††thanks: On leave of absence from Rzhanov Institute of Semiconductor Physics
and the Novosibirsk State University, Russia

# How to detect the spacetime curvature without rulers and clocks

A. V. Nenashev<EMAIL_ADDRESS>Department of Physics and Material
Sciences Center, Philipps-Universität Marburg, D-35032 Marburg, Germany  S. D.
Baranovskii<EMAIL_ADDRESS>Department of Physics and
Material Sciences Center, Philipps-Universität Marburg, D-35032 Marburg,
Germany  Department für Chemie, Universität zu Köln, Luxemburger Straße 116,
50939 Köln, Germany

###### Abstract

We demonstrate how one can distinguish a curved 4-dimensional spacetime from a
flat one, when it is possible, relying only on the causality relations between
events. It is known that it is possible only for spacetimes that are not
conformally flat. We prove that if a spacetime is not conformally flat, then
its non-flatness can be verified by only a few (sixteen) measurements of
causal relations. Therefore the results of this paper clarify what can be said
about flatness or non-flatness of the spacetime after a finite number of
measurements of causal relations.

##  I Introduction

What is the flat space and what is the curved space? The usual answer is: the
space is flat if it can be mapped (at least piecewise) onto a flat Euclidean
space with preserving the metric. Otherwise, the space is curved.

In more details, if one wants to know whether the space is flat or curved, one
have to choose a coordinate system—a collection of coordinates for every
point—and measure all the distances between neighboring points. These
measurements provide the metric tensor $g_{ik}$ that generally depends on
coordinates, i. e. on radius-vector $\vec{r}$ [1, §13.2]. Function
$g_{ik}(\vec{r})$ (a metric) contains all information about geometry of the
space. Then, if the same function $g_{ik}(\vec{r})$ can be achieved in the
flat space, by choosing a proper coordinate system there, then the geometry of
the space under study is also flat. And if it is impossible, then the
properties of the given space differ from that of the flat one, meaning that
the space is curved.

A simplest example of the curved space is the sphere, and a customary
coordinate system on it is defined by the latitude and the longitude. One
cannot draw, on a flat piece of paper, a latitude-longitude net that
reproduces the same net on the sphere without any distortions—and this exactly
means that the sphere is not flat.

All the above is valid also for the _spacetime_. It is flat (curved) if its
metric can (cannot) be mapped onto the Minkowski space. To determine the
metric tensor of the spacetime, one have to measure both distances and time
intervals between events, and thus have to be equipped by some sorts of rulers
and clocks. In principle, a finite number of measurement is enough, as in a
“five-point curvature detector” considered by Synge [2, Chapter XI, §8]. An
alternative way of finding out the spacetime curvature consists in looking at
neighboring word lines of freely falling test particles, and measuring how
fast initially parallel world lines diverge in the course of time [1, Chapter
11]. The “gravity gradiometer” for measuring the curvature of spacetime [1,
§16.5] is a construction made of rigid rods. Detection of gravitational waves,
i. e. curvature variations, is based on interferometric measurements of
distances between stationary objects in now-operating observatories such as
LIGO [3], Virgo [4] and KAGRA [5], or between freely moving objects
(satellites) in proposed LISA detector [6]. Also laser ranging of the Moon and
artificial satellites, as well as binary pulsar timing, may be used for
observation of low-frequency gravitational waves [7]. All these methods also
imply usage of rulers and/or clocks. Even the “ideal rods and clocks built
form geodesic world lines” (the Marzke-Wheeler clocks) [1, §16.4] are based on
the method of Schild’s ladder, which demands finding midpoints of geodesic
line segments, i. e. using rods or clocks. Quantum mechanics allows to measure
the difference between proper times along two world lines in interferometric
experiments with massive bodies, which leads to so-called gravitational
Aharonov-Bohm effect [8]. Recently, this effect was observed in an experiment
with freely falling rubidium atoms [9]. This experiment can be considered as a
probe of space-time curvature, where atoms act as quantum clocks [10].

But what if we do not have any rulers and clocks? This question arises from
the fact that the spacetime, by itself, does not contain any built-in length
or time standards (apart from Planck units that are too small for having any
relation to “normal”, non-quantum-gravity physics). Any information about
lengths and times may come only from material objects, such as rigid rods,
oscillating balance wheels or quartz crystals, and so on.

It is even not evident, whether there are appropriate rulers (clocks) at any
spatial (temporal) scale. According to the “desert” hypothesis in particle
physics, there are no particles with masses from $\simeq 10^{12}$ eV to
$10^{25}$ eV. Then, it might be no physical objects with lengths between
$\simeq 10^{-18}$ m (the currently probed length scale) and $10^{-31}$ m (the
Grand Unified Theory length scale). In such a case, what would be the meaning
of spacetime curvature within this range of scales?

The spacetime itself, however, possesses very fundamental relations between
its points (events)—namely the _causal relations_. Roughly speaking, causal
relations between two events $A$ and $B$ reflect where is event $B$ located
with respect to the _light cone_ of event $A$. Event $B$ is located on the
light cone of event $A$, if one can send a light signal at one of these events
(the earliest one), and the signal will arrive exactly at the other event. If
it is not the case, then event $B$ is located either in the _absolute future_
with respect to $A$ (i. e. information from $A$ can reach $B$), of in the
_absolute past_ (i. e. information from $B$ can reach $A$), or _elsewhere_ (i.
e. it is impossible to send information neither from $A$ to $B$, nor from $B$
to $A$). All these concepts—light cone, absolute future/past, elsewhere—are
causal relations, and they do not rely on any measuring tools like rulers and
clocks. It is worth noting that the term “light cone” does not actually refer
to light as a concrete physical phenomenon—an electromagnetic wave. The word
“light” here just means a fastest possible carrier of information,
irrespective of its physical nature.

Let us consider a gedanken experiment, in which only causal relations between
events are involved. There are four lamps: red, yellow, green and blue, and
four observers: Alice, Bob, Charlie and Daniel, see Fig. 1a. Each lamp was
initially switched off, and switches on at some moment (these moments can be
different for different lamps). Imagine that Alice saw the light from all four
lamps _simultaneously_. Also Bob saw all four switchings on simultaneously.
And so did Charlie. It is also known that Daniel saw simultaneously the light
from red, yellow and green lamps. Then, the question arises: _did Daniel see
the blue light at the same moment of time as he saw other three lights?_

Figure 1: A thought experiment with light signals: (a) in the flat Minkowski
space; (b) in the space with a massive body (a star) near one of light rays;
(c) a sketch of world lines of light signals; (d) a similar sketch for an
alternative version of the experiment.

In the flat Minkowski spacetime, where light travels with a constant velocity
$c$, the answer to this question is “yes” (provided that all lamps and
observers are located in different places). This answer follows from Theorem 1
that is formulated and proved below. In the terminology introduced in Section
II, answer “yes” is guarenteed by _well-stitchedness_ of the Minkowski
spacetime.

Let us repeat this thought experiment in the curved spacetime of general
relativity. It is possible that a light ray from the blue lamp to Daniel
passed close to a massive body—a star, see Fig. 1b. Then the ray underwent
gravitational deflection of light, and arrived to Daniel later then in the
case when the star is absent (so-called Shapiro time delay effect [1, §40.4]).
In this circumstance, Daniel did not see all four lamps simultaneously, and
the answer to the above-formulated question is “no”.

Hence, it is possible to figure out that the spacetime is curved by testing
causal relations only, without any clocks and rulers. Namely, if the
experiment considered above gave the result “no” at least once, then the
existence of spacetime curvature is proven. This is one of the principal
results of the present article.

But what if this experiment, being repeated for all possible positions of
lamps and observers and all moments of switching on, always gives the result
“yes”? Does it mean that the spacetime is flat? No, it does not. It is well
known that causal relations carry some fraction of information about the
spacetime metric, but not the full information. Light cones are conserved
under a _conformal transformation_ that acts on the metric tensor $g_{ik}$ as
follows:

$g_{ik}(\vec{r})\to\Omega^{2}(\vec{r})g_{ik}(\vec{r}),$ (1)

where $\Omega$ is an arbitrary non-zero function of the spacetime coordinates
[11, Sec. 6.13]. Therefore causal relations also remain unchanged under
transformation (1). This means that one cannot distinguish between two
metrics, that turn into each other by conformal transformations, on the basis
of causal relations only. In particular, let us consider a _conformally flat_
spacetime, i. e. that can be mapped to flat one by a conformal transformation.
An example of such a conformally flat spacetime is a manifold of constant
curvature. It clearly follows from the above, that a conformally flat
spacetime is indistinguishable from a flat one by examining causal relations.

And what if the spacetime under study is not conformally flat? This question
can be answered on the basis of Theorem 2 that is formulated and proved in the
present article. It follows from Theorem 2 that each non-conformally-flat
4-dimensional spacetime can be distinguished from a flat (Minkowski) one by
testing a finite set of causal relations. This set resembles very closely that
of the thought experiment considered above; it includes 16 relations between 8
events.

To summarize: the results of this paper clarify what can be said about
flatness or non-flatness of the spacetime after a finite number of
measurements of causal relations between events. If a 4-dimensional spacetime
is not conformally flat, then its non-flatness can be verified by as few as 16
measurements of causal relations. And if a spacetime is conformally flat,
then, as is known, it cannot be distinguished from a flat spacetime by any
number of such measurements.

In this paper we focus on the usual, four-dimensional spacetime that possesses
three spatial and one temporal coordinates.

The rest of this paper is organized as follows. In Section II, we introduce
the concept of “well-stitched” spacetime, defined solely in terms of causal
relations. We formulate Theorems 1 and 2 that constitute the main result of
this article. In Section III, we discuss various issues regarding to the
results of this paper, in particular, possibility of practical detection of
the spacetime curvature using the proposed scheme. In Section IV, we prove
Theorem 1 that applies to light cones in the flat Minkowski spacetime. In
Section V, we consider a spacetime with non-vanishing Weyl tensor, and prove
Theorem 2. The proof is based on constructing a concrete example of eight
events that demonstrates violation of well-stitchedness for such a spacetime.

##  II Concept of a well-stitched spacetime

Let us denote as $A\leftrightarrowtriangle B$ the situation when event $A$
belongs to the light cone of event $B$ (or, equivalently, $B$ belongs to the
light cone of $A$). In a flat 4-dimensional spacetime (i. e. a Minkowski
spacetime), $A\leftrightarrowtriangle B$ means that

$(x_{A}-x_{B})^{2}+(y_{A}-y_{B})^{2}+(z_{A}-z_{B})^{2}=c^{2}(t_{A}-t_{B})^{2},$
(2)

where $x$, $y$ and $z$ are Cartesian coordinates, $t$ is the time, and $c$ is
the speed of light. In a curved (pseudo-Riemannian) spacetime,
$A\leftrightarrowtriangle B$ means that events $A$ and $B$ are connected by a
light-like geodesic line.

This relation allows to express the conformal flatness of a 4-dimensional
spacetime in a very simple way. Namely, we will show that conformal flatness
is equivalent to “well-stitchedness” of a spacetime defined as follows:

######  Definition 1 (well-stitched spacetime).

A four-dimensional spacetime is well-stitched iff for any eight events $A$,
$B$, $C$, $D$, $E$, $F$, $G$ and $H$, that are all different from each other,
from relations $A\leftrightarrowtriangle E$, $A\leftrightarrowtriangle F$,
$A\leftrightarrowtriangle G$, $A\leftrightarrowtriangle H$,
$B\leftrightarrowtriangle E$, $B\leftrightarrowtriangle F$,
$B\leftrightarrowtriangle G$, $B\leftrightarrowtriangle H$,
$C\leftrightarrowtriangle E$, $C\leftrightarrowtriangle F$,
$C\leftrightarrowtriangle G$, $C\leftrightarrowtriangle H$,
$D\leftrightarrowtriangle E$, $D\leftrightarrowtriangle F$ and
$D\leftrightarrowtriangle G$ follows relation $D\leftrightarrowtriangle H$.

Table 1 clarifies this set or relations. The relation between $D$ and $H$,
denoted by the question mark in the table, follows from other 15 relations
denoted by symbols $\leftrightarrowtriangle$.

| $A$ | $B$ | $C$ | $D$
---|---|---|---|---
$E$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$
$F$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$
$G$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$
$H$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $\leftrightarrowtriangle$ | $?$
Table 1: Illustration of the set of relations in Definition 1.

