Is T = T^-1 in Lorentz Transformations for 4 Vectors?

In summary, the conversation revolved around the question of whether T^v_u equals its own inverse, T^{-1}, for a Lorentz transformation of a 4 vector. The participants discussed the matrix representation of this transformation and the possibility of a typo in the assignment given by the lecturer. It was concluded that T and T^{-1} are not equal in this case.
  • #1
yukcream
59
0
I am not sure wheather
matrix T^v_u = {T^u_v}^-1
or T^v_u = {T^v_u} is true?
T is a lorentz transformation for 4 vector~

Yuk
 
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  • #2
Start by writing down the matrix representation of a general Lorentz transformation.

yukcream said:
I am not sure wheather
matrix T^v_u = {T^u_v}^-1

In matrix language, that is [itex]T^T=T^{-1}[/itex]. In other words, you are asking if the transpose of [itex]T[/itex] equals the inverse of [itex]T[/itex].

Try writing it out explicitly and see if it does.

or T^v_u = {T^v_u} is true?

Is there a typo here? Because here you just seem to be asking if [itex]T^{\mu}_{\nu}[/itex] equals itself, unless there is something else implied by the { } braces.
 
  • #3
It is great that you can understand my notation~ you are so smart!

It is my fault that I want to ask is T^v_u = {T^v_u} -1?

thanks for help
yuk
 
  • #4
yukcream said:
It is great that you can understand my notation~ you are so smart!

Your notation is a lot like LaTeX, which is what I used to make my math symbols. So, I'm not really that smart. :tongue2:

It is my fault that I want to ask is T^v_u = {T^v_u} -1?

OK, so now you're asking if [itex]T[/itex] is equal to its own inverse. Again, try to write down the matrix and see if that is so.
 
  • #5
Actually T is a special transformation~ called Lorentz transformation. In 4 vector case they are not equal~ but my lecturer asked me to prove T = T inverse~ I wonder is there is any typing error in his assignment given? or the position of the indexs may affect the result? They are not the same in fact!

yuk yuk
 

1. What is the Lorentz Transformation?

The Lorentz Transformation is a mathematical formula that describes how space and time coordinates change between two reference frames that are moving at constant velocities relative to each other. It is a key concept in Einstein's theory of special relativity.

2. Why is the Lorentz Transformation important?

The Lorentz Transformation is important because it allows us to understand how measurements of time and space are affected by motion and helps us reconcile the differences between the classical laws of physics and the laws of special relativity.

3. What is the difference between the Lorentz Transformation and Galilean Transformation?

The Galilean Transformation is a mathematical formula that describes how measurements of time and space change between two reference frames that are moving at constant velocities relative to each other, but it does not take into account the effects of relativity. The Lorentz Transformation, on the other hand, includes the effects of relativity and is a more accurate description of how measurements change between reference frames.

4. How is the Lorentz Transformation derived?

The Lorentz Transformation is derived from two postulates: the principle of relativity, which states that the laws of physics are the same in all inertial reference frames, and the constancy of the speed of light, which states that the speed of light in a vacuum is the same for all observers regardless of their relative motion. Using these postulates, the Lorentz Transformation can be mathematically derived.

5. What are some practical applications of the Lorentz Transformation?

The Lorentz Transformation has many practical applications, including in the fields of astrophysics, particle physics, and GPS technology. It is used to correct for the effects of time dilation and length contraction in high-speed objects, and to accurately calculate the positions of satellites in GPS systems.

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