Difference between revisions of "4-vectors"
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Where
Where is the Lorentz transformation matrix for motion in the z direction.
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<center><math>x'^{\mu}=\sum_{\nu=0}^3 (\Lambda_{\nu}^{\mu})x^{\nu}</math></center> | <center><math>x'^{\mu}=\sum_{\nu=0}^3 (\Lambda_{\nu}^{\mu})x^{\nu}</math></center> | ||
| − | <center><math> | + | |
| − | + | <center>Where <math>\Lambda</math> is the Lorentz transformation matrix for motion in the z direction. | |
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| + | <center><math>(ds)^2\equiv | ||
| + | \begin{bmatrix} | ||
| + | x_0 & -x_1 & -x_2 & -x_3 | ||
| + | \end{bmatrix} \cdot | ||
| + | \begin{bmatrix} | ||
| + | x^0 \\ | ||
| + | x^1 \\ | ||
| + | x^2 \\ | ||
| + | x^3 | ||
| + | \end{bmatrix}=(x^0)^{'2}-(x^1)^{'2}-(x^2)^{'2}-(x^3)^{'2}= (x^0)^{2}-(x^1)^2-(x^2)^2-(x^3)^2</math></center> | ||
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Revision as of 01:55, 6 June 2017
4-vectors
Using index notation, the time and space coordinates can be combined into a single "4-vector" , that has units of length, i.e. ct is a distance.
Using the Lorentz transformations and the index notation,
This can be expressed in matrix form as
Letting the indices run from 0 to 3, we can write
We can express the space time interval using the index notation