- #1
Rococo
- 67
- 9
I understand how contravariant 4-vectors transform under a Lorentz transformation, that is:
##x'^μ= \Lambda^\mu~_\nu x^\nu## [1]
and how covariant 4-vectors transform:
##x'_\mu=(\lambda^{-1})^\nu~_\mu x_\nu##. [2]
Now, I have come across the following relations:
##\Lambda^\mu~_\nu = \frac{∂x'^\mu}{∂x^\nu} = \frac{∂x_\nu}{∂x'_\mu}##
and
##(\lambda^{-1})^\nu~_\mu=\frac{∂x'_\mu}{∂x_\nu}=\frac{∂x^\nu}{∂x'^\mu}##.
It is clear to me from [1] that ##\Lambda^\mu~_\nu = \frac{∂x'^\mu}{∂x^\nu} ##. But I cannot see how this is then also equal to ##\frac{∂x_\nu}{∂x'_\mu}##.
Similarly, it is easy to see from [2] that ##(\lambda^{-1})^\nu~_\mu=\frac{∂x'_\mu}{∂x_\nu}##. But I cannot understand how this is also equal to ##\frac{∂x^\nu}{∂x'^\mu}##.
Any help on understanding how the indices are manipulated would be appreciated.
##x'^μ= \Lambda^\mu~_\nu x^\nu## [1]
and how covariant 4-vectors transform:
##x'_\mu=(\lambda^{-1})^\nu~_\mu x_\nu##. [2]
Now, I have come across the following relations:
##\Lambda^\mu~_\nu = \frac{∂x'^\mu}{∂x^\nu} = \frac{∂x_\nu}{∂x'_\mu}##
and
##(\lambda^{-1})^\nu~_\mu=\frac{∂x'_\mu}{∂x_\nu}=\frac{∂x^\nu}{∂x'^\mu}##.
It is clear to me from [1] that ##\Lambda^\mu~_\nu = \frac{∂x'^\mu}{∂x^\nu} ##. But I cannot see how this is then also equal to ##\frac{∂x_\nu}{∂x'_\mu}##.
Similarly, it is easy to see from [2] that ##(\lambda^{-1})^\nu~_\mu=\frac{∂x'_\mu}{∂x_\nu}##. But I cannot understand how this is also equal to ##\frac{∂x^\nu}{∂x'^\mu}##.
Any help on understanding how the indices are manipulated would be appreciated.