Journal of Modern Physics, 2012, 3, 1639-1646
http://dx.doi.org/10.4236/jmp.2012.330201 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
Radial Electric Field in Tokamak Plasmas as a Physical
Consequence of Ehrenfest’s Paradox
ITER Domestic Agency—“ITER Project Center”, 123182, pl. Academician Kurchatov, 1, p. 3, Moscow, Russia
Email: a.romannikov @ iterrf.ru
Received August 28, 2012; revised September 27, 2012; accepted October 3, 2012
A simplified form and some possible theoretical resolutions of the so-called Ehrenfest’s Paradox are described. A rela-
tion between physical consequences of this relativistic paradox and charge density ρ of tokamak plasma is shown.
Plasma experiments which could resolve the Ehrenfest’s Paradox are presented.
Keywords: Tokamak; Ehrenfest’s Paradox
Ehrenfest’s Paradox was presented in  for the first
time in 1909. Detailed historical, physical, and geomet-
rical descriptions of Ehrenfest’s Paradox can be found in
 and references therein. For our experimental purposes,
let’s present Ehrenfest’s Paradox in the following simpli-
fied form. Consider two thin rings with radii 1 and
2 (and 12
). The second ring is accelerated by an
external force so that any point on the ring has linear
velocity . The observers in the laboratory frame
measure circumferences of these rings (1 and 2) and
radii. The question is: what are the results of these mea-
surements taking into account relativistic effects?
The analysis of relativistic contraction and of related
effects in non-inertial rotating frame (including geomet-
rical arguments, purely kinematical and dynamical
grounds) is very complex. But physical consequences of
these complex analyses are sufficiently simple. We can
basically present three possible theoretical hypotheses for
the results of circumference measurements.
The first hypothesis, which is not widely accepted,
will only be mentioned here. Pursuant to this hypothesis,
both the radius of the rotating ring 2 and its circum-
ference 2 contract in the laboratory frame by relativis-
tic effects, so that their ratio remains equal
and 2π2πRR L R RL
2π [3,4]. From the contemporary point
of view, this resolution of the Ehrenfest’s Paradox is
highly unlikely see, for example .
For most researchers, a more acceptable resolution to
Ehrenfest’s Paradox in the laboratory frame is as follows:
We shall refer to it as “hypothesis 1”. Though the nu-
ances of “hypothesis 1” are not so important for our dis-
cussion, we will consider this hypothesis in more detail.
There are two possible ways to obtain this result. Let’s
introduce the circumference and radius 2
rotating ring in the rotating reference frame (with the
linear velocity at the radius ).
The first and less accepted approach is based on the
assumption of “no Lorentz contraction for rotating refer-
ence frame” [5-8]. This would mean that
The second and more widespread approach is based, in
a simplified form, on the following ideas. According to
, and . We shall
refer to it as “condition (1)”. The authors (see, for exam-
ple, [10-16]), through the analysis of a metric tensor for a
rotating reference frame, have come to the following
conclusion. The condition (1) is fulfilled but the rotating
observers would see the increase of the circumference in
the form of 1
. The laboratory observers would
see a relativistic contraction of a rotating ring in the form
, and 122
RR . It is
important to emphasize that the majority of the advo-
cated of the standard resolution of the Ehrenfest’s Para-
dox a priori believe that the geometry of the rotated ring
in the laboratory inertial frame is Euclidean, 22
and they analyze mainly the geometry of the rotating ring
in the rotating frame.
Let’s follow the foregoing logic of the more wide-
spread approach, but assume that the circumference
measured by the observers in the rotating reference frame
opyright © 2012 SciRes. JMP
does not change due to the rotation and is equal to the
initial length of the non-rotating ring 1 (see, for exam-
ple, [17-19]). Then, the laboratory observers would con-
We shall use the Equation (2) refer to it as “the hy-
pothesis 2”. It is necessary to emphasize, that in the case
of “hypothesis 2” the geometry of the rotating ring in the
laboratory frame is Non-Euclidean.
