Show that a nonlinear transformation preseves velocity

[SevenHells]

Homework Statement

I have a particle moving with uniform velocity in a frame $S$, with coordinates $$ x^\mu , \mu=0,1,2,3. $$
I need to show that the particle also has uniform velocity in a frame $S'$, given by
$$x'^\mu=\dfrac{A_\nu^\mu x^\nu + b^\mu}{c_\nu x^\nu + d}, $$
with $A_\nu^\mu,b^\mu,c_\nu x^\nu,d$ constant.

Homework Equations

I don't think these are very relevant because they're not the transformations for the question but
$$\Delta x = \gamma(\Delta x' + v\Delta t')$$
$$\Delta t = \gamma(\Delta t' + v\Delta x'/c^2)$$
$$\Delta x' = \gamma(\Delta x - v\Delta t)$$
$$\Delta t' = \gamma(\Delta t - v\Delta x/c^2)$$

The Attempt at a Solution

I wrote the $S'$ coordinates out and using $x'^0=t'$,$x'^1=x'$,$x'^2=y'$,$x'^3=z'$, try to calculate the velocities but I don't think it's right. I'm not sure how to show a transformation preserves the particle velocity. Could anyone point me how to show this for the Lorentz transformations, and then I could try to do it for my transformations?

 

[BvU]

Oh, and for inline math we use $\#\#$ to start and end, not ##$##

 

[SevenHells]

Oh, and for inline math we use $\#\#$ to start and end, not ##$##

I've fixed that now, thanks. Did you have something typed before "Oh,"?

 

[BvU]

Ummm, no ...

 

[strangerep]

I have a particle moving with uniform velocity in a frame $S$, with coordinates $$ x^\mu , \mu=0,1,2,3. $$
I need to show that the particle also has uniform velocity in a frame $S'$, given by
$$x'^\mu=\dfrac{A_\nu^\mu x^\nu + b^\mu}{c_\nu x^\nu + d}, $$
with $A_\nu^\mu,b^\mu,c_\nu x^\nu,d$ constant.

"Uniform velocity" means zero acceleration. So you must show that if $$\frac{dv^i}{dt} \equiv \frac{d^2 x^i}{dt^2} = 0$$then
$$\frac{dv'^i}{dt'} \equiv\frac{d^2 x'^i}{dt'^2} = 0$$ (where $i=1,2,3$).

BTW, what is the context of this problem? It's actually a classic -- the fractional linear transformations are known to be the most general transformations which preserve inertial motion. :-)

Not sure how much of a hint I should give you, so I'll start with this:

Work with the differentials, i.e., find $dx^i$ and $dt$ separately, then take their quotient to find an expression for $v$. Take differential $dv$ similarly, and take its quotient with $dt$ to find the acceleration.

Further hints: use the chain rule to compute the various differentials.