\section{Introduction}

If a body moves with a constant velocity, then, as is well known,
the body is relativistically contracted in the direction of motion,
whereas its length in the normal direction is unchanged. A naive
generalization to a rotating disc leads to the conclusion that the
circumference of the disc is contracted, whereas the radius of the
disc is unchanged. This paradox is known as the Ehrenfest paradox.
Obviously, the paradox is a consequence of the application of the
constant-velocity result to a system with a nonconstant velocity.

The standard resolution (see \cite{gron1}, \cite{gron2} and references
therein)
of the Ehrenfest paradox is as follows: One introduces the coordinates
of the rotating frame $S'$
\begin{equation}\label{eq1}
 \varphi'=\varphi -\omega t \; , \;\;\;\; r'=r \; , \;\;\;\; z'=z
  \; , \;\;\;\; t'=t \; ,
\end{equation}
where $\varphi$, $r$, $z$, $t$ are cylindrical coordinates of the
inertial frame $S$ and $\omega$ is the angular velocity. The metric in $S'$ is
given by
\begin{equation}\label{eq2}
 ds^2=(c^2-\omega^2 r'^2)dt'^2 -2\omega r'^2 \, d\varphi' dt' -dr'^2
 -r'^2 \, d\varphi'^2 -dz'^2 \; .
\end{equation}
It is generally accepted that the space line element should be
calculated by the formula \cite{land}
\begin{equation}\label{eq3}
 dl'^2=\gamma'_{ij}dx'^i dx'^j \; , \;\;\; i,j=1,2,3 \; ,
\end{equation}
where
\begin{equation}\label{eq4}
 \gamma'_{ij}=\frac{g'_{0i}g'_{0j}}{g'_{00}}-g'_{ij} \; .
\end{equation}
This leads to the circumference of the disc
\begin{equation}\label{eq5}
 L'=\int_{0}^{2\pi} \frac{r' d\varphi'}{\sqrt{1-\omega^2 r'^2/c^2}}
   =\frac{2\pi r'}{\sqrt{1-\omega^2 r'^2/c^2}}
   \equiv\gamma(r')2\pi r' \; .
\end{equation}
The circumference of the same disc as seen from $S$ is $L=2\pi r=2\pi r'$.
If the disc is constrained  to have the same radius $r$ as the same
disc when it does not rotate, then $L$ is not changed by the rotation, but
the proper circumference $L'$ is larger than the proper circumference of
the nonrotating disc. This implies that there are tensile stresses in the
rotating disc.

However, there is something wrong with this standard resolution
of the Ehrenfest paradox. Consider a slightly simpler situation;
a rotating ring in a rigid nonrotating
circular gutter with the radius $r=r'$.
The statement that (\ref{eq5}) represents the
proper circumference implies that the {\em proper} frame of the
rotating ring is given by (\ref{eq1}). This
means that an observer on the ring sees that the
circumference is $L'=\gamma L$. The circumference of the gutter
seen by him
cannot be different from the circumference of the ring
seen by him, so the
observer on the ring sees that the circumference of the
relatively moving gutter is {\em larger} than the proper
circumference of the gutter, whereas we expect that he should
see that it is smaller.
This leads to another paradox. It cannot be resolved by saying that
the observer on the ring accelerates, because one can consider a limit
$r\rightarrow\infty$, $\omega\rightarrow 0$, $r\omega\equiv u=$constant,
which implies that the acceleration $a=r\omega^2$ becomes zero,
whereas the paradox remains.

Before explaining how we resolve this paradox, we give some
general notes on the physical meaning of various coordinate frames in
the theory of relativity. In practice, one usually uses the
coordinates that simplify the technicalities of the
physical problem considered.
For example, when one describes physical effects in a
rigid body, it may be convenient to use a comoving coordinate frame,
i.e., a frame in which all particles of the rigid body have
constant spatial coordinates. The coordinates of $S'$ in
(\ref{eq1}) may be interpreted in this way.
However, {\em the choice of the
coordinate frame is more than a matter of convenience}.
The main lesson we have learned from Lorentz
coordinate frames is the fact that what an observer observes
(time intervals, space intervals, components of a tensor, ...)
depends on how the observer moves. The main purpose of
theoretical physics is to predict what will be {\em observed} under
given circumstances. Therefore, unless stated otherwise,
in this paper {\em by a
coordinate frame we understand a coordinate frame that
is inherent to an observer}, not to a set of physical particles.
Our criticism of some earlier treatments originates from
such an interpretation of coordinate frames. To avoid a possible
misunderstanding, we note that coordinate frames do not
necessarily need to be interpreted in this way, in which case
our criticism does not apply.

We resolve the paradox by recognizing that,
according to our interpretation, the frame defined by
(\ref{eq1}) is the proper frame {\em only} of the observer at
$r=r'=0$. This observer has no velocity relative to $S$,
so the corresponding coordinate transformation (\ref{eq1})
does not depend on any velocity. As will become clear from the discussion
of Section 2, the frame defined by (\ref{eq1}) is actually the Fermi
frame of an observer who rotates, but has no velocity with respect
to the frame $S$.
Observers at different positions
on the rotating disc have different velocities, so one has to use a
different coordinate transformation for each of them. In other
words, {\em although there is no relative motion among different points on a
rotating disc, each point belongs to a different noninertial
frame.}
This is not strange to those who
are familiar with the theory of Fermi coordinates \cite{mtw}, \cite{synge},
but it seems that many relativity-theorists are not.

Note also that since we do not interpret the coordinates of $S'$ in
(\ref{eq1}) as something inherent to the disc as a whole, $r'$ can be
arbitrarily large in (\ref{eq2}), although
there is a coordinate singularity at $r'=c/\omega$.
It resembles
the Schwarzschild singularity of a black hole, where the radial coordinate
is not restricted to be
larger than the Schwarzschild radius. However, to avoid a possible
misunderstanding, note that the coordinate singularity in (\ref{eq2}) does
not correspond to an event horizon, because a rotating observer at
$r'=0$ {\em can} receive information from $r'\geq c/\omega$.

There is also another paradox connected with the standard approach to
rotating frames. Let us consider how the nonrotating gutter looks like to
a rotating observer in the center. His proper frame
{\em is} given by (\ref{eq1}). If (\ref{eq3}) is the correct definition
of the space line element, then he should see that the circumference
of the gutter is larger
than the proper circumference of the gutter by a factor $\gamma(r')$.
However, $\omega r'/c$ can be
arbitrarily large, so $\gamma(r')$ can be not only
arbitrarily large, but also even imaginary. On the other hand, we know from
everyday experience that the apparent velocity $\omega r'$ of stars,
due to our rotation,
can exceed the velocity of light, but we see neither a contraction,
nor an elongation of the stars observed.

We resolve this paradox by examining the assumptions
under which formula (\ref{eq3}) is obtained. We find that this formula
should be used with great care and show that it is not applicable
in our case.
The correct definition of the space line element depends
on how it is measured, and we find that, in our case, $\gamma'_{ij}$ should be
replaced by $-g'_{ij}$ in (\ref{eq3}).

It is fair to note that there are also some other ``nonstandard"
approaches to the Ehrenfest paradox (see \cite{tart2}, \cite{kla},
and references
therein), but none of these approaches is similar to ours.
In particular, the crucial fact that
each point of the rotating ring belongs to a different frame has
not been taken into account in any of these approaches.
Formula (\ref{eq3}) has already been criticized \cite{kla}, but our criticism
of (\ref{eq3}) is quite different and more general.

The paper is organized as follows: In Section 2 we find the correct
coordinate transformation that leads to the
frame of an observer moving in flat spacetime.
In Section 3 we explain why (\ref{eq3})
is not always a correct definition of a space line element and
show that in a frame that corresponds to an observer in flat spacetime
it is more appropriate to calculate the space line element by
$-g'_{ij}$. We also make some general remarks on the physical meaning of
general coordinate transformations.
In Section 4 we study the relativistic contraction as seen by various
observers and resolve the Ehrenfest paradox. In Section 5 we study the rate
of clocks at various positions, as seen by various observers.
In Section 6 we discuss the velocity of light as seen by various observers.
In Section 7 we discuss our results, resolve some additional physical
problems, and give some generalizations. Section 8 is
devoted to concluding remarks, where the relevance of our results
to general relativity is emphasized.

