Lorentz Transformation Explained: What Is It?

[captain]

i am confused what a lorentz transformation really is or is it just a transformation described by the boost transformation.

 

[bernhard.rothenstein]

i am confused what a lorentz transformation really is or is it just a transformation described by the boost transformation.

A Lorentz transformation establishes a relationship between the space-time coordinates of the sameevent detected from two inertial reference frames in relative motion and in the standard arrangement and with well defined initial conditions. The two events (E(x,y,z,t) and E'(x',y',z',t') take place at the same point in space when the synchronized clocks (a la Einstein) of the two inertial frames, located at that point, read t and t'.

 

[neutrino]

A Lorentz transformation establishes a relationship between the space-time coordinates of the sameevent detected from two inertial reference frames in relative motion and in the standard arrangement and with well defined initial conditions.

[emphasis(underlined) mine]

Is that a necessary condition? (By "standard arrangement" I assume you are referring to the standard configuration of inertial coordinate systems.)

 

[bernhard.rothenstein]

[emphasis(underlined) mine]

Is that a necessary condition? (By "standard arrangement" I assume you are referring to the standard configuration of inertial coordinate systems.)

I assume the same thing as you do. Parallel axes, overlapped OX(O'X') axes, motion of I' relative to I in the positive direction of the overlapped axes, coincidence of the origins at the origin of time. I am not very familiar with the standard English terms.
Thanks for your question and help

 

[neutrino]

I assume the same thing as you do. Parallel axes, overlapped OX(O'X') axes, motion of I' relative to I in the positive direction of the overlapped axes, coincidence of the origins at the origin of time. I am not very familiar with the standard English terms.
Thanks for your question and help

But you still haven't answered my question.

Is it necessary for the relative motion be along the x-axes (y- and z-axes parallel) and the origins to coincide at t = t' =0 for the transformations to be valid?

I ask this because the OP didn't specifically mention
$$x' = \gamma\left(x - \beta ct)$$
$$ct' = \gamma\left(ct - \beta x)$$
(or its inverse).

 

[dextercioby]

A Lorentz transformation establishes a relationship between the space-time coordinates of the sameevent detected from two inertial reference frames in relative motion and in the standard arrangement and with well defined initial conditions.

A Lorentz transformation is simply an element of the Lorentz group, the group of the linear, homogenous and orthogonal transformations of the Minkowski space ($\mathbb{M}_{4}=\left(\mathbb{R}^{4},\eta\right)$), group shown to be isomorphic to $O(1,3,\mathbb{R})$. That's the mathematical definition.

The physical one is contained in your post: <<A Lorentz transformation establishes a relationship between the space-time coordinates of the same event detected from two inertial reference frames in relative motion>>.

 

[robphy]

The Lorentz Transformations include more than the boost transformations... spatial rotations [and possibly reflections] are included.

 

[bernhard.rothenstein]

But you still haven't answered my question.

Is it necessary for the relative motion be along the x-axes (y- and z-axes parallel) and the origins to coincide at t = t' =0 for the transformations to be valid?

I ask this because the OP didn't specifically mention
$$x' = \gamma\left(x - \beta ct)$$
$$ct' = \gamma\left(ct - \beta x)$$
(or its inverse).

The way in which you present the LT, they hold in the case when the I' frame where the event involved in the transformation is E'(x',y',t') moves with speed v in the positive direction of the overlapped OX(O'X') axes relative to the I frame where the same event is E(x,y,t) in a two space dimensions approach. You can easy test the condition x'=0 for t=0 and x=0. You can add y=y'=0. That is the scenario that leads to the simplest shaped LT. Further questions?