The thought experiment shown in Fig. 1a,b is a specific case of the set of
relations that appears in the definition of a well-stitched spacetime. In this
experiment, as shown schematically in Fig. 1c, events $E$, $F$, $G$, $H$ of
switching lamps on precede events $A$, $B$, $C$, $D$ of their perceptions by
the observers. Other timelines are also possible. For example, Fig. 1d
demonstrates another version of the thought experiment, in which two lamps are
switched on at events $B$ and $D$; then the light signals are percepted at
events $E$, $F$, $G$, $H$ and retranslated further from these four events;
finally, retranslated signals reach events $A$ and $C$. The setting shown in
Fig. 1d will be used for the proof of Theorem 2 in Section V.

In order to establish the equivalence between well-stitchedness and conformal
flatness, we will prove two theorems.

######  Theorem 1.

The 4-dimensional Minkowski spacetime is well-stitched.

This theorem will be proven in Section IV.

A conformal transformation (1), which maps a conformally flat spacetime onto
Minkowski one, conserves causal relations. Therefore it conserves well-
stitchedness as well, because this property relies on causal relations only.
Hence, Theorem 1 implies the following result:

######  Corollary 1.1.

Any 4-dimensional conformally flat spacetime is well-stitched.

Then, in order to close our reasoning, it is necessary to consider _non_
-conformally-flat spacetimes, and prove that they _are not_ well-stitched. In
this way, the following theorem is helpful:

######  Theorem 2.

If a 4-dimensional spacetime has a non-zero Weyl tensor at some point $O$,
then well-stitchedness violates in some vicinity of $O$.

The Weyl tensor here is a traceless part of the Riemann curvature tensor [12,
§92]. We will prove this theorem in Section V.

It is well known that a 4-dimensional spacetime is conformally flat if and
only if its Weyl tensor is equal to zero everywhere [11, Sec. 6.13]. Let us
consider a spacetime that is not conformally flat. According to the above-
mentioned property, there is some point at which the Weyl tensor is different
from zero. Therefore, Theorem 2 entails the following

######  Corollary 2.1.

If a 4-dimensional spacetime is not conformally flat, then it is not well-
stitched.

Corollaries 1.1 and 2.1 together consist the main result of this paper:
A 4-dimensional spacetime is conformally flat if and only if it is well-
stitched.

##  III Possible implications

Let us discuss, whether it is possible in practice (rather than in theory) to
detect the spacetime curvature produced by the Earth by finding out a
violation of well-stitchedness. The curvature is described by the Riemann
tensor $R_{iklm}$ that has 20 degrees of freedom. Ten of them are components
of the Ricci tensor $R_{ik}$, and in an empty space they vanish according to
Einstein equations. Other ten degrees of freedom are independent components of
the Weyl tensor $C_{iklm}$. Moreover, in an empty space, where $R_{ik}=0$, the
Riemann and Weyl tensors are equal to each other, see Eq. (39). Hence, in the
empty space (e. g. on an orbit around the Earth) a full characterization of
the curvature is just the same as measuring the Weyl tensor.

Consider a set of satellites orbiting around the Earth and forming a suitable
configuration for detecting the spacetime curvature. An example of such a
configuration is shown in Fig. 2. Satellite $A$ simultaneously sends optical
signals towards satellites $E$, $F$, $G$ and $H$. Satellite $B$ does the same
in another moment of time, so that each of satellites $E$, $F$, $G$ and $H$
receives signals from $A$ and from $B$ simultaneously. As soon as satellite
$E$ receives the signals, it emits its own signals towards satellites $C$ and
$D$, and so do satellites $F$, $G$ and $H$. Satellite $C$ is positioned such
that all four signals towards $C$ are received simultaneously. Similarly,
satellite $D$ is set at the point where signals from $E$, $F$ and $G$ comes at
the same time. Then, to reveal the spacetime curvature, one has to detect the
difference in time $\delta t_{HD}$ between the moment when signals from $E$,
$F$ and $G$ reach satellite $D$, and the moment when the signal from $H$
reaches there.

Figure 2: Possible arrangement of a swarm of satellites near the Earth,
enabling one to detect the curvature of the space due to the Earth’s gravity.
One group of satellites is located on a circle (violet dashed line), another
one — on the axis of symmetry of this circle (green dashed line). Yellow
arrows show the directions of optical signals between the satellites. $L$ is
the size of the swarm, and $R$ is the distance from the Earth’s center.

How large is the time difference $\delta t_{HD}$ that is needed to be
detected? One can apply an estimate from Subsection V.7 and find that

$\delta t_{HD}\simeq\frac{L^{3}C}{c}\,,$ (3)

where $L$ is a characteristic distance between the satellites, $C$ is a
suitable component (or a linear combination of components) of the Weyl tensor,
and $c$ is the speed of light. If the circle $EFGH$ lies in the $yz$-plane,
and satellites $A$, $B$, $C$ and $D$ are located at equal distances from the
center of the circle, then the configuration of events $A,\ldots,H$ is the
same as used in Section V, up to relabeling. In these case, quantity $C$ in
the latter equation is component $C_{tyxy}$, as follows from Subsection V.7.
Some other configurations of events can be obtained from that considered in
Section V by Lorentz boosts and/or spatial rotations of the satellite group.
They enable us to get access to other components of the Weyl tensor and,
eventually, obtain the full information about the spacetime curvature by
measuring time delays in 10 different configurations of satellites.

The Riemann tensor scales with the distance $R$ from the center of a massive
body as $\simeq r_{g}/R^{3}$, where $r_{g}$ is the gravitational radius of the
body (see [12, §100] for the Riemann tensor in spherical coordinates).
Substituting this expression as $C$ into Eq. (3), we obtain

$\delta t_{HD}\simeq\frac{r_{g}}{c}\left(\frac{L}{R}\right)^{3}\,.$ (4)

In the limiting case $L\simeq R$, which corresponds to a setting shown in Fig.
1b, quantity $\delta t_{HD}$ turns into the Shapiro time delay that has the
order of magnitude $\simeq r_{g}/c$.

Near the Earth ($r_{g}\approx 1$ cm), Eq. (4) gives rise to an estimate
$\delta t_{HD}\simeq 0.3\,{\rm ns}\times(L/R)^{3}$. Hence, detection of the
curvature related to the Earth gravity demands picosecond resolution for
optical signal detection, which is routinely accessed at present. The same
level of resolution is necessary for adjusting positions of satellites by
optical signaling, in order to fulfill fifteen relations
$A\leftrightarrowtriangle E$, $A\leftrightarrowtriangle F$ and so on (or,
equivalently, to set fifteen time delays $\delta t_{AE}$, $\delta t_{AF}$,
etc. to zero). For a solar orbit, estimate (4) with the gravitational radius
of the Sun $r_{g}\approx 3$ km provides the time delay $\delta t_{HD}$ of
order of magnitude $10\,\mu{\rm s}\times(L/R)^{3}$.

It is important to note that, even in absence of exact positioning, one can
get an information about spacetime curvature from sixteen time delays $\delta
t_{AE},\delta t_{AF},\ldots,\delta t_{HD}$. It can be done by mere replacement
of $\delta t_{HD}$ in Eq. (3) with a suitable linear combination of time
delays $\delta t_{AE},\delta t_{AF},\ldots,\delta t_{HD}$, which is stable
with respect to small variations of satellite positions. Existence of such a
stable linear combination is established in Subsection V.4 for a special
configuration of events.

We wish to emphasize two distinctive features of the proposed method of
detecting the spacetime curvature. First, it provides a direct access to the
local value of the curvature tensor. This is in contrast to other tests of the
general relativity, which deal with integrated characteristics such as Mercury
orbit precession, light deflection by the Sun, and so on. Second, it does not
demand precise instruments, like atomic clocks, drag-free satellite control,
etc. [13] To obtain the spacetime curvature by our method with some relative
accuracy $\eta$, one have to measure time delays with the same order $\eta$ of
relative accuracy.

Regardless the possibility of testing spacetime curvature in practice, the
results of this study raise some fundamental issues. Theorems 1 and 2 state
that, for 4-dimensional pseudo-Riemannian spaces, conformal flatness is
equivalent to well-stitchedness. Conformal flatness is a concept of
differential geometry, whereas a new notion of well-stitchedness, introduced
in Definition 1, belongs to _incidence geometry_. Hence, Theorems 1 and 2
“translate” conformal flatness from the language of differential geometry to a
more primitive language of incidence geometry. In the latter language, a set
of events connected by null geodesic lines, as in Definition 1, consists a
configuration of type $(8_{4}16_{2})$, which means a collection of 8 points
and 16 lines, with 4 lines at each point and 2 points at each line [14,
Chapter III]. Especially significant are those configurations, in which the
last incidence is self-fulfilling, as for example in Pappus $(9_{3})$ and
Desargues $(10_{3})$ configurations. Configurations that have this property
express some geometrical theorems [14]. The configuration $(8_{4}16_{2})$
related to in Definition 1 also has this property in the 4-dimensional
Minkowski spacetime, that is expressed in Theorem 1.

There is an approach to quantum gravity, in which the spacetime is discrete
and is represented as a _causal set_ — a collection of discrete points,
partially ordered by causality relations [15, 16, 17]. In this perspective,
the concept of a well-stitched spacetime may provide a convenient tool for
distinguishing between a (conformally) flat causal set and a curved one. The
fraction of violations of well-stitchedness in a causal set may serve as a
measure of the Weyl curvature (cf. a method of scalar curvature evaluation in
Ref. [18]).

It is known since 1970s that the _causal structure_ of a pseudo-Riemannian
spacetime is in one-to-one correspondence with its _conformal structure_ [19,
20]. That is, if a spacetime can be mapped to another one with conserving
causal relations, then the map between the spacetimes is conformal (i. e.
obeys Eq. (1)), and vice versa [15, Sec. 2.8]. The present study reveals the
primitive elements of which the causal structure of a conformally flat
4-dimensional spacetime consists—these elements are configurations
$(8_{4}16_{2})$, such as ones shown in Fig. 1.

Another view to a spacetime without involving rulers and clocks is its
_geodesic (projective) structure_ , i. e. the set of geodesic lines, or in
physical terms—the set of possible world lines of freely falling test
particles [21, 22]. Remarkably, geodesic and conformal structures together
determine the _Weyl structure_ [23, 24], that in the Riemannian geometry
allows also to reconstruct the metric (e. g. using the methods of Refs. [25,
26]). Spacetimes are called geodesically equivalent if they can be mapped into
each other with preserving the geodesic lines. According to Beltrami’s theorem
[27, Sec. 40], the flat Minkowski spacetime is geodesically equivalent to
other spacetimes with a constant curvature, and only to them: namely, to de
Sitter (positive curvature) and anti-de Sitter (negative curvature)
spacetimes. In the context of the present study, it is natural to ask: whether
it is possible to detect the curvature of a spacetime by pointing out to a
finite set of geodesic lines? We will show that it is indeed possible, unless
the spacetime has a constant curvature and therefore is geodesically
equivalent to the flat one. Consider four geodesic lines $OAB$, $OCD$, $AD$
and $BC$ that form the so-called _complete quadrilateral_ , shown in Fig. 3.
Point $O$, two geodesic lines $a$ and $b$ passing through $O$, two points $A$,
$B$ on line $a$, and two points $C$, $D$ on line $b$ are chosen arbitrarily,
with the only restriction that point $A$ lies between $O$ and $B$, and point
$C$ — between $O$ and $D$. In the flat spacetime, where geodesics are just
straight lines, five points $O$, $A$, $B$, $C$ and $D$ lie in one plane, and
therefore lines $AD$ and $BC$ must intersect at some point $E$. The same
argument is true in a spacetime of a constant curvature. But in a general
curved spacetime lines $AD$ and $BC$ may not intersect. This occurs, for
example, if geodesic line $BC$ passes near a massive body that deflects this
line out of plane $OABCD$. Therefore spacetime curvature can be detected by
the observation that lines $AD$ and $BC$ do not intersect each other.

Figure 3: A complete quadrilateral, or a configuration $(6_{2}4_{3})$,
composed of four geodesic lines $OAB$, $OCD$, $AD$ and $BC$. In a flat
spacetime, as well as in a spacetime of a constant curvature, lines $AD$ and
$BC$ must intersect at some point $E$ (the left part). But in a generic curved
spacetime, these lines may not intersect, for instance, if geodesic line BC
passes near a massive body (the right part).

