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\begin{document}
\title{Inertial frames, special relativity  and consistency}
\author{Radhakrishnan Srinivasan\thanks{R \& D Group -- Exports services,
IBM Global Services India Pvt.\ Ltd., 5th floor, Golden Enclave,
Airport road, Bangalore~560017, India.
E-Mail:~\texttt{sradhakr@in.ibm.com}}}
\date{}
\maketitle

\begin{abstract}
The theory of special relativity (SR) is considered in the framework
of classical first-order logic. The four axioms of SR are:
The existence of a flat four-dimensional spacetime
continuum, the existence of global inertial frames of reference and
Einstein's two postulates. The propositions permitted are those
involving the kinematics and dynamics of SR as formulated
from an inertial frame; these give a complete description of
the evolution of the universe.
Assuming SR as consistent, there must exist a model $M$
for SR in which $F$ is an inertial frame in a non-trivial universe
$U(IBC)$ of material objects, with appropriate initial-boundary
conditions $IBC$ specified. Let $P$ be defined as
``$F$ is an inertial frame in $U(IBC)$''.
Using G\"{o}del's second incompleteness theorem, it is argued that
$P$ is undecidable in SR. Let $Q$ be any proposition such that
$(Q \Rightarrow P)$ is not a theorem of SR. If, in addition,
$(P \Rightarrow Q)$ is a theorem of SR
then $Q$ must necessarily be true in $M$.
It follows that there must exist a model~$N$ for
SR in which $Q$ is true and $P$ is false, i.e., $F$ is an
accelerated frame in $U(IBC)$. The philosophical
and mathematical implications of this result for the consistency
of SR are discussed.
\end{abstract}

\section*{Introduction}
It is well known that in the special theory of relativity
the identification of a suitable inertial frame of reference
is problematic. An excellent summary of the difficulties is available
in~\cite{RR}, and one can do no better than quote extensively the
relevant portions from this reference:
\begin{quote}
In a sense, special relativity is analogous to ``naive set theory''
in mathematics. By this I mean that special relativity is based
on certain plausible-sounding premises which actually are
quite serviceable for treating a wide class of problems, but
which on close examination are susceptible to self-referential
antinomies. This is most evident with regard to the assumption
of identifiability of inertial frames. As Einstein remarked,
``in the special theory of relativity there is an inherent
epistemological defect'', namely, that the preferred class
of reference frames on which the theory relies is circularly
defined. Special relativity asserts that the lapse of proper
time between two (timelike-separated) events is greatest
along the inertial worldline connecting these two events ---
a seemingly interesting and useful assertion --- but if we ask
which of the infinitely many paths connecting those two events
is the ``inertial'' one, we can only answer that it is the
one with the greatest lapse of proper time. If we simply accept
this uncritically, and we are willing to naively rely on the
testimony of accelerometers as unambiguous indicators of
``inertia'', we have a fairly solid basis on which to do
physics, and we can certainly work out correct answers to many
questions. However, the epistemological defect was worrisome
to Einstein, and caused him (in a remarkably short time) to
abandon special relativity and global Lorentz invariance as
a suitable conceptual framework for the formulation of
physics. \newline  \newline
The naive reliance on accelerometers as unambiguous indicators
of global inertia in the context of special relativity is
immediately undermined by the equivalence principle, because
we're then required to predicate \emph{any} application of
special relativity on the absence (or at least the negligibility)
of irreducible gravitational fields, and this condition is
simply not verifiable within special relativity itself, because
of the circularity in the principle of inertia. This circularity
genuinely troubled Einstein, and was one of the major motivations
(along with the problem of reconciling mass-energy equivalence
with the Equivalence Principle) that led him to abandon
special relativity.
\end{quote}
Elsewhere in~\cite{RR}, it is asserted that:
\begin{quote}
Suppose we naively superimpose special relativity on Newtonian
physics, and adopt a naive definition of ``inertial worldline'',
such as a worldline with no locally sensible acceleration. On
that basis we find that there can be multiple distinct
``inertial'' worldlines connecting two given events (e.g.,
intersecting elliptical orbits of different eccentricities),
which conflicts with the special relativistic principle of unique
inertial interval between any pair of timelike separated events.
\dots Of course, with special relativity (as with set theory)
we can easily block such specific conundrums --- once they are
pointed out --- by imposing one or more restrictions on the
definition of ``inertial'' (or the definition of a ``set''),
and in so doing we make the theory somewhat less naive, but
the experience raises legitimate questions about whether we
can be sure we have blocked all possible escapes.
\end{quote}
The author of~\cite{RR} later comments:
\begin{quote}
It might be argued that relativity is a purely formalistic theory,
which simply assumes an inertial frame is specified, without telling
how to identify one. Certainly we can completely insulate special
relativity from any and all conflict by simply adopting this
strategy, i.e., asserting that special relativity avers no mapping
at all between it's elements and the objects of our experience.
However, although this strategy effectively blocks conflict,
it also renders the theory quite unfalsifiable and
phenomenologically otiose.
\end{quote}

