Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The set of Lorentz boosts and space rotations form a group

  1. Dec 2, 2011 #1
    Ok. I understand that the set of Lorentz boosts and space rotations is equivalent to the set of Lorentz transformations. I understand that they form a group, but what I cannot seem to grasp is this. What the explicit form of such 4x4 matrices? One needs to know this in order to show that the properties of a group hold. The way I thought they were represented is as follows:

    [itex]L_{x}[\beta]=\left(\begin{array}{cccc} \gamma & -\beta \gamma & 0 & 0 \\-\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)[/itex]

    The Lorentz boosts in the y and z directions would have similar elements in different entries of the matrix. Is this all I have to work with to show that the Lorentz transformations form a group?
  2. jcsd
  3. Dec 3, 2011 #2

    Vanadium 50

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2017 Award

    You don't absolutely need the matrix form to prove things are a group. If you still want to do it this way, you need to find the matrix for an arbitrary boost, not just the boost along one axis, which is a mess - why you usually don't see it.
  4. Dec 3, 2011 #3


    User Avatar
    Science Advisor
    Gold Member

    If you really want to see the general form, you can find it on Wikipedia at Lorentz transformation#Boost in any direction. See also the end of the next section "Composition of two boosts" which gives the same result using 3-vectors.

    (Note that this is a boost in any arbitrary direction, but doesn't include any rotation of the spatial axes.)
  5. Dec 5, 2011 #4
    So this general form represents a boost in any direction. If we multiply that by some arbitrary rotation along the spatial axes will we in turn find the boost in some specified direction? I feel like I am confusing myself.
  6. Dec 5, 2011 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    "boost in any arbitrary direction" means "boost in some specified direction". The specified direction is the direction of the velocity (3-)vector [itex]\vec v[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook