🚀 go-pugleaf

 >>> Opposite charges are attracted to each other and like charges are
 >>> repelled.  Why?

-*- Thus Spake Connie Patten:
 C> Every question could be answered by saying "it just does", but
 C> usually in physics we go to the highest level model so far
 C> devised.  In earlier days (pre-Franklin) we could have said
 C> "things just attract".  But now we can go a bit further and look
 C> at all forces as exchanges of particles, such as gravitons and
 C> photons, and distinguish between attractive and repulsive
 C> interactions with the help of Feynmann diagrams.

Indeed. The deepest resason I know of as to why like charges repel is
due to the spin of the exchanged photon. Basically photons have spin 1
and particles interacting via spin 1 particles repel if their charge
is the same. For the exchange of spin 0 particles (such as the
effective theory of the nuclear interaction of protons and neutrons
via pions), like charges attract.

The reasoning comes from relativistic field theory (See for
instance, S.L. Shapiro and S.A. Teukolsky, "Black Holes, White Dwarfs,
and Neutron Stars", Appendix D):

In the following d_x = d/dx and d_t = d/dt,... (d_x)^2 = d^2/dx^2,...
...and and the speed of light, c = 1; of course. 

In these units Maxwell's eqs. involving charge, where E = (E_x, E_y,
E_z), B = (B_x,...)  rho = charge density, j = (j_x,...) = current
density, are:

  div E = 4 pi rho  
            --> d_x E_i(x,y,z,t) + d_y E_i + d_z E_i = 4 pi rho(x,y,z,t)

  curl B - d_t E = j

In relativistic form these are:

  -(d_t)^2 A_i + (d_x)^2 A_i + (d_y)^2 A_i (d_z)^2 A_i  = -4 pi J_i   (1)

where A = (A_t, A_x, A_y, A_z), and J = (rho, j_x, j_y, j_z), and the
above is satisfies for each i = (t,x,y,z). A is the relativistic
electromagnetic potential and is a 4-vector.  A_t is just the
electrostatic potential Phi so that E_i = -d_i A_t - d_t A_i

Eq. (1) can be derived from deeper principles, namely the Lagrangian
formalism for a spin 1 vector field satisfying a gauge symmetry. Sorry
for this statement, but it is necessary so that someone can't say that
I just put the minus in front fo the 4 pi (which is in fact the source of
the answer to the original question) to get the right answer.

In the absence of charges or currents, J = 0, eq (1) is just the free
wave equation the (vector) field A_i, ie. each component of A_i
satisfies

  -(d_t)^2 A_i + (d_x)^2 A_i + (d_y)^2 A_i (d_z)^2 A_i = 0   (2)

whose solutions are A_i = const sin(x+y+z - t), or const cos(x+y+z -
t), describing waves of velocity 1 [remember c = 1]. These are photons
(quantum mechanically A_i(x,t) is the probability wave function of the
photon).

If we take a static situation (d_t A = 0), with no magnetic fields
(for exmaining the electrostatic energy between two charges), then we
can take A_i = (Phi, 0, 0, 0). A charge at the origin has a current
density J = (rho(x), 0, 0, 0), where rho(x) is a density function
which is highly peaked at x=y=z=0, say a very narrow gaussian or bell
curve centered at the origin.

Then eq. (1) becomes

  [ (d_x)^2 + (d_y)^2 + (d_z)^2 ] Phi(x) = -4 pi rho(x)      (3)

which can be solved for Phi(x) in the limit that rho(x) becomes
infinitely thin and satisfies 
 +infty  
    /
    | dxdydz rho(x) = q
    /
 -infty

Then

  Phi(x) = q/sqrt(x^2 + y^2 + z^2)  = q/r

the usual electrostatic law. Now the electrostatic energy of another
point charge q' in the field of the charge q above is

  V_qq' = q' Phi  = q q'/r

which is POSITIVE for LIKE charges q and q'. Thus increasing r by
moving them apart, decreases the electrostatic energy. This is what
the charges want to do (minimize their energy), so they repel. In
other words it requires energy to push them together.

