Lorentz transformations on spacetime

[helpcometk]

Homework Statement

A3. Show that the Lorentz transformations on a spacetime 4-vector can be written as
x'μ = (Lμν)*(χν)
. Find the matrix L. Prove that (in matrix notation) Lτ gL = g where g is
the Minkowski spacetime metric.

Homework Equations

Any help suggesting at least equations will be appreciated. μ , τ and ν are exponents.

The Attempt at a Solution

 

[tiny-tim]

welcome to pf!

hi helpcometk! welcome to pf!

(try using the X² and X₂ icons just above the Reply box )

Show that the Lorentz transformations on a spacetime 4-vector can be written as
x'μ = (Lμν)*(χν)

you should be able to write out L just by looking at the standard Lorentz transformation equations

(and btw, they're not exponents, they're superscripts, or just indices )

 

[helpcometk]

Thanks for the reply Tim.
Specifically which expressions i have to take into account?

 

[tiny-tim]

The standard Lorentz equations …

the ones with x' y' z' t' on the left and x y z t on the right!

 

[paranormal]

The Lorentz transformations are mathematical equations that describe how the coordinates of an event in one inertial reference frame are related to the coordinates of the same event in another inertial reference frame. These transformations are based on the principles of special relativity and are used to account for the effects of time dilation and length contraction.

The equation x'μ = (Lμν)*(χν) represents the transformation of a spacetime 4-vector, where x'μ is the transformed vector, Lμν is the transformation matrix, and χν is the original vector. To find the matrix L, we can use the Lorentz transformation equations:

x'0 = γ(x0 - vt)
x'1 = x1
x'2 = x2
x'3 = x3

where γ is the Lorentz factor and v is the relative velocity between the two reference frames. These equations can be written in matrix form as:

x' = Lx

where x = [x0, x1, x2, x3] and x' = [x'0, x'1, x'2, x'3]. This gives us the following transformation matrix:

L = [γ -γv 0 0;
-γv γ 0 0;
0 0 1 0;
0 0 0 1]

To prove that Lτ gL = g, we can multiply the matrices:

Lτ gL = [γ -γv 0 0;
-γv γ 0 0;
0 0 1 0;
0 0 0 1]
[1 0 0 0;
0 -1 0 0;
0 0 -1 0;
0 0 0 -1]
[γ -γv 0 0;
-γv γ 0 0;
0 0 1 0;
0 0 0 1]

= [γ -γv 0 0;
γv -γ 0 0;