Understanding Lambda Mu Nu Prime and Its Indices

[DiracPool]

I'm trying to do some GR self-instruction through a variety of video lectures and thought this would be a good place to seek clarification on the inevitable thorny issues. I've tried this before and didn't make it too far, but I'm trying to get back on the horse, so to speak, and give it another go. So far this time I have been able to get farther through the course(s), but progress again has started to stall. For me personally, it always starts to bog down when I get "index overload" and my brain simply shuts down. That is, I can only juggle the lambda mu nu eta zeta beta alpha gamma prime "rhyme" for so many hours and iterations until the gears stop turning and the whiteboard just looks like a mass of greek symbols. Even so, I'm determined to persevere and hopefully get through these rough patches.

The first rough patch I've come across so far deals with the introduction of a mathematical object Maloney calls lambda mu nu prime, presumably a 4x4 matrix. I don't know what this is or how it fits with the rest of the discussion. In addition, it includes an upper and lower index which he doesn't explain how to read. The lower index is primed and the upper is not. What does that mean? What kind of matrix is this, why doesn't he write it out? Is it related to the flat space-time Minkowski zeta matrix?

In any case, please fast forward to 36:40:



Another video I came across this Lambda term in with similar confusion is Here at 10:50:



So, perhaps here the idea is that any mathematical physicists worth his slide rule just of course can look at this lambda object and know exactly what it is, which is why neither of these guys takes a minute to explain what they are. Not me though, and any help would be appreciated.

 

[PeterDonis]

The lower index is primed and the upper is not. What does that mean?

It means this is a Lorentz transformation matrix; it tells you how the components of a 4-vector in one frame transform into another frame. The primed index refers to one frame; the unprimed index refers to the other frame.

This Wikipedia article may help:

http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_forms

 

[DiracPool]

It means this is a Lorentz transformation matrix; it tells you how the components of a 4-vector in one frame transform into another frame. The primed index refers to one frame; the unprimed index refers to the other frame.

So is it just a generic form of, say, a boost or rotation matrix? Or is it some kind of other specific, unique matrix?

 

[Matterwave]

So is it just a generic form of, say, a boost or rotation matrix? Or is it some kind of other specific, unique matrix?

I can't view the videos you posted (youtube is not allowed here lol) but from your description, it seems Peter's assertion that it represents a Lorentz transformation is completely right. This means that the Lambda represents the generic form of a Lorentz transformation which could be a boost matrix, or a rotation matrix, or a combination of both (or any number of boosts/rotations).

 

[PeterDonis]

is it just a generic form of, say, a boost or rotation matrix?

Yes.

 

[DiracPool]

Ok, cool, I got it, thanks to responders.