Unfortunately, it is practically impossible to resolve
the Ehrenfest’s Paradox by observing the real rotating
disks or rings because of centrifugal forces that lead to
significant deformation of the rings. Thus, it is difficult
to measure very small relativistic effects against the
background of the centrifugal deformation at accessible
rotation velocities and size of rings. Note that the first
experimental studies of the measurement of physical
consequences of the Ehrenfest’s Paradox were presented
only in , 2011.
2. Physical Consequences of the Ehrenfest’s
Paradox in Tokamak Plasma
Recently, in [19,20] the effect of relativistic contraction
of an “electron ring” circumference in steady state toka-
mak plasma rotating in toroidal direction with current
velocity e has been analyzed. Let be the minor
radius of a tokamak magnetic surface, see Figure 1. The
minor tokamak radius was assumed to be much less
than the major radius , where
, and electron
toroidal rotation velocity was assumed to be moderate, so
that it would be possible to exclude centrifugal forces in
the momentum balance of plasma . The toroidal rota-
tion velocity of “the ion ring” i, as a general rule, is
much less than the toroidal rotation velocity of the “elec-
tron ring”, a fact known from experiment . It is as-
sumed that at the moment of plasma creation (with no
current) from neutral gas (hydrogen or deuterium), the
electron density and the ion density
Figure 1. A sketch of tokamak.
equal, and the difference between the total number of
electrons and total number of ions does not vary during a
discharge and is equal to 0 in the tokamak chamber. We
can use the last assumption because neutral gas is in-
jected into the tokamak chamber, and neutral gas is
pumped from the tokamak. Electrons and ions can move
and can be redistributed in the minor radius direction of
tokamak plasma after the occurrence of a current.
One can note that the peak value of the experimentally
measured radial electric field r in tokamak coin-
cides with the occurrence in plasma a small difference
between electron density e and ion density
in the laboratory frame [19,20], of the order of
; and for the ohmic modes:
. We shall refer to it as “con-
Let us assume at the beginning, that the total number
of electrons and ions does not vary during a discharge,
electron density and ion density, ion toroidal rotation
velocity and electron rotation velocity values are constant
and do not depend on the tokamak minor radius . In
this case, initial electron density (before plasma current)
is and ion density is , where .
Therefore, this can be considered as two thin rings (the
electron ring and the ion ring), originally having the
same circumference , which are
brought to different toroidal rotation velocities, e and
i, where ie
V. The situation is similar to the one
considered above in the context of Ehrenfest’s Paradox.
Following the ideas summarized by E. L. Feinberg in
, it is not necessary to investigate in detail the proc-
ess of electron ring and ion ring acceleration to given
velocities. One can compare only an initial and a final
states. In this case, measurement of the part of the elec-
tric field which can arise, for example, within the frame
of “the hypothesis 2”, is much easier than the investiga-
tion of the deformations of a rotating rigid ring. The rea-
son is as follows: on one hand, the current electron ve-
locity can reach hundreds km/s, on the other hand, possi-
ble deformations of the “electron ring” due to centrifugal
force will lead only to the occurrence of dipole compo-
nents in the electric field associated with minor change of
radius of the rotating ring. Relativistic contraction of the
ring circumference without change of radius and conser-
vation of total electron number (under the “the hypothe-
sis 2”) can lead to the occurrence of a monopole compo-
nent in electric field which is relatively easy to measure,
as it will be shown below.
Following , let us consider the change in density
in tokamak plasma under “hypothesis 1”
Copyright © 2012 SciRes. JMP
A. ROMANNIKOV 1641
and “hypothesis 2”.
It is obvious from Equation (1) that tokamak charge
is not changed after the appearance of the
toroidal rotating electron (current) ring and ion ring in
plasma, that is:
Following , it is possible to show that in this case
rotation creates a finite charge density
in a tokamak
plasma.. If we ignore higher-order terms in
sion, we can write:
where e and i are: the electron density in the rotat-
ing frame with velocity e
V and the ion density in the
rotating frame with the velocity i, respectively. We
have taken into account the framework of “hypothesis 2”:
Change of the charge density in this case is only asso-
ciated with relativistic change of the denominator in the
expression for the density. Let us note that
on parameters measured in the laboratory frame: the cur-
rent density , the electron density and the ion tor-
oidal rotation velocity .