We will show in Appendix H that the distance between lines $AD$ and $BC$,
projected to a vector perpendicular to both lines, is proportional to the cube
of the quadrilateral’s size multiplied by a linear combination of the Riemann
tensor components. Considering different such quadrilaterals around point $O$,
one can measure 19 of 20 degrees of freedom of the Riemann tensor at point
$O$: namely, 10 independent components of the Weyl tensor $C_{iklm}$ and 9
components of the traceless part $(R_{ik}-\frac{1}{4}R{{}^{l}}_{l}\,g_{ik})$
of the Ricci tensor $R_{ik}$ (see Appendix H). If in any quadrilateral $OABCD$
its lines $AD$ and $BC$ intersect, then the above-mentioned 19 degrees of
freedom vanish at each point, i. e. the spacetime is both conformally-flat
($C_{iklm}=0$) and Einstein ($R_{ik}=\frac{1}{4}R{{}^{l}}_{l}\,g_{ik}$). It
follows from these properties [28, Sec. 5.2] that the spacetime has a constant
curvature, i. e. is either flat or de Sitter or anti-de Sitter. On the other
hand, if in some such quadrilateral the lines $AD$ and $BC$ do not intersect,
then the spacetime is not of a constant curvature. Hence, the finite set of
geodesic lines shown in Fig. 3 allows us to discriminate between spacetimes of
a constant curvature and all other spacetimes.

In the quantum realm, geodesic lines loose their physical meaning of
trajectories of freely falling particles. But the fundamental role of causal
relations still persists in quantum theory. One can thus expect that the
property of well-stitchedness may be important in the context of quantum field
theory in a curved spacetime. As an illustrative example, let us consider a
Feynman diagram shown in Fig. 4, where sixteen lines connect eight vertices
$A,\ldots,H$ in the same manner as causal relations connect eight events in
Definition 1. Such a diagram may appear in a quantum field with a four-
particle interaction, for example in the $\varphi^{4}$-model, and represent a
correction to the vacuum energy. Remarkably, gravity-induced four-_fermion_
interaction naturally appears in the framework of Einstein-Cartan theory with
torsion [29, 30, 31].

Figure 4: An eight-vertex Feynman diagram that may describe a correction to
the vacuum energy sensitive to the Weyl tensor.

The propagators in the position space, related to lines $AE$, $AF$, …, and
$DH$, diverge when the corresponding relations $A\leftrightarrowtriangle E$,
$A\leftrightarrowtriangle F$, …, and $D\leftrightarrowtriangle H$ are
satisfied. Therefore, in a well-stitched spacetime, divergence of fifteen
propagators leads to divergence of the rest (sixteenth) propagator. In a not-
well-stitched spacetime it is not the case. One may thus expect that the
contribution $\Delta\mathcal{L}$ of the diagram shown in Fig. 4 to the vacuum
Lagrangian density would depend on whether the spacetime is well-stitched. On
the other hand, presence or absence of well-stitchedness is determined by Weyl
tensor $C_{iklm}$, due to Theorems 1 and 2. Hence, it is natural to suppose
that contribution $\Delta\mathcal{L}$ is a function of $C_{iklm}$. Considering
the expansion of $\Delta\mathcal{L}$ in a power series over components of the
Weyl tensor, one can easily see that linear terms vanish because there is no
scalar linear combination of components $C_{iklm}$. One can however construct
of the Weyl tensor a scalar quadratic form, namely $C^{iklm}C_{iklm}$. For
this reason, one can expect that the first non-vanishing curvature-dependent
term in $\Delta\mathcal{L}$ is quadratic in $C_{iklm}$:

$\Delta\mathcal{L}(C_{iklm})-\Delta\mathcal{L}(0)\propto
C^{iklm}C_{iklm}\sqrt{-g},$ (5)

where factor $\sqrt{-g}$ accounts for the invariant volume element, $g$ being
the determinant of the metric tensor. That is, the curvature-related action
$S_{C}$ that arises from the diagram shown in Fig. 4 may be of the following
form:

$S_{C}=\alpha_{g}\int C^{iklm}C_{iklm}\sqrt{-g}\,{\rm d}t\,{\rm d}x\,{\rm
d}y\,{\rm d}z,$ (6)

where $\alpha_{g}$ is some constant. In a system of units where $\hbar=c=1$,
$\alpha_{g}$ is a dimensionless quantity. The strength of the four-boson
interaction $\lambda$ is also dimensionless, and it enters into $\alpha_{g}$
as $\lambda^{8}$ according to the number of vertices in the diagram. Hence
$\alpha_{g}\simeq\lambda^{8}$ up to a numerical factor. In physical units, the
latter relation has a form

$\alpha_{g}\simeq\lambda^{8}\hbar c.$ (7)

Remarkably, $\alpha_{g}$ does not depend on the particle’s mass. Of course, a
thorough calculation of constant $\alpha_{g}$ must take renormalization of
$\lambda$ into account.

Action $S_{C}$ in Eq. (6) is known as Weyl action. It consists the basis of
so-called conformal gravity theory. Though Weyl action substantially differs
from Einstein-Hilbert action $S_{EH}$ of general relativity,
$S_{EH}\propto\int R\sqrt{-g}\,{\rm d}t\,{\rm d}x\,{\rm d}y\,{\rm d}z$ (where
$R$ is the scalar curvature), there are intriguing relations between conformal
gravity and standard Einstein’s general relativity. In particular, conformal
gravity appeared in the context of supergravity [32], twistor-string theory
[33], ultraviolet regularization of gravity [34], astrophysical observations
[35, 36] as an origin of Planck and electroweak scales [37], as a more
fundamental theory beyond Einstein’s gravity [38, 39, 40, 41], and even as a
candidate alternative to Einstein gravity [35]. Conformal gravity also
considered to induce a Starobinsky-type cosmological inflation [42].

The idea that the action of the gravitational field stems from dynamics of the
matter in a curved spacetime was formulated by Sakharov as early as in 1967
(see reprints [43, 44] of his 1967 work). This idea is often referred as
“induced gravity” [45, 46, 47]. In a closely related concept of “emergent
gravity”, gravitational field arises along with spacetime itself from some
more fundamental constructs [48, 15, 49, 50, 51]. Calculations by many
research groups showed that the induced gravity appears even in the
approximation of non-interacting quantum fields, from a diagram containing
only one loop [52, 53, 45, 54, 46, 55, 47]. However, a much more complicated
diagram shown in Fig. 4 might be interesting in the aspect that it naturally
leads to the induced _conformal_ gravity.

## Conclusions

In this work, we have found a _discrete_ equivalent of a conformally flat
4-dimensional pseudo-Riemannian spacetime. This equivalent is a new notion of
a “well-stitched” spacetime, inspired by the thought experiment depicted in
Fig. 1, and further developed in Section II. The well-stitched spacetime is
defined solely in terms of light signals between a finite number of events
(see Definition 1), without using rulers or clocks of any sort.

We have proved two theorems about well-stitched spacetimes. Theorem 1 states
that any conformally flat 4-dimensional pseudo-Riemannian spacetime is well-
stitched. Theorem 2, conversely, states that any _non_ -conformally-flat
4-dimensional pseudo-Riemannian spacetime is _non_ -well-stitched. Hence, the
differential-geometric concept of conformal flatness has been “translated” to
the language of discrete geometry.

This new look onto the spacetime geometry opens different perspectives (see
Section III), ranging from the curvature detection by time-delay measurements
with satellites to the search for an origin of gravity.

###### Acknowledgements.

A.N. thanks the Faculty of Physics of the Philipps Universität Marburg for the
kind hospitality during his research stay.

## Methods

##  IV Causal structure of the flat spacetime: proof of Theorem 1

In this Section, we consider points (events) in the 4-dimensional Minkowski
space. For simplicity, we set speed of light $c$ equal to 1. The interval
$ds^{2}_{PQ}$ between two points $P=(t_{P},x_{P},y_{P},z_{P})$ and
$Q=(t_{Q},x_{Q},y_{Q},z_{Q})$ is defined as

$ds_{PQ}^{2}=-(t_{P}-t_{Q})^{2}+(x_{P}-x_{Q})^{2}+(y_{P}-y_{Q})^{2}+(z_{P}-z_{Q})^{2},$
(8)

and relation $P\leftrightarrowtriangle Q$ just means that $ds_{PQ}^{2}=0$. The
scalar product of two contravariant vectors
$\vec{a}=(a_{t},a_{x},a_{y},a_{z})$ and $\vec{b}=(b_{t},b_{x},b_{y},b_{z})$ is

$\vec{a}\cdot\vec{b}=-a_{t}b_{t}+a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}\,.$ (9)

Our proof of Theorem 1 is based on three lemmas formulated below.

######  Lemma 3.

For any four points $P$, $Q$, $R$, $S$, if relations $P\leftrightarrowtriangle
R$, $P\leftrightarrowtriangle S$, $Q\leftrightarrowtriangle R$ and
$Q\leftrightarrowtriangle S$ are fulfilled, then vector $\overrightarrow{PQ}$
is perpendicular to vector $\overrightarrow{RS}$ (that is,
$\overrightarrow{PQ}\cdot\overrightarrow{RS}=0$).

######  Lemma 4.

For any three different points $P$, $Q$, $R$, if relations
$P\leftrightarrowtriangle Q$, $P\leftrightarrowtriangle R$ and
$Q\leftrightarrowtriangle R$ are fulfilled, then these points lie on one line.

######  Lemma 5.

For any four points $P$, $Q$, $R$, $S$, if relations $P\leftrightarrowtriangle
S$, $Q\leftrightarrowtriangle S$ and $R\leftrightarrowtriangle S$ are
fulfilled, and points $P$, $Q$, $R$ lie on one line and are different from
each other, then this line is light-like (that is, $P\leftrightarrowtriangle
Q$, $P\leftrightarrowtriangle R$ and $Q\leftrightarrowtriangle R$).

These lemmas are true in the Minkowski space. Proofs of them are given in
Appendices A, B and C.

Now we return to Theorem 1. Let eight points $A$, $B$, $C$, $D$, $E$, $F$,
$G$, $H$ be all different from each other, and 15 relations
$A\leftrightarrowtriangle E,\ldots,D\leftrightarrowtriangle G$ listed in
Definition 1 are fulfilled. First, we consider the generic case, when points
$A$, $B$ and $C$ do not lie on one line, and also points $E$, $F$ and $G$ do
not lie on one line. The opposite case will be discussed later.

Due to Lemma 3, $\overrightarrow{AB}\cdot\overrightarrow{EF}=0$,
$\overrightarrow{AB}\cdot\overrightarrow{EG}=0$,
$\overrightarrow{AC}\cdot\overrightarrow{EF}=0$, and
$\overrightarrow{AC}\cdot\overrightarrow{EG}=0$. Therefore any vector parallel
to plane $ABC$ is perpendicular to any vector parallel to plane $EFG$.

Since $A\leftrightarrowtriangle E$, $A\leftrightarrowtriangle F$,
$D\leftrightarrowtriangle E$ and $D\leftrightarrowtriangle F$, it follows from
Lemma 3 that vector $\overrightarrow{AD}$ is perpendicular to vector
$\overrightarrow{EF}$. Similarly, as a consequence of relations
$A\leftrightarrowtriangle E$, $A\leftrightarrowtriangle G$,
$D\leftrightarrowtriangle E$ and $D\leftrightarrowtriangle G$, vector
$\overrightarrow{AD}$ is perpendicular to $\overrightarrow{EG}$. Therefore,
vector $\overrightarrow{AD}$ is perpendicular to plane $EFG$ and hence is
parallel to plane $ABC$. This means that points $A$, $B$, $C$ and $D$ lie on
one plane. Analogous reasoning shows that points $E$, $F$, $G$ and $H$ lie on
one plane.

Hence, all the eight points lie in two planes $ABCD$ and $EFGH$, and each
vector of the first plane is perpendicular to each vector in the second one.

There are three options: (a) plane $ABCD$ contains a time-like vector; (b)
plane $EFGH$ contains a time-like vector; (c) none of these planes contains a
time-like vector. We will consider them separately.

In option (a), we can choose axis $t$ along a time-like vector lying in plane
$ABCD$. Then, any vector in plane $EFGH$ is perpendicular to axis $t$. Let us
choose axes $y$ and $z$ along some pair of mutually perpendicular vectors of
plane $EFGH$. Axis $x$ is then perpendicular to plane $EFGH$ and therefore
must be parallel to plane $ABCD$. Planes $ABCD$ and $EFGH$ intersect at some
point $O$. Let us choose this point as the origin of the coordinate axes.

