# Generalized Lorentz Transformation

Summary: The problem is to generalize the Lorentz transformation to two dimensions.

Relevant Equations

Lorentz Transformation along the positive x-axis:
$$\begin{pmatrix} \bar{x^0} \\ \bar{x^1} \\ \bar{x^2} \\ \bar{x^3} \\ \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} {x^0} \\ {x^1} \\ {x^2} \\ {x^3} \\ \end{pmatrix}$$
Lorentz Transformation along the positive y-axis:
$$\begin{pmatrix} \bar{x^0} \\ \bar{x^1} \\ \bar{x^2} \\ \bar{x^3} \\ \end{pmatrix} = \begin{pmatrix} \gamma & 0 & -\gamma \beta & 0 \\ 0 & 1 & 0 & 0 \\ -\gamma \beta & 0 & \gamma & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} {x^0} \\ {x^1} \\ {x^2} \\ {x^3} \\ \end{pmatrix}$$
Velocity transformations from S to S' where S' moves along the positive x-axis at speed v relative to S:
$$u'_x = \frac{u_x - v}{1 - \frac{u_xv}{c^2}} \\ u'_y = \frac{u_y}{\gamma(1 - \frac{u_xv}{c^2})}$$

Problem Statement

$$\bar{S} \text{ moves at velocity } \\ \vec{v} = \beta c(Cos\phi \hat{x} + Sin\phi \hat{y}) \text{ relative to S, with axes parallel and origins coinciding at time } \\ t = \bar{t} = 0. \text{ Find the Lorentz transformation from S to } \bar{S}$$

Attempt at a solution
$$\text{Let S' move at velocity } \beta cCos\phi\hat{x} \text{ relative to S. We can find the velocity of } \bar{S} \text { relative to S' using the velocity transformations. } \\ u'_x = 0 \\ u'_y = \gamma_x\beta cSin\phi \\ where \, \gamma_x = \frac{1}{(1-\beta^2Cos^2\phi)^\frac{1}{2}}$$

$$\text{We can express the coordinates in } \bar{S} \text{ given the coordinates in S' by the y-axis Lorentz transformation, with } \\ \bar{\beta} = \beta Sin\phi \gamma_x \, \\ \bar{\gamma} = \frac{\gamma}{\gamma_x} \\ where \, \gamma = \frac{1}{(1 - \beta^2)^\frac{1}{2}}$$

$$\text{And we can express the coordinates in S' given the coordinates in S by the x-axis transformation with } \\ \beta' = \beta Cos\phi \\ \gamma' = \gamma_x$$

$$\text{ Therefore we can express the coordinates in } \bar{S} \text { given the coordinates in S as follows }$$

$$\begin{pmatrix} \bar{x^0} \\ \bar{x^1} \\ \bar{x^2} \\ \bar{x^3} \\ \end{pmatrix} = \begin{pmatrix} \frac{\gamma}{\gamma_x} & 0 & -\frac{\gamma}{\gamma_x}\beta Sin\phi \gamma_x & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{\gamma}{\gamma_x}\beta Sin\phi \gamma_x & 0 & \frac{\gamma}{\gamma_x} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \gamma_x & -\gamma_x\beta Cos\phi & 0 & 0 \\ -\gamma_x \beta Cos\phi & \gamma_x & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x^0 \\ x^1 \\ x^2 \\ x^3 \end{pmatrix} =$$
\begin{pmatrix}
\gamma & -\gamma \beta Cos\phi & -\gamma \beta Sin\phi & 0 \\
-\gamma_x\beta Cos\phi & \gamma_x & 0 & 0 \\
-\gamma \gamma_x\beta Sin\phi & \gamma \gamma_x\beta^2Sin\phi Cos\phi & \frac{\gamma}{\gamma_x} & 0 \\
0 & 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix}
x^0 \\
x^1 \\
x^2 \\
x^3
\end{pmatrix}

This final matrix represents the solution to the problem. But, this solution is wrong, as the correct matrix is given an follows.

\begin{pmatrix}
\gamma & -\gamma \beta Cos\phi & -\gamma \beta Sin\phi & 0 \\
-\gamma \beta Cos\phi & (\gamma Cos^2\phi + Sin^2\phi) & (\gamma - 1)Sin\phi Cos\phi & 0 \\
-\gamma \beta Sin\phi & (\gamma - 1)Sin\phi Cos\phi & (\gamma Sin^2\phi + Cos^2\phi) & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}

I am unable to determine what is wrong with this solution, any suggestions would be appreciated.

PeterDonis
Mentor
2020 Award
Moderator's note: Moved to advanced physics homework forum.

