# direct and inverse Lorentz transformation

by bernhard.rothenstein
Tags: inverse, lorentz, transformation
 P: 997 When we state that: Knowing the direct Lorentz transformation we obtain the inverse one by changing the corresponding primed physical quantities with unpriomed ones and changing the sign of the relative velocity, we are based on what? Thanks
 Sci Advisor P: 1,883 On our knowledge that v'=-v. Everything else is just relabeling.
 PF Patron Sci Advisor Emeritus P: 16,094 If you want to verify that a Lorentz boost with velocity v and a Lorentz boost with velocity -v are inverse operations, there is always the mindless approach: apply both and see if you get the identity transformation.
P: 997

## direct and inverse Lorentz transformation

 Quote by Hurkyl If you want to verify that a Lorentz boost with velocity v and a Lorentz boost with velocity -v are inverse operations, there is always the mindless approach: apply both and see if you get the identity transformation.
English is not my first language, but I think, after looking for the meaning of "mindless" , that people who are ready to share knowledge will never use such a word!
Soft words and hard arguments
Mentor
P: 27,565
 Quote by bernhard.rothenstein English is not my first language, but I think, after looking for the meaning of "mindless" , that people who are ready to share knowledge will never use such a word! Soft words and hard arguments
You need to consider that you have misinterpreted the context of the usage of that word in what Hurkyl's wrote. Nothing in there was disparaging.

Zz.
P: 16
 Quote by bernhard.rothenstein When we state that: Knowing the direct Lorentz transformation we obtain the inverse one by changing the corresponding primed physical quantities with unpriomed ones and changing the sign of the relative velocity, we are based on what? Thanks
In my opinion it's as Ich said. It simply comes from the fact that all inertial frames are equivalent. Hence, it doesn't matter which one we call S and which S', the Lorentz transformation should look the same. Changing all primed quantities with unprimed ones (or viceversa) is valid simply because we're free to label each frame however we want.

About the sign of the velocity: you technically don't need to change it if you just stick to the rule of "changing all primed quantities with unprimed ones". It's just that it turns out that v' = -v, so you might as well not change the velocity and flip its sign, it's all the same.

I hope it made sense :).
 P: 25 Its all relative, duh
 P: 25 Thought experiment. In Star Trek, a 450 crew starship travelling at the speed of light or greater has to stop on a dime so Captain Kirk can see his girlfriend. How far in advance would they have to make that decision?
P: 105
 Quote by Peter McKenna Thought experiment. In Star Trek, a 450 crew starship travelling at the speed of light or greater has to stop on a dime so Captain Kirk can see his girlfriend. How far in advance would they have to make that decision?
Well, assuming the length of a dime is 2.0136296 10 ^ -18 lightyears, and the ship has instantaneous acceleration (which it doesn't but for the sake of simplicity), they don't have plan in advance at all :)
Sci Advisor
P: 8,430
 Quote by bernhard.rothenstein When we state that: Knowing the direct Lorentz transformation we obtain the inverse one by changing the corresponding primed physical quantities with unpriomed ones and changing the sign of the relative velocity, we are based on what? Thanks
Just plain algebra. If you write the Lorentz equations like this:

$$x' = \gamma (x - vt)$$
$$y' = y$$
$$z' = z$$
$$t' = \gamma (t - vx/c^2)$$
with $$\gamma = 1/\sqrt{1 - v^2/c^2}$$

Then you can just solve these equations for x, y, z, and t to get the inverse transformation.
P: 997
 Quote by JesseM Just plain algebra. If you write the Lorentz equations like this: $$x' = \gamma (x - vt)$$ $$y' = y$$ $$z' = z$$ $$t' = \gamma (t - vx/c^2)$$ with $$\gamma = 1/\sqrt{1 - v^2/c^2}$$ Then you can just solve these equations for x, y, z, and t to get the inverse transformation.
Thanks. I.e. combining the first and the last equation I obtain the inverse one?
Sci Advisor
P: 8,430
 Quote by bernhard.rothenstein Thanks. I.e. combining the first and the last equation I obtain the inverse one?
Right, combining the x' equation and the t' equation will give you the equation for x and t.
P: 997
 Quote by JesseM Right, combining the x' equation and the t' equation will give you the equation for x and t.
Tanks again. My problem is which are the roots of that fact: reciprocity, symmetry...
When I want to save the derivation of the inverse transformation stating the rule in my thread what should I say more to the learner?
Sci Advisor
P: 8,430
 Quote by bernhard.rothenstein Tanks again. My problem is which are the roots of that fact: reciprocity, symmetry... When I want to save the derivation of the inverse transformation stating the rule in my thread what should I say more to the learner?
Well, the fact that the equations come out looking identical except for the v being replaced by -v can be explained in terms of reciprocity, the laws of physics should be the same in every inertial frame so if two observers have a set of physical rulers and clocks synchronized by the Einstein synchronization convention which they use to define their respective coordinate systems, they'd better get the same set of equations for transforming their coordinates to the other coordinate system. And don't be confused by the fact that one uses v and the other uses -v, in fact they are both using exactly the same general transformation, it's just that the velocity of the (x',t') system as seen in the (x,t) system is opposite to the velocity of the (x,t) system as seen in the (x',t') system...you can think of both of them using this general transformation:

$$x_{other} = \gamma (x_{mine} - V t_{mine})$$
$$t_{other} = \gamma (t_{mine} - V x_{mine} /c^2)$$

where V represents the velocity of the "other" coordinate system's origin along "my" x-axis. If "my" coordinate system is the (x,t) system and the "other" is (x',t') then V=v, while if "my" coordinate system is the (x',t') system and the "other" is (x,t) then V=-v.
 PF Patron HW Helper Sci Advisor P: 4,083 Strictly speaking, as Hurkyl mentioned, this discussion only applies to a pure boost. A spatial rotation is technically a Lorentz Transformation, and so is a boost-with-spatial-rotation. So, in these last cases, flipping the sign of the relative velocity "V" is not sufficient to obtain the inverse. As I often suggest for questions like the OP's question, one could study the Euclidean analogue of the question.

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