About a Lorentz matrix and its inverse

[BarbaraDav]

Hi you all.
Please forgive my poor English;
I will try to put my best foot forward!

I'm studying special relativity mostly on Naber's
"The geometry of Minkowski spacetime". Just after
introducing the concept of Lorentz matrix L
(by means of "M" I point its inverse) through
the well known relation

L^c_a L^d_b g_cd = g_ab

he states (pag. 22 equation following 1.3.10) that

L^4_i / L^4_4 = - M^i_4 / M^4_4

As far as I see no other assumption is involved in.
Please, can you give any hint about establishing
this equation ?

Best Regards
Barbara Da Vinci

 

[Fredrik]

If I understand your notation correctly, that equation is equivalent to saying that if the velocity of frame F' in F is $\vec v$, then the velocity of frame F in F' is $-\vec v$. I don't know how to motivate that statement other than by saying that it makes some sense to think of it as an interpretation of Einstein's first postulate. ("The laws of physics are the same in all inertial frames").

Let me know if you need more information to understand why those two statements are the same. Hint: What do you get when a Lorentz transformation acts on (t,x,y,z)=(1,0,0,0)?

 

[George Jones]

Welcome to Physics Forums!

Use equation (1.2.11).

 

[BarbaraDav]

Thanks for replying: I got it !
Have a great day !