With this choice of coordinates, points $A$, $B$, $C$ and $D$ lie in the plane
spanned onto axes $t$ and $x$. Similarly, points $E$, $F$, $G$ and $H$ lie in
the plane spanned onto axes $y$ and $z$. That is,

$\displaystyle y_{A}=z_{A}=0,$ (10) $\displaystyle y_{B}=z_{B}=0,$ (11)
$\displaystyle y_{C}=z_{C}=0,$ (12) $\displaystyle y_{D}=z_{D}=0,$ (13)
$\displaystyle t_{E}=x_{E}=0,$ (14) $\displaystyle t_{F}=x_{F}=0,$ (15)
$\displaystyle t_{G}=x_{G}=0,$ (16) $\displaystyle t_{H}=x_{H}=0.$ (17)

Condition $A\leftrightarrowtriangle E$ therefore acquires the form

$-t_{A}^{2}+x_{A}^{2}+y_{E}^{2}+z_{E}^{2}=0.$ (18)

Let us denote the distance from point $E$ to the origin of coordinates as $r$:

$y_{E}^{2}+z_{E}^{2}=r^{2}.$ (19)

Then, it follows from Eqs. (18) and (19) that

$t_{A}^{2}-x_{A}^{2}=r^{2}.$ (20)

Similarly, conditions $B\leftrightarrowtriangle E$, $C\leftrightarrowtriangle
E$, $D\leftrightarrowtriangle E$ along with Eq. (19) lead to

$\displaystyle t_{B}^{2}-x_{B}^{2}=r^{2},$ (21) $\displaystyle
t_{C}^{2}-x_{C}^{2}=r^{2},$ (22) $\displaystyle t_{D}^{2}-x_{D}^{2}=r^{2},$
(23)

and conditions $A\leftrightarrowtriangle F$, $A\leftrightarrowtriangle G$,
$A\leftrightarrowtriangle H$ and Eq. (20) lead to

$\displaystyle y_{F}^{2}+z_{F}^{2}=r^{2},$ (24) $\displaystyle
y_{G}^{2}+z_{G}^{2}=r^{2},$ (25) $\displaystyle y_{H}^{2}+z_{H}^{2}=r^{2}.$
(26)

Summarizing, we see that points $A$, $B$, $C$ and $D$ lie on the hyperbola

$t^{2}-x^{2}=r^{2},\quad y=0,\quad z=0,$ (27)

and points $E$, $F$, $G$ and $H$ lie on the circle

$y^{2}+z^{2}=r^{2},\quad t=0,\quad x=0.$ (28)

Such an arrangement of points is shown in Fig. 5a.

Figure 5: Four possible layouts of points (events) $A$, …, $H$ in the flat
spacetime, shown in specially chosen coordinate frames: (a) points $A$, $B$,
$C$, $D$ lie on a hyperbola in plane $xt$, and points $E$, $F$, $G$, $H$ — on
a circle in plane $yz$; (b) points $E$, $F$, $G$, $H$ lie on a hyperbola in
plane $xt$, and points $A$, $B$, $C$, $D$ — on a circle in plane $yz$; (c)
points $A$, $B$, $C$, $D$ lie on a parabola in plane $t-x=0$, $z=0$, and
points $E$, $F$, $G$, $H$ — on a parabola in plane $t-x=s$, $y=0$; (d) all
eight points lie on the same light-like line $\ell$. Point $O$ is the
intersection point of planes $ABCD$ and $EFGH$. Axis $u$ is a bisector of the
angle between axes $x$ and $t$.

Finally, Eqs. (23) and (26) can be combined into

$-t_{D}^{2}+x_{D}^{2}+y_{H}^{2}+z_{H}^{2}=0,$ (29)

that means $D\leftrightarrowtriangle H$. Hence, within option (a), Theorem 1
is proven.

Option (b) can be reduced to option (a) by mere relabeling $A\leftrightarrow
E$, $B\leftrightarrow F$, $C\leftrightarrow G$, $D\leftrightarrow H$, see Fig.
5b. Thus the above proof is valid also in option (b).

In option (c), as we show in Appendix D, one can choose the coordinate system
in such a way that plane $ABCD$ is defined as

$t-x=0,\quad z=0,$ (30)

and plane $EFGH$ is defined as

$t-x=s,\quad y=0,$ (31)

with some parameter $s$. In these coordinates, relation
$P\leftrightarrowtriangle Q$, where $P\in\\{A,B,C,D\\}$ and
$Q\in\\{E,F,G,H\\}$, has the form

$-(t_{P}-t_{Q})^{2}+(x_{P}-x_{Q})^{2}+y_{P}^{2}+z_{Q}^{2}=0.$ (32)

Expressing here $t_{P}$ through $x_{P}$ by Eq. (30), and $t_{Q}$ through
$x_{Q}$ by Eq. (31), one obtains

$2sx_{P}+y_{P}^{2}=2sx_{Q}-z_{Q}^{2}+s^{2}.$ (33)

Let us denote as $w$ the following quantity:

$2sx_{A}+y_{A}^{2}=w.$ (34)

Using Eq. (33), one can easily deduce from relations $A\leftrightarrowtriangle
E$ and $A\leftrightarrowtriangle H$ and Eq. (34) that

$\displaystyle 2sx_{E}-z_{E}^{2}+s^{2}=w,$ (35) $\displaystyle
2sx_{H}-z_{H}^{2}+s^{2}=w.$ (36)

Then, it follows from relation $D\leftrightarrowtriangle E$ and Eq. (35) that

$2sx_{D}+y_{D}^{2}=w.$ (37)

Points $A$, $B$, $C$, $D$ lie on a parabola $2sx+y^{2}=w$, and points $E$,
$F$, $G$, $H$ — on a parabola $2sx-z^{2}+s^{2}=w$, see Fig. 5c.

Left-hand sides of Eqs. (37) and (36) are equal to each other:

$2sx_{D}+y_{D}^{2}=2sx_{H}-z_{H}^{2}+s^{2}.$ (38)

According to Eq. (33), this means that relation $D\leftrightarrowtriangle H$
is fulfilled in option (c). Q.E.D.

We have completed the proof in the generic case, when points $A$, $B$ and $C$
do not lie on one line, and also points $E$, $F$ and $G$ also do not lie on
one line. What remains is to prove Theorem 1 in the opposite case. Let, for
definiteness, points $A$, $B$ and $C$ lie on the same line $\ell$. Applying
Lemma 5 with $P=A$, $Q=B$, $R=C$ and $S=E$, one can ensure that line $\ell$ is
light-like, and that relations $A\leftrightarrowtriangle B$,
$A\leftrightarrowtriangle C$ and $B\leftrightarrowtriangle C$ are satisfied.
Applying Lemma 4 five times: with $P=A$, $Q=B$, $R=E$, with $P=A$, $Q=B$,
$R=F$, with $P=A$, $Q=B$, $R=G$, with $P=A$, $Q=B$, $R=H$, and then with
$P=D$, $Q=E$, $R=F$, one can see that all eight points $A,B,C,D,E,F,G$ and $H$
lie on the same light-like line $\ell$ (see Fig. 5d). Since points $D$ and $H$
are connected by a light-like line, then relation $D\leftrightarrowtriangle H$
is fulfilled. This completes the proof of Theorem 1.

##  V Causal structure of a spacetime with nonzero Weyl tensor: proof of
Theorem 2

According to the premise of Theorem 2, we suppose that Weyl tensor differs
from zero at some point $O$. The Weyl tensor $C_{iklm}$ is a traceless part of
Riemann curvature tensor $R_{iklm}$. By definition, Weyl tensor is equal to
[12, §92]

$C_{iklm}=R_{iklm}-\frac{1}{2}R_{il}\,g_{km}+\frac{1}{2}R_{im}\,g_{kl}\\\
+\frac{1}{2}R_{kl}\,g_{im}-\frac{1}{2}R_{km}\,g_{il}+\frac{R}{6}\left(g_{il}\,g_{km}-g_{im}\,g_{kl}\right),$
(39)

where $R_{ik}={R^{l}}_{ilk}$ is the Ricci tensor, and $R={R^{i}}_{i}$ is the
scalar curvature.

For convenience, we set the speed of light $c$ to 1 in this section. The
signature adopted for the metric tensor is $(-+++)$.

###  V.1 Choice of a reference frame

In this section, we will use so-called _Riemann normal coordinates_ [1, §11.6]
with the origin at point $O$. In these coordinates, metric tensor $g_{ik}$ at
the origin is the same as that of the special relativity, and the first
derivatives of $g_{ik}$ vanish at the same point:

$\displaystyle g_{ik}=\eta_{ik}\text{ at }O,$ (40)
$\displaystyle\frac{\partial g_{ik}}{\partial r^{l}}=0\text{ at }O,$ (41)

where

$\eta_{ik}=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}.$ (42)

The second derivatives of $g_{ik}$ are determined by the curvature tensor
$R_{iklm}$, such that [56]

$g_{ik}(\vec{r})=\eta_{ik}-\frac{1}{3}R_{ilkm}(O)\,r^{l}r^{m}+o\left(|\vec{r}|^{2}\right),$
(43)

where $o(...)$ is the Landau little-o symbol, and
$|\vec{r}|^{2}=t^{2}+x^{2}+y^{2}+z^{2}$.

Additionally, we demand that

$C_{tyxy}(O)\neq 0.$ (44)

If the latter inequality does not hold, then we perform an appropriate Lorentz
transformation of the coordinates, after which inequality (44) becomes
satisfied. The existence of such a transformation is proven in Appendix E.

###  V.2 Intervals in a neighborhood of point $O$

In a flat spacetime, relation $P\leftrightarrowtriangle Q$ means that the
interval between points $P$ and $Q$ is equal to zero. If a spacetime is
curved, the algebraic definition of the interval, as in Eq. (8), becomes
meaningless. However, the interval can be determined by integration of the
line element $(g_{ik}dr^{i}dr^{k})^{1/2}$ along the geodesic line that
connects two given points. Let us define interval $\delta s^{2}_{PQ}$ between
points $P$ and $Q$ as

$\delta s^{2}_{PQ}=\pm\left[\int_{P}^{Q}\sqrt{\pm
g_{ik}\frac{dr^{i}}{d\lambda}\,\frac{dr^{k}}{d\lambda}}\;d\lambda\right]^{2},$
(45)

where integration is performed over the geodesic line segment between $P$ and
$Q$, $\lambda$ is a parameter along this geodesic line, $r^{i}(\lambda)$ are
coordinates of a point on the geodesic line, and two signs $\pm$ are equal to
each other and chosen such that the expression under the square root is non-
negative.

The function $\delta s^{2}_{PQ}$ defined by Eq. (45) is nothing else but
Synge’s world function $\Omega(PQ)$ [2] multiplied by 2. If the geodesic line
that connects $P$ and $Q$ is timelike, than $\delta
s^{2}_{PQ}=-(\tau_{PQ})^{2}$, where $\tau_{PQ}$ is the proper time passed
between events $P$ and $Q$ along the geodesic. If this geodesic line is
spacelike, than $\delta s^{2}_{PQ}$ is the squared length of the geodesic line
segment between $P$ and $Q$. And if this geodesic line is lightlike, than
$\delta s^{2}_{PQ}=0$.

Hence, $\delta s^{2}_{PQ}=0$ if and only if relation $P\leftrightarrowtriangle
Q$ is fulfilled.

In the Minkowski space, where $g_{ik}\equiv\eta_{ik}$, quantity $\delta
s^{2}_{PQ}$ is the usual special-relativity interval:

$\delta
s^{2}_{PQ}=\eta_{ik}(r_{Q}^{i}-r_{P}^{i})(r_{Q}^{k}-r_{P}^{k}).\quad\text{(flat
spacetime)}$ (46)

In a curved spacetime that is described by Eq. (43), there must be corrections
to Eq. (46). Let us define $\epsilon$-neighborhood of point $O$ as a set of
such points $(t,x,y,z)$ that

$t^{2}+x^{2}+y^{2}+z^{2}<(2\epsilon)^{2}.$ (47)

If both points $P$ and $Q$ belong to the $\epsilon$-neighborhood of point $O$,
then [56]

$\delta s^{2}_{PQ}=\eta_{ik}(r_{Q}^{i}-r_{P}^{i})(r_{Q}^{k}-r_{P}^{k})\\\
-\frac{1}{3}R_{iklm}(O)\,r_{P}^{i}\,(r_{Q}^{k}-r_{P}^{k})\,r_{P}^{l}(r_{Q}^{m}-r_{P}^{m})+o(\epsilon^{4}).$
(48)

Later, we will use expansion (48) in order to construct a set of points that
demonstrates violation of well-stitchedness.

