Relativistic Invariance Part Two

[Ed Quanta]

I am asked to prove that the d'Alembertian operator (the 4 dimensional Laplacian operator) |_|^2 is a lorentz invariant operator. Do I just multiply the Lorentz transformation matrix by the second partial derivatives with respect to four space?

 

[turin]

Originally posted by Ed Quanta
I am asked to prove that the d'Alembertian operator (the 4 dimensional Laplacian operator) |_|^2 is a lorentz invariant operator. Do I just multiply the Lorentz transformation matrix by the second partial derivatives with respect to four space?

I suppose you could do that. But I would just show that it is a scalar. Show that it is a contraction of a 1^st rank tensor with itself. Use the Minkowski metric tensor to raise the index of one of the partial derivatives and it should be obvious.

 

[GSXR750]

To prove that the d'Alembertian operator is Lorentz invariant, we need to show that it remains unchanged under Lorentz transformations. This means that if we apply a Lorentz transformation to the input of the operator, the output should remain the same.

To do this, we can start by considering the d'Alembertian operator in its covariant form:

Δ = ημν ∂^2/∂xμ∂xν

where ημν is the Minkowski metric and xμ represents the four space coordinates.

Now, we can apply a Lorentz transformation to the coordinates xμ, which can be represented by a 4x4 matrix Λ. This transformation can be written as:

x'μ = Λμνxν

where x'μ represents the transformed coordinates.

Substituting this into the covariant form of the d'Alembertian operator, we get:

Δ' = ημν ∂^2/∂(Λμαxα)∂(Λνβxβ)

= ημν ΛμαΛνβ ∂^2/∂xα∂xβ

Since the Lorentz transformation matrix Λ is constant, it can be taken outside the derivatives. Also, since the Minkowski metric ημν is symmetric, we can rewrite it as:

ημν = ηνμ

Substituting these into the above equation, we get:

Δ' = ηνμ ημν ΛμαΛνβ ∂^2/∂xα∂xβ

= ημν ημν ΛμαΛνβ ∂^2/∂xα∂xβ

= ημν ημν ∂^2/∂xα∂xβ

= Δ

Thus, we can see that the transformed d'Alembertian operator is equal to the original operator, proving that it is Lorentz invariant. This also means that the operator is independent of the choice of coordinates and is applicable in all inertial frames of reference.

In conclusion, to prove that the d'Alembertian operator is Lorentz invariant, we need to show that it remains unchanged under Lorentz