The product of two pure boost in different directions wii not be a pure boost anymore.
$$\exp(\zeta^1X_1)\cdot\exp(\zeta^2X_2)\neq\exp(\zeta^1X_1+\zeta^2X_2)$$
Because the boosts in different directions do not commute with each other.

The product of two pure boost in different directions wii not be a pure boost anymore.
$$\exp(\zeta^1X_1)\cdot\exp(\zeta^2X_2)\neq\exp(\zeta^1X_1+\zeta^2X_2)$$
Because the boosts in different directions do not commute with each other.

I noticed that of course. But, isn't it true that the matrix gives S(bar) in terms of S? I don't see how the logic could be wrong. We give S(bar) in terms of S' and S' in terms of S, thus S(bar) in terms of S. I agree the formula is not a boost, but isn't it still correct?

PeroK
Homework Helper
Gold Member
2020 Award
I noticed that of course. But, isn't it true that the matrix gives S(bar) in terms of S? I don't see how the logic could be wrong. We give S(bar) in terms of S' and S' in terms of S, thus S(bar) in terms of S. I agree the formula is not a boost, but isn't it still correct?

What you have is a valid transformation. For example, the time dilation effect will be correct.

But, if you assume that the frame ##\bar S## is moving with velocity ##\vec v## in frame ##S##, then frame ##S## is not moving with velocity ##-\vec v## in your frame ##\bar S##.

In fact, that requirement is usually an unstated assumption in this problem. The assumption that you want symmetry of relative velocity between ##S## and ##\bar S##.

Your approach does not achieve this, but transforms to a rotated version of the "required" coordinates for ##\bar S##.

What you have is a valid transformation. For example, the time dilation effect will be correct.

But, if you assume that the frame ##\bar S## is moving with velocity ##\vec v## in frame ##S##, then frame ##S## is not moving with velocity ##-\vec v## in your frame ##\bar S##.

In fact, that requirement is usually an unstated assumption in this problem. The assumption that you want symmetry of relative velocity between ##S## and ##\bar S##.

Your approach does not achieve this, but transforms to a rotated version of the "required" coordinates for ##\bar S##.

If what I have is a valid transformation, then I'm confused about what is calculating. Given the coordinates in S, does it give the coordinates in S(bar)?

PeroK
Homework Helper
Gold Member
2020 Award
If what I have is a valid transformation, then I'm confused about what is calculating. Given the coordinates in S, does it give the coordinates in S(bar)?
How do you define the coordinates in ##\bar S##? There's no single orientation of axes. The axes can be in any directions you want.

Take a simple boost in the x-direction from ##S## to ##S'##. There's nothing to say that ##S'## must take the same x-axis as ##S##. It's perfectly valid for ##S'## to define the direction of motion as the ##y## direction. Or, to rotate its coordinates in any way.

It's by convention that the coordinates in ##S'## are defined in a certain way with respect to the ##S## coordinates.

If you have ##\bar S## moving in both the x and y directions relative to ##S##, then it's not so obvious how you want to define the x,y axes in ##\bar S## with respect to the x, y axes in ##S##.

The only thing that must be true is that the magnitude of the velocity of ##S## must be correct. But, any orientation of axes where this holds is a valid coordinate system for ## \bar S##.

However, the coordinate system where the velocity of ##S## is ##- \vec v## is big convention the one given for the 2D boost.

PeroK
Homework Helper
Gold Member
2020 Award
If what I have is a valid transformation, then I'm confused about what is calculating. Given the coordinates in S, does it give the coordinates in S(bar)?
PS note that for a boost along the x-axis, all the axes in ##S'## can be taken to be parallel to those in ##S##. I.e. the two sets of axes coincide when the origins coincide.

But, if the boost is not along a single coordinate axis, then the mutually orthogonal x, y axes in one frame cannot coincide with mutually orthogonal x', y' axes in the other.

It's a good exercise to check this out.

I noticed that of course. But, isn't it true that the matrix gives S(bar) in terms of S? I don't see how the logic could be wrong. We give S(bar) in terms of S' and S' in terms of S, thus S(bar) in terms of S. I agree the formula is not a boost, but isn't it still correct?
Perhaps you can think of the problem in such a way:
$$\exp(\zeta^1 X_1)\exp(\zeta^2 K_2)=\lim_{n\rightarrow \infty}\prod^{n}_{i=1}\exp\left(\frac{\zeta^1 X_1}{i}\right) \prod^{n}_{j=1}\exp\left(\frac{\zeta^2 X_2}{j}\right)$$
and
$$\exp(\zeta^1 X_1+\zeta^2 K_2)=\lim_{n\rightarrow \infty}\prod^{n}_{i=1}\exp\left(\frac{\zeta^1 X_1}{i}\right)\exp\left(\frac{\zeta^2 X_2}{i}\right)$$
Obviously the two formulas describe different physical processes

vanhees71