###  V.3 Combination of intervals and Weyl tensor

In this subsection, we will establish a relation between intervals $\delta
s^{2}_{PQ}$, on the one side, and the Weyl tensor, on the other side. Let us
first express component $C_{tyxy}$ of the Weyl tensor at point $O$ through the
Riemann curvature tensor $R_{iklm}$, using Eqs. (39), (40), and the skew
symmetry properties $R_{iklm}=-R_{kilm}=-R_{ikml}$:

$C_{tyxy}(O)=\frac{R_{tyxy}(O)-R_{tzxz}(O)}{2}\,.$ (49)

Then we choose a spatial scale $\epsilon$, small enough such that expansion
(48) makes sense, and consider a typical example of eight points (events) $A$
– $H$ that obey sixteen relations $A\leftrightarrowtriangle
E,\ldots,D\leftrightarrowtriangle H$ in the Minkowski space:

$\displaystyle A$ $\displaystyle=$
$\displaystyle(\;\;\,\sqrt{2}\epsilon,\;\;\,\epsilon,\;\;\,0,\;\;\,0),$ (50a)
$\displaystyle B$ $\displaystyle=$
$\displaystyle(-\sqrt{2}\epsilon,-\epsilon,\;\;\,0,\;\;\,0),$ (50b)
$\displaystyle C$ $\displaystyle=$
$\displaystyle(\;\;\,\sqrt{2}\epsilon,-\epsilon,\;\;\,0,\;\;\,0),$ (50c)
$\displaystyle D$ $\displaystyle=$
$\displaystyle(-\sqrt{2}\epsilon,\;\;\,\epsilon,\;\;\,0,\;\;\,0),$ (50d)
$\displaystyle E$ $\displaystyle=$
$\displaystyle(\;\;\;\;\;0,\;\;\;\;\;0,\;\;\,\epsilon,\;\;\,0),$ (50e)
$\displaystyle F$ $\displaystyle=$
$\displaystyle(\;\;\;\;\;0,\;\;\;\;\;0,-\epsilon,\;\;\,0),$ (50f)
$\displaystyle G$ $\displaystyle=$
$\displaystyle(\;\;\;\;\;0,\;\;\;\;\;0,\;\;\,0,\;\;\,\epsilon),$ (50g)
$\displaystyle H$ $\displaystyle=$
$\displaystyle(\;\;\;\;\;0,\;\;\;\;\;0,\;\;\,0,-\epsilon),$ (50h)

where coordinates of the points are listed in order $(t,x,y,z)$. This set of
points is depicted in Fig. 6, and is a particular case of the arrangement
shown in Fig. 5a.

Figure 6: Set of points (events) $A$, …, $H$ defined by Eqs. (50).

Points $A$, $B$, $C$, and $D$ lie on hyperbola

$t^{2}-x^{2}=\epsilon^{2},\quad y=z=0,$ (51)

whereas points $E$, $F$, $G$, and $H$ lie on circle

$y^{2}+z^{2}=\epsilon^{2},\quad t=x=0,$ (52)

in accordance with Eqs. (27) and (28) of Section IV. Let $P$ be one of points
$\\{A,B,C,D\\}$, and $Q$ be one of points $\\{E,F,G,H\\}$. Then, due to Eqs.
(51) and (52),

$\eta_{ik}(r_{Q}^{i}-r_{P}^{i})(r_{Q}^{k}-r_{P}^{k})\\\
\equiv-(t_{Q}-t_{P})^{2}+(x_{Q}-x_{P})^{2}+(y_{Q}-y_{P})^{2}+(z_{Q}-z_{P})^{2}\\\
=-t_{P}^{2}+x_{P}^{2}+y_{Q}^{2}+z_{Q}^{2}=0,$ (53)

that is, the first term in the right-hand side of expansion (48) vanishes.
Hence, due to Eq. (48), the interval $\delta s^{2}_{PQ}$ is proportional to
the curvature tensor $R_{iklm}(O)$, up to the residual term $o(\epsilon^{4})$.
This fact can be expressed as follows:

$\delta s^{2}_{PQ}=R_{iklm}(O)\,M_{PQ}^{iklm}+o(\epsilon^{4}),$ (54)

where tensor $M_{PQ}^{iklm}$ is the coefficient at $R_{iklm}(O)$ in Eq. (48):

$M_{PQ}^{iklm}=-\frac{1}{3}\,r_{P}^{i}\,(r_{Q}^{k}-r_{P}^{k})\,r_{P}^{l}(r_{Q}^{m}-r_{P}^{m}).$
(55)

Let us consider the following combination $\Sigma$ of sixteen intervals
$\delta s^{2}_{PQ}$, where $P\in\\{A,B,C,D\\}$, and $Q\in\\{E,F,G,H\\}$:

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle\delta s^{2}_{AE}+\delta
s^{2}_{AF}-\delta s^{2}_{AG}-\delta s^{2}_{AH}$ (56) $\displaystyle+$
$\displaystyle\delta s^{2}_{BE}+\delta s^{2}_{BF}-\delta s^{2}_{BG}-\delta
s^{2}_{BH}$ $\displaystyle-$ $\displaystyle\delta s^{2}_{CE}-\delta
s^{2}_{CF}+\delta s^{2}_{CG}+\delta s^{2}_{CH}$ $\displaystyle-$
$\displaystyle\delta s^{2}_{DE}-\delta s^{2}_{DF}+\delta s^{2}_{DG}+\delta
s^{2}_{DH}\,.$

The choice of signs in this definition enables a remarkable property of
stability of $\Sigma$ under small variations of points $A$ – $H$. We will
establish this property in subsection V.4.

It is evident from Eqs. (54) and (56) that quantity $\Sigma$ is also
proportional to $R_{iklm}(O)$, up to the residual term:

$\Sigma=R_{iklm}(O)\,M^{iklm}+o(\epsilon^{4}),$ (57)

where $M^{iklm}$ is a combination of tensors $M_{PQ}^{iklm}\,$:

$\displaystyle M$ $\displaystyle=$ $\displaystyle M_{AE}+M_{AF}-M_{AG}-M_{AH}$
(58) $\displaystyle+$ $\displaystyle M_{BE}+M_{BF}-M_{BG}-M_{BH}$
$\displaystyle-$ $\displaystyle M_{CE}-M_{CF}+M_{CG}+M_{CH}$ $\displaystyle-$
$\displaystyle M_{DE}-M_{DF}+M_{DG}+M_{DH}$

(we have omitted tensor superscripts “$iklm$” in Eq. (58) for brevity).

Tensor $M^{iklm}$ can be calculated straightforwardly, by substituting of
equations (50a)–(50h) and (55) into Eq. (58). Such a calculation however
demands evaluation of the right-hand side of Eq. (55) for each $P$, $Q$, $i$,
$k$, $l$ and $m$, i. e. $4\times 4\times 4\times 4\times 4\times 4=4096$
times. We have done this with the aid of a computer (the Matlab code is
presented in Appendix F), and obtained the following results:

$\displaystyle M^{tzxz}=M^{xztz}$ $\displaystyle=$
$\displaystyle\frac{8\sqrt{2}}{3}\epsilon^{4},$ (59) $\displaystyle
M^{tyxy}=M^{xyty}$ $\displaystyle=$
$\displaystyle-\frac{8\sqrt{2}}{3}\epsilon^{4},$ (60)

and all other components of tensor $M^{iklm}$ are equal to zero. Substituting
these expressions for the components of $M^{iklm}$ into Eq. (57), one can find
that

$\Sigma=\frac{8\sqrt{2}}{3}\epsilon^{4}\Big{[}R_{tzxz}(O)+R_{xztz}(O)\\\
-R_{tyxy}(O)-R_{xyty}(O)\Big{]}+o(\epsilon^{4}),$ (61)

that can be simplified using the permutation symmetry $R_{iklm}=R_{lmik}$:

$\Sigma=\frac{16\sqrt{2}}{3}\epsilon^{4}\left[R_{tzxz}(O)-R_{tyxy}(O)\right]+o(\epsilon^{4}).$
(62)

It is clearly seen from Eq. (49) that the expression in square brackets is
equal to $-2C_{tyxy}(O)$. Hence, the linear combination $\Sigma$ of intervals
is expressed through the Weyl tensor component $C_{tyxy}$:

$\Sigma=-\frac{32\sqrt{2}}{3}\,\epsilon^{4}\,C_{tyxy}(O)+o(\epsilon^{4}).$
(63)

###  V.4 Stability of combination $\Sigma$ under small shifts of points $A$ –
$H$

The aim of this subsection is to find out how quantity $\Sigma$ changes under
small variations of points $A,\ldots,H$. Let us choose eight points
$A^{\prime},\ldots,H^{\prime}$ located near points $A,\ldots,H$ so that the
difference between coordinates of the corresponding points is small with
respect to $\epsilon^{2}$:

$\displaystyle|t_{A^{\prime}}-t_{A}|=o(\epsilon^{2}),\quad|x_{A^{\prime}}-x_{A}|=o(\epsilon^{2}),$
$\displaystyle|y_{A^{\prime}}-y_{A}|=o(\epsilon^{2}),\quad|z_{A^{\prime}}-z_{A}|=o(\epsilon^{2}),$
(64)

and so on. Then, we replace points $A,\ldots,H$ with their corresponding
points $A^{\prime},\ldots,H^{\prime}$ in definition (56) of quantity $\Sigma$,
and denote the result of this replacement as $\Sigma^{\prime}$:

$\displaystyle\Sigma^{\prime}$ $\displaystyle=$ $\displaystyle\delta
s^{2}_{A^{\prime}E^{\prime}}+\delta s^{2}_{A^{\prime}F^{\prime}}-\delta
s^{2}_{A^{\prime}G^{\prime}}-\delta s^{2}_{A^{\prime}H^{\prime}}$ (65)
$\displaystyle+$ $\displaystyle\delta s^{2}_{B^{\prime}E^{\prime}}+\delta
s^{2}_{B^{\prime}F^{\prime}}-\delta s^{2}_{B^{\prime}G^{\prime}}-\delta
s^{2}_{B^{\prime}H^{\prime}}$ $\displaystyle-$ $\displaystyle\delta
s^{2}_{C^{\prime}E^{\prime}}-\delta s^{2}_{C^{\prime}F^{\prime}}+\delta
s^{2}_{C^{\prime}G^{\prime}}+\delta s^{2}_{C^{\prime}H^{\prime}}$
$\displaystyle-$ $\displaystyle\delta s^{2}_{D^{\prime}E^{\prime}}-\delta
s^{2}_{D^{\prime}F^{\prime}}+\delta s^{2}_{D^{\prime}G^{\prime}}+\delta
s^{2}_{D^{\prime}H^{\prime}}\,.$

In this subsection, we will show that the difference between $\Sigma$ and
$\Sigma^{\prime}$ is small in comparison with $\epsilon^{4}$.

Let $P$ be any of points $\\{A,B,C,D\\}$, and $Q$ be any of points
$\\{E,F,G,H\\}$. Let us find the difference between intervals $\delta
s^{2}_{P^{\prime}Q^{\prime}}$ and $\delta s^{2}_{PQ}$ with the aid of
expansion (48). First, we note that the second term in the right-hand side
(rhs) of Eq. (48) is proportional to $\epsilon^{4}$. Therefore a change of
this term, due to a small shift of points $P$ and $Q$, must be small with
respect to $\epsilon^{4}$. Then, let us consider the first term in the rhs of
Eq. (48). Essentially, this term is a Minkowski-space scalar product
$(\vec{r}_{Q}-\vec{r}_{P})\cdot(\vec{r}_{Q}-\vec{r}_{P})\equiv(\vec{r}_{Q}-\vec{r}_{P})^{2}$.
Replacing $P$ and $Q$ with $P^{\prime}$ and $Q^{\prime}$ changes this term by
amount of

$(\vec{r}_{Q^{\prime}}-\vec{r}_{P^{\prime}})^{2}-(\vec{r}_{Q}-\vec{r}_{P})^{2}\\\
=2(\vec{r}_{Q}-\vec{r}_{P})\cdot(\overrightarrow{\Delta
r}_{Q}-\overrightarrow{\Delta r}_{P})+(\overrightarrow{\Delta
r}_{Q}-\overrightarrow{\Delta r}_{P})^{2},$ (66)

where

$\overrightarrow{\Delta
r}_{P}=\vec{r}_{P^{\prime}}-\vec{r}_{P}\,,\qquad\overrightarrow{\Delta
r}_{Q}=\vec{r}_{Q^{\prime}}-\vec{r}_{Q}\,.$ (67)

According to our choice of points $P^{\prime}$ and $Q^{\prime}$, each
component of vectors $\overrightarrow{\Delta r}_{P}$ and
$\overrightarrow{\Delta r}_{Q}$ is small compared to $\epsilon^{2}$, see Eq.
(64). For this reason, the last term in Eq. (66) is as small as
$o(\epsilon^{4})$. Therefore, the difference between intervals $\delta
s^{2}_{P^{\prime}Q^{\prime}}$ and $\delta s^{2}_{PQ}$ is determined only by
the first term in the rhs of Eq. (66), up to $o(\epsilon^{4})$:

