Completely General Lorentz Transformation

In summary: Thanks very much for that. You've just saved a good fraction of my life:redface:In summary, the matrix form of the Lorentz Transformation does exist, but it is very cumbersome to write out.
  • #1
OniLink++
12
0
Does anyone have the matrix form of the completely general Lorentz Transformation, with rotations AND boosts, or does it not exist?
 
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  • #2
It can certainly be written out, but it would be very cumbersome. The exact form would depend on how you decided to parametrize it.
 
  • #3
bcrowell said:
It can certainly be written out, but it would be very cumbersome. The exact form would depend on how you decided to parametrize it.

Could you specify, exactly, what you mean? When you parametrize it, it should just be 3 rotations and 3 boosts, correct?
 
  • #4
OniLink++ said:
Could you specify, exactly, what you mean? When you parametrize it, it should just be 3 rotations and 3 boosts, correct?

E.g., you could parametrize the boost vector by its magnitude and two angles giving its direction. Or you could parametrize it by its three components.

Since boosts and rotations don't commute, you could parametrize by doing the operations in either order.
 
  • #5
bcrowell said:
E.g., you could parametrize the boost vector by its magnitude and two angles giving its direction. Or you could parametrize it by its three components.

Since boosts and rotations don't commute, you could parametrize by doing the operations in either order.
Oh, ok then. It just got a lot more complicated. Hrmm... how about the Lorentz Transformations with the 3 components and the rotations applied before the boosts?
 
  • #6
Why don't you write out a boost matrix and a rotation matrix and just multiply them together? They do form a group. However the order in which boosts and rotations happen is critical. It sounds like another tedious calculation.
 
  • #7
If you really want to see it written out in all its ugly glory, I'd suggest using symbolic math software. Maxima is free and open-source, and I have some material in this book http://www.lightandmatter.com/genrel/ on how to apply it to relativity. See section 2.5.3 for some similar examples.
 
  • #8
bcrowell said:
If you really want to see it written out in all its ugly glory, I'd suggest using symbolic math software. Maxima is free and open-source, and I have some material in this book http://www.lightandmatter.com/genrel/ on how to apply it to relativity. See section 2.5.3 for some similar examples.

Thanks very much for that. You've just saved a good fraction of my life:redface:
 

1. What is a Completely General Lorentz Transformation?

A Completely General Lorentz Transformation (CGLT) is a mathematical equation that describes how coordinates and physical quantities change when viewed from different reference frames in special relativity. It includes all possible transformations, making it a more comprehensive version of the standard Lorentz Transformation.

2. What is the difference between a CGLT and a standard Lorentz Transformation?

A standard Lorentz Transformation only considers transformations between inertial frames of reference, while a CGLT includes transformations between any type of reference frame, such as accelerating or rotating frames. This makes it a more universal and accurate tool for studying special relativity.

3. How is a CGLT derived?

A CGLT is derived using the principles of special relativity, which state that the laws of physics should be the same for all observers in inertial frames of reference. It involves mathematical equations that account for the effects of time dilation, length contraction, and simultaneity.

4. What are some practical applications of CGLTs?

CGLTs are used in many areas of physics and engineering, such as in particle accelerators, GPS systems, and spacecraft navigation. They are also important in theoretical studies of general relativity and cosmology.

5. Are there any limitations to CGLTs?

While CGLTs are a powerful tool for understanding special relativity, they do not account for the effects of gravity or acceleration. In these cases, more complex transformations, such as those in general relativity, must be used. Additionally, CGLTs only apply to the special case of flat spacetime and cannot be used for curved spacetime.

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