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Now we lower the indices, by expressing the upper-index coordinates (contravariant components) by lower-index coordinates (covariant components), in order to find the Lorentz transformation for the covariant components:

After raising the indices, we lower the indices again:

Prove that .

By index renaming, , the question becomes

Prove that .

Denote as . Then the question is simplified to

Prove that .

The right hand side has 64 terms.

Since the spacetime interval is Lorentz-invariant, . So the left hand side can be replaced by .

Note that the 4 terms on the left side also appear on the right hand side.

Since this equation is true for any coordinates, it is an identity. By comparing coefficients, we have:

1. For any terms with , such as and ,

So

2. For any terms with .

So

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Denoting as is misleading, because that presupposes that is directly related to the matrix .

To avoid this bug, instead, we denote as . So

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— Me@2020-09-12 09:33:00 PM

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2020.09.13 Sunday (c) All rights reserved by ACHK