General Lorentz transformation, spatial vector components

[JD96]

Hello,

a derivation of the lorentz transformation for an arbitrary direction of the relative velocity often makes use of writing the spatial position vector of an event as the sum of its component parallel and perpendicular to the velocity vector in one inertial frame and then transforming both components in the following way:
$\vec x'_{⊥}= \vec x_{⊥}$
$\vec x'_{\parallel}= γ(\vec x_{\parallel}-\vec v t)$
Now, I understand that the magnitude of the perpendicular component stays the same in both frames and I also know both inertial frames have to agree on the magnitude of the relative velocity. Unfortunately I can't think of arguments why the unit vector of the velocity vector and the perpendicular component should be identical in the two frames, that is why it can't be the case that some kind of rotation of the spatial axis gets involved when performing a general lorentz transformation. But in order to justify the above transformations I need to be confident that the direction and orientation of those vectors are indeed the same.
This is the only part in the derivation that troubles me and I would appreciate any help. Hopefully I stated my problem clear enough so that maybe someone can come up with an argument adressing these issues.

Thanks in advance!

 

[PeterDonis]

why it can't be the case that some kind of rotation of the spatial axis gets involved when performing a general lorentz transformation

A general Lorentz transformation does include spatial rotations. What you've written down is a more restricted transformation that is properly called a "boost", although the term "Lorentz transformation" is often used to describe this kind of transformation as well. A boost is defined as a transformation that does not change the direction of any spatial axes; all it does is change velocity. A transformation that included rotation of the spatial axes would be described by a more complicated equation, not the one you've written down.

 

[JD96]

A transformation that included rotation of the spatial axes would be described by a more complicated equation, not the one you've written down.

Intuitively I would think that a set of three rulers that are orthogonal to each other in one frame are not in another one or does one have to distinguish between a rotation that is brought in manually with a rotation matrix and one that is "induced" by relativistic effects?

 

[PAllen]

Intuitively I would think that a set of three rulers that are orthogonal to each other in one frame are not in another one or does one have to distinguish between a rotation that is brought in manually with a rotation matrix and one that is "induced" by relativistic effects?

What you say is true, in general. However, if relative motion is parallel to one of rulers, the orthogonality is preserved.

 

[bcrowell]

You might also want to look at Thomas precession.

 

[JD96]

@bcrowell So from what I understand using the graphics in the wiki article: If I perform three lorentz transformations (where in each one the relative velocity is aligned with one of the axes), I have not simply transformed the coordinates between two frames whose relative velocity is arbitrarily oriented, but also whose coordinate axes are rotated with respect to each other. If I would like to derive the general lorentz boost this way, is it possible to account for this effect without additional assumptions?

 

[bcrowell]

@bcrowell So from what I understand using the graphics in the wiki article: If I perform three lorentz transformations (where in each one the relative velocity is aligned with one of the axes),

Two Lorentz transformations does it, and it's not relevant whether they're aligned with the coordinate axes, as long as they're not along the same axis as each other.

If I would like to derive the general lorentz boost this way, is it possible to account for this effect without additional assumptions?

There are many different ways to derive the Lorentz transformations, and it can be done using many different choices of the initial set of assumptions. See ch. 2 of my SR book for a survey: http://www.lightandmatter.com/sr/ . What set of assumptions did you have in mind?

There is also the question of what your goal is in terms of the language and form in which you want to end up displaying the transformations. Coordinates and matrices? Tensors and index gymnastics? Probably the more concrete the form you aim for, the uglier it's going to be. I think in general when people want to organize their thoughts about Lorentz transformations and do it in an elegant way, they work in the language of group theory. You have a certain set of boosts and rotations, and you describe their commutators.

 

[JD96]

@bcrowell I would like to derive the 3+1d lorentz boost in matrix form using the one dimensional lorentz transformation (I haven't yet learned about differential geometry and group theory). Apart from the method mentioned in my OP what other ways are there? I now know that using several different frames and doing a row of lorentz boosts will cause problems since several lorentz boosts will include rotations. Thats why I was wondering whether one could correct for the thomas precession to fix this way of deriving the 3+1d lorentz boost. Hopefully I was able to formulate my problem clear enough :-)
Thanks for all the responses so far!

 

[bcrowell]

@bcrowellThats why I was wondering whether one could correct for the thomas precession to fix this way of deriving the 3+1d lorentz boost.

It's not a correction. If you compose two boosts, a rotation is simply part of the result.

Re your other questions, please see the info and link I gave above.

 

[JD96]

I will have a look at the link you gave me when I have a little bit more time to study it in greater detail.

With correction I meant that when aiming to derive the 3+1d lorentz boost in matrix form as given here https://en.wikipedia.org/wiki/Lorentz_transformation#Boost_in_any_direction, but instead using the procedure of performing several 1+1d lorentz boosts and then establishing a relation between the coordinates of the initial and final inertial frame, a rotation will be an "unwanted byproduct" and won't lead me to the form given in the wiki article (or am I wrong here?). So I guess I would have to redo the rotation in order to obtain a pure boost.

 

[PeterDonis]

or am I wrong here?

A single 3+1d Lorentz boost, like what the Wiki page describes, is not the same as a sequence of 1+1d Lorentz boosts in different directions. The former is just a single boost where, for whatever reason, you haven't aligned any of the coordinate axes in the direction of the boost; the only issue involved is finding an expression for the single transformation in this case.

The latter is two (or more) boosts in different directions, and the issue is finding a generalized transformation that is equivalent to the two (or more) boosts applied one after the other. It's only that latter case that results in a rotation being included as part of the result; i.e., the generalized transformation that results from composing two or more boosts in different directions is not a pure boost; it's a boost plus a rotation. The "3+1d Lorentz boost" formula you gave does not describe this kind of transformation.