…
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
.
Denoting as
is misleading, because that presupposes that
is directly related to the matrix
.
To avoid this bug, instead, we denote as
. So
…
— Me@2020-09-12 09:33:00 PM
.
.
2020.09.13 Sunday (c) All rights reserved by ACHK