In this paper, we adopt the strategy suggested in the above quote,
that is, we treat special relativity as a ``purely formalisitic
theory~\dots''. We then argue that the above conclusion
of~\cite{RR}, namely, ``Certainly we can completely insulate
special relativity from any and all conflict by adopting
this strategy~\dots'', is questionable. The problem is caused by
the fact that in a formalisation in first-order predicate logic
of the theory of special relativity (which treats only inertial
frames), the self-referential nature of any proposition like
``$F$ is an inertial frame in universe $U$'' causes it to be
formally undecidable in the theory, assumed consistent.
Therefore the consistency of the theory demands the existence
of a model in which the negation of the above assertion holds,
i.e.,``$F$ is an accelerated frame in universe $U$''. In fact
in this model, one can require a broad class of logical consequences
of ``$F$ is an inertial frame in universe $U$'' to also hold.
We conclude with a discussion of the philosophical and
mathematical implications of this result
for the consistency of the theory of special relativity.

\section*{The postulates of special relativity}
The emphasis in this paper is on presenting the basic ideas without
burying them in a detailed mathematical treatment
(which is reserved for future work).
The four postulates of the formal theory of special relativity (SR)
that will be considered here are adapted from~\cite{AH}, as follows:
\\ \\
P1. The geometry of spacetime.  \\
Statement: ``Space and time form an infinite four-dimensional continuum''.
\\ \\
P2. The existence of globally inertial frames.  \\
Statement: ``There exist global spacetime frames with respect to
which unaccelerated objects move in straight lines at constant
velocities''.
\\ \\
P3. The speed of light is constant. \\
Statement: ``The speed of light $c$ is a universal constant, the
same in any inertial frame''.
\\ \\
P4. The principle of special relativity. \\
Statement: ``The laws of physics are the same in any inertial frame,
regardless of position or velocity''.\\ \\

Note that P2 implies that the geometry of spacetime is Euclidean (flat),
``light'' in P3 means electromagnetic waves in general (propagating
in a vacuum) and ``laws'' in P4 refer to (a re-formulation of) Newton's
laws of mechanics and Maxwell's laws of electromagnetism.
We consider a suitable formalization of the theory SR in
the classical first-order predicate logic. A non-trivial universe
$U(IBC)$ containing a large number of material objects is specified;
here the argument $IBC$ represents well-posed initial-boundary conditions
consistent with P1--P4 and sufficient for a complete determination
of the evolution of $U(IBC)$ in spacetime. One could take $U(IBC)$
to be that part of the ``real'' universe that is sufficiently far
away from massive celestial objects, so that it is essentially free
from gravitational fields and spacetime is flat for all practical
purposes; the objects in $U(IBC)$ would have to be sufficiently
small for this approximation to be valid. Of course, in a purely
formal treatment of SR, one need not necessarily have to deal
with reality and this is the viewpoint adopted here (as noted
in the Introduction). The propositions permitted
in SR are all (and only) those relating to the kinematics
and dynamics of special relativity as determined
by the above postulates and involve both
mechanics and electromagnetism. Thus if SR is consistent, a model for
SR will assign truth values to all these propositions which will
provide a complete and detailed description of the evolution of
$U(IBC)$ in spacetime. Note that $SR$ is sufficiently strong for
G\"{o}del's incompleteness theorems~\cite{Gd} to apply.

\section*{Main results}
It is extremely important to observe that P2 not only
affirms the existence of globally inertial frames of reference
\emph{non-constructively} (i.e., without specifying how to identify
an inertial frame); in fact P2 \emph{does not even give a complete,
unambiguous definition of an inertial frame}. The reason for this
ambiguity is that the term ``unaccelerated'' in P2 has meaning only
\emph{after} the concept of ``inertial frames'' is in place.
So it is not possible to \emph{define} inertial frames by first trying
to identify unaccelerated objects. This is also precisely why
accelerometers cannot serve the purpose of defining inertial
frames from a formal point of view. 

