radonballoon
- 20
- 0
Homework Statement
Show that a^{\mu}b_{\mu} \equiv -a^0b^0 + \vec{a} \bullet \vec{b} is invariant under Lorentz transformations.
Homework Equations
\Lambda_{\nu}^{\mu} \equiv \left(<br /> \begin{array}{cccc}<br /> \gamma & -\beta \gamma & 0 & 0 \\<br /> -\beta \gamma & \gamma & 0 & 0 \\<br /> 0 & 0 & 1 & 0 \\<br /> 0 & 0 & 0 & 1<br /> \end{array} \right)<br />
<br /> b^0 = -b_0<br />
The Attempt at a Solution
So the only way I could get it to be invariant was performing a lorentz transformation on each a^{\mu}b_{\mu}, squaring each, and then setting the first term negative and adding them. I don't know why this works, however.:
<br /> \Lambda_{\nu}^{\mu}a^{\mu}b_{\mu} = \left(<br /> \begin{array}{cccc}<br /> \gamma & -\beta \gamma & 0 & 0 \\<br /> -\beta \gamma & \gamma & 0 & 0 \\<br /> 0 & 0 & 1 & 0 \\<br /> 0 & 0 & 0 & 1<br /> \end{array} \right) \left(<br /> \begin{array}{c}<br /> a^0b_0 \\<br /> a^1b_1 \\<br /> a^2b_2 \\<br /> a^3b_3 <br /> \end{array} \right) = \left(<br /> \begin{array}{c}<br /> \gamma (a^0b_0 - \beta a^1b_1) \\<br /> \gamma (a^1_b1 - \beta a^0b_0) \\<br /> a^2b_2 \\<br /> a^3b_3 <br /> \end{array} \right)<br />
Squaring each term, negating first term squared, and summing:
<br /> \gamma^2(-(a^0b_0)^2-\beta^2 (a^1b_1)^2 + 2\beta a^1b_1a^0b_0 + (a^1b_1)^2+\beta^2 (a^0b_0)^2 - 2\beta a^1b_1a^0b_0) + (a^2b_2)^2 + (a^3b_3)^2<br />
Doubled terms cancel, and the others can be grouped so that they are multiplied by 1-\beta^2 which cancels the \gamma^2, and you are left with the original equation again, albeit with terms squared.
Any help would be great!