To prove that a three-dimensional volume element is invariant under Lorentz transformations, we need to show that the determinant of the transformation matrix is equal to 1. Lorentz transformations are a specific type of coordinate transformation that preserve the spacetime interval in special relativity.
Let's consider a Lorentz transformation in four-dimensional spacetime with coordinates (x, y, z, t) and (x', y', z', t'). The transformation can be expressed as:
x' = γ(x - vt) y' = y z' = z t' = γ(t - vx/c^2)
where γ represents the Lorentz factor, v is the relative velocity between the reference frames, and c is the speed of light.
Now, let's define a volume element in three-dimensional space as dV = dx dy dz. To determine its transformation under Lorentz transformations, we need to find the corresponding expression in the primed coordinates, dV'.
We can express dV' as follows:
dV' = dx' dy' dz' = (d/dx')(d/dy')(d/dz')(x', y', z') = (d/dx')(d/dy')(d/dz')(γ(x - vt), y, z) = (d/dx')(d/dy')(d/dz')(γx - γvt, y, z) = γ(d/dx')(d/dy')(d/dz')(x - vt, y, z) = γ(dx dy dz) = γdV
where we have used the chain rule for partial derivatives and the fact that the derivatives of y and z with respect to themselves are equal to 1.
Now, we can compute the determinant of the transformation matrix to determine if the volume element is invariant:
det(J) = det(dV'/dV) = det(γ) = γ^3
For the volume element to be invariant, we require det(J) = γ^3 = 1. This implies that γ = 1, which corresponds to the case when the relative velocity between the reference frames is zero. In other words, when the velocity is zero, the volume element is invariant under Lorentz transformations.
This result indicates that under Lorentz transformations, the volume element undergoes a contraction or expansion factor of γ^3 depending on the relative velocity between the frames. Only when the relative velocity is zero does the volume element remain unchanged.