About a Lorentz matrix and its inverse

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    Inverse Lorentz Matrix
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Discussion Overview

The discussion revolves around the properties of Lorentz matrices and their inverses in the context of special relativity, specifically referencing Naber's "The geometry of Minkowski spacetime." Participants explore a specific equation relating the components of a Lorentz matrix and its inverse.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • Barbara Da Vinci seeks clarification on the equation L^4_i / L^4_4 = - M^i_4 / M^4_4, as presented in Naber's text, and notes that no other assumptions seem to be involved.
  • One participant interprets the equation as a statement about the relationship between the velocities of two inertial frames, suggesting it reflects Einstein's first postulate regarding the equivalence of physical laws in all inertial frames.
  • A suggestion is made to consider the implications of a Lorentz transformation acting on the coordinates (t,x,y,z)=(1,0,0,0) to further understand the relationship.
  • Another participant references a specific equation (1.2.11) from the text as potentially relevant to the discussion.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants explore different interpretations and implications of the equation without resolving the underlying assumptions or motivations.

Contextual Notes

Participants do not clarify the assumptions behind the equation or the specific context in which it applies, leaving some aspects unresolved.

BarbaraDav
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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
 
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If I understand your notation correctly, that equation is equivalent to saying that if the velocity of frame F' in F is [itex]\vec v[/itex], then the velocity of frame F in F' is [itex]-\vec v[/itex]. 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)?
 
Welcome to Physics Forums!

Use equation (1.2.11).
 
Thanks for replying: I got it !
Have a great day !
 

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