Our contention is that the true, constructive definition of
inertial frames is in P3 and P4. Note that one could merely
state P2 as ``There exist globally inertial spacetime frames
as defined in P3 and P4''; the straight line motion of
unaccelerated objects with respect to inertial frames
is Newton's first law and is part of P4.
Observe that P1--P4 say nothing at all about the laws of physics
in accelerated frames. So an inertial frame cannot be identified
non-constructively in SR, i.e., by an argument of the form: ``Suppose
$F$ were an accelerated frame. Then such-and-such contradiction
would arise (such as, an accelerometer giving a non-zero reading).
Therefore $F$ is an inertial frame''.
\begin{definition}[Inertial frame]
The only way to confirm that a given frame $F$ can be inertial in $U(IBC)$
is to first \emph{assume} it to be one and then verify that \emph{P3} and
\emph{P4} are satisfied globally in spacetime. In other words the
proposition $P$, defined as ``$F$ is an inertial frame in $U(IBC)$''
can be true only if the theory \emph{SR+}$P$ is consistent.
\end{definition}
Note that the above definition asserts the implication
\begin{equation}\label{defIFR}
P \Rightarrow \mbox{con}(\mbox{SR}+P),
\end{equation}
where con(SR+$P$) is G\"{o}del's formalization of the proposition
``SR+$P$ is consistent''.
To see why our definition of inertial frame demands (\ref{defIFR}),
note that \emph{any} inconsistency in SR+$P$ would automatically
translate into a proof (by contradiction) of $\neg P$ in SR, because
such inconsistency would contradict P3 and/or P4.
``Inconsistency'' here means that SR+$P$ proves
both $T$ and $\neg T$ for some legitimate proposition $T$ of SR.
If one is able to deduce such an inconsistency in SR+$P$,
the laws of physics certainly do not hold as seen from $F$.
A non-zero reading of an accelerometer from $F$ would imply
that P4 does not hold and so would rule out $F$ as an inertial
frame; but a zero reading is only one instance of P4 holding from
$F$, and does not rule out the possibility of some other contradiction
arising elsewhere in spacetime. So a zero reading of an accelerometer
from $F$ does not confirm it as an inertial frame, by our definition.

It is important to recognize that the
reverse implication in (\ref{defIFR}) need not hold~--- there is
nothing in the \emph{formal} theory SR that implies uniqueness
of possible inertial frames. In fact if SR+$P$ is consistent then
the reverse implication \emph{cannot} hold, as the following metatheorem
asserts:
\begin{theorem}
If \emph{SR} is consistent, then P is not a theorem of \emph{SR}.
\end{theorem}
\begin{proof}
Since $\mbox{con}(\mbox{SR}+P) \Rightarrow \mbox{con}(\mbox{SR})$,
it follows from (\ref{defIFR}) that
$P \Rightarrow \mbox{con}(\mbox{SR})$. G\"{o}del's second
incompleteness theorem then implies that if SR is consistent,
P is unprovable in SR. One concludes that if SR
is consistent, then there must exist a model for SR in which P is false,
i.e., $F$ is an accelerated frame in $U(IBC)$. The metatheorem
follows.
\end{proof}
\begin{remark}\label{rm1}
Suppose SR+$P$ is consistent. By our definition of inertial frame,
$P$ can be true in SR and there must exist a model~$M$ for SR
in which $F$ is an inertial frame in $U(IBC)$. By the metatheorem,
it follows that one \emph{must also} be able to take $P$ as false,
in a model~$T$ for SR. One concludes that if SR is consistent,
it \emph{cannot} uniquely identify a given frame $F$ as inertial in
$U(IBC)$. Of course, one could argue that the evolution of $U(IBC)$
in the models $M$ and $T$ need not be identical and so this result
does not necessarily imply that SR is inconsistent. But a
stronger result in Remark~\ref{rm2} below seems to increase
the probability of an inconsistency in SR. It is interesting to
note that in the non-Aristotelian finitary logic (NAFL) proposed by
the author~\cite{SR}, this non-uniqueness immediately translates
into an inconsistency in SR. So the theory of special relativity
cannot be formalized in any consistent NAFL theory.
\end{remark}
\begin{remark}\label{rm2}
Suppose SR+$P$ is consistent. Let $Q$ be such that
$(Q \Rightarrow P)$ is \emph{not} a theorem of SR; this is possible
because $P$ is not a theorem of SR, by the metatheorem.
If, in addition, $Q$ is a logical consequence of $P$,
i.e., if $(P \Rightarrow Q)$ is a theorem of SR,
then $Q$ must necessarily hold in model~$M$ of Remark~\ref{rm1}.
For example, $Q$ cannot be identical to $P$,
but $Q$ could stand for the null result of an accelerometer
experiment performed from $F$. $Q$ could also be the conjunction
of an arbitrarily large (but finite) number of propositions provable
in SR+$P$, subject only to the above restriction. It follows
that there must exist a model $N$ for SR in which $Q$ is true
and $P$ is false.
\end{remark}
\begin{remark}
Some care would be needed in reformulating $IBC$ and $Q$ so as to hold
in the models $T$ and $N$ of Remarks~\ref{rm1}~and~\ref{rm2}.
This is best illustrated with a simple example.
Suppose A and B are two bodies undergoing an initial
relative acceleration with respect to each other. Now $IBC$ for the
model $M$ of Remark~\ref{rm1} might be
that $A$ is inertial and $B$ is accelerating
with respect to it, i.e., there is a force on $B$. When reformulating
$IBC$ to hold in $T$ or $N$, where, for instance, $B$ is
an inertial frame, obviously the force will now act on $A$.
Note that force is simply an imposed acceleration and
this will obviously change if inertial frames
are switched; this does not mean that $IBC$ has been invalidated
in these models. What should not change, if one wishes to retain
the same $IBC$, is the \emph{relative} acceleration between A and B.
To summarize, $IBC$ should be specified in such a way that
$(IBC \Rightarrow P)$ is not be provable in SR (as in the case of
$Q$). Of course, similar considerations apply when specifying
$Q$ in model $N$.
\end{remark}

