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Lorentz matrix

 

Another common way of expressing the Lorentz transformation is in matrix form:

 Recalling the rules for matrix multiplication we see that:

 

 

 We have found exacly the same Lorentz transformation equations as described in The Lorentz transformations Part I - Presentation

The two ways of expressing the equations are strictly equivalent.

We can write an even more compact form by using the index notation (see tab Index Notation)

Index notation

 We can write the Lorentz matrix in a even more compact notation, using index notation, in the form:

 

where the indices μ and ν take the values of the number of spacetime dimensions, ie 0 to 3.

So the components of x'μ are (x'0, x'1, x'2, x'3) = (ct', x', y', z')

And those of xμ are (x0, x1, x2, x3) = (ct, x, y, z)

Concerning the matrix,

the μ index refers to the μth row and the ν index refers to the νth column

 

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"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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