How to intuitively think of translations and Galilean boosts commuting?

[binbagsss]

how to think of translations and Galilean boosts commuting intuitively

 

[Ibix]

How fast something is going and in what direction doesn't depend on where you are.

 

[ergospherical]

Depends on your definition of intuitive, but if a translation is T(X) = X + A and a boost is B(X) = X + Vt, then TB(X) = BT(X) = X + Vt + A because addition commutes.

 

[PeterDonis]

Moderator's note: Thread moved to the Classical Physics forum since that is the proper context for discussion of Galilean boosts.

 

[PeterDonis]

translations

Note that this has to mean spatial translations for your statement in the OP to be true. Galilean boosts and time translations do not commute.

 

[Vanadium 50]

One builds intuition by working through problems. There is no magic trick to it.

 

[haushofer]

Actually, they don't commute in the Bargmann algebra :P See e.g.

https://arxiv.org/abs/1011.1145

 

[meopemuk]

Imagine yourself in a car. You press accelerator pedal (this is boost), then you wait for some time (=time translation). Obviously, after these two transformations you'll find yourself far from the place you've started from.

In reverse order: You wait for some time (=time translation), then you step on accelerator (=boost). You have not moved.

Conclusion: time translations and boosts do not commute.
Eugene.

 

[weirdoguy]

press accelerator pedal (this is boost)

How?

 

[meopemuk]

By definition, boost is a transformation that changes velocity of reference frame.

When I press accelerator pedal, velocity of my car changes. Then the inertial reference frame associated with me and my car experiences a boost.

Eugene.

 

[haushofer]

By definition, boost is a transformation that changes velocity of reference frame.

When I press accelerator pedal, velocity of my car changes. Then the inertial reference frame associated with me and my car experiences a boost.

Eugene.

During the acceleration itself you're not experiencing what physicists would call a 'boost'. Inertial observers are connected by 'boosts'. During acceleration you're not an inertial observer.

 

[weirdoguy]

By definition, boost is a transformation that changes velocity of reference frame.

No, boost changes inertial reference frame to a different inertial one.

 

[meopemuk]

No, boost changes inertial reference frame to a different inertial one.

True, during acceleration I am not an inertial observer. But after I released the accelerator pedal, I move with a constant speed and I may regard myself as an inertial observer, which is boosted with respect to my previous state.

I agree that this is not a perfect analogy for boost, but not a bad one if we disregard the (short) time during which I am stepping on the gas pedal.

Perhaps, a better analogy would be to jump (mentally) to another car that passes nearby.

Eugene.

 

[binbagsss]

Actually, they don't commute in the Bargmann algebra :P See e.g.

https://arxiv.org/abs/1011.1145

From the abstract I see the Bargmann algebra is assoicated with the centrally extended Galilean algebra. Could you give, very briefly, implications of what this means compared to the question I asked which was on, what I assume can be referred to as the unextended Galilean group. thanks

 

[binbagsss]

Note that this has to mean spatial translations for your statement in the OP to be true. Galilean boosts and time translations do not commute.

ofc.

 

[PeroK]

True, during acceleration I am not an inertial observer. But after I released the accelerator pedal, I move with a constant speed and I may regard myself as an inertial observer, which is boosted with respect to my previous state.

I agree that this is not a perfect analogy for boost, but not a bad one if we disregard the (short) time during which I am stepping on the gas pedal.

Perhaps, a better analogy would be to jump (mentally) to another car that passes nearby.

Eugene.

A reference frame is not an object like a car that has a single trajectory. A boost is not a physical process like acceleration of an object. A boost is a mapping or transformation from one reference frame to another.