\section*{Discussion}
For appropriate $P$, $Q$ and $U(IBC)$, any proof in SR that the
model $N$ of Remark~\ref{rm2} cannot exist implies that SR
is inconsistent. In the model $N$, let the appropriate inertial
frame be $G$, which must be different from $F$. Not only must
$IBC$ be satisfied in $N$; the requirement that $Q$ must also
hold heavily over-determines the governing differential equations
of the system and the genuine question that arises is whether
$N$ can exist at all, given that $Q$ can be chosen to be an
arbitrary logical consequence of $P$. This is a mathematical
issue that may be difficult to settle. As an example, suppose that
the ``real'' world we live in is free from gravitational fields, so
that SR applies. No matter how many facts we collect in support of the
conjecture $P$, the conjunction of all these facts (i.e., $Q$) must hold
in a model $N$ for SR in which $P$ is false, assuming SR to be
consistent. In other words, no matter how many experiments
(numerical simulations) support our conjecture that $F$ is an inertial
frame in our universe, the proposition that
``there exists some experiment (numerical simulation) whose
results will violate this conjecture'' cannot be refuted in SR
(assumed consistent). Note that this proposition asserts the indicated
existence non-constructively; this is permitted in classical logic.
This is very similar to G\"{o}del's assertion that con(SR) is
unprovable in SR (assumed consistent), and so there must exist a model
for SR in which con(SR) is false.

Philosophically, one could ask whether classical logic is indeed
the appropriate logic in which to formulate SR. Is it possible to
avoid the above difficulty by formulating SR in intuitionistic or
constructivistic logic, where non-constructive existence would be banned?
Note that the paradox is caused by the self-referential nature of
the definition of ``inertial frames'' and ``acceleration''.
Is it valid to assert that an object is ``really'' accelerating
with respect to the ``truly'' inertial frame, when the theory~SR
fails to prove such an assertion? In the logic~NAFL proposed by
the author~\cite{SR}, the answer is an emphatic ``no''; the
law of the excluded middle does not apply to a proposition
undecidable in the theory in which it is formulated, and any
``reality'' independent of the human mind does not exist.

A particularly interesting consequence of this paper is for the well-known
twin paradox~\cite{TP}. The twins A and B are together at some particular
instant and are of identical age; B then takes off on a round
trip consisting of straight-line motion away from A and back.
Let $P$ denote ``A is attached to an inertial
frame in $U(IBC)$'', and let $Q$ stand for ``B is younger
than A when they meet again after the round trip by B''.
Then by Remark~\ref{rm2}, there must exist a model $N$ for
SR (assumed consistent) in which $P$ is false and $Q$ is true
for an \emph{identical} round trip by B in
(observably) identical conditions. What would make this paradox
even more interesting is if we posit that the two twins are the
\emph{only} objects in $U(IBC)$. Of course, the first issue would
be whether G\"{o}del's second incompleteness theorem, used in our
result, would apply in this case. Regardless of the answer,
it is our contention that SR still fails to identify an inertial
frame uniquely; the model $N$ must still exist,
if SR is consistent. If A and B are the only two identical material
objects in $U(IBC)$, it will be impossible \emph{in principle} to
identify which of the two is ``really'' accelerating. Neither A nor B
can be attached to an inertial frame in $N$, and we would have to point
to empty space to posit an inertial frame. The question is, does
this not amount to postulating absolute space, denied in SR?
It should be emphasized that these paradoxes arise only in the
theory SR as formulated in this paper; they may possibly be
satisfactorily resolved in other formulations of special relativity
in which accelerated frames of reference are treated.

\section*{Dedication}
The author dedicates this research to his son R.~Anand and wife
R.~Jayanti.

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\bibitem{SR}
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\bibitem{TP}
Baez, John. \\
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\end{thebibliography}
\end{document}