For a pion field which is a spin 0 field, as I mentioned above, the
resulting eq for Phi would be

     Phi = - q q'/r

Hence like nuclear charges attract, in the pion exchange approximation.

As long as I'm here, let me point out an interesting effect of the
`real' theory of nuclear forces, namely quantum chromodynamics, or the
theory of quarks and gluons. In comlete analogy with the above
eqs. for photons and electrons (or electric charges), the quarks
interact via the color (stong) force, which is mediated via gluon
exchange. However a VERY important difference is that the gluon
potential field which we'll call G_t (analogous to A_t = Phi)
satisfies the following eq. where the quark charge density is Rho(x):

  [(d_x)^2 + (d_y)^2 + (d_z)^2] G_t(x)  + q M G_t(x)  = -4 pi Rho(x)

M is a certain matrix. (Actually G_t is a 3x3 matrix corresponding to
the 8 different gluons) [compare eq (3): (d_x)^2 Phi +... = -4 pi rho,
for photons].

In the absence of quarks, Rho(x) = 0, and the equation for G_t is:

   [(d_x)^2 + (d_y)^2 + (d_z)^2] G_t(x)  + q M G_t(x)  = 0

or 

   [(d_x)^2 + (d_y)^2 + (d_z)^2] G_t(x)  =  q M G_t(x)    (4)

Since the volume integral of [(d_x)^2 + (d_y)^2 + (d_z)^2] G_t [or in
electromagnetism [(d_x)^2 + (d_y)^2 + (d_z)^2] Phi] is the flux of E
thru the surface of the volume and equals the charge contained in that
volume (Gauss' law!), eq. (4) for quarks says that gluons themselve
generate "color" electic flux. Thus gluons are colored (or charged),
completely unlike photons, which are electrically neutral. By analogy,
an electrically charged particle produces electric field lines which
radiate from the charge, but do not interact themselves. Quarks on the
other hand generate color electric field lines, which are themselves
colored and hence eminate their own field lines. These in turn eminate
more field lines, ad infinitum... This makes QCD a seriously
non-linear force. This above effect means that there is a large amount
of mass-energy (via E=mc^2) in the gluon field surrounding the quarks
inside protons and neutrons. In fact a large fraction (>50%, and the
subject of current computation) of the mass of each proton and neutron
comes from the mass-energy of the gluon (color electromagnetic) fields
holding the constituent quarks together.

Gravity is attractive for similar reasons; basically in the weak field
limit (ie. away from black holes and neutron stars, etc.), only the
temporal part of the spacetime metric is appreciably curved, ie. space
remains flat, while geodesics (light-lines) are curved toward the
massive object in the timelike direction. Hence as we move forward in
time, following the geodesic, we curve (in the time direction) into
the massive object, seeming to fall toward it. Deriving the
gravitational potential eqs. as done above for electrostatics yields
a pion like equation, ie. like charges attract. This plus, the fact
mass is always positive, means gravity is an attractive force.

It's not always true though that gravity attracts as you might think. For
instance the gravitational field produced by a very (infinitely) long,
thin, massive thread (cosmic string), is such that it makes the
spacetime conical around it, with the string lying along the apex of
the spacetime cone. This has the effect that the spacetime is FLAT
around the thread, ie. no gravitational attraction toward the thread,
even for very very massive threads. However circumnavigating the
thread at a distance R, astronauts would notice a deficit angle,
ie. the distance traveled would not be 2 pi R, but 2 pi R - 8 pi G m R
where G is Newton's const, and m is the mass/length density of the
thread. Thus one wouldn't fall toward a cosmic string; this subtle point
of general relativity was missed in a certain Star Trek (NG) episode I
once saw!

Cheers,   --Jim


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      James E. Hetrick            Institute for Theoretical Physics
     hetrick@phys.uva.nl               University of Amsterdam