\section{The frame of an observer moving in flat spacetime}

The generalized Lorentz transformations for a local Fermi
frame of an observer
that has arbitrary time-dependent velocity and angular velocity
in flat spacetime are found in
\cite{nels}. We present the final results, using
slightly different notation. Let $S$ be an inertial frame
and let $S'$ be the frame of the observer whose velocity and angular velocity
are $u^i(t')$ and $\omega^i(t')$, respectively, as seen by an observer in $S$.
The coordinate transformation between these two frames is given by
\begin{equation}\label{er1}
 x^i =-A_{j}^{\; i}(t')x'^j +\int_{0}^{t'} \gamma(t')u^i(t')\, dt' +
 \frac{1}{\mbox{\bf{u}}^2(t')} [\gamma(t')-1][u^k(t')A_{jk}(t')x'^j]u^i(t') \; ,
\end{equation}
\begin{equation}\label{er2}
 t=\int_{0}^{t'} \gamma(t')\, dt' +
 \frac{1}{c^2}\gamma(t')[u^k(t')A_{jk}(t')x'^j] \; ,
\end{equation}
where $\gamma(t')=1/\sqrt{1-\mbox{\bf{u}}^2(t')/c^2}$ and
$A_{ji}(t')=-A_{j}^{\; i}(t')$ is the
rotation matrix evaluated at $\mbox{\bf{x}}'=0$. The rotation matrix
satisfies the differential equation
\begin{equation}\label{er4}
 \frac{d A_{ij}}{dt}=-A_{i}^{\; k}\omega_{kj} \; ,
\end{equation}
where $\omega_{ik}=\varepsilon_{ikl}\omega^{l}$, $\varepsilon_{123}=1$.
The metric tensor in $S'$ is
\begin{eqnarray}\label{metric}
 & g'_{ij}=-\delta_{ij} \; , \;\;\;\;\;
   g'_{0j}=-(\mbox{\boldmath $\omega$}'\times\mbox{\bf{x}}')_j \; , &
\nonumber \\
 & g'_{00}=c^2 \left(
1+\displaystyle\frac{\mbox{\bf{a}}'\cdot\mbox{\bf{x}}'}{c^2}
    \right)^2 -(\mbox{\boldmath $\omega$}'\times\mbox{\bf{x}}')^2 \; , &
\end{eqnarray}
where
\begin{equation}
 \omega'^i =\gamma (\omega^i -\Omega^i) \; , \;\;\;\;\;
 a'^i =\gamma^2 \left[a^i +\frac{1}{\mbox{\bf{u}}^2}(\gamma
  -1)(\mbox{\bf{u}}\cdot\mbox{\bf{a}})u^i\right] \; ,
\end{equation}
$\Omega^{i}$ is the time-dependent Thomas precession frequency
\begin{equation}
\Omega_{i}=\frac{1}{2\mbox{\bf{u}}^2}(\gamma -1)\varepsilon_{ikj}
 (u^k a^j -u^j a^k) \; ,
\end{equation}
and $a^i=du^i/dt$ is the time-dependent acceleration.
The transformations (\ref{er1})-(\ref{er2}) are chosen such that
the space origins of $S$ and $S'$ coincide for $t=t'=0$. If $\mbox{\bf{u}}$
is time independent and $\mbox{\boldmath $\omega$}=0$, then
(\ref{er1})-(\ref{er2}) reduce to the well-known ordinary Lorentz
boosts. If $\mbox{\bf{u}}=0$ and $\mbox{\boldmath $\omega$}$
is time independent, then (\ref{er1})-(\ref{er2}) reduce to (\ref{eq1}).
It is important to emphasize that $\mbox{\bf{u}}(t')$ is the velocity of the
{\em space origin} $\mbox{\bf{x}}'=0$ of $S'$. If $S'$ is a rotating frame, then
other space points of $S'$ have a different velocity. (Remind that rotation
is {\em not} a motion along a circle, but rather a change of orientation
of the axes with respect to an inertial frame.) Therefore,
in general, $S'$ is the proper frame {\em only}
of the observer at $\mbox{\bf{x}}'=0$.
Note also that $g'_{\mu\nu}=\eta_{\mu\nu}$ only at $\mbox{\bf{x}}'=0$,
which is another confirmation that $S'$ is the frame of the observer at
$\mbox{\bf{x}}'=0$ only. The metric (\ref{metric}) is also consistent
with a more general theory of Fermi coordinates \cite{mtw},
which are coordinates of an observer arbitrarily moving in curved spacetime,
and also have the property that $g_{\mu\nu}=\eta_{\mu\nu}$ at the
space origin, i.e., at the position of the observer. Note also that if
$\mbox{\bf{a}}'$ and $\mbox{\boldmath $\omega$}'$ vanish, then (\ref{metric})
is a metric of an inertial frame and is equal to $\eta_{\mu\nu}$
everywhere, so, in this case, $S'$ can be considered as a frame of an observer at
{\em arbitrary} constant $\mbox{\bf{x}}'$.

It is interesting to note that the geometrical
construction of Fermi coordinates is well established \cite{mtw},
\cite{synge}, but no analog of (\ref{er1})-(\ref{er2}) is
known for curved spacetime. The transformations (\ref{er1})-(\ref{er2})
are obtained by summation of infinitesimal Lorentz transformations (and
rotations). It is not so easy to find an analog of Lorentz transformations
in curved spacetime, because they correspond to the coordinate
transformation between Fermi frames of two different free-falling observers.
We can, however,
write the transformations (\ref{er1})-(\ref{er2})
in a more elegant form, which could be illuminating
for a generalization to curved spacetime. Let
\begin{equation}\label{el1}
x^{\mu}=f^{\mu}(t',\mbox{\bf{x}}';\mbox{\bf{u}})
\end{equation}
denote the ordinary Lorentz transformations, i.e., the transformations
between two inertial frames specified by the relative velocity
$\mbox{\bf{u}}$, which can be considered as the relative velocity
between two inertial (free-falling)
observers at the instant when they have the same position. The differential
of (\ref{el1}) is
\begin{equation}\label{el2}
dx^{\mu}=f^{\mu}_{\; ,\nu} (t',\mbox{\bf{x}}';\mbox{\bf{u}}) dx^{\nu} \; .
\end{equation}
The transition to a noninertial frame introduces a time-dependent
velocity: $\mbox{\bf{u}} \rightarrow \mbox{\bf{u}}(t')$.
The transformations (\ref{er1})-(\ref{er2}) may be obtained by integrating
(\ref{el2}) in the following way:
\begin{equation}\label{el3}
x^{\mu}=\int_{0}^{t'}f^{\mu}_{\; ,0} (t',0;\mbox{\bf{u}}(t')) dt' +
\int_{C}
f^{\mu}_{\; ,i} (t',\mbox{\bf{x}}';\mbox{\bf{u}}(t')) dx'^{i} \; ,
\end{equation}
where $C$ is an arbitrary curve with constant $t'$, starting from $0$
and ending at $-A_{j}^{\; i}(t')x'^{j}$.
The subintegral function in the second term
of (\ref{el3}) is a total derivative, so this term does not depend on the
curve $C$ and can be easily integrated. The time derivative in the
first term is taken with $\mbox{\bf{u}}(t')$ kept fixed, so
$f^{\mu}_{\; ,0}$ in this term is not a total derivative.

Let us now apply the general formalism described in this section to a
uniformly rotating ring. We assume that the ring is put in a rigid
nonrotating circular gutter with the radius $R$,
which provides that the radius of the rotating ring
is the same as the radius of the same ring when it does not rotate,
and is equal to $R$,
as seen by an observer in $S$. This allows us not to worry about
the complicated dynamical forces that tend to change the radius of the
ring as seen by the observer in $S$, and pay all our attention to the
kinematic effects resulting from the transformations
(\ref{er1})-(\ref{er2}).

The ring can be considered as a series of independent short rods,
uniformly distributed along the gutter. (By a short rod we
understand a rod with a length much shorter than $R$.) We assume
that the gutter is placed at the $z=0$ plane. We put the
space origin of $S$ at a fixed point on the gutter, such that
the $y$-axis is tangential to the gutter and the $x$-axis is perpendicular
to the gutter at $\mbox{\bf{x}}=0$. (In the rest of this section, as well as
in Sections 4 and 5,
$\mbox{\bf{x}}\equiv
( x,y)$ and the $z$-coordinate is suppressed.)
We study a single short rod initially placed at $\mbox{\bf{x}}=0$ and
uniformly moving along the gutter in the counterclockwise direction.
(This mimics a uniform motion of an electron in a
synchrotron). The gutter causes a torque that provides that the rod is
always directed tangentially to the gutter. Therefore,
$\omega=u/R$, where $u=\sqrt{\mbox{\bf{u}}^2}$ is time independent.
Now, $\gamma=1/\sqrt{1-\omega^2 R^2/c^2}$ is also time independent.
Since a clock in $S'$ is at $\mbox{\bf{x}}'=0$, the clock rate between
a clock in $S$ and a clock in $S'$ is given by $t=\gamma t'$, as seen
by an observer in $S$. We assume that, initially, the axes $x'$, $y'$ are
parallel to the axes $x$, $y$, respectively. Therefore the velocity
\begin{equation}
\mbox{\bf{u}}(t')=\omega R (-\sin \gamma\omega t', \cos \gamma\omega t')
\end{equation}
is always in the $y'$-direction and the solution of (\ref{er4}) is
\begin{equation}
A_{ij}(t')=\left(
 \begin{array}{cc}
  \cos \gamma\omega t' & \sin \gamma\omega t' \\
  -\sin \gamma\omega t' & \cos \gamma\omega t'
 \end{array} \right) \; .
\end{equation}
The transformations (\ref{er1})-(\ref{er2}) become
\begin{equation}\label{er1'}
 \left( \begin{array}{c} x \\
                         y
        \end{array} \right)=
 \left( \begin{array}{cc}
   \cos \gamma\omega t' & -\gamma\sin \gamma\omega t' \\
   \sin \gamma\omega t' & \gamma\cos \gamma\omega t'
  \end{array} \right)
 \left( \begin{array}{c} x' \\
                         y'
        \end{array} \right)
 +R \left( \begin{array}{c} \cos \gamma\omega t' -1 \\
                            \sin \gamma\omega t'
           \end{array} \right) \; ,
\end{equation}
\begin{equation}\label{er2'}
t=\gamma t' +\frac{\gamma}{c^2}\omega R y' \; .
\end{equation}
In particular, at $t'=0$ these transformations become
\begin{equation}\label{t=0}
 x=x' \; , \;\;\;\; y=\gamma y' \; , \;\;\;\; t=\frac{\gamma u}{c^2}y' \; ,
\end{equation}
which coincide with the ordinary Lorentz boost at $t'=0$ for the velocity in
the $y$-direction.

