Lorentz Transformation Matrix: Tensor of Order 2?

[Ben Geoffrey]

Is the Lorentz transformation matrix Λ_μ^ν a tensor of order two and does it transform like a tensor ?

 

[stevendaryl]

Is the Lorentz transformation matrix Λ_μ^ν a tensor of order two and does it transform like a tensor ?

Yes, but it's a little subtle. A transformation matrix simultaneously has two different interpretations:

1. As a "passive" transformation: If you have a vector $V$ and its components in one coordinate system are $V^\mu$, then its components in a second coordinate system (related to the first through a Lorentz transformation) will be given by: $V^\nu = \sum_\mu \Lambda_\mu^\nu V^\mu$.
2. As an "active" transformation: If $V$ is one vector, then we can defined a second, boosted vector $V'$ via: $(V')^\nu = \sum_\mu \Lambda_\mu^\nu V^\mu$.

The distinction is a little subtle: In the first case, you have the same vector described in two different coordinate systems, and in the second case, you two different vectors described in the same coordinate system.

Viewed as an active transformation, $\Lambda_\mu^\nu$ is a tensor.

 

[Ben Geoffrey]

Thank you for that. I have another question. If its a tensor then can we say its written in covariant formulation ? Is there anything like a GTR formulation of STR ?

 

[Ben Geoffrey]

Sir my question is more along the lines of we write GTR in tensor notation right ? so if we write Lorentz transformation matrix in tensor notation does it mean we've combined STR and GTR and written it as one theory ?

 

[PeterDonis]

Is there anything like a GTR formulation of STR ?

if we write Lorentz transformation matrix in tensor notation does it mean we've combined STR and GTR and written it as one theory ?

STR and GTR are one theory. STR is just the special case of GTR for which spacetime is flat. This is true regardless of what notation you use. You can do GTR without using tensor notation (though it's a lot more tedious).

 

[stevendaryl]

Sir my question is more along the lines of we write GTR in tensor notation right ? so if we write Lorentz transformation matrix in tensor notation does it mean we've combined STR and GTR and written it as one theory ?

You can formulate any theory of physics (including Newtonian physics) as a tensor theory, as far as I know. You don't really change the theory when you change the mathematical formulation. SR in tensor notation is still SR.