I Variant and Invariant Physical Quantities....

[fog37]

TL;DR Summary

Variant and Invariant quantities

Hello,

In non-relativistic physics (where things move slower than the speed of light), the following physical quantities are invariant and variant (or relative) i.e. vary in value depending on the chosen frame of reference:

Variant quantities: time $t$, velocity $v$, momentum $p$, position $r(x,y,z)$, energy $E$, kinetic energy $KE$, temperature $T$, pressure $P$, etc.

Invariant quantities: mass $m$, time intervals $\Delta T$, distances or displacements $\Delta r$, potential energy $U$ since it depends on $\Delta r$...

In relativistic mechanics, the listed invariant quantities also vary because time and space intervals are also depend on the selected reference frame. Is that correct?

Are there any other important variant or invariant quantity that should be included?

Thank you!

 

[etotheipi]

Variant quantities: ...temperature T, pressure P, etc.

I would have said, that any scalar field like temperature or pressure is by definition frame invariant, since those fields must satisfy $\phi(x) = \phi'(x')$. Relabelling the space does not change the temperature anywhere, for instance!

 

[sysprog]

@fog37 If I order a pizza in Chicago and the pizzeria says I'll get it in one hour, and then I fly to Paris, and the delivery guy takes the next flight, and I get the pizza an hour after my flight lands, then that guy's getting a $2.5k tip $-$ maybe that's a weak attempt at humor (it is an attempt at humor), but I don't get what you mean by "vary in value depending on the chosen frame of reference" $-$ it seems to me that no they don't $-$ as @etotheipi gently pointed out (using a different way of saying it), an hour is still an hour no matter which 'time zone' you're in.

 

[etotheipi]

but I don't get what you mean by "vary in value depending on the chosen frame of reference" − it seems to me that no they don't

Various quantities certainly do change depending on your frame of reference! The position vector of a particle depends on the origin of your frame, the velocity of that particle is frame-dependent, the time coordinate of an event varies under temporal translations, and likewise kinetic energy, total energy, and many other quantities differ between frames!

I was only objecting to pressure and temperature, because these are examples of scalar fields which are (by definition) frame invariant.

an hour is still an hour no matter which 'time zone' you're in.

In Galilean spacetime time intervals are indeed invariant. But the absolute time assigned to an event, is only determined up to an additive constant, so is coordinate dependent.

 

[sysprog]

Various quantities certainly do change depending on your frame of reference! The position vector of a particle depends on the origin of your frame, the velocity of that particle is frame-dependent, the time coordinate of an event transforms via time-translations, and likewise kinetic energy, total energy, and many other quantities differ between frames!

I was only objecting to pressure and temperature, because these are examples of scalar fields which are (by definition) frame invariant.

The OP said "non-relativistic" $-$ is an hour longer or shorter in Chicago or London than in Paris? $-$ I understand that for the photons that are presenting the clock time, the time doesn't change, but for the clock it does, at certifiably the same rate, regardless of which town we observe it in.

 

[etotheipi]

Please re-read the end of my post #4. I do not want to involve myself in an argument about something as self-explanatory as this; it's got nothing to do with special relativity or photons or Chicago or whatever.

 

[sysprog]

Please re-read the end of my post #4. I do not want to involve myself in an argument about something as self-explanatory as this; it's got nothing to do with special relativity or photons or Chicago or whatever.

ok, but you added to that post after I read it $-$ I think that I was making a very limited non-SR claim $\dots$

 

[suremarc]

In Galilean spacetime time intervals are indeed invariant. But the absolute time assigned to an event, is only determined up to an additive constant, so is coordinate dependent.

Isn’t this confusing the map and the territory? You could define an absolute time T(t) which satisfies T’(t’)=T(t). While there is not a unique choice of T, any such choice of T is frame invariant.

 

[italicus]

There is another important quantity, which is invariant in SR : the 4-interval between two events :
$$ (c\Delta t)^{2} - \Delta x^{2} = (c\Delta t')^{2} - \Delta x'^{2} = ...$$ (similar for other I.O. )
I’m considering a boost in the x direction only.

Given a particle of invariant mass $m$ , the magnitude of its 4-impulse :
$$ P = (\gamma m c, \gamma m v) $$
is $mc$ , which is an invariant too, doesn’t depend on the inertial frame where it is observed.

 

[etotheipi]

Isn’t this confusing the map and the territory? You could define an absolute time T(t) which satisfies T’(t’)=T(t). While there is not a unique choice of T, any such choice of T is frame invariant.

Hmm, I think maybe you are right and I was wrong, but isn't that an unnecessary layer of abstraction? I was just referring to the fact that time-translations are a parameter of the Galilean transformation.

The coordinates of some event $\mathcal{P}$ with respect to a basis of Galilean spacetime are indeed themselves scalars, but I think it seems obvious to say that this tuple of numbers $[\mathbf{x}_{\mathcal{P}}]_{\mathcal{F}} = (x,y,z,t)$ describing an event is dependent on the choice of the origin and basis, so are 'frame dependent'.

 

[fog37]

Thanks for the comments.

The way I see it is that a reference frame $F_1$ has an origin $O$ and possibly moves (either at constant speed or in accelerated motion) relative to another reference frame $F_2$.

Nonrelativistically, for example, the position $r(t)$, speed $v(t)$, momentum $p(t)$, acceleration $a(t)$ of a particle will have different numerical values at each different time instant $t$ in the two different reference frames. Time intervals $\Delta t= t_2-t_1$ and position differences $\Delta r= r_2-r_1$ are instead the same in both frames.

If the frames are inertial, then the acceleration $a(t)$ will be the same in both frames...

Kinetic energy $KE$ and its variations $\Delta KE$ both vary when calculated from different reference frames (inertial or noninertial).

This is the type of variance/invariance I am referring to. I assume we use the same coordinate system (ex. Cartesian) in the different reference frames.