\section{General coordinate transformations and the space
line element in a non-time-orthogonal frame}

A non-time-orthogonal frame is a frame in which $g'_{0i}$ is different
from zero. It is generally accepted that the space line element in such a frame
is given by (\ref{eq3}). However, if we assume that this formula can be
applied to calculate the space distance as seen by a local observer,
then, as we have found in Section 1, Eq. (\ref{eq3})
leads to an imaginary length of a distant unaccelerated object as seen
by a rotating observer. In order to resolve this puzzle, we examine the
assumptions under which formula (\ref{eq3}) is derived.

In \cite{land}, formula (\ref{eq3}) is derived by assuming that the space
distance between two points is measured by measuring the time $\Delta t'$ that
light needs to travel from point $A$ to point $B$  and then back to
point $A$. It is also assumed that the time is measured by a clock
that does not change its position $x'^i$.
The definition of the space distance $l'=c\,\Delta t'/2$ leads to
(\ref{eq3}).

In order to perform the described measurement in a
rotating frame, the clock must be
positioned at point $A$. However, according to our interpretation of
(\ref{eq1}), this point can be faraway
from the center of the rotation, so the required velocity
of point $A$ can exceed $c$, as seen in $S$. Therefore,
in general, such a measurement cannot be performed.

In practice, we measure space distances between distant objects in a
completely different way, namely, by measuring the angles under which we see the
objects. (We assume that we know the radial distance of these objects from
us. The radial distance is not problematic in the theoretical sense, because
$g'_{0r}=0$ in (\ref{eq2})).
Our rotation does not influence this angle. Therefore,
the apparent velocity of distant objects
can exceed the velocity of light owing to our rotation,
but a pure rotation (without velocity)
will not lead to relativistic contraction, nor to elongation. The effect is
that, in a rotating frame, it is more appropriate to calculate the space
line element as
\begin{equation}\label{dl}
dl'^2=-g'_{ij}dx'^{i}dx'^{j} \; ,
\end{equation}
despite the fact
that $g'_{0i}$ is different from zero. This formula should be used
to calculate the space distance between two arbitrary points
which have the same $t'$ coordinate, no matter
how far these points are from the observer at $x'^i =0$.
Of course, if these points are end points of a body,
then, in general, the distance calculated in this way
will not be equal to the proper length of the body, but merely to
the length seen by the observer.
Formula (\ref{dl})
is also correct for
frames that are both accelerated and rotating, defined by
(\ref{er1})-(\ref{er2}).

To clarify the meaning of formula (\ref{eq3}) completely, note that in
\cite{mol} this formula is derived in a completely different way,
without referring to any particular method of measurement. However,
what is actually derived in \cite{mol} is the fact that the
quantity (\ref{eq3}) does not change under coordinate transformations
of the form
\begin{equation}\label{sameframe}
 t''=f^0 (t',x'^1,x'^2,x'^3) \; , \;\;\;\; x''^i=f^i (x'^1,x'^2,x'^3) \; .
\end{equation}
We refer to such transformations as {\em internal transformations}.
Obviously, (\ref{eq1}) is not an internal transformation.
Regular internal transformations form a subgroup of the group of all
regular coordinate transformations.
Note that the invariant quantity
$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ can always be written as
\begin{equation}\label{logun}
ds^2=d\eta^2 -\gamma_{ij}dx^i dx^j \; ,
\end{equation}
where
\begin{equation}\label{logun2}
d\eta^2 =\left[ \frac{g_{0\mu}dx^{\mu}}{\sqrt{g_{00}}} \right]^2 \; ,
\end{equation}
so $d\eta^2$ also does not change under internal
transformations. The quantity $d\eta^2$ is nothing else but a time line element
\cite{land}, defined by a measuring procedure similar to
the measuring procedure used to define
the space line element (\ref{eq3}).

Let us illustrate the power of (\ref{eq3}), (\ref{sameframe}), and
(\ref{logun2}) on the example that has already been discussed at some
length in \cite{mol}. The Galilei transformation
$t''=t$, $x''=x-ut$ can also serve as a correct coordinate transformation
needed
to describe the relativistic effects related to a frame moving with a constant
velocity $u$. The metric in these coordinates is given by
\begin{equation}\label{mol1}
ds^2=c^2(1-u^2/c^2)dt''^2 -2udx''dt''-dx''^2 \; ,
\end{equation}
where it has been assumed that the metric of $S$ is given by
$ds^2=c^2\, dt^2 -dx^2$. From (\ref{eq3}) and $dt=0$ one can
obtain the relativistic contraction $dl=dx=dl''/\gamma$,
where $\gamma =1/\sqrt{1-u^2/c^2}$. Similarly,
from (\ref{logun2}) and $dx''=0$ one can obtain $dt=\gamma d\eta''$.
The frame $S''$ is physically equivalent to the frame
$S'$ which would be obtained from $S$ by the ordinary Lorentz
transformations, in the sense that $S''$ and $S'$ are connected by an
internal coordinate transformation
\begin{equation}\label{mol2}
x'=\gamma x'' \; , \;\;\;\;\; t'=t''/\gamma -\gamma u x''/c^2 \; .
\end{equation}
Note that the non-time-orthogonal metric (\ref{mol1}), unlike (\ref{eq2}) and
(\ref{metric}), can be transformed to a time-orthogonal metric by an
{\em internal} transformation. Note also that the metric (\ref{mol1}), unlike
(\ref{eq2}) and (\ref{metric}), is not a metric of a Fermi frame.

In \cite{mol}, internal transformations are interpreted as transformations
that correspond to a redefinition of the coordinates of the same {\em
physical} observer. However,
there is something unphysical about internal transformations; if $t'$ is a measure
of the physical time for the observer in $S'$, then $t''$ is not, because
it corresponds to a ``time" of the same observer which depends on the
space point
$x'^i$. Therefore, we introduce a more restrictive class of coordinate
transformations, which could be better suited to interpret them
as transformations that correspond to a redefinition of the coordinates of
the same physical observer:
\begin{equation}\label{veryweak}
t''=f^0 (t') \; , \;\;\;\; x''^i=f^i (x'^1,x'^2,x'^3) \; .
\end{equation}
We refer to such transformations as {\em restricted internal transformations}.
Regular restricted internal transformations form a subgroup of the group of all
regular internal transformations.
The quantities $g'_{00}dt'^2$ and (\ref{dl}) do not change under
restricted internal transformations.

Now we have two definitions of the space line element,
(\ref{eq3}) and (\ref{dl}), and related to this, two types
of restricted coordinate transformations, internal and restricted internal.
The space line element (\ref{eq3}) reduces to (\ref{dl}) if
$g_{0i}=0$. However, as we have shown in this section,
(\ref{dl}) is more appropriate in some cases,
even if $g_{0i}\neq 0$. How to
know in general what is the suitable definition of the
space line element?

We can immediately formulate one rule which is certainly
suitable: {\em If the metric of a frame can be transformed to a
time-orthogonal frame by an internal transformation, then the
space line element should be calculated by (\ref{eq3})}.

According to the results of this section, we can also formulate
another rule: {\em If the metric of a frame in flat spacetime can be
obtained from $g_{\mu\nu}=\eta_{\mu\nu}$ by a transformation of the form of
(\ref{er1})-(\ref{er2}) followed by an arbitrary restricted internal transformation, then
the space line element should be calculated by (\ref{dl}).}
Such coordinate transformations can be interpreted as the most general coordinate
transformations in flat spacetime that correspond to a physical observer
who has a positive mass.

We still do not know a general rule. However, one can be satisfied
to have a rule for Fermi frames only, or for Fermi frames modified
by an arbitrary restricted internal transformation, because only such frames have a direct
physical interpretation. One can be tempted to guess that
for all such frames the space line element should be calculated by
(\ref{dl}), but such a conjecture requires further investigation.

For the sake of completeness, let us make a few remarks on general
coordinate transformations in curved spacetime.
The most general coordinate transformation
that corresponds to a physical observer who has a positive mass is a
transformation that leads
to Fermi coordinates, followed by an arbitrary restricted internal transformation.
Other coordinate frames may be useful for some physical calculations,
for example, because it is easier to solve some covariant equations of motion
in these coordinates. However, if one is interested in how
the physical system
looks like to a physical observer, one must transform the results to the
coordinates specific for this observer.

To summarize this section, we conclude that the correct definition
of the space line element depends on how it is measured. Formula
(\ref{eq3}) is not incorrect, but its applicability is limited and
it should be used with great care. In our case of
accelerated, rotating frames,
it is more appropriate to calculate the space line element with
$-g'_{ij}$ instead of with $\gamma'_{ij}$.

\section{Relativistic contraction}

In Section 2 we have found the coordinate transformation that describes
the frame of a short rod uniformly moving along the circular gutter.
Let as assume for a while that the length of
the rod is infinitesimally small and
that the rod is rigid (i.e., its proper length $dL'$ is equal to the
proper length of the same rod when it does not accelerate).
Let us determine the relativistic contraction of the rod, as seen by
an observer in $S$. The observer in $S$ sees both ends of the rod
at the same instant, so $dt=0$.
From symmetry it is obvious that the relativistic contraction cannot depend on
$t$, so, in order to simplify the calculations, we evaluate this at $t=0$.
Since the rod is at $x'=y'=0$, (\ref{er2'}) implies that $t'=0$. Taking the
differential of (\ref{er1'}) and (\ref{er2'}) with respect to space and
time coordinates, and then putting $x'=y'=t'=dt=0$, we find that the
observer in $S$ sees the length
\begin{equation}\label{inf}
 dL=dy=\frac{dy'}{\gamma}=\frac{dL'}{\gamma} \; ,
\end{equation}
which is the expected relativistic contraction.

