The Invariance Principle: Understanding the Essence of Special Relativity

[Master J]

I am just beginning graduate self-study of Special and General Relativity, so forgive me if my question seems niave.

I have found the beautiful line " The essence of Special Relativity is that the laws of physics are Poincaré invariant" - Modern Mathematical Physics, Szekeres.

The space time interval s is invariant under Poincaré transformations. So then, s must appear everywhere in physics forumlae, right? Is that true? I haven't come across any as of yet, but that's what I get from this. And what kinds of formulae does it appear in? What are some good examples?

 

[mfb]

So then, s must appear everywhere in physics forumlae, right?

Why do you expect that?
If s appears somewhere, it stays constant if you go to a different coordinate system. This does not mean that every formula has to have an s inside.
Some formulas can be frame-dependent (you have to change them if you change the reference frame), and some formulas simply do not use spacetime intervals at all.

 

[Naty1]

when getting started, it's rather difficult to draw general conclusions from statements as you read and learn what they mean. In fact, interpretating explanations often remains so.

Another reason not to expect 's' all over the place is that Poincare invariance applies to Minkowski spacetime...flat spacetime where gravitational curvature is negligible.

 

[cosmic dust]

"s" is just an invariant 1-D measure: it is the invariant length of a curve (which is a 1-D object in 4-D spacetime). There are many invariant measures in SR (such as 4-volume d⁴x=dtdxdydz, which is invariant under proper orthochronous tranfosrmations). "s" and other invariant measures don't have to appear in every formula of physics, that's not the meaning of Poincare invariance. The meaning is that the laws of physics should be covariant, that is that they have the same form in every poincare-transformed coordinate system.

 

[DrGreg]

If an interval ds is timelike then |ds|=c|dτ| where τ is "proper time". You'll find proper time mentioned quite a lot.

 

[Chestermiller]

I am just beginning graduate self-study of Special and General Relativity, so forgive me if my question seems niave.

I have found the beautiful line " The essence of Special Relativity is that the laws of physics are Poincaré invariant" - Modern Mathematical Physics, Szekeres.

The space time interval s is invariant under Poincaré transformations. So then, s must appear everywhere in physics forumlae, right? Is that true? I haven't come across any as of yet, but that's what I get from this. And what kinds of formulae does it appear in? What are some good examples?

If s represents a 4D position vector drawn from an arbitrary origin in flat space-time to a particle, the derivative of s with respect to proper time is the 4 velocity of the particle, and the second derivative of s with respect to proper time is the 4 acceleration of the particle. The 4 acceleration of the particle is pretty important in the 4D relativistic version of Newton's second law.

 

[Meir Achuz]

The key to Szekeres's quote is in the 'Mathematical' title of his book.
As a (mathematical) physicist, I would say that the essence is Galileo's definition of 'invariance' in his own words (rather than his equations):
"Any two observers moving at constant speed and direction with respect
to one another will obtain the same results for all [mechanical]
experiments. The laws of physics are the same in a uniformly moving
room as they are in a room at rest.", just extended to
remove the word 'mechanical'. Then it would include electromagnetism, about which Galileo knew very little.