I Derivative operators in Galilean transformations

  • Thread starter TomServo
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Summary
I'm confused about how the derivative with respect to time transforms under a Galilean transformation.
I'm studying how derivatives and partial derivatives transform under a Galilean transformation.

On this page:

http://www.physics.princeton.edu/~mcdonald/examples/wave_velocity.pdf

Equation (16) relies on ##\frac{\partial t'}{\partial x}=0## but ##\frac{\partial x'}{\partial t}=-v##

But this seems like a contradiction to me. If you swap primed/unprimed you get ##\frac{\partial t}{\partial x'}=0## but ##\frac{\partial x}{\partial t'}=v##, in which case you have ##x=vt+x_0## and ##t=t'=\frac{x-x_0}{v}##. Thus ##\frac{dt'}{dx}=\frac{\partial t'}{\partial x}=\frac{1}{v}##, in violation of Eq. (16).

So where have I gone wrong? Thanks.
 

Orodruin

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##t## is not given by ##(x-x_0)/v##, that can be true only for a very particular world-line and that is not what you are considering, you are considering the transformation of coordinates. ##\partial x/\partial t’ = v## is a partial differential, not a total differential.
 

vanhees71

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The Galilei transformation reads
$$t'=t, \quad \vec{x}'=\vec{x}-\vec{v} t.$$
You consider ##t## and ##\vec{x}## as independent variables when it comes to (non-relativistic) field equations. Thus you have
$$\frac{\partial t'}{\partial t}=1, \quad \vec{\nabla} t'=0, \quad \partial_t \vec{x}'=-\vec{v}, \quad \vec{\nabla} \otimes \vec{x}=\hat{1}.$$
 
##t## is not given by ##(x-x_0)/v##, that can be true only for a very particular world-line and that is not what you are considering, you are considering the transformation of coordinates. ##\partial x/\partial t’ = v## is a partial differential, not a total differential.
Could you further explain what you mean here? I know what worldlines are, but it seems to me (just algebraically) that the ##t=\frac{x-x’}{v}## relation holds in general. After all, I’m just solving the transformation equation for t. I know this is wrong, but I’m trying to understand why the algebra leads me astray (or seems to).
 
And where I wrote ##x_0## originally I meant ##x’##.
 

Orodruin

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Then you are treating x’ as a constant when differentiating with respect to x. That is incorrect. What is being kept constant when you take the partial with respect to x is t, not x’.
 

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