5.1 Covariant Formulation of Charge Integral

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In the usual Minkowski space formulation with charge density _(~x; t), the

charge in a volume V is written as

Q =

Z

V

dV _ =

Z

V

dV ~r _ ~E by Maxwell's eqs. (5.1)

Q =

I

@V

d~S _ ~E by Gauss' law (5.2)

where surface integral is over boundary of V . Note that,

~r

_ ~E =

1 p

(3)g

@i

q

(3)gEi; dV = d3x

q

(3)g (5.3)

where

(3)

g is the determinant of the 3-metric, so

Z

dV ~r _ ~E =

Z

d3x @i

_q

(3)gEi

_

=

Z

dSiEi : (5.4)

The Lorentz covariant formulation uses the similar result

1 p

􀀀(4)g

@_

_q

􀀀(4)gF__

_

= D_F__ : (5.5)

The volume V is replaced by an arbitrary spacelike hypersurface _ (partial

Cauchy surface) with boundary @_. The volume element on _ is a non-

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spacelike co-vector (1-form) dS_. Given the current density 4-vector j_(x)

we write

Q =

Z

_

dS_j_ (5.6)

We can choose _ (at least locally) to be t = constant, in which case dS_ =

(dV;~0). Since j0 = _, we recover the previous expression for Q. Now use

Maxwell's equations. D_F__ = j_ to rewrite Q as

Q =

Z

_

dS_D_F__ (5.7)

=

1

2

I

@_

dS__F__ by Gauss' law (5.8)

where dS__ is the area element of @_. When _ is t = constant the only

non-vanishing components of dS__ are

dS0i = 􀀀dSi0 _ dSi (5.9)

in which case

Q =

I

@_

dSi F0i (5.10)

But F0i = 􀀀Fi0 = Ei, so we recover the previous formula.

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