$\delta s^{2}_{P^{\prime}Q^{\prime}}-\delta
s^{2}_{PQ}=2(\vec{r}_{Q}-\vec{r}_{P})\cdot(\overrightarrow{\Delta
r}_{Q}-\overrightarrow{\Delta r}_{P})+o(\epsilon^{4}).$ (68)

The difference between quantities $\Sigma^{\prime}$, Eq. (65), and $\Sigma$,
Eq. (56), is a combination of expressions (68), taken with appropriate signs.
After a simple algebra, one can represent this difference as follows:

$\Sigma^{\prime}-\Sigma=\\\
-2(\vec{r}_{A}+\vec{r}_{B}-\vec{r}_{C}-\vec{r}_{D})\cdot(\overrightarrow{\Delta
r}_{E}+\overrightarrow{\Delta r}_{F}-\overrightarrow{\Delta
r}_{G}-\overrightarrow{\Delta r}_{H})\\\
-2(\vec{r}_{E}+\vec{r}_{F}-\vec{r}_{G}-\vec{r}_{H})\cdot(\overrightarrow{\Delta
r}_{A}+\overrightarrow{\Delta r}_{B}-\overrightarrow{\Delta
r}_{C}-\overrightarrow{\Delta r}_{D})\\\ +o(\epsilon^{4}).$ (69)

Finally, looking at definitions of points $A$ – $H$, Eqs. (50a) – (50h), one
can see that

$\vec{r}_{A}=-\vec{r}_{B},\quad\vec{r}_{C}=-\vec{r}_{D},\quad\vec{r}_{E}=-\vec{r}_{F},\quad\vec{r}_{G}=-\vec{r}_{H}.$
(70)

Substitution of these relations into Eq. (69) gives rise to a simple result:

$\Sigma^{\prime}=\Sigma+o(\epsilon^{4}).$ (71)

Remarkably, all terms in the difference $\Sigma^{\prime}-\Sigma$ cancel each
other, except for the residual term $o(\epsilon^{4})$. This cancellation
became possible as a result of the special choice of signs in the definition
of quantity $\Sigma$.

Hence, a shift of points $A$ – $H$ by distances $o(\epsilon^{2})$ results in
as small change of quantity $\Sigma$ as $o(\epsilon^{4})$.

###  V.5 Making fifteen intervals equal to zero

Generally, all sixteen intervals $\delta s^{2}_{AE},\ldots,\delta s^{2}_{DH}$,
that contribute to $\Sigma$, can be different from zero. But, as we will show
in this subsection, it is possible to set _fifteen_ of them to zero by small
shifts of points $A$ – $H$. This will be done in three steps described below.
Shifted points are denoted as $A^{\prime}$ – $H^{\prime}$.

Step 1. We do not shift points $A$, $B$, and $C$. That is, $A^{\prime}=A$,
$B^{\prime}=B$, and $C^{\prime}=C$.

Step 2. For each of points $E$, $F$, $G$ and $H$, we leave one of coordinates
unchanged, namely $z_{E}$, $z_{F}$, $y_{G}$ and $y_{H}$. Then we adjust other
three coordinates such that three intervals between the given point and points
$A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ vanish. Therefore we obtain
shifted points $E^{\prime}$, $F^{\prime}$, $G^{\prime}$ and $H^{\prime}$ that
obey relations

$\displaystyle\delta s^{2}_{A^{\prime}E^{\prime}}=\delta
s^{2}_{A^{\prime}F^{\prime}}=\delta s^{2}_{A^{\prime}G^{\prime}}=\delta
s^{2}_{A^{\prime}H^{\prime}}$ $\displaystyle=$ $\displaystyle\delta
s^{2}_{B^{\prime}E^{\prime}}=\delta s^{2}_{B^{\prime}F^{\prime}}=\delta
s^{2}_{B^{\prime}G^{\prime}}=\delta s^{2}_{B^{\prime}H^{\prime}}$
$\displaystyle=$ $\displaystyle\delta s^{2}_{C^{\prime}E^{\prime}}=\delta
s^{2}_{C^{\prime}F^{\prime}}=\delta s^{2}_{C^{\prime}G^{\prime}}=\delta
s^{2}_{C^{\prime}H^{\prime}}$ $\displaystyle=0.$ (72)

Step 3. For point $D$, we fix coordinate $t_{D}$, and adjust other three
coordinates such that three intervals between this point and points
$E^{\prime}$, $F^{\prime}$ and $G^{\prime}$ vanish:

$\delta s^{2}_{D^{\prime}E^{\prime}}=\delta
s^{2}_{D^{\prime}F^{\prime}}=\delta s^{2}_{D^{\prime}G^{\prime}}=0,$ (73)

where $D^{\prime}$ is the shifted point $D$.

As shown in Appendix G, differences between initial points $A$ – $H$ and their
corresponding shifted points $A^{\prime}$ – $H^{\prime}$ do not exceed ${\cal
O}(\epsilon^{3})$ along each coordinate. This allows us to apply the results
of Subsection V.4 to this set of points.

###  V.6 Completion of the proof

Let us recall the results obtained above. We consider a curved spacetime with
a nonzero Weyl tensor at some point $O$. In Subsection V.1, we have chosen a
local inertial (at point $O$) reference frame, in which component $C_{tyxy}$
of the Weyl tensor is nonzero at $O$. In Subsection V.2, we have introduced
such a definition of interval $\delta s^{2}_{PQ}$ between two points $P$ and
$Q$ in a curved spacetime, that $\delta s^{2}_{PQ}=0$ if and only if relation
$P\leftrightarrowtriangle Q$ is fulfilled. In Subsection V.3, we have defined
eight points $A$ – $H$ within the $\epsilon$-neighborhood of point $O$. Then
we have found a linear combination $\Sigma$ of intervals between these points,
which is proportional to $C_{tyxy}$ up to terms $o(\epsilon^{4})$. In
Subsection V.4, we have shown that this combination $\Sigma$ remains unchanged
under small ($o(\epsilon^{2})$) variations of coordinates of points $A$ – $H$,
up to $o(\epsilon^{4})$. Finally, we have found in Subsection V.5 such a
variation of coordinates that enable to set fifteen intervals $\delta
s^{2}_{A^{\prime}E^{\prime}},\ldots,\delta s^{2}_{D^{\prime}G^{\prime}}$ to
zero.

Collected together, these results consist a proof of Theorem 2. To see this,
let us choose a small spatial scale $\epsilon$, then define eight points $A$ –
$H$ in the $\epsilon$-neighborhood around point $O$ according to Subsection
V.3, and find shifted points $A^{\prime}$ – $H^{\prime}$ by the procedure
described in Subsection V.5. As a result, all intervals, that define quantity
$\Sigma^{\prime}$ in Eq. (65), vanish, except for $\delta
s^{2}_{D^{\prime}H^{\prime}}$. Therefore

$\Sigma^{\prime}=\delta s^{2}_{D^{\prime}H^{\prime}}\,.$ (74)

Taking into account stability of $\Sigma$ under small shifts, Eq. (71), one
can see that

$\delta s^{2}_{D^{\prime}H^{\prime}}=\Sigma+o(\epsilon^{4}).$ (75)

Then, expressing quantity $\Sigma$ through the Weyl tensor according to Eq.
(63), one obtains

$\delta
s^{2}_{D^{\prime}H^{\prime}}=-\frac{32\sqrt{2}}{3}\,\epsilon^{4}\,C_{tyxy}(O)+o(\epsilon^{4}).$
(76)

In the chosen reference frame, $C_{tyxy}(O)\neq 0$, see Subsection V.1. When
spatial scale $\epsilon$ is sufficiently small, the term containing
$\epsilon^{4}\,C_{tyxy}(O)$ exceeds (in absolute value) the residual term
$o(\epsilon^{4})$ in Eq. (76). Therefore, for such value of $\epsilon$,

$\delta s^{2}_{D^{\prime}H^{\prime}}\neq 0.$ (77)

The set of points $A^{\prime}$ – $H^{\prime}$ obeys Eqs. (72) and (73).
According to Subsection V.2, this means that fifteen relations

$\displaystyle A^{\prime}\leftrightarrowtriangle E^{\prime},$
$\displaystyle\;\;A^{\prime}\leftrightarrowtriangle
F^{\prime},\;\;A^{\prime}\leftrightarrowtriangle
G^{\prime},\;\;A^{\prime}\leftrightarrowtriangle H^{\prime},$ $\displaystyle
B^{\prime}\leftrightarrowtriangle E^{\prime},$
$\displaystyle\;\;B^{\prime}\leftrightarrowtriangle
F^{\prime},\;\;B^{\prime}\leftrightarrowtriangle
G^{\prime},\;\;B^{\prime}\leftrightarrowtriangle H^{\prime},$ $\displaystyle
C^{\prime}\leftrightarrowtriangle E^{\prime},$
$\displaystyle\;\;C^{\prime}\leftrightarrowtriangle
F^{\prime},\;\;C^{\prime}\leftrightarrowtriangle
G^{\prime},\;\;C^{\prime}\leftrightarrowtriangle H^{\prime},$ $\displaystyle
D^{\prime}\leftrightarrowtriangle E^{\prime},$
$\displaystyle\;\;D^{\prime}\leftrightarrowtriangle
F^{\prime},\;\;D^{\prime}\leftrightarrowtriangle G^{\prime}$ (78)

are fulfilled.

If the spacetime were well-stitched, relation
$D^{\prime}\leftrightarrowtriangle H^{\prime}$ would follow from relations
(78). But Eq. (77) means that relation $D^{\prime}\leftrightarrowtriangle
H^{\prime}$ is not fulfilled. Thus, points $A^{\prime}$ – $H^{\prime}$ violate
well-stitchedness.

In conclusion, provided that the Weyl tensor differs from zero at some point
$O$, there are eight points $A^{\prime}$ – $H^{\prime}$ that violate well-
stitchedness. Q.E.D.

###  V.7 Estimate of the time delay

As a by-product of the proof presented above, we have obtained the value of
$\delta s^{2}_{D^{\prime}H^{\prime}}$, which is a measure of violation of
relation $D^{\prime}\leftrightarrowtriangle H^{\prime}$. Inequality $\delta
s^{2}_{D^{\prime}H^{\prime}}\neq 0$ means that a light signal emitted at event
$D^{\prime}$ does not pass through event $H^{\prime}$. We denote as $\delta
t_{D^{\prime}H^{\prime}}$ the time delay of this signal, i. e. the difference
between the moment of time, when this signal reaches point $H^{\prime}$, and
the moment of time of event $H^{\prime}$ itself. We are going to find the
value of this time delay.

For the sake of generality, let us consider two events $P$ and $Q$, and denote
as $\widetilde{Q}$ the event, in which a light signal from $P$ reaches the
spatial point of event $Q$ (see Fig. 7). One can make the reference frame
locally-inertial (i. e. set Christoffel symbols to zero) along line
$P\widetilde{Q}$ by a small distortion of coordinates [12, §85]. In such a
frame, the light travels from $P$ to $\widetilde{Q}$ with the constant
velocity $c$, whence

Figure 7: Illustration of time delay $\delta t_{PQ}$. A light signal emitted
at event $P$ passes in a vicinity of event $Q$. The event when this signal
comes to the point of space of event $Q$ is denoted as $\widetilde{Q}$. Time
delay $\delta t_{PQ}$ is the difference in time between events $Q$ and
$\widetilde{Q}$.

$\ell_{PQ}=c(t_{PQ}+\delta t_{PQ}),$ (79)

where $\ell_{PQ}$ and $t_{PQ}$ are differences between $P$ and $Q$ in space
and in time, correspondingly. Interval $\delta s^{2}_{PQ}$ is expressed in
this frame just as in special relativity:

$\delta s^{2}_{PQ}=\ell_{PQ}^{2}-c^{2}t_{PQ}^{2}.$ (80)

Substituting Eq. (79) into Eq. (80) and neglecting the higher-order
infinitesimal $\delta t_{PQ}^{2}$, one can express time delay $\delta t_{PQ}$
as follows (cf. Ref. [57]):

$\delta t_{PQ}\approx\frac{\delta s^{2}_{PQ}}{2c^{2}t_{PQ}}\equiv\frac{\delta
s^{2}_{PQ}}{2c^{2}(t_{Q}-t_{P})}\,.$ (81)

Taking $\delta s^{2}_{D^{\prime}H^{\prime}}$ from Eq. (76) and
$t_{H^{\prime}}-t_{D^{\prime}}\approx t_{H}-t_{D}=\sqrt{2}\epsilon/c$ from Eq.
(50), one can obtain with the help of Eq. (81) that

$\delta
t_{D^{\prime}H^{\prime}}\approx-\frac{16\epsilon^{3}\,C_{tyxy}(O)}{3c}\,.$
(82)

This is the time delay, detection of which reveals the spacetime curvature.

