Derive Lorentz transformations in perturbation theory

[Hill]

Homework Statement

See the image below.

Relevant Equations

Lorentz transformations

I've arrived to an expected answer, but I am not sure at all that the process was what the problem statement wants.
First, I considered $0=(t+\delta t)^2-(x+vt)^2-(t^2-x^2) \approx 2t \delta t - 2xvt - v^2t^2$. Ignoring $O(v^2)$ gives $\delta t=vx$, i.e., $t \rightarrow t+vx$.
Keeping $O(v^2)$ gives $t \rightarrow t+vx+\frac 1 2 v^2t$, which is the correct expansion of the full transformation to the second order.
Now, taking $x \rightarrow x+ \delta x, t \rightarrow t+vx$ gives by the similar calculation, $x \rightarrow x+vt+\frac 1 2 v^2x$.

Is it what the exercise means?

 

[Costco Physicist]

What you do is expand both terms in the dementor as a Taylor series. Why this can be done is that it needs to be assumed that v/c << 1 so that the rest of the terms in the Taylor series will die out as (v/c)^n when n>1. For the units you are working with, v<<c. The application makes more sense when working with units that don't have v as a percentage of the speed of light. I'll let you do the same one for a time.

Then plug in your x' and t' terms as dx' and dt' into your wolrd line to prove that 1+1 space-time inference still holds, even when the approximation worked out is applied.

Then do the same for second-order approximation

 

[Hill]

What you do is expand both terms in the dementor as a Taylor series

Yes, that is how I knew that the $\frac 1 2 v^2$ terms are correct.

 

[Costco Physicist]

Yes, that is how I knew that the $\frac 1 2 v^2$ terms are correct.

I added more.

 

[Hill]

I added more.

I see.
But you start with the full transformation, while the exercise wants to derive the approximation of the correct transformation using perturbation theory, and only to check the agreement with the full transformation to the second order.

 

[Costco Physicist]

I see.
But you start with the full transformation, while the exercise wants to derive the approximation of the correct transformation using perturbation theory, and only to check the agreement with the full transformation to the second order.

Oh ****. Your right. I'm working on it. I didn't read the whole problem

 

[Costco Physicist]

Oh ****. Your right. I'm working on it. I didn't read the whole problem

Try plugging in t+dt into vt' for t+dt=t'

 

[Hill]

Try plugging in t+dt into vt' for t+dt=t'

Isn't it what I've done in the OP?

 

[Costco Physicist]

Isn't it what I've done in the OP?

Looks like you did.

Isn't it what I've done in the OP?

Looks like you did it. You can get your new second-order terms because have your new terms from your first-order approximation,

 

[strangerep]

[...] expand both terms in the dementor as a Taylor series

 

[Costco Physicist]

?