How can a 4x4 matrix be a Lorentz transformation and what are its properties?

[astoria]

Homework Statement

What does it mean for a 4x4 matrix to be a Lorentz transformation? What does it mean for it to be proper and orthochronous?

Show that for the standard Lorentz transformation, c^2t^2-x^2-y^2-z^2=c^2t'^2-x'^2-y'^2-z'^2

What is the significance of this statement when quantity is zero? What is the significance of this statement when this quantity is positive? Show that the observers agree on the direction of time and the spatial orientation.

In an inertial co-ordinate system O, a rod whose length according to O is L is aligned parallel to x-axis and moves with 3 velocity (v,w,0) with gamma=3. Show that in the frame of O' the x separation of the end of the rods is 3L but that the rod now makes an angle with the x-axis which should be found. (O' is moving at speed v in the x direction relative to O)

Also, in a similar question there is a rod of length 2l with end points (+-l,a,0) in O'. The first part involves showing that O measures the length of the rod to be 2l/gamma which is fine. The next part says show that the observer O sees the light rays from the two ends of the rod coming in at angles that lead to an apparent length of 2l*gamma.

The Attempt at a Solution

I think a Lorentz transformation is one in which g=L(transpose)gL. I think proper means a determinant of 1 and orthochronous means top left entry is positive. Are these the proper definitions or just results?

I can show that statement and know that it implies that the light-cone for an event is the same for all observers. I'm not really sure what the significance is at 0 or positive though. At 0 is it the light cone for the event that happened for both at (0,0,0,0)?

Showing direction of time is the same is fine, not sure about spatial orientation.

To do this I've said that (0,L) is equivalent to (t',x')=(-vR/c^2,R) solving gives R=3L. Have I done this correctly or should I have done it the other way round and left t'=0 and worked out two values for t, I presume it doesn't matter? To work out the angle I presume that the rod is 3L in the x direction and -3Lvw/c^2 in the y direction, so angle is arctan(w/c^2). I'm not sure whether the rod makes a positive angle or negative angle with the x axis. Have I done this question correctly, if so, in the neatest way?

No idea how to do the last part or why it occurs, I guess it must be that O has two light rays coming in that he doesn't observe to be simultaneous.

Thanks

 

[astoria]

I've found the formal definition of Lorentz transformation and proper and orthochronous, so ignore that bit.