So, we have calculated the charge density
point inside a stationary tokamak chamber in the labora-
tory frame. One can “forget” about the particular nature
of the charge density
relating to the Non-Euclidean
geometry of a rotating electron (ion) ring of the tokamak
plasma in the laboratory frame, and can instead use the
Poisson equation (being the relativity theory equation in
the laboratory frame) with
taken from Equation (4)
to calculate the electrostatic radial electric field in toka-
mak plasma. Hence, r in a simple tokamak plasma
is created by two relativistic terms in the density of
, Equation (4), which appeared in the labora-
In case of a real tokamak, plasma parameters depend
on the minor radius of magnetic surfaces. Consideration
of such dependence for the purpose of calculation of to-
kamak plasma charge density is shown in [19,20] in de-
tail for 1
=. The principal point here is the consid-
eration of each nested magnetic surface with plasma dis-
tributed in a thin, hollow ring.
Following , we can rewrite Equation (3) in the
case of “hypothesis 1” so as to take into account the
processes of redistribution of electrons and ions on the
minor radius by plasma diffusion (convection):
Due to electron and ion diffusion (change of the nu-
merator in the expression for the density), additional
volume charge densities in plasma can arise, and this can
be expressed by the term
en rn r
Hence, only the diffusion (convection) of ions and elec-
trons could create
In the case of “hypothesis 2”, we can rewrite the
Equation (4) in the following form:
in tokamak plasma.
en rn r
Equation (6) has two relativistic terms, and “condition
(2)” is not merely casual coincidence in this case. Let us
emphasize again that
depends on the plasma pa-
rameters measured in the laboratory frame: the current
nr, the electron density e, the ion tor-
oidal rotation velocity
i and the diffusion (convec-
tion) term. So, we have calculated the charge density
in each point inside a stationary tokamak chamber in the
laboratory frame, Equation (6). As emphasized above,
one can “forget” about the particular nature of the charge
relating partially to the Non-Euclidean ge-
ometry of a rotating electron (ion) ring of the tokamak
plasma in the laboratory frame, and can use the Poisson
taken from Equation (6) to calculate
the electrostatic radial electric field in the tokamak
plasma. In our consideration, the diffusion (convection)
term is not determined. We can mention one integral
property of the diffusion (convention) term, which is a
consequence of the physical assumption that the differ-
ence between the total number of electrons and total
number of ions does not vary during a discharge and is
equal to 0 in a tokamak chamber. It is:
en rn rV
Copyright © 2012 SciRes. JMP
where ch is the volume of the toroidal tokamak cham-
ber. The diffusion (convention) term can be determined
within the frame of different approaches, see .
3. T-11M Tokamak Experiment
Having accepted “hypothesis 2”, we have seen that
plasma current creates relativistic volume charge density
. The second relativistic term on
the right-hand side of Equation (6) for plasma usually is
more than five times smaller than
third term is the symmetrical redistribution of charges by
the diffusion (convection) along the minor radius in a
plasma chamber. Thus
can be crucial in
the creation of r, especially at the beginning of a
discharge, and if plasma has modulated current. For to-
kamak plasma contained in a metallic chamber,
can modify the electric potential of the chamber
jr en r
with respect to the ground; see the Equation (7). The
electric potential of the chamber is proportional in this
case to the volume of plasma, the averaged value of
, the electric capacitance of a closed metal-
lic tokamak chamber C, and relates to chamber RC time.
If one wants to measure the potential of the tokamak
(t) during the discharge, one can expect two
options. In the case of “hypothesis 1”, there would be no
(t) due to the plasma current, i.e.
(t) = 0,
see the Equation (7); in the case of “hypothesis 2”, the
potential of the chamber will change proportional to
Thus, measurement of tokamak chamber potential
(t) during discharges could resolve Ehrenfest’s Para-
dox in principle.