Let us now turn our attention to the concept of the proper length of a
body. Traditionally, it is defined as a length of the body as seen
from the proper frame of the body. However, as we have seen,
in general, there is no
such thing as a proper frame of the body as a whole. Such a thing
exists only for a nonrotating, inertially moving body in flat
spacetime. The concept of a proper length
of a large body does not have any fundamental meaning,
simply because a ``large body" is not actually one object, but a
set of many interacting particles. However,
the proper length of an infinitesimally small part of a body is well
defined. Therefore, we can define the proper length of a whole
body as the sum of the proper lengths of its infinitesimal parts.
Applying this to (\ref{inf}), we see
that the relativistic contraction of a short (but not infinitesimal)
rigid rod uniformly moving along the circular gutter
is given by $L=L_0/\gamma$, as seen by the
observer in $S$. Here
$L_0$ is the proper length defined as above.

Now, as in Section 2, assume that the rotating ring is a series of
independent short rods, uniformly distributed along the gutter.
Each rod is relativistically contracted, but the ring is not.
This means that the
distances between the neighboring ends of the neighboring rods
are larger than those for a nonrotating ring, so
the proper length of the ring is also larger than that of a nonrotating
ring. This is concluded
also in \cite{gron2}. This situation mimics a more realistic
ring made of elastic material, where atoms play the role of
short rigid rods. Owing to the rotation the distances between
neighboring atoms
increase, so there are tensile stresses in the material.
However, it is important to emphasize that the rotation is not
essential for understanding of the origin of these tensile forces,
because a similar effect also occurs in a linear relativistic
motion \cite{dew}.

The same relativistic contraction of
short rods will be seen by a
rotating observer in the center, because his frame is given by the
Galilei transformation (\ref{eq1}) and the lengths are calculated by
$g_{ij}$, as explained in Section 3.

Let us now study how the nonrotating gutter looks like
from the point of view of an observer on the rotating ring.
Without
losing on generality, we evaluate this at $t'=0$. We calculate the
length of an infinitesimal part of the gutter lying near the
observer, so $x=y=0$. Both ends are seen at the same instant, so
$dt'=0$. Taking the
differential of (\ref{er1'}) with respect to space coordinates,
and then putting $t'=0$, we find that the
observer in $S'$ sees the length
\begin{equation}\label{inf2}
 dL'=dy'=\frac{dy}{\gamma}=\frac{dL}{\gamma} \; ,
\end{equation}
which is the expected relativistic contraction.

It is important to emphasize that (\ref{inf2}) is correct only in the
infinitesimal form. The observer on the ring will not see other distant
parts of the gutter contracted in the same way; for him, the gutter and
the ring do not look azimuthally symmetric.
In the following
we study how other parts of the ring look like from the point of view
of the observer on the ring. We introduce polar
coordinates $r$, $\varphi$, defined by
\begin{equation}
y=r \sin \varphi \; , \;\;\;\; R+x=r \cos \varphi \; ,
\end{equation}
which are new space coordinates for $S$, with the origin in the center
of the circular gutter. The angle $\varphi$ is a good label of the position
of any part of the ring even in $S'$. (To visualize this, one can draw
angular marks on the gutter. The number of marks separating two points
on the gutter or on the ring is a measure of the ``angular distance" in
any frame.)
Let $S''$ be the frame of another
part of the ring. The position of that part of the ring is
$x''=y''=0$. The relative position of the space origin of $S''$ with respect
to that of $S'$ is given by the constant relative angle $\Delta\varphi_0$,
as seen by an observer in $S$. In analogy
with (\ref{er1'})-(\ref{er2'}), we find that $S''$ is determined by
\begin{equation}\label{er1''}
 \left( \begin{array}{c} x \\
                         y
        \end{array} \right)=
 \left( \begin{array}{cc}
   \cos (\gamma\omega t''+\Delta\varphi_0) & -\gamma\sin (\gamma\omega
          t''+\Delta\varphi_0) \\
   \sin (\gamma\omega t''+\Delta\varphi_0) & \gamma\cos (\gamma\omega
          t''+\Delta\varphi_0)
  \end{array} \right)
 \left( \begin{array}{c} x'' \\
                         y''
        \end{array} \right)
 +R \left( \begin{array}{c} \cos (\gamma\omega t''+\Delta\varphi_0) -1 \\
                            \sin (\gamma\omega t''+\Delta\varphi_0)
           \end{array} \right) \; ,
\end{equation}
\begin{equation}\label{er2''}
t=\gamma t'' +\frac{\gamma}{c^2}\omega R y'' \; .
\end{equation}
The observer in $S'$ will see the other part of the ring at the
relative ``angular distance" $\Delta\varphi$, which, owing to the relativistic
effects, differs from $\Delta\varphi_0$. Let the labels $A$, $B$ denote the
coordinates of the part of the ring that lie at $S'$ and $S''$,
respectively. Since the rotation is uniform,
the relative ``angular distance"
\begin{equation}\label{Dphi}
\Delta\varphi=\varphi_{B}(t''_{B})-\varphi_{A}(t''_{A})
 = \Delta\varphi_0 +\gamma\omega t''_{B}-\gamma\omega t'_{A} \; ,
\end{equation}
cannot depend on $t'$, so without losing on generality, we evaluate this at
$t'=0$. Since the observer sees both parts of the ring at the same
instant, we have $t'_{A}=t'_{B}=0$. Since $x''_{B}=y''_{B}=0$, from
(\ref{er1''}) we find
\begin{equation}\label{y2}
y_{B}=R\sin (\gamma\omega t''_{B}+\Delta\varphi_0) \; ,
\end{equation}
and from (\ref{er2''})
\begin{equation}\label{t2}
t_{B}=\gamma t''_{B} \; .
\end{equation}
From $t'_{B}=0$ and (\ref{t=0})
it follows $t_{B}=\omega R y_{B}/c^2$, which, because
of (\ref{t2}), can be written as $\gamma t''_{B}=\omega R y_{B}/c^2$.
This, together with (\ref{y2}), leads to the equation that determines
$t''_{B}$:
\begin{equation}\label{eqgron1}
\gamma\omega t''_{B}=\beta^2\sin (\gamma\omega
t''_{B}+\Delta\varphi_0) \; ,
\end{equation}
where $\beta^2 \equiv \omega^2 R^2 /c^2$.
From $t'_{A}=0$ and (\ref{Dphi}) we see that $\Delta\varphi=\gamma\omega
t''_{B}+\Delta\varphi_0$, so (\ref{eqgron1}) can be written as
\begin{equation}\label{eqgron1'}
\Delta\varphi -\Delta\varphi_0 =\beta^2\sin \Delta\varphi \; .
\end{equation}

Equation (\ref{eqgron1'}) determines the relative ``angular
distance" $\Delta\varphi$
between two points on the ring as seen by the observer at one of the
points, if the relative angle between these two points, as seen by the
observer in $S$, is $\Delta\varphi_0$. In other words, (\ref{eqgron1'}) determines
how the ring looks like to the observer on the ring. For an inertial
observer whose instantaneous
position and velocity are equal to that of the observer on the ring,
the same equation (\ref{eqgron1'}) is found in \cite{gron1}, where
the solution is graphically depicted. This means, contrary to the
conclusion of \cite{gron1}, that the inertial and the noninertial
observers see the ring in the same way.

If the two points on the ring are very close to each other, then
$\Delta\varphi_0$ and $\Delta\varphi$ are very small. By expanding
equation (\ref{eqgron1'}) for small angles we find the approximative
solution $\Delta\varphi=\gamma^2 \Delta\varphi_0$. The factor $\gamma^2$ is
easy to understand; one factor of $\gamma$ appears because the
part of the gutter close to the observer on the ring looks shorter
for that observer than it really is, and the other factor of $\gamma$
appears because the part of the ring close to the observer on the ring
is longer than that of the same ring when it does not rotate.

\section{The rate of clocks}

Assume that there are two clocks at different positions on the
ring. Assume also that they show the same time, as seen by an
observer in $S$. Then, as shown in Section 2, both clocks
show the time $t'=t/\gamma$, as seen from $S$.

These two clocks do not show the same time as seen by an observer
on the ring. If the position of the observer coincides with the
position of one of the clocks, then the time-shift of the other
clock is given by (\ref{eqgron1}).

