that experimental evidence supports the existence of non-trivial symmetries.
  • A participant questions the relevance of invariance when modifying Maxwell's equations, seeking clarification on whether specific variations maintain symmetry.
  • One participant asserts that everyday observations, such as the invariance of shape during motion, provide a form of experimental evidence for invariance under transformations.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of invariance in physical laws, with no consensus reached on whether invariance is required or merely beneficial. The discussion remains unresolved regarding the implications of modifying Maxwell's equations.

Contextual Notes

Participants reference various transformation types and their implications, but the discussion does not resolve the mathematical conditions under which invariance may or may not hold.

ralqs
Messages
97
Reaction score
1
I didn't know where to post this question, but the general math forum seemed like the best idea.

Newton's Laws are invariant with respect to the Galilean transformations. Maxwell's equations are invariant with respect to the Lorentz transformations. My question is, is it necessary, mathematically, for a law of laws of physics to be invariant with respect to some transformation? Put another way, if we took Maxwell's equations and started adding and removing terms, will the resulting equations have to be invariant with respect to an appropriate transformation rule?
 
Mathematics news on Phys.org
It doesn't HAVE to be invariant under some transformation, but PHYSICALLY, it is better that it is invariant under some transformation.
 
ralqs said:
My question is, is it necessary, mathematically, for a law of laws of physics to be invariant with respect to some transformation? Put another way, if we took Maxwell's equations and started adding and removing terms, will the resulting equations have to be invariant with respect to an appropriate transformation rule?

Strictly speaking, there's always the trivial "identity transformation".

As dalcde says, I don't think that the laws of physics (and therefore mathematical encodings of them) has to necessarily have non-trivial symmetries... for example, physics in an Aristotelian-type world.

However, experiment has suggested that there are non-trivial symmetries in the physical world (so, e.g., we don't seem to live in an Aristotelian- or even Galilean-type world). So, it seems that it fruitful to seek out symmetries and express physical laws with mathematics which reflects those symmetries.

You might be interested in a lecture on symmetry by Feynman .
 
Last edited by a moderator:
ralqs said:
...Put another way, if we took Maxwell's equations and started adding and removing terms, will the resulting equations have to be invariant with respect to an appropriate transformation rule?

No. But why would this be interesting, or are you alluding to a particular variation of maxwell's equations and want to know if the variation still obeys some particular symmetry. If so, I would be interested in such variant.

Symmetry rules are made fragile to better ideas.
 
The fact is that when I walk from here to there my shape does not change. That is, immediately, an "invariance with respect to some transformation". So, yes, "experimental evidence" (in this case, the experiment is just observing what happens when something moves) says that motion is invarient with respect to some transformation.
 
Okay, thanks every one.
 

Similar threads

Undergrad Questions about the general principle of relativity
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Principle of relativity: active vs passive point of view
  • · Replies 29 ·
Replies
29
Views
3K
Undergrad Principle of relativity and Galileo's group
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Is the Principle of Relativity different for different theories?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad The wave equation interpretation of special relativity
  • · Replies 10 ·
Replies
10
Views
1K
Undergrad Noether's second theorem: two questions
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Need for Lorentz transformation in pre-relativity period
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Understanding General Relativity
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Apparent Violations of Principle of Relativity?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Special relativity vs Lorentz invariance
  • · Replies 29 ·
Replies
29
Views
4K