##  Appendix A Proof of Lemma 3

Lemma 3. For any four points $P$, $Q$, $R$, $S$, if relations
$P\leftrightarrowtriangle R$, $P\leftrightarrowtriangle S$,
$Q\leftrightarrowtriangle R$ and $Q\leftrightarrowtriangle S$ are fulfilled,
then vector $\overrightarrow{PQ}$ is perpendicular to vector
$\overrightarrow{RS}$ (that is,
$\overrightarrow{PQ}\cdot\overrightarrow{RS}=0$).

Proof. Let $\vec{r}_{P}$, $\vec{r}_{Q}$, $\vec{r}_{R}$ and $\vec{r}_{S}$ be
radius-vectors of points $P$, $Q$, $R$ and $S$, correspondingly. Then,
$\overrightarrow{PQ}=\vec{r}_{Q}-\vec{r}_{P}$, and
$\overrightarrow{RS}=\vec{r}_{S}-\vec{r}_{R}$. Relations
$P\leftrightarrowtriangle R$, $P\leftrightarrowtriangle S$,
$Q\leftrightarrowtriangle R$ and $Q\leftrightarrowtriangle S$ mean that

$\displaystyle(\vec{r}_{P}-\vec{r}_{R})\cdot(\vec{r}_{P}-\vec{r}_{R})=0,$ (83)
$\displaystyle(\vec{r}_{P}-\vec{r}_{S})\cdot(\vec{r}_{P}-\vec{r}_{S})=0,$ (84)
$\displaystyle(\vec{r}_{Q}-\vec{r}_{R})\cdot(\vec{r}_{Q}-\vec{r}_{R})=0,$ (85)
$\displaystyle(\vec{r}_{Q}-\vec{r}_{S})\cdot(\vec{r}_{Q}-\vec{r}_{S})=0.$ (86)

Summing up Eqs. (84) and (85), and subtracting Eqs. (83) and (86) from them,
and expanding all brackets, one can get

$2\vec{r}_{P}\cdot\vec{r}_{R}-2\vec{r}_{P}\cdot\vec{r}_{S}-2\vec{r}_{Q}\cdot\vec{r}_{R}+2\vec{r}_{Q}\cdot\vec{r}_{S}=0,$
(87)

that is,

$2(\vec{r}_{Q}-\vec{r}_{P})\cdot(\vec{r}_{S}-\vec{r}_{R})\equiv
2\overrightarrow{PQ}\cdot\overrightarrow{RS}=0,$ (88)

Q.E.D.

##  Appendix B Proof of Lemma 4

Lemma 4. For any three different points $P$, $Q$, $R$, if relations
$P\leftrightarrowtriangle Q$, $P\leftrightarrowtriangle R$ and
$Q\leftrightarrowtriangle R$ are fulfilled, then these points lie on one line.

Proof. Let us choose the coordinate system with the origin at point $P$. It
follows from relation $P\leftrightarrowtriangle Q$ that point $Q$ lies on the
light cone with the vertex at $P$:

$-t_{Q}^{2}+x_{Q}^{2}+y_{Q}^{2}+z_{Q}^{2}=0.$ (89)

Let us turn the axes $x$, $y$, $z$ such that point $Q$ lie in plane $tx$.
Therefore, according to Eq. (89),

$x_{Q}=t_{Q}\neq 0,\quad y_{Q}=z_{Q}=0.$ (90)

[Another option is $x_{Q}=-t_{Q}$, but in this case we will flip the direction
of axis $x$ and return to Eq. (90).] Due to relations
$P\leftrightarrowtriangle R$ and $Q\leftrightarrowtriangle R$, point $R$ must
lie on both light cones—with vertices at $P$ and $Q$:

$\displaystyle-t_{R}^{2}+x_{R}^{2}+y_{R}^{2}+z_{R}^{2}=0,$ (91)
$\displaystyle-(t_{R}-t_{Q})^{2}+(x_{R}-t_{Q})^{2}+y_{R}^{2}+z_{R}^{2}=0.$
(92)

Subtracting Eq. (91) from Eq. (92) yields

$2t_{Q}(t_{R}-x_{R})=0,$ (93)

i. e. $x_{R}=t_{R}$. Substituting $x_{R}=t_{R}$ into Eq. (91), one can find
that $y_{R}=z_{R}=0$.

Hence, all three points $P$, $Q$, $R$ belong to the same (light-like) line,
which equation is

$x=t,\quad y=0,\quad z=0.$ (94)

Q.E.D.

##  Appendix C Proof of Lemma 5

Lemma 5. For any four points $P$, $Q$, $R$, $S$, if relations
$P\leftrightarrowtriangle S$, $Q\leftrightarrowtriangle S$ and
$R\leftrightarrowtriangle S$ are fulfilled, and points $P$, $Q$, $R$ lie on
one line and are different from each other, then this line is light-like (that
is, $P\leftrightarrowtriangle Q$, $P\leftrightarrowtriangle R$ and
$Q\leftrightarrowtriangle R$).

Proof. The line, that joins points $P$, $Q$ and $R$, may be time-like, space-
like or light-like. Let us suppose first that this line is time-like. In such
a case, this line can be chosen as $t$-axis. With this choice of coordinates,
relations $P\leftrightarrowtriangle S$ and $Q\leftrightarrowtriangle S$ obtain
the following form:

$\displaystyle x_{S}^{2}+y_{S}^{2}+z_{S}^{2}=(t_{S}-t_{P})^{2},$ (95)
$\displaystyle x_{S}^{2}+y_{S}^{2}+z_{S}^{2}=(t_{S}-t_{Q})^{2}.$ (96)

Therefore $|t_{S}-t_{P}|=|t_{S}-t_{Q}|$. Taking into account that $t_{P}\neq
t_{Q}$ (since points $P$ and $Q$ are different), one can resolve the latter
equality as

$t_{S}=\frac{t_{P}+t_{Q}}{2}.$ (97)

Similarly, one can deduce from relations $P\leftrightarrowtriangle S$ and
$R\leftrightarrowtriangle S$ that

$t_{S}=\frac{t_{P}+t_{R}}{2}.$ (98)

It is clearly seen from Eqs. (97) and (98) that $t_{Q}=t_{R}$. But this
contradicts to the proposition that points $Q$ and $R$ are different.
Consequently, the line that passes through $P$, $Q$ and $R$ cannot be time-
like.

Next, we suppose that this line is space-like. Let us choose this line as
$x$-axis. Then, almost the same reasoning, as in the case of a time-like line,
leads from relations $P\leftrightarrowtriangle S$, $Q\leftrightarrowtriangle
S$ and $R\leftrightarrowtriangle S$ to equations

$\displaystyle x_{S}=\frac{x_{P}+x_{Q}}{2},$ (99) $\displaystyle
x_{S}=\frac{x_{P}+x_{R}}{2},$ (100)

and to conclusion that $x_{Q}=x_{R}$. And again, this contradicts to the
proposition that $Q$ and $R$ are different. This means that the line that
connects points $P$, $Q$, $R$ is not space-like.

Hence, this line is neither time-like nor space-like. It is therefore light-
like, Q.E.D.

##  Appendix D The case when neither plane $ABCD$ nor plane $EFGH$ contains a
time-like vector

Let $\alpha$ and $\beta$ be 2-dimensional planes in 4-dimensional Minkowski
space, and any vector in $\alpha$ is orthogonal to any vector in $\beta$. (In
the main text, $\alpha$ is plane $ABCD$, and $\beta$ is plane $EFGH$.) We
suppose also that neither $\alpha$ nor $\beta$ contains a time-like vector.
The aim of this Appendix is to show that, in a properly chosen coordinate
frame, planes $\alpha$ and $\beta$ are defined by Eqs. (30) and (31),
respectively.

Let us choose two non-collinear, mutually perpendicular vectors $\vec{a}$ and
$\vec{b}$ in plane $\alpha$. Each of these vectors can be either space-like,
or light-like. (Time-like vectors are forbidden by supposition.) There are
three cases: (i) both vectors are space-like, (ii) both are light-like, and
(iii) one vector is space-like and the other one is light-like.

In case (i), one can choose directions of coordinate axis $y$ and $z$ along
vectors $\vec{a}$ and $\vec{b}$, correspondingly. With this choice, plane
$\alpha$ is parallel to $yz$-plane, and consequently any vector of plane
$\beta$ is perpendicular to $yz$-plane, i. e. parallel to $tx$-plane.
Therefore plane $\beta$ is parallel to plane $tx$. But it contradicts to the
supposition that plane $\beta$ does not contain a time-like vector.

Case (ii) contradicts to the choice of vectors $\vec{a}$ and $\vec{b}$ as non-
collinear and mutually perpendicular. Indeed, two light-like vectors in the
Minkowski space can be perpendicular to each other only if they are collinear.

Therefore cases (i) and (ii) lead to contradictions, and thus one of the
vectors is light-like and the other is space-like. Let $\vec{a}$ be a light-
like vector, and $\vec{b}$ be a space-like one. One can choose the coordinate
axes such that vector $\vec{a}$ lie in $tx$-plane, and its $t$\- and
$x$-components be the same:

$x_{a}=t_{a}\neq 0,\quad y_{a}=z_{a}=0.$ (101)

It follows from Eq. (101) and from orthogonality of vectors $\vec{a}$ and
$\vec{b}$ that $x_{b}=t_{b}$. By rotation of $yz$-part of the coordinate
system, one can put $z$-component of vector $\vec{b}$ to zero, while
components of vector $\vec{a}$ remain unchanged. After this, components of
vector $\vec{b}$ obeys the following conditions:

$x_{b}=t_{b},\quad z_{b}=0,\quad y_{b}\neq 0$ (102)

(the latter inequality follows from non-collinearity of vectors $\vec{a}$ and
$\vec{b}$). Equations (101) and (102) show that vectors $\vec{a}$ and
$\vec{b}$ are linear combinations of the following two vectors:

$\vec{c}=(1,1,0,0)\text{ and }\vec{d}=(0,0,1,0),$ (103)

where the components are listed in order $(t,x,y,z)$. Therefore, radius-vector
$\vec{r}_{\alpha}$ of an arbitrary point on plane $\alpha$ can be expressed as

$\vec{r}_{\alpha}=\vec{r}_{\alpha 0}+\lambda\vec{c}+\mu\vec{d},$ (104)

where $\lambda$ and $\mu$ are real parameters that vary from point to point,
and $\vec{r}_{\alpha 0}$ is a radius-vector of some fixed point on $\alpha$.
Similarly, radius-vector $\vec{r}_{\beta}$ of an arbitrary point on plane
$\beta$ depends on two real parameters $\lambda^{\prime}$ and $\mu^{\prime}$
as follows:

$\vec{r}_{\beta}=\vec{r}_{\beta
0}+\lambda^{\prime}\vec{e}+\mu^{\prime}\vec{f},$ (105)

where $\vec{r}_{\beta 0}$ is a radius-vector of some fixed point on plane
$\beta$, and each of two non-collinear vectors $\vec{e}$ and $\vec{f}$ is
perpendicular to both $\vec{c}$ and $\vec{d}$. It is convenient to choose
vectors $\vec{e}$ and $\vec{f}$ as

$\vec{e}=(1,1,0,0)\text{ and }\vec{f}=(0,0,0,1).$ (106)

One can easily check that
$\vec{c}\cdot\vec{e}=\vec{c}\cdot\vec{f}=\vec{d}\cdot\vec{e}=\vec{d}\cdot\vec{f}=0$.