Let us calculate the time-shift of the clock at the fixed position
$(x,y)$, as seen by the observer in $S'$. From (\ref{er1'})
we express $y'$ as a function of $x$, $y$, and $t'$, and put this
in (\ref{er2'}). The result is
\begin{equation}\label{txy}
t=\gamma t' +
 \frac{\omega R}{c^2}[y \cos \gamma \omega t'
 -(x+R) \sin \gamma \omega t'] \; .
\end{equation}
For comparison, if (\ref{er1'}) and (\ref{er2'}) are replaced by
the ordinary Lorentz boosts for a constant velocity in the $y$-direction,
then (\ref{txy}) should be replaced by
\begin{equation}\label{txyo}
t=\frac{t'}{\gamma}+\frac{u}{c^2}y \; .
\end{equation}

To understand the physical meaning of (\ref{txy}),
we explore some special cases.
If $\gamma \omega t'=2k\pi$, then $t=\gamma t'+\omega R y/c^2$. In this
case, the rate of clocks $\Delta t/\Delta t'=\gamma$ is the same as
that for the observer
in $S$. This can also be understood as a time-averaged rate,
because the oscillatory functions in (\ref{txy}) vanish when
they are averaged over time. Therefore, the observer in $S'$ agrees
with the observer in $S$ that the clock in $S'$ is slower, but only in
a time-averaged sense.
At some instants the observer in $S'$ sees that
the clock in $S$ is slower than his clock.
For example, by putting $x=0$ and
expanding (\ref{txy}) for small $t'$, we recover formula (\ref{txyo}),
with $u=\omega R$. If the clock in $S$ is in the center, which corresponds
to $x=-R$, $y=0$, then (\ref{txy}) gives $t=\gamma t'$, so in this case
there is no oscillatory behavior.

\section{Velocity of light}

Let us also make some comments on the velocity of light. The Sagnac effect
is usually interpreted as a dependence of the velocity of light
on the direction of light propagation in a rotating frame
(see, for example, \cite{post}, \cite{tart} and references therein).
However, such an interpretation is based on the interpretation
of the frame $S'$ defined by (\ref{eq1}) as a proper frame of all
observers on a rotating platform. Now we know that each observer
belongs to a different local Fermi frame, and from (\ref{metric}) we see that
in the {\em vicinity} of any observer the metric is equal to the
Minkowski metric $\eta_{\mu\nu}$. This implies that {\em
for any local observer the velocity of light is isotropic and is
equal to $c$, providing that it is measured by propagating a light
beam in a {\bf small} neighborhood of the observer,
using Einstein synchronized clocks}. This is
also true for an observer in curved spacetime, because his
proper frame is given by the appropriate Fermi coordinates, which
also have a property that $g_{\mu\nu}=\eta_{\mu\nu}$ at the position
of the observer. The phrases
``local" and ``small" denote spatial dimensions inside which
the metric tensor does not change significantly.

Of course, the velocity
of light does not have to be equal to $c$ for an observer which is not
at the same position as the light. However, this is not only a property
of non-time-orthogonal frames. For example, if the acceleration
of an uniformly accelerated observer and the propagation of light
are both in the $x'$-direction, then from (\ref{metric}) one can
find that the accelerated observer sees the velocity of light as
$|dx'/dt'|=c\sqrt{1+a'x'/c^2}$, being equal to $c$ only at
$x'=0$. A similar effect occurs for a radial
motion of light in the vicinity of the Schwarzschild radius of a black hole,
as seen by a static observer faraway from the Schwarzschild radius.

Concerning the Sagnac effect, we do not claim that the standard prediction
for the phase shift is incorrect. It can also be derived by performing
calculations in the nonrotating frame $S$ \cite{post}, and such a
derivation, based on the well-understood Minkowski
spacetime, is perfectly correct. We have nothing new to say about
the phase shift, which appears when clockwise and counterclockwise
propagated
light beams finally meet. However, as seen by an observer on the rim of a
rotating disc, the velocity of the light beam will be a complicated function
of time $t'$, or equivalently, of the position $(x',y')$ of the beam.
The trajectory of
the light beam expressed in $S$-coordinates takes a simple form
\begin{equation}\label{sag}
y=R \sin \omega_L t \; , \;\;\;\;\;
x=R(-1+\cos \omega_L t) \; ,
\end{equation}
where $\omega_L =\pm c/R$. The plus and minus signs refer to the
counterclockwise and clockwise propagated beams, respectively.
Using (\ref{er1'}), (\ref{er2'}), and (\ref{sag}), one can eliminate
$x,y,t$ and express $x',y'$ as functions of $t'$. The speed of
light as seen by the observer in $S'$ is
\begin{equation}\label{sag2}
v'_L =\sqrt{\left( \frac{dx'}{dt'}\right)^2 +
\left( \frac{dy'}{dt'}\right)^2 } \; .
\end{equation}
Expanding (\ref{er1'}) and (\ref{sag}) for small $t'$ and $t$, respectively,
one can easily find $y'=\pm ct'+{\cal O}(t'^2)$, $x'={\cal O}(t'^2)$,
which means that the observer sees the velocity
of light equal to $c$ when the light is at the same position
as the observer, just as expected.

\section{Discussion}

From the experience acquired by careful calculations in the preceding
sections, we can generalize some of the results without much effort,
using qualitative and intuitive arguments.

If an observation in $S$ is performed at the instant $t$, then the
solution of (\ref{er4}) can always be chosen such that at $t$
the axes $x'^i$ are parallel to the corresponding axes $x^i$. Therefore,
for a small range of values of $t'$, the transformations
(\ref{er1})-(\ref{er2}) can be approximated by the ordinary Lorentz boosts (see
(\ref{t=0})).
From this fact we conclude that
if a moving rigid body is short enough, then its relativistic contraction
in the direction of the instantaneous velocity, as seen from $S$,
is simply given by $L(t)=L'/\gamma(t)$, i.e., it depends
only on the instantaneous velocity, not on its acceleration and rotation.
(``Short enough" means that $L'\ll c^2/a'_{\|}$, where $a'_{\|}$ is the
component of the proper acceleration parallel to the direction
of the velocity \cite{nikol}).

By a similar argument we may conclude that
an arbitrarily accelerated and rotating
observer sees equal lengths of other differently
moving objects as an inertial observer whose
instantaneous position and velocity are equal to that of the
arbitrarily accelerated and rotating observer.

So far we have studied a rotating ring. A rotating disc is a more
complicated object, with some additional dynamical effects
related to elastic and inertial forces. However, a disc can be modeled
as a series of concentric rings, each of them being constrained to have a
fixed radius. In this case, the analysis of a rotating disc becomes
essentially the same as that of a rotating ring.

Let us also give some additional arguments why our resolution of the
Ehrenfest paradox is correct. Our method, based
on coordinate transformations (\ref{er1})-(\ref{er2}), is really
a generalization of the well-known derivation of the Lorentz contraction
for constant velocities. In our
approach the origin of the relativistic contraction lies in
the non-Galilean transformation, not in the nontrivial metric, whereas
in the standard approach the transformation is Galilean and
the contraction is due to the nontrivial metric (\ref{eq2}).
Note finally that our approach allows a generalization to a more
complicated motion, whereas the standard approach does not.

Finally, let us make some comments on the observability of the relativistic
contraction. In principle, it could be observed by photographing a rod
with a very short exposition, such that both ends are observed
at the same instant. Since the velocity of the incoming information
(velocity of light) is finite, both ends of the rod should be
positioned at the same distance from the observer. Therefore, the ideal
setup for such a measurement is a rod in a uniform circular
motion and a camera in the center, providing that we can achieve a
short enough exposition. It is assumed that in this experiment the
only object that moves circularly is a rod (with two ends); there is
neither a rotating disc, nor a rotating ring.

An indirect, but easier-to-perform experimental
verification of the relativistic contraction could perhaps be obtained
by measuring the velocity of a rotating ring in a rigid circular
gutter, needed to achieve the break of the ring, and comparing it with
the elongation needed to achieve the break of the ring
caused by ordinary stretching.

Of course, in both types of experiments
the problem is to achieve a relativistic velocity
of macroscopic objects, so these can be
considered merely as {\it gedanken} experiments.

\section{Conclusion}

In this paper a new resolution of the Ehrenfest paradox has been
provided by taking into consideration the fact that although there is no
relative motion
among different points on a rotating disc, each point belongs to a
different noninertial local Fermi frame.
If a rotating ring (or a disc) is constrained to have a
 fixed radius from the point of view of an inertial observer, it has been
 found that there are tensile stresses in the disc, in agreement with the
 prediction of the standard approach. However, contrary to the
 prediction of the standard approach, it has been found that an
 observer on the rim of the disc will see equal lengths of other differently
moving objects
 as an inertial observer whose instantaneous
 position and velocity are equal to that of the observer on the rim,
providing that the observations of different events are simultaneous.
This also generalizes to observers arbitrarily moving in
flat spacetime.

The paper deals mainly with flat spacetime, with particular
attention paid to circular motion. However, it gives several
results which are of very general relevance, not only for arbitrary
motion in flat spacetime, but also for general relativity and
curved spacetime.

First, it has been demonstrated that
the generally accepted formula (\ref{eq3}) is not always correct.
The correct definition of the space line element depends on how it
is measured, so (\ref{eq3}) should be used with great care.
In some cases, the ``naive" formula (\ref{dl}) is more appropriate.
One such case is a metric of a frame in flat spacetime that can be
obtained from $g_{\mu\nu}=\eta_{\mu\nu}$ by a transformation of the form of
(\ref{er1})-(\ref{er2}), followed by an arbitrary restricted internal transformation.
Further investigation is needed in order to generalize this result.

Second, the paper demonstrates the importance of the use of
Fermi coordinates. One of the consequences of their use is the result
that for any local observer the velocity of light is isotropic and is
equal to $c$, providing that it is measured by propagating a light
beam in a small neighborhood of the observer.
This fact should be
used for a correct treatment of the Sagnac effect
if one wants to explore the general relativistic corrections.
Fermi coordinates should also be used in order to understand the
physical effects related to a rotating black hole, to give
a correct treatment of the Hawking radiation, as well as
for any other
physical effect, whenever intended to describe the world
how it looks like to a particular observer.