The first series of special experiments for electric po-
tential measurements at several points in a tokamak
chamber were carried out at T-11M tokamak (main
plasma parameters in presented shots were: deuterium
plasma, the average steady-state electron density <ne>
~1013 cm−3, the plasma current Ip ~ 50 kA, r = 20 cm, R =
70 cm) with modulated current [19,23]. The example of
the typical measurement is shown on Figure 2. For the
purpose of calculation of the theoretical dependence (tri-
angles and dashed curve in Figure 2), we have used: 1)
experimental data for plasma current and electron density;
2) experimental chamber resistivity R = ~4 MOhm; 3)
experimental chamber RC time ~2.5 ms. Electron density
diagnostics did not give us adequate information for sev-
eral milliseconds at the beginning of discharge. We have
extrapolated the electron density growth during the first
~8 ms by a linear function.
One can see satisfactory coincidence of theoretical
calculation results based on “ hypothesis 2” with the ex-
4. Physical Consequences of Equation (6)
and Some Tokamak Experiments
Radial electric field
r plays an important role in
various modes of improved plasma confinement in a to-
kamak . Some of those modes will be used in the
thermonuclear reactor ITER , which is currently un-
Because “condition 2” exists in tokamak plasma, we
usually cannot use the Poisson equation for the calcula-
r—it is not possible to measure or even to
calculate independently e and in the labo-
ratory frame with required accuracy.
Unfortunately, another approach—the ambipolarity
equation for radial flows—cannot be used as well, since
the ambipolarity emerges automatically from the toroidal
symmetry of the considered configuration [26,27]. For
those reasons, more complex approaches to the estima-
r are used. There are many successful
methods of calculation of r, see, for example, [21,
28,29] and references in [21,28]. Historically, the first
r was presented in . The value
r was calculated by taking into account the
higher orders in the expansion in small parameter of the
plasma kinetic theory. One use common approach for
rthat takes into account the influence of a small
fraction of the locally trapped ions in the ripples of the
toroidal magnetic field on the formation of
32]. It is necessary to emphasize that ion radial diffusion
is often considered the most important determinant of
r, see, for example . Effects of viscosity be-
tween main ions and neutrals in tokamak plasma  can
sometimes self-consistently estimate on the pe-
Copyright © 2012 SciRes. JMP
Figure 2. Time dependence of plasma current Ip(t) and the tokamak chamber electric potential
(t) during the discharge,
T-11M tokamak . Solid curve is the experimental dependence of
(t); triangles and dashed curve are the theoretical
right © 012 SciRes.
riphery of the plasma. The influence of large gradients of
the radial electric field on the shape of ion trajectories
 sometimes allows qualitative explanation of some of
the issues of the formation of r in modes with
internal thermal barriers. Some authors introduce ano-
malous viscosities in the standard neo-classical theory
It is necessary to note that it is very difficult to mea-
sure accurately the radial electric field profile in a toka-
This complexity gives rise to situations where experi-
mental results concerning
r are difficult to explain
using simple approaches, see for example [36-40]. There-
fore, it is often necessary to use complex contemporary
theories. It will be shown below that same experimental
results can be explained by enough simple approach
based on Equation (6), too.
Let us assume that the charge density, Equation (6),
creates r in tokamak plasma. To compare the re-
sults of this approach to rwith the results of some
actual experiments, we take into account the additional
plasma equation, see, for example,  (which is derived
from radial equilibrium of forces on a magnetic surface):
Vr B rVr B
Vr in the article
are the velocities of the poloidal and toroidal rotation of
plasma ions, respectively, and hence, of the plasma as a
whole (the velocities are low enough so the centrifugal
effect may be omitted); is the speed of light, is
the electron charge;
i and are the density
and pressure of plasma ions; p
poloidal and toroidal magnetic fields. To establish the
main features of relations between r and
as well as other plasma parameters, we may ignore the
weak poloidal dependence of parameters in Equation (8)
R, where is the major radius of the to-
is the poloidal angle; 1
). A relation of
type (8) is always true when the plasma is in the steady
state (only these states are considered below). For the
sake of simplicity, let’s assume that magnetic surfaces
are nested cylinders with small radii , and the plasma
consists of electrons and, for example, deuterium ions.
Let’s also assume that the velocity of poloidal rotation
may be taken from experiments or derived from neo-
We can express
r and t independently
using Equation (6), the Poisson equation and Equation
VrV rV r
Br enrBrrBr enrBr
We take into account the assumption that the toroidal
rotation velocity at the plasma boundary with
close to zero . One can find an example of the deri-
vation procedure and interpretation of equations similar
to Equations (9) and (10) in .
