- #1
HJ Farnsworth
- 128
- 1
Greetings,
I have been having trouble deriving the equation for the general Lorentz boost for velocity in an arbitrary direction. It seems to me that given the 1D Lorentz transformations...
matrix for Lorentz transformation in x-direction, X:
{{1/sqrt(1-v^2), -v/sqrt(1-v^2), 0, 0},
{-v/sqrt(1-v^2), 1/sqrt(1-v^2), 0, 0},
{0, 0, 1, 0},
{0, 0, 0, 1}}
matrix for Lorentz transformation in y-direction, Y:
{{1/sqrt(1-v^2), 0, -v/sqrt(1-v^2), 0},
{0, 1, 0, 0},
{-v/sqrt(1-v^2), 0, 1/sqrt(1-v^2), 0},
{0, 0, 0, 1}}
matrix for Lorentz transformation in z-direction, Z:
{{1/sqrt(1-v^2), 0, 0, -v/sqrt(1-v^2)},
{0, 1, 0, 0},
{0,0,1,0},
{-v/sqrt(1-v^2), 0, 0, 1/sqrt(1-v^2)}}
...all I should have to do is replace each v in X with vx, each v in Y with vy, and each v in Z with vz, and then multiply the matrices to get ZYX as the general Lorentz boost equation. This should sequentially compute the Lorentz transformation in the x-direction with the x-velocity, then the y-direction with the y-velocity, then the z-direction with the z-velocity. However, when I do that, the answer I get only vaguely resembles the general Lorentz boost equation found on Wikipedia (http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_form).
I can't find the derivation for the general boost matrix anywhere, and more importantly, I have no idea why the simple math I used is wrong - it seems conceptually correct to me.
Please tell me what the flaw in my logic is, and how to correct it.
Thanks for any help you can give.
-HJ Farnsworth
I have been having trouble deriving the equation for the general Lorentz boost for velocity in an arbitrary direction. It seems to me that given the 1D Lorentz transformations...
matrix for Lorentz transformation in x-direction, X:
{{1/sqrt(1-v^2), -v/sqrt(1-v^2), 0, 0},
{-v/sqrt(1-v^2), 1/sqrt(1-v^2), 0, 0},
{0, 0, 1, 0},
{0, 0, 0, 1}}
matrix for Lorentz transformation in y-direction, Y:
{{1/sqrt(1-v^2), 0, -v/sqrt(1-v^2), 0},
{0, 1, 0, 0},
{-v/sqrt(1-v^2), 0, 1/sqrt(1-v^2), 0},
{0, 0, 0, 1}}
matrix for Lorentz transformation in z-direction, Z:
{{1/sqrt(1-v^2), 0, 0, -v/sqrt(1-v^2)},
{0, 1, 0, 0},
{0,0,1,0},
{-v/sqrt(1-v^2), 0, 0, 1/sqrt(1-v^2)}}
...all I should have to do is replace each v in X with vx, each v in Y with vy, and each v in Z with vz, and then multiply the matrices to get ZYX as the general Lorentz boost equation. This should sequentially compute the Lorentz transformation in the x-direction with the x-velocity, then the y-direction with the y-velocity, then the z-direction with the z-velocity. However, when I do that, the answer I get only vaguely resembles the general Lorentz boost equation found on Wikipedia (http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_form).
I can't find the derivation for the general boost matrix anywhere, and more importantly, I have no idea why the simple math I used is wrong - it seems conceptually correct to me.
Please tell me what the flaw in my logic is, and how to correct it.
Thanks for any help you can give.
-HJ Farnsworth