\section*{Acknowledgment}

The author is grateful to Damir Stoi\'{c} for motivating
discussions.
This work was supported by the Ministry of Science and Technology of the
Republic of Croatia under Contract No. 00980102.


\section*{Motivation}

Current physical theory predicts that at small scales the
conventional picture of spacetime as a 4-dimensional differential
manifold breaks down to something more discrete, finitary and
quantum. This inadequacy of the smooth spacetime manifold is on one
hand due to the ideal character of event determinations of a
classical observer, on the other due to the appearance of
singularities.

To deal with the first shortcoming of the manifold model,
we insist that realistic models of measurement should be pragmatic:
we actually perform a finite number of observations and record a
finite number of events.  Thus, the conventional infinitude of
events that we adopt to model classical spacetime structure seems
to be a gross generalization of little operational value: we have
no actual experience of a continuous infinity of events and
their infinitesimal differential separation can not be recorded
in the laboratory.

However, due to the success of the classical model of observation
at large scales one expects a connection between the
realistic models and their ideal counterpart. The anticipated
connection could be formulated in terms of a correspondence
principle. That is to say, the structure of ideal observations
arises in some kind of limit of the structures of pragmatic
measurements. The aim of this paper is to provide a physical
account for this correspondence.

The central object of the correspondence will be the {\em classical
event} living in the limit manifold. This is modelled by a point
there and a classical observer records no quantum interference of
events. But in the pragmatic regime we interpret the event as a
{\em pure state of spacetime} and we admit coherent quantum
superposition between events. In this sense
the quantum substratum of pragmatic events 'decoheres' to the classical
point event of the limit manifold and the physical meaning of
the correspondence principle is the usual quantum one due to
Bohr.

We use the finitary substitutes proposed by Sorkin (1991) to model
combinatorial relations between events in realistic measurements.
The incidence algebras due to Rota (1968) are employed to
accomplish the same thing but operationally, that is,
algebraically. There is a duality implicit here that is pregnant to
familiar notions about the duality of quantum dynamics. In our
treatment the Sorkin model is held to represent an evolution of
states much like the Schr\"odinger picture of quantum systems,
while the Rota model recalls the evolution of operators similar to
the Heisenberg picture. Our approach lies with
the latter picture although one is always able to switch
back to the Sorkin scheme (Zapatrin, 1998). In brief, we propose
that posets describe the dynamical evolution of events when the
algebras describe the dynamical evolution of event determinations
(operations).

Our algebraic approach is constructive, that is, we provide a
matrix representation for the algebras employed. The latter possess
preferred elements that represent the pragmatic observations that
in the ideal limit are expected to yield the irreducible elements
of classical observations: the manifold point events. They
constitute abelian subalgebras of the incidence algebras and are
coined {\em stationaries}. Of course, actual pragmatic observations
are expected to effect dynamical transitions between quantum states
of spacetime (stationaries). These are modelled by non-commuting
operations in the algebra of pragmatic events and are called {\em
transients}.  We anticipate that stationaries will correspond to
classical point events in the limit manifold (in Section \ref{s4} we
show how sequences of station{\it a}ries become station{\it e}ry
recording classical events), while transients to some kind of tangent
structure at an event.

To deal with the second shortcoming of the manifold model we note
that at the pragmatic level of observations there are no
points, but only algebras. We call this feature {\em
alocality}.  Nevertheless, we require the classical correspondence
limit to yield the familiar local structure of spacetime: the point
event and the space tangent to it. Since any point of the limit
manifold can be the host of a singularity of some important
physical field, the alocal quantum pragmatic substratum may prove
to be an effective resolution of spacetime into finite quantum
elements. The quantum substratum is asingular (finite) because
alocal. Thus, from our perspective, locality, that is, the
assumption of a differential continuum for spacetime (Einstein,
1924), is the prime reason for
singularities, so that at the pragmatic level of observation we
abandon it. We suggest a plausible quantum theory of spacetime
structure with a strong operational and finitistic character.
Then, based on the algebraic models of pragmatic observations we
may develop a non-commmutive differential geometry to erect a
quantum theory of gravity on it (Parfionov and Zapatrin, 1995).
The correspondence limit suggested in the present paper may also
be employed in this context to recover the classical algebra of
spacetime observables and the conventional differential geometry
of the spacetime manifold (used to describe general relativity)
from the pragmatic non-commutative quantum substratum.

It should be mentioned that our algebraic approach is rather
flexible in the following sense: alternatively to the novel notion
of alocality in the pragmatic regime we can formulate
the notion of {\em nearest neighbour} connections. The
latter was assumed by Finkelstein (1985)
to be the principal characteristic of the physical causal topology in
the quantum deep so as to localise in some sense a causality relation
between events (Bombelli {\em et al.}, 1987). This causality relation
was modelled by a partial order. Thus, if we physically interpret
Sorkin's finitary substitutes as causal sets (Sorkin, 1995), a recent
result (Zapatrin, 1998) allows us to represent the 'nearest neighbour'
causal connection between events algebraically in the pragmatic
regime thus vindicate Finkelstein's demand for an algebraic
representation of immediate causality (Finkelstein, 1985). There, in
turn, we have the advantage of interpreting this connection
operationally and study its quantum properties. The question we are
confronted with is:  what is the physical meaning in the pragmatic
algebraic regime of the Sorkin--Finkelstein local causality? As an
answer we expect a formulation of local causality in operational terms
with a quantum interpretation, something that is missing in Sorkin's
picture which is dual to ours.  Affine to this question is the
following one: how our pragmatic event determinations accord with and
be adequate models of the causal structure of the world at small
scales?

Finally, inspired by the Sorkin (1991) approach, we contend that
pragmatic measurements can be subjected to refinements. In passing
to the dual picture we deal with algebras and, in accordance with
the correspondence limit, the ideal ultimate refinement corresponds
to what is known as the algebra of classical observables
(coordinates) and the manifold supporting them.

\section{Finitary preliminaries}\label{s1}

A finitary topological space is defined in (Sorkin, 1991) as a
space with any bounded region in it consisting of a finite number
of points. This seems to be a reasonable model for actual
measurements involving a finite number of events during experiments
of finite spatiotemporal extent.

Any finitary topological space $\mmm$ can be equivalently pictured as
poset. Introduce the relation "$\to$" between points of $\mmm$

\[
p\to q \quad \Leftrightarrow \quad \mbox{ the constant sequence }
\{p,p,\ldots,p,\ldots\} \quad \mbox{ tends to } q
\]

\noindent using the standard definition of convergence: a sequence
$\{p_1,p_2,\ldots \}\to q$ iff for any open set $U$ containing $q$
there is a number $N_U$ such that $p_n\in U$ for any $n\ge N_U$.

The obtained relation "$\to$" is always reflexive ($p\to p$) and
transitive ($p\to q$, $q\to r$ imply $p\to r$). Vice versa, any
quasiordered set $(\mmm, \to)$ acquires a topology defined through the
closure operator on subsets $P\subseteq M$:

\[
{\rm\bf Cl} P =
\{q\,:\, \exists p \in P \quad p\to q \}
\]

For technical reasons (see Section \ref{s2}) we employ the
Alexandrov (1956) construction of {\em nerves} to substitute the
continuous topology.  Recall that the nerve ${\cal K}$ of a covering
$\uuu$ of a manifold $\mmm$ is the simplicial complex whose
vertices are the elements of $\uuu$ and whose simplices are formed
according to the following rule. A set of vertices (that is,
elements of the covering) $\{U_0, \ldots , U_k\}$ form a
$k$-simplex of ${\cal K}$ if and only if they have nonempty
intersection:

\[
\{U_0, \ldots , U_k\} \in {\cal K} \: \Leftrightarrow \:
U_0 \cap U_1 \cap \ldots \cap U_k \neq \emptyset
\]

\noindent Any nerve ${\cal K}$ being a simplex can be as well treated
as a poset, denoted also by ${\cal K}$. The points of the poset ${\cal K}$
are the simplices of the complex ${\cal K}$, and the arrows are drawn
according to the rule:

\[
p \to q \: \Leftrightarrow \: p \mbox{ is a face of } q
\]

\medskip

In the nondegenerate cases the posets associated with nerves and
those produced by Sorkin's (1991) `equivalence algorithm' are the
same. We choose nerves because their specific algebraic structure
makes it possible to build the dual algebraic theory.

\section{Incidence algebras}\label{s2}

The notion of incidence algebra of a poset was introduced by Rota
(1968) in a purely combinatorial context. Let $P$ be a poset.
Consider the set of formal symbols $\ketbra{p}{q}$ for all $p, q\in
P$ such that $p\le q$ and its linear span

\be{e38a}
\Omega = {\rm span}\{\ketbra{p}{q}\}_{p\le q}
\end{equation}

\noindent and endow it with the operation of multiplication

\be{38}
\ketbra{p}{q} \cdot \ketbra{r}{s} =
\ket{p} \braket{q}{r} \bra{s} =
\braket{q}{r} \cdot \ketbra{p}{s} =
\left\lbrace \begin{array}{rcl}
\ketbra{p}{s} &,& \mbox{if } q=r \cr
0 && \mbox{otherwise}
\end{array} \right.
\end{equation}

The correctness of this definition of the product, that is, the
existence of $\ketbra{p}{s}$ when $q=r$ is due to the transitivity
of the partial order. The obtained algebra $\Omega$ with the
product (\ref{38}) is called {\em incidence algebra} of the poset
$P$.

