Is the Reissner-Nordström repulsion real?

[bcrowell]

The googleable literature on the Reissner-Nordström repulsion is leaving me confused. This seems to be a contentious topic. I've found the following:

1. Grøn, "Non-existence of the Reissner-Nordström repulsion in general relativity," 1983, "The radial motion of a free neutral test particle in the Reissner-Nordström space-time is studied. It is shown that when the contribution of the electrical field energy to the mass of the central body is properly taken account of, there is no repulsion of the test particle inside the inner horizon of the space-time." http://adsabs.harvard.edu/abs/1983PhLA...94..424G

2. Qadir, "Reissner-Nordstrom repulsion," "It is pointed out that there was an error in the recent paper by Grøn claiming that there is no electro-gravitic repulsion in the Reissner-Nordstrom geometry. It is concluded that the earlier result of Mahajan, Qadir and Valanju still stands." http://adsabs.harvard.edu/abs/1983PhLA...99..419Q

3. Grøn, "Poincaré stress and the Reissner-Nordström repulsion," GRG 20 (1988) 123, " The Reissner-Nordström repulsion is shown to be a consequence of Poincaré stresses in a static, charged object. The strong energy condition implies that the Reissner-Nordström repulsions do not stop a neutral test particle, falling freely from rest at infinity in the fields of a charged body, before it hits the surface of the body. However, if the particle falls from rest at a sufficiently small height above the surface of the body, it will not reach the surface due to the Reissner-Nordström repulsion."

4. "Einstein's general theory of relativity: with modern applications in cosmology," Øyvind Grøn, Sigbjørn Hervik, 2007
[homework problem] "Consider a radially infalling neutral particle in the Reissner-Nordström spacetime with M>|Q|. Show that when the particle comes inside the radius r=Q2/m it will feel a repulsion away from r=0 (i.e. that d2r/dtau2<0 for tau the proper time of the particle). Is this inside or outside the outer horizon r+? Show further that the particle can never reach the singularity at r=0."

Unfortunately 1-3 are not freely accessible online. Are 3 and 4 showing that Grøn has changed his mind (possibly being won over by the arguments in 2)? But the specific statements made in 3-4 do not clearly contradict the ones made in 1.

It's not obvious to me that "repulsion" has a unique definition. The acceleration referred to in 4 is a coordinate acceleration, not a proper acceleration, so it's coordinate-dependent. It seems to me that on a general manifold, "repulsion" clearly can't be well defined. E.g., if a body at the north pole of a sphere causes a body at the south pole to accelerate in a certain direction, is that an attraction, or a repulsion? But I suppose it's possible to define "repulsion" in a more unabmiguous way if your spacetime is asymptotically flat...?

 

[sheaf]

Not really answering your question directly, but a related discussion which may be useful - namely the apparently repulsive field of a charged mass point is discussed by Pekeris:

http://www.pnas.org/content/79/20/6404.full.pdf"

 

[bcrowell]

Not really answering your question directly, but a related discussion which may be useful - namely the apparently repulsive field of a charged mass point is discussed by Pekeris:

http://www.pnas.org/content/79/20/6404.full.pdf"

Thanks, for the reply, Sheaf. However, when I click on the link I get an error. Do you have a reference so that maybe I could find an abstract by googling?

-Ben

 

[sheaf]

Thanks, for the reply, Sheaf. However, when I click on the link I get an error. Do you have a reference so that maybe I could find an abstract by googling?

-Ben

Bizarre, I can't get to it now either, but I could this morning. Sorry about that. Maybe there's a server problem. A google search which will bring up that link is "gravitational field point charge pekeris". If you don't have any luck and can't get to it, PM me and I can mail a copy.

Basically he discusses the surprising repulsive force that you get when M=0 in Reissner Nordstrom, then argues that this is unrealistic because with the charge present, there will be always an equivalent electromagnetic mass, consequently the field ends up being attractive !

 

[bcrowell]

I see, thanks!

But I guess this doesn't say anything about the nonzero mass case.

 

[sheaf]

I see, thanks!

But I guess this doesn't say anything about the nonzero mass case.

No, I guess it doesn't. It seems to be a restatement of your abstract number 1 above. His abstract states (sorry, the copy from the pdf screws up the mathematical notation, and I'm too lazy to Latex it !)

Adopting, with Schwarzschild, the Einstein gauge
(lg_mu_nul = -1), a solution of Einstein's field equations for a charged
mass point of mass M and charge Q is derived, which differs from
the Reissner-Nordstr0m solution only in that the variable r is replaced
by R = (r^3 + a^3)^1/3, where a is a constant. The Newtonian
gravitational potential psi = (2/c^2)(l - goo) obeys exactly the Poisson
equation (in the R variable), with the mass density equal to
(E^2/4mc^2), E denoting the electric field. Psi also obeys a second
linear equation in which the operator on psi, is the square root of
the Laplacian operator. The electrostatic potential phi (= Q/R),
A, and all the components of the curvature tensor remain finite
at the origin of coordinates. The electromagnetic energy of the
point charge is finite and equal to (Q2/a). The charge Q defines
a pivotal mass M* = (Q/G^1/2). IfM < M*, then the whole mass
is electromagnetic. If M > M*, the electromagnetic part of the
mass M_em equals [M - (M^2 - M*^2)^1/2], whereas the material part
of the mass M_em equals (M^2 - M*^2)^1/2. When M > M*, the constant
a is determined following Schwarzschild, by shrinking the
"Schwarzschild radius" to zero. When M < M*, a is determined
so as to make the gravitational acceleration vanish at the origin.

Good point you make about defining repulsion in a covariant manner.