# About a Lorentz matrix and its inverse

In summary, the conversation is about special relativity and the equation L^4_i / L^4_4 = - M^i_4 / M^4_4, which states that the velocity of frame F in F' is equal to the negative of the velocity of frame F' in F. The equation is derived from the interpretation of Einstein's first postulate and the use of equation (1.2.11). The conversation also includes a warm welcome to a new member of Physics Forums.
Hi you all.
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.
this equation ?

Best Regards
Barbara Da Vinci

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)?

Welcome to Physics Forums!

Use equation (1.2.11).

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

## 1. What is a Lorentz matrix?

A Lorentz matrix is a mathematical matrix used in physics to describe the effects of special relativity on space and time. It is a 4x4 matrix that represents the transformation between two frames of reference moving at a constant velocity relative to each other.

## 2. How is a Lorentz matrix calculated?

A Lorentz matrix is calculated using the Lorentz transformation equations, which take into account the relative velocity between the two frames of reference, the speed of light, and the dimensions of the matrix. These equations involve a combination of trigonometric and hyperbolic functions.

## 3. What is the significance of a Lorentz matrix in physics?

A Lorentz matrix is significant in physics because it allows for the description of the effects of special relativity on events and measurements in different frames of reference. It is essential for understanding phenomena such as time dilation and length contraction.

## 4. What is the inverse of a Lorentz matrix?

The inverse of a Lorentz matrix is another Lorentz matrix that can be used to transform coordinates and measurements from one frame of reference to another in the opposite direction. It is calculated by taking the transpose of the original matrix and changing the sign of the spatial components.

## 5. How is a Lorentz matrix used in practical applications?

A Lorentz matrix is used in many practical applications, including particle physics, astrophysics, and engineering. It is crucial for understanding and predicting the behavior of high-speed particles and objects, such as those in particle accelerators or traveling near the speed of light.