The incidence algebra $\Omega$ is obviously associative, but not
commutative in general. Namely, it is commutative if and only if
the poset $\ppp$ contains no arrows.

Let us split $\Omega$ into two subspaces

\[
\Omega = {\cal A} \oplus {\cal R}
\]

\noindent where

\be{e37a}
{\cal A} = {\rm span}\{\ketbra{p}{p}\}_{p\in P}
\end{equation}

\noindent and call

\[
{\cal R} = {\rm span} \{ \ketbra{p}{q} \}_{p<q}
\]

\noindent the {\em module of differentials} of the poset $P$. It is
a fact that ${\cal R}$ is a bimodule over ${\cal A}$.

As we refine the poset, the limit space is intended to be a
manifold. The incidence algebras are dual objects to posets,
therefore their behavior should be similar to that of differential
forms in classical geometry. The algebra ${\cal A}$ is intended to play
the r\^ole of classical coordinates, while ${\cal R}$ should be graded
being an analogue of the module of differential forms.

For this aim we consider only simplicial complexes which are
treated as posets. $p\le q$ means that $p$ is a face of $q$. The
elements of simplicial comlexes are naturally graded. Then
any basic element $\ketbra{p}{q}$ of the incidence algebra $\Omega$
acquires a degree being the difference of the degrees of its
constituents:

\be{e40a}
{\rm deg}\ketbra{p}{q} = \mbox{'the difference of cardinalities of
$p$ and $q$'}
\end{equation}

\noindent splitting $\Omega$ into linear subspaces

\be{40}
\Omega = \Omega^0 \oplus \Omega^1 \oplus \ldots
\end{equation}

\noindent with

\[
\begin{array}{rcl}
\Omega^0 &=& {\rm span}\{\ketbra{p}{p}\} \: = \: {\cal A} \cr
\ldots & \ldots & \ldots \cr
\Omega^n &=& {\rm span}\{\ketbra{p}{q}\}_{\deg\ketbra{p}{q} = n} \cr
\ldots & \ldots & \ldots
\end{array}
\]

\noindent making $\Omega$ graded algebra:

\[
\forall \omega \in \Omega^m, \omega' \in \Omega^n \qquad
\omega\om' \in \Omega^{m+n}
\]

\noindent and therefore making the module of differentials
${\cal R}$ graded ${\cal A}$-bimodule:

\[
{\cal R} = \Omega^1 \oplus \Omega^2 \oplus \ldots
\]

This grading makes the incidence algebras discrete differential
manifolds (Dimakis and M\"uller-Hoissen, 1999) as they posssess both
commutative scalars (the subalgebra ${\cal A}$) and differentials over it
(the module ${\cal R}$). For a more detailed account the reader is referred
to (Breslav and Zapatrin, 1999).

\section{Rota topology and the duality}\label{s2a}

In this section we establish a duality between a certain class of
finitary substitutes and their incidence algebras. We select
this class in such a way that canonical mappings between the
points admit conjugate homomorphisms of incidence algebras
making the correspondence between posets and algebras
functorial.

As it was shown in the previous section, with any poset its
non-commutative incidence algebra can be associated. It was proved by
Stanley (1968) that if two posets have isomorphic incidence algebras
then they are isomorphic. The reverse procedure building a poset
$P(\Omega)$ from an arbitrary finite-dimensional algebra  $\Omega$ was
suggested in (Zapatrin, 1998).  Let us briefly describe the
construction.

The elements of the poset $P(\Omega)$ are the irreducible representations
of the algebra $\Omega$. Building the partial order on $P(\Omega)$ cosists
of two steps. First, the nearest neighbour connections $p\to q$ are
built according to the following rule: let $p,q$ are two irreducible
representations of $\Omega$, denote by $p^0,q^0$ their kernels:

\[
p^0 = p^{-1}(0) \quad ; \quad q^0 = q^{-1}(0)
\]

\noindent which are ideals in $\Omega$. Then define the nearest
neighbours $p\to q$:

\be{e93}
p\to q \quad \Leftrightarrow \quad
p^0q^0 \neq p^0\cap q^0
\end{equation}

\noindent where the left-hand side $p^0q^0$ is understood as the
product of subsets of $\Omega$. The resulting partial order on the set
$P(\Omega)$ is obtained as the transitive closure of the relation
(\ref{e93}). The topology associated with this partial order is
referred to as {\em Rota topology}.

When the algebra $\Omega$ is commutative, the Rota topology is discrete
(no linked pairs $p\to q$). The obtained topology becomes non-trivial
only when $\Omega$ is non-commutative.

\paragraph{Remark.} When all irreducible representations of $\Omega$ are
one-dimensional we can build two topologies on the set $P(\Omega)$
Gel'fand and Rota ones, and it is interesting to compare them. The
result is the following: the Gel'fand topology is always discrete,
while the Rota topology may be non-trivial.

The possibility of mutual transitions between between finitary
topological spaces and algebras is based on the following theorem
(Zapatrin, 1998):

{\em
If the algebra $\Omega$ is the incidence algebra $\Omega(\ppp)$ of a poset
$\ppp$ then the resulting poset is isomorphic to $\ppp$:
}

\[
\ppp \simeq \ppp(\Omega(\ppp))
\]

As it was mentioned, Stanley (1968) theorem claims that

\[
\Omega(\ppp) \simeq \Omega(\qqq)
\Leftrightarrow
\ppp \simeq \qqq
\]

\noindent and one could expect that a poset homomorphism, that
is, a continuous  mapping of appropriate finitary topological
spaces, should give rise to a homomorphism of their incidence
algebras. Alas, this is not the case for general posets $\ldots$

To gather functoriality we have to restrict somehow the class of
posets we are dealing with and the mappings between them. We did
it already in the previous section in order to make the
incidence algebras graded. Namely, we restricted ourselves to
simplicial complexes. To make incidence algebras dual objects,
we, following Alexandrov (1956), restrict the mappings between
simplicial complexes to simplicial mappings only. Recall that a
mapping $\omega: {\cal K}_\alpha \to {\cal K}_{\alpha'}$ between two
simplicial complexes ${\cal K}_\alpha$ and ${\cal K}_{\alpha'}$ is said
to be simplicial if

\begin{itemize}
\item the $\omega$-image of any vertex in ${\cal K}_\alpha$ is a vertex
in ${\cal K}_{\alpha'}$

\be{e77}
\omega({\cal K}^0_\alpha) \subseteq {\cal K}^0_{\alpha'}
\end{equation}

\item $\omega$ is completely defined by its values on the vertices
of ${\cal K}_\alpha$.

\item $\omega$ preserves simplices
\end{itemize}

Under this condition the correspondence between posets and their
incidence algebras becomes functorial. With any $\omega:
{\cal K}_\alpha \to {\cal K}_{\alpha'}$ its adjoint $\omega^*:
\Omega({\cal K}_{\alpha'}) \to \Omega({\cal K}_\alpha)$ is defined in the
following way. Let $\ketbra{P}{Q}$ be a basic element of
$\Omega({\cal K}_{\alpha'})$. Then $Q$ may be represented as a
disjoint sum $Q = P+S$ and we put

\be{e79}
\omega^*(\ketbra{P}{Q}) =
\sum\{ \ketbra{p}{q} \:: \omega(p) = P;\, q=p+s,\, \omega(s)=S \}
\end{equation}

The direct verification shows that the so-defined $\omega^*$ is
always a homomorphism of the incidence algebras. As a result, we
have the duality between simplicial complexes and their
incidence algebras: if $\omega$ is surjective its adjoint $\omega^*$
is injective and vice versa.

\section{Physical interpretation of incidence algebras}\label{s3}

The differential manifold model of spacetime is an ill-founded
assumption and a gross generalisation of what we actually
experience as spacetime. It is essentially based on the
non-operational supposition that we can pack an infinity of
events in an infinitesimal spacetime volume element when, in
fact, we only record a finite number of them during experiments
of finite duration in laboratories of finite size. It is exactly
due to this trait of the manifold model that at small scales our
theories of quantum spacetime structure and dynamics are plagued
by non-renormalizable infinities of the values of many important
physical fields. On the other hand, the requirement that the laws
of nature are local almost mandates the assumption of smoothness
for spacetime and we seem to get back to square one. However, the
success the manifold has enjoyed in picturing the local dynamics of
matter should not mask the unphysicality of its character,
especially at small scales. In particular, quantum theory, when
applied to investigate the structure and dynamics of spacetime in
the small, is simply incompatible with a classical,
non-operational ideal of a continuous infinity of events labelled
by commutative coordinates. Pragmatic measurements of quantum
spacetime are finite and inevitably induce uncontrollable
dynamical perturbations to it. Thus, the requirement for
operationality and finiteness as well as the success that a quantum
theory of matter has had when formulated algebraically motivate us
to formulate an algebraic theory of pragmatic finite measurements
of spacetime at quantum scales.

The local structure of the differential manifold is the point event
and its infinitesimal differential neighbours in the tangent space.
As mentioned above, this classical geometric structure serves us
well in casting dynamical laws as differential equations (classical
locality), but is rather inadequate for picturing actual quantum
spacetime measurements that are finite hence free from infinities
(singularities). Especially in the quantum deep this classical
conception of locality can not survive. We propose to revise it by
substituting the geometrical point events and the space of
directions tangent to them by finitely generated algebras affording
a cogent quantum spacetime interpretation for their structure. In
this sense our scheme is alocal and finitary and more likely to
evade the infinities of the differential manifold. Of course, the
'naturalness' of our substituting quantum alocality for the
classical differential locality will be vindicated if we are able
to recover the limit manifold with its classical observables and
differential structure by some kind of correspondence principle
applied to the alocal algebras of pragmatic measurements. We carry
this out in the section \ref{s4}.

