 #1
 114
 17
 Homework Statement:
 Prove the invariance of the charge in any inertial frame of reference
 Relevant Equations:

Heaviside's function
Fourcurrent
I followed a demonstration in one of my electromagnetism books, but it is not clear to me.
My problem is at the starting point.
The book begins by considering the office defined in the following way:
$$Q=\int d^4xJ^\alpha(x)\partial_\alpha\theta(\eta_\beta x^\beta)$$
where ##\eta_\beta=(1,0,0,0)##
How do I relate to the definition I know?
$$Q=\int d^3x J^0(x)$$
I know that the derivative of the Heaviside function is the Dirac delta, but evaluated where? Are the following accounts I have done correct?
$$Q=\int d^4xJ^\alpha(x)\delta(t)$$
My problem is at the starting point.
The book begins by considering the office defined in the following way:
$$Q=\int d^4xJ^\alpha(x)\partial_\alpha\theta(\eta_\beta x^\beta)$$
where ##\eta_\beta=(1,0,0,0)##
How do I relate to the definition I know?
$$Q=\int d^3x J^0(x)$$
I know that the derivative of the Heaviside function is the Dirac delta, but evaluated where? Are the following accounts I have done correct?
$$Q=\int d^4xJ^\alpha(x)\delta(t)$$