Below we present important quantitative and qualita-
tive correlations of theoretical results (using Equations
(6), (9) and (10)) with tokamak experimental data [20,36-
The first important conclusion from Equation (10): if
is not close to zero
in the plasma core then a toroidal ion beam is created in
that region (its velocity logarithmically tending to infin-
ity [20,23]). So, we have to suppose that
at the core of
Consequently, one can write the Poisson equation for
and calculate the radial electric field profile:
in the plasma periphery; and has the minimum (negative
It is possible to derive Equation (11) from Equation (9)
directly, see .
As very good know from experiments, see, for exam-
ple, , the radial electric field for tokamak Ohmic
modes is equal zero in the plasma center; is close to zero
value in several kV/m) in the middle region of the
as calculated using Equation (11) is presented
stigation of so-called “locked” mode is another
3 for typical ohmic discharge in TCV tokamak
. One can see satisfactory agreement of theoretical
calculation based on Equation (11) with experimental re-
antitative example of this approach. In this mode, plas-
ma stops rotating in the toroidal direction on all magnetic
surfaces, and experimental
becomes close to zero
Figure 3. Example of radial electric field Er(r) profile ca
mental toroidal rotation plasma velocity profile.
culation in a typical ohmic discharge in TCV tokamak .
is the effective radius of the tokamak magnetic surface.
Solid curve is the experimental Er(r) profile; squares are the
Er(r) values calculated with Equation (11) using the experi-
Copyright © 2012 SciRes. JMP
A. ROMANNIKOV 1645
everywhere . For example, obtaining the result
0Er with the condition that
ri is difficult
to explain with approaches based only on Equation (8),
but it is the trivial consequence of Equation (11).
Polarity and typical values of toroidal rotation velocity
in plasma cores in most ohmic modes and some modes
with ICRH (if ion toroidal rotation velocity is opposite to
the plasma current and is equal to 10 ÷ 100 km/s [21,39])
are correctly described by Equation (10) with
jr en n
for real ion pressure profiles [20,23].
If the sum of
incomes negative a plasma periphery
Equation (10) be
(this often indicates the suppression of electron losses at
plasma periphery) then the plasma core begins rotating in
the current direction (which correlates with experimental
An important consequence of Equations (9) and (10) is
the fact that plasma confinement is better in the mode
with co-current plasma rotation and positive
than in the mode with the counter-current plasma rotation
Er  (if other plasma parameters
are similar and two modes differ from each other only in
the direction oftation and
Er polarity). This is
confirmed by experimental data .
The integral relation between plasma parameters,
Equations (9) and (10), leads us to the
following fact: a
local variation of plasma parameters in a given region,
for example on a region of peripheral magnetic surface,
leads to the “instantaneous” total change of toroidal rota-
tion plasma velocity and
Er on the whole magnetic
surfaces inside the perturbed magnetic surface. Such
non-diffusive penetration of perturbations was observed
in experiments; see, for example, .
1) The presented
relativistic theory of radial electric
d formation, based on Equation (6), can sometimes
plain quantitatively and more often qualitatively, many
experimental tokamak results for
Specific examples are presented in the article.
2) Tokamak plasma can be a useful tool for -
search of possible physical consequences of E
Paradox. Measurement of the tokamak chamber potential
(t) with respect to ground during discharges could
resolve that Paradox in principle. The plasma discharge
created from initially neutral gas inside a metallic toka-
mak chamber can affect
(t) in two ways:
a) The most expected effect is
(t) = 0. In this case
Ehrenfest’s Paradox should be resolved
amework of “hypothesis 1”.
b) Another possibility is
In this case Ehrenfest’s Paradox should esolved
within the framework of “hypothesis 2”.
 P. Ehrenfest, “ und Relativitätsth-
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: An Operational Approach,” Foundation of
Available experimental measurements of
were described above, and comparison
sults, based on Equation (6), see the first item of Con-
clusion, show, that the Ehrenfest’s Paradox potentially
should be resolved within the frame of “hypothesis 2”, as
result of Non-Euclidean geometry of the rotating elec-
tron (ion) rings in the laboratory should be.
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