We give the following physical meaning to the elements of $\Omega$ in
(\ref{e37a}):

\begin{enumerate}
\item ${\cal A}$ constitutes the space of stationaries. The latter can
be thought of as elementary acts of determination of the pure
states of quantum spacetime. We interpret them as quantum
spacetime events. The algebraic connective '$+$' between them is
interpreted as coherent superposition between quantum events. The
commutativity of stationaries foreshadows the compatibility of the
determinations of the coordinates of events in the classical
manifold regime (Section \ref{s4}). In the dual (poset) picture the
sationaries correspond to self-incidences $p\to p$.

\item $\Omega^1$ constitutes the space of transients. These can be
thought of as elementary quantum dynamical processes between
stationaries, thus they represent discrete one-step transitions
between quantum spacetime events. Transients do not commute with
each other and this foreshadows the Lie structure of covectors in
the limit space.

\item $\Omega^i$ ($i\ge 2$) constitute the spaces of paths which are
thought of as composites of transients. If we associate with a
transient a quantum of an additive physical
quantity like energy (or its dual time), then the total grade of the
appropriate element of the algebra corresponds to the total energy
associated with it (or to the duration of the whole transition process).

\end{enumerate}

In the Motivation we alluded to the Sorkin poset scheme as being an
analog of the Schr\"odinger picture of quantum dynamics while our
algebraic approach as being the simile of the Heisenberg picture:
this is based on the duality of the two approaches (Section
\ref{s2a}). In an analogous way quantum states are the linear duals
of the operators in the conventional algebraic approach to quantum
mechanics.

Here too any finitary substitute is associated with an incidence
algebra in such a way that the topology of the poset is the same as
that encoded in the algebra. This resembles the fact that the
Schr\"odinger and the Heisenberg pictures encode the same
information about quantum dynamics.  Furthermore, the arrows
between point events in the Sorkin scheme can be thought of as the
directed dynamical transitions of spacetime event-states while in
our picture such dynamical connections are between pragmatic
operations. The topology in both schemes is physically interpreted
as dynamical connections between events although our picture being
algebraic naturally affords a quantum interpretation.

\section{Limiting procedure and the correspondence principle
}\label{s4}

When spacetimes are subsituted by finitary topological spaces,
we may consider finer or coarser experiments. That is why we
have to formalize the notion of refined experiment. Within the
Sorkin discretization procedure (Section \ref{s1}) a refinement
means passing to an inscribed covering of the manifold. In this
case any element of the finer covering is contained in an
element of the coarser one. Since we are dealing with nerves and
simplicial mappings between them we have to take care of the
condition (\ref{e77}). Recall that a vertex of the nerve is
associated with an element of the covering. In general it
may happen that a small region of the fine covering can belong
to two elements of a coarser one. So, we have have to require
for any element of the fine covering to keep track of its origin
in order for (\ref{e77}) to hold.

Each step of a limiting procedure, that is, a refined covering,
gives rise to a projection of appropriate complexes: the finer
one is projected to the coarser one. In the dual framework
we have an injection of the smaller algebra associated with a
coarser measurement to the bigger one.

In general, limiting procedures for approximating systems (whatever
they be, posets or algebras) are organised using the notion of
converging nets. Namely, each pragmatic observation is labelled by
an index $\alpha$ and we have the relation of refinement $\succ$ on
observations: $\alpha \succ \alpha'$ means that the observation
$\alpha$ is a refinement of $\alpha'$.

When we are dealing with posets with each pair $\alpha,\alpha'$
such that $\alpha \succ \alpha'$ a canonical projection
$\omega_{\alpha'}^\alpha : {\cal K}_\alpha \to {\cal K}_{\alpha'}$ is defined
such that for any $\alpha \succ \alpha' \succ \alpha''$

\[
\omega_{\alpha''}^\alpha =
\omega_{\alpha''}^{\alpha'}\omega_{\alpha'}^\alpha
\]

\noindent and we introduce the set of threads. A thread is a
collection $\{t_\alpha\}$ of elements $t_\alpha \in {\cal K}_\alpha$
such that

\[
t_{\alpha} = \omega_{\alpha}^{\alpha'} t_{\alpha'}
\]

\noindent whenever $\alpha \succ \alpha'$. Denote by $\tttt$ the
set of all threads.

The next step is to make $\tttt$ a topological space which is done
in a standard way (Alexandrov, 1956): $\tttt$ is a subspace of the
total cartesian product

\[
\tttt_0 = \times_\alpha {\cal K}_\alpha
\]

\noindent while each of ${\cal K}_\alpha$ is a topological space.
Endow $\tttt_0$ with the product Tikhonov topology, then $\tttt$
being a subset of $\tttt_0$ becomes topological space.
Finally we obtain the limit space $X$ as the collection of all
closed threads from $\tttt$. This procedure is described in detail
in (Sorkin, 1991).

\medskip

The scheme for building limit algebras is exposed in (Landi and
Lizzi, 1998). As mentioned above, with any pair $\alpha \succ
\alpha'$ of pragmatic observations we have a canonical injection

\[
\omega^{*\,\alpha'}_{\alpha} : \Omega({\cal K}_{\alpha'}) \to
\Omega({\cal K}_{\alpha})
\]

Moreover, due to the requirement (\ref{e77}) the restriction of
each $\omega^*$ on commutative subalgebras ${\cal A} = \Omega^0 \subseteq
\Omega$ is well defined. Now we first consider the set of all
sequences

\[
{\bf \Omega} = \times_\alpha \Omega({\cal K}_\alpha) =
\{ \{a_\alpha\}\,|\, a_\alpha \in \Omega({\cal K}_\alpha) \}
\]

\noindent and select the set of {\em converging} sequences in the
following way. Note that ${\cal A}$ is an algebra. Introduce a norm
$||\cdot||_\alpha$ in each finite-dimensional algebra
$\Omega({\cal K}_\alpha)$, then a sequence $\{a_\alpha\}$ converges if
and only if for any $\epsilon > 0$ there exists a filter
$\fff_\epsilon$ of indices $\alpha$ such that

\[
\forall \alpha, \alpha'\in \fff_\epsilon
\quad \alpha \succ \alpha' \Rightarrow
||\omega^{*\,\alpha}_{\alpha'}a_\alpha - a_{\alpha'}||_{\alpha'} <
\epsilon
\]

Since any element of the limit algebra is a net we may consider the
coupling between the limit algebra and the limit space which
consists of nets. The result of this coupling is a converging net
of numbers whose limit is thought of as the value of an element of
the limit algebra at a point of the limit space.

The Sorkin scheme recovers the manifold in the limit of
refinements of finitary posets. Our dual picture aspires to the
same in the limit of resolution of pragmatic event
determinations. Since our algebraic scheme affords a quantum
spacetime interpretation, this limit can be thought of as a
correspondence principle linking the finitary quantum spacetime
substrata with the smooth classical spacetime manifold. The
alocal, algebraic quantum spacetime determinations of the substrata
converge to the local geometric spacetime point and its cotangent
space. This is to be contrasted for instance with the Bombelli {\em et
al.} (1987) causal set scenario where the limiting procedure may be
thought of as a `random sprinkling' of events according to some
appropriate distribution so that the `limit spacetime manifold', with
its topological, differential and Lorentz-causal structure, arises as a
statistical average of causal sets, thus it is essentially of {\em
thermodynamic} nature.

On the other hand, our correspondence limit is well-defined in the
{\em quantum} (rather than statistical) sense as the well-known
correspondence principle: the pragmatic quantum stationaries
decohere to the point events of the limit manifold, while the
non-commuting transients to covectors.

\section*{Concluding remarks}

In the present paper we gave quantum spacetime interpretation to
the incidence algebras induced by posets which, in turn,
correspond to finitary topological spaces. Sorkin's limit for
recovering the manifold as a maximal refinement of finitary
posets is cast here as Bohr's correspondence principle. Still,
due to the implausibility of any notion of pre-existing space in
the quantum dynamical deep, we would rather give a more
physical, causal or temporal interpretation to the posets'
partial order (Sorkin, 1995), so that we can link our algebraic scheme
with Bombelli {\em et al.} (1987) causal set approach to quantum
gravity. Our quantum interpretation of the incidence algebras
induced by causal sets is a first step into yet another attempt
at quantizing causality (Finkelstein, 1969). It is one of the
authors' previous result (Zapatrin, 1998) and Finkelstein's
(1985) claim for immediate causal links between
events to represent the physical causal topology that caught our
attention and motivated us to try to link the present work with causal
sets.  This project, however, is still at its birth.

\paragraph{Acknowledgments.} The work was carried out within the
program "T\^ete-\`a-t\^ete in St.Petersburg" under the auspices of the
Euler Mathematical Institute. The authors express their
gratitude to the participants of the Friedmann seminar
(St.Petersburg, Russia) for their kind suggestions.
One of the authors (IR) wholeheartedly acknowledges
numerous exchanges on quantum spacetime structures with Professor
Anastasios Mallios. The second author (RRZ) acknowledges the
support from the RFFI research grant and the research grant
"Universities of Russia".

\medskip

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