Eur. Phys. J. C (2014) 74:3098 Page 3 of 14 3098
such a frame is non-Euclidean.
2
But, as we will show below,
that does not seem to be leading to any kind of resolution
of the paradox but to a somewhat similar paradox from the
rotating frame’s point of view.
As pointed out earlier, in the case of a rotating disk one should
distinguish between the observer at the center of the disk
(called the centrally rotating observer/frame) whose spa-
tial coordinates, measured in the non-rotating (laboratory)
frame, are fixed and those at different nonzero radii which
are non-inertial due to the centrifugal force felt by them and
called orbiting observers/frames. Einstein calls them eccen-
tric observers“relative to whom a gravitational field prevails”
[11]. In other words these observers, by the equivalence prin-
ciple, find themselves and anything fixed with respect to the
disk in a gravitational field. Later, elaborating on this matter,
it will be shown that rotating observers at nonzero radii are
of central importance in our discussion of RRTs but for the
purpose of Ehrenfest’s Paradox we only deal with the rotat-
ing observer/frame at the center of the disk. From a rotating
observer’s point of view the above-mentioned non-Euclidean
character of the disk geometry could be obtained from con-
sidering the metric of flat spacetime in the rotating frame, as
it is the spatial geometry (metric), defined through spacetime
metric, which accounts for spatial distances including that of
the disk circumference. Using the differential GRT (2), the
flat spacetime metric in the non-rotating frame
ds
2
= c
2
dt
2
− dr
2
−r
2
dφ
2
− dz
2
(4)
transforms into [15,16]
ds
2
=(c
2
−
2
r
2
)dt
2
−2r
2
dtdφ
−dr
2
−r
2
dφ
2
− dz
2
(5)
in the rotating frame. It is seen that this metric is applicable
for radii less than c/, corresponding to the so-called light
cylinder, beyond which g
00
becomes negative (with the cor-
responding points having velocities greater than c) and hence
from the physical point of view is not of interest [15,16].
The famous result, based on special relativistic arguments
made by Einstein, that a rotating clock at nonzero radius
r = R runs slower than that sitting at the center of the disk
(or very close to it) [11,12] is clearly encoded in the above
metric, from which we have dτ =
1 −
2
R
2
c
2
dt where dt is
the world time recorded by the inertial/laboratory clocks as
well as the one at the center of the disk. The above spacetime
metric plays the same role for a centrally rotating observer
as the Rindler spacetime metric
2
Actually it seems that Theodor Kaluza should be credited with the first
assignment of non-Euclidean geometry to a rotating disk [14], though
he has not provided any mathematical details to support his idea.
ds
2
= η
ab
dx
a
dx
b
= (1 + a ¯x
1
)
2
(d ¯x
0
)
2
−(d ¯x
1
)
2
− (d ¯x
2
)
2
− (d ¯x
3
)
2
(6)
with
x
0
= (a
−1
+¯x
1
) sinh(a ¯x
0
); x
2
=¯x
2
x
1
= (a
−1
+¯x
1
) cosh(a ¯x
0
); x
3
=¯x
3
(7)
plays for a uniformly accelerating observer with 3-acceleration
a = (a, 0, 0). In other words the Rindler metric in the limit
¯x
1
1 (i.e. for points infinitesimally close to the world line
of the observer) is equivalent to the Fermi metric [17], at first
order (i.e. O( ¯x
l
) ), in the absence of rotation (i.e. = 0),
while (5) in the limit r 1 (i.e. infinitesimally close to the
centrally rotating observer) is equivalent to the Fermi metric,
at the same order, in the absence of linear acceleration (i.e.
a = 0) [9]. It should be noted that the spacetime in a rotating
observer’s frame (5), like Rindler spacetime, is the flat space-
time in a coordinate system which is not maximally extended
due to the existence of a light cylinder in the former and the
horizon in the latter. On the other hand, unlike Rindler space-
time, it is a stationary spacetime (reflected in the presence of
its cross term dtdφ) and so one needs to employ a spacetime
decomposition formalism to define spatial distances and time
intervals, and on their basis to prescribe suitable measure-
ment procedures. In what follows we will employ the 1 + 3
or threading formulation of a spacetime decomposition [15]
which is essentially based on sending and receiving light sig-
nals between nearby observers (refer to the appendix for a
brief introduction). Although we are not going to discuss the
spacetime measurement procedure here, the employment of
the 1 +3 formulation makes it clear that, in principle, we are
using light signals to measure the relevant physical quanti-
ties, namely spatial distance and time intervals. Based on a
1 +3 formulation, the spatial line element for the metric (5)
is given by [15]
dl
2
= dr
2
+ dz
2
+
r
2
dφ
2
1 −
2
r
2
c
2
. (8)
Now for a circle of radius r = r
= R in the z =constant
plane the circumference is given by
P
=
2π
0
dl =
2π R
1 −
2
R
2
/c
2
=
P
1 −
2
R
2
/c
2
, (9)
so that P
> P with P the circumference of a non-rotating
disk. Therefore from the rotating observer’s point of view P
and P
are also not equal, but the relation between the two
quantities is just the opposite of that found by the inertial
(laboratory) observer based on Lorentz contraction.
The interpretation of the above results goes as follows:
Although the transformed spacetime is the flat spacetime in
123
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