Now we will choose points $\vec{r}_{\alpha 0}$ and $\vec{r}_{\beta 0}$. First,
we take randomly chosen points of planes $\alpha$ and $\beta$ as
$\vec{r}_{\alpha 0}$ and $\vec{r}_{\beta 0}$. Then we shift point
$\vec{r}_{\alpha 0}$ along $y$-axis (i. e. along $\vec{d}$), until its
$y$-coordinate becomes equal to the $y$-coordinate of point $\vec{r}_{\beta
0}$. Next, we shift point $\vec{r}_{\beta 0}$ along $z$-axis (i. e. along
$\vec{f}$), until its $z$-coordinate becomes equal to that of point
$\vec{r}_{\alpha 0}$. And finally, we shift point $\vec{r}_{\beta 0}$ along
vector $\vec{e}$, until its $x$-coordinate becomes equal to that of point
$\vec{r}_{\alpha 0}$. In the end of these manipulations, points
$\vec{r}_{\alpha 0}$ and $\vec{r}_{\beta 0}$ differ from each other only by
their $t$-coordinates. Let us put the origin of coordinates to point
$\vec{r}_{\alpha 0}$. Therefore

$\vec{r}_{\alpha 0}=(0,0,0,0),\quad\vec{r}_{\beta 0}=(s,0,0,0),$ (107)

where $s$ is the difference between $t$-coordinates of these points.

One can see from Eqs. (103), (104) and (107) that coordinates $(t,x,y,z)$ of
any point on plane $\alpha$ are

$t=\lambda,\quad x=\lambda,\quad y=\mu,\quad z=0,$ (108)

where $\lambda$ and $\mu$ are real parameters. This means that plane $\alpha$
is defined by Eq. (30).

Similarly, Eqs. (105), (106) and (107) express the coordinates of any point on
plane $\beta$ through real parameters $\lambda^{\prime}$ and $\mu^{\prime}$:

$t=\lambda^{\prime}+s,\quad x=\lambda^{\prime},\quad y=0,\quad
z=\mu^{\prime},$ (109)

that defines plane $\beta$ according to Eq. (31).

Hence, equations (30) and (31) for planes $\alpha$ and $\beta$ are
established.

##  Appendix E Possibility of choosing a reference frame, in which
$C_{tyxy}\neq 0$

In this Appendix, we consider Weyl tensor $C_{iklm}$ at some given point $O$.
We suppose that the reference frame is local inertial at this point, and
therefore the metric tensor is equal to $\eta_{ik}$, i. e. to the metric
tensor of special relativity.

All possible realizations of the Weyl tensor consist an irreducible
representation $\cal R$ of the Lorentz group [58]. Suppose that there are such
non-zero realizations of the Weyl tensor, in which its component $C_{tyxy}$ is
equal to zero, and remains zero after any Lorentz transformation. A set of
such realizations is also some representation ${\cal R}_{0}$ of the Lorentz
group. By definition ${\cal R}_{0}\subset\cal R$. Also ${\cal R}_{0}\neq\cal
R$, since a realization with $C_{tyxy}\neq 0$ belongs to $\cal R$ but not to
${\cal R}_{0}$. But this contradicts to irreducibility of $\cal R$.

It is therefore proven by contradiction, that any non-zero Weyl tensor
$C_{iklm}$ can be such Lorentz-transformed that inequality $C_{tyxy}\neq 0$
will hold after the transformation.

##  Appendix F MATLAB code for calculation of tensor $M^{iklm}$

    components = ’txyz’;

    r_ABCD = [ sqrt(2),  1, 0, 0;
              -sqrt(2), -1, 0, 0;
               sqrt(2), -1, 0, 0;
              -sqrt(2),  1, 0, 0];

    r_EFGH = [ 0, 0,  1,  0;
               0, 0, -1,  0;
               0, 0,  0,  1;
               0, 0,  0, -1];

    sign_PQ = [ 1,  1, -1, -1;
                1,  1, -1, -1;
               -1, -1,  1,  1;
               -1, -1,  1,  1];

    for i = 1:4
     for k = 1:4
      for l = 1:4
       for m = 1:4
        M_iklm = 0;
         for P = 1:4
          for Q = 1:4
           r_P = r_ABCD(P,:);
           r_Q = r_EFGH(Q,:);
           M_PQ_iklm = -1/3 * r_P(i) ...
            * (r_Q(k)-r_P(k)) * r_P(l) ...
            * (r_Q(m)-r_P(m));
           M_iklm = M_iklm ...
            + sign_PQ(P,Q) * M_PQ_iklm;
          end
         end
        if abs(M_iklm) > 1e-10
         disp(sprintf( ’M_%s = %f’, ...
          components([i,k,l,m]), M_iklm ));
        end
       end
      end
     end
    end

##  Appendix G Estimation of distances between points $A$ – $H$ and points
$A^{\prime}$ – $H^{\prime}$

In Subsection V.5, we suggested a method of finding such a set of “shifted”
points $A^{\prime}$ – $H^{\prime}$ that fifteen intervals listed in Eqs. (72)
and (73) vanish. Here we discuss how close each “shifted” point is to its
“unshifted” counterpart $A$ – $H$.

For points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, this question is
trivial because they coincide with $A$, $B$ and $C$, correspondingly.

Let us consider point $E^{\prime}$. Its position is defined by a system of
three equations

$\delta s^{2}_{A^{\prime}E^{\prime}}=\delta
s^{2}_{B^{\prime}E^{\prime}}=\delta s^{2}_{C^{\prime}E^{\prime}}=0$ (110)

with three unknowns—namely, three coordinates $t_{E^{\prime}}$,
$x_{E^{\prime}}$ and $y_{E^{\prime}}$ of point $E^{\prime}$. The fourth
coordinate $z_{E^{\prime}}$ is fixed:

$z_{E^{\prime}}=z_{E}.$ (111)

Equations (110) are nonlinear, but can be approximately linearized when the
deviation of point $E^{\prime}$ from $E$ is small in comparison with the
spatial scale $\epsilon$. Omitting the residual (nonlinear) term
$o(\epsilon^{4})$ in Eq. (68), and taking into account that
$\vec{r}_{A^{\prime}}-\vec{r}_{A}=\vec{r}_{B^{\prime}}-\vec{r}_{B}=\vec{r}_{C^{\prime}}-\vec{r}_{C}=0$,
one can get the following expressions for intervals $\delta
s^{2}_{A^{\prime}E^{\prime}}$, $\delta s^{2}_{B^{\prime}E^{\prime}}$ and
$\delta s^{2}_{C^{\prime}E^{\prime}}$:

$\displaystyle\delta s^{2}_{A^{\prime}E^{\prime}}$ $\displaystyle\approx$
$\displaystyle
2(\vec{r}_{E}-\vec{r}_{A})\cdot(\vec{r}_{E^{\prime}}-\vec{r}_{E})+\delta
s^{2}_{AE}\,,$ (112) $\displaystyle\delta s^{2}_{B^{\prime}E^{\prime}}$
$\displaystyle\approx$ $\displaystyle
2(\vec{r}_{E}-\vec{r}_{B})\cdot(\vec{r}_{E^{\prime}}-\vec{r}_{E})+\delta
s^{2}_{BE}\,,$ (113) $\displaystyle\delta s^{2}_{C^{\prime}E^{\prime}}$
$\displaystyle\approx$ $\displaystyle
2(\vec{r}_{E}-\vec{r}_{C})\cdot(\vec{r}_{E^{\prime}}-\vec{r}_{E})+\delta
s^{2}_{CE}\,.$ (114)

Substituting Eqs. (111) – (114) into (110), we obtain a system of linearized
equations. It is convenient to represent this system in a matrix form:

$\hat{A}\begin{pmatrix}t_{E^{\prime}}-t_{E}\\\ x_{E^{\prime}}-x_{E}\\\
y_{E^{\prime}}-y_{E}\end{pmatrix}+\begin{pmatrix}\delta s^{2}_{AE}\\\ \delta
s^{2}_{BE}\\\ \delta s^{2}_{CE}\end{pmatrix}\approx\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},$ (115)

where $\hat{A}$ is a $3\times 3$ matrix of coefficients:

$\hat{A}=2\begin{pmatrix}t_{A}-t_{E}&x_{E}-x_{A}&y_{E}-y_{A}\\\
t_{B}-t_{E}&x_{E}-x_{B}&y_{E}-y_{B}\\\
t_{C}-t_{E}&x_{E}-x_{C}&y_{E}-y_{C}\end{pmatrix}=2\epsilon\begin{pmatrix}\;\;\sqrt{2}&-1&1\\\
-\sqrt{2}&\;\;1&1\\\ \;\;\sqrt{2}&\;\;1&1\\\ \end{pmatrix}$ (116)

Matrix $\hat{A}$ is non-degenerate, and therefore system of equations (115) is
consistent. Its solution reads:

$\begin{pmatrix}t_{E^{\prime}}-t_{E}\\\ x_{E^{\prime}}-x_{E}\\\
y_{E^{\prime}}-y_{E}\end{pmatrix}\approx-\hat{A}^{-1}\begin{pmatrix}\delta
s^{2}_{AE}\\\ \delta s^{2}_{BE}\\\ \delta s^{2}_{CE}\end{pmatrix}.$ (117)

The right-hand side of the latter equation is proportional to $\epsilon^{3}$.
Indeed, coefficients of the inverse matrix $\hat{A}^{-1}$ are proportional to
$\epsilon^{-1}$. Intervals $\delta s^{2}_{AE}$, $\delta s^{2}_{BE}$ and
$\delta s^{2}_{CE}$ are proportional to $\epsilon^{4}$, that can be seen in
Eq. (48). Hence,

$\displaystyle|t_{E^{\prime}}-t_{E}|={\cal
O}(\epsilon^{3}),\quad|x_{E^{\prime}}-x_{E}|={\cal O}(\epsilon^{3}),$
$\displaystyle|y_{E^{\prime}}-y_{E}|={\cal
O}(\epsilon^{3}),\quad|z_{E^{\prime}}-z_{E}|=0.$ (118)

Similar considerations for point $F^{\prime}$ lead to a linearized system of
three equations with three unknowns $t_{F^{\prime}}$, $x_{F^{\prime}}$ and
$y_{F^{\prime}}$. This system can be written in the same form as Eq. (115),
where $E,E^{\prime}$ are replaced with $F,F^{\prime}$, and matrix $\hat{A}$ is
slightly different:

$\hat{A}=2\epsilon\begin{pmatrix}\;\;\sqrt{2}&-1&-1\\\ -\sqrt{2}&\;\;1&-1\\\
\;\;\sqrt{2}&\;\;1&-1\\\ \end{pmatrix}.$ (119)

As a result, one can get the same estimates for coordinate differences between
points $F^{\prime}$ and $F$ as in Eq. (118).

For points $G^{\prime}$ and $H^{\prime}$, the analysis is exactly the same as
for points $E^{\prime}$ and $F^{\prime}$, up to replacements $E\to G$,
$E^{\prime}\to G^{\prime}$, $F\to H$, $F^{\prime}\to H^{\prime}$, and
$y\leftrightarrow z$. It leads to the conclusion that points $E^{\prime}$,
$F^{\prime}$, $G^{\prime}$, $H^{\prime}$ differ from their corresponding
“unshifted” points $E$, $F$, $G$, $H$ by at most ${\cal O}(\epsilon^{3})$
along each coordinate.

The last point to consider is $D^{\prime}$. This point is defined by the
system of three equations

$\delta s^{2}_{D^{\prime}E^{\prime}}=\delta
s^{2}_{D^{\prime}F^{\prime}}=\delta s^{2}_{D^{\prime}G^{\prime}}=0$ (120)

with respect to three unknowns—coordinates $x_{D^{\prime}}$, $y_{D^{\prime}}$
and $z_{D^{\prime}}$, whereas the fourth coordinate $t_{D^{\prime}}$ is fixed:
$t_{D^{\prime}}=t_{D}$. One can express these intervals via Eq. (68),
neglecting the nonlinear residual term $o(\epsilon^{4})$:

$\displaystyle\delta s^{2}_{D^{\prime}E^{\prime}}\approx
2(\vec{r}_{D}-\vec{r}_{E})\cdot(\vec{r}_{D^{\prime}}-\vec{r}_{D}+\vec{r}_{E}-\vec{r}_{E^{\prime}})+\delta
s^{2}_{DE}\,,$ $\displaystyle\delta s^{2}_{D^{\prime}F^{\prime}}\approx
2(\vec{r}_{D}-\vec{r}_{F})\cdot(\vec{r}_{D^{\prime}}-\vec{r}_{D}+\vec{r}_{F}-\vec{r}_{F^{\prime}})+\delta
s^{2}_{DF}\,,$ $\displaystyle\delta s^{2}_{D^{\prime}G^{\prime}}\approx
2(\vec{r}_{D}-\vec{r}_{G})\cdot(\vec{r}_{D^{\prime}}-\vec{r}_{D}+\vec{r}_{G}-\vec{r}_{G^{\prime}})+\delta
s^{2}_{DG}\,.$

These expressions provide a linearized form of the system of equations (120):

$\hat{B}\begin{pmatrix}x_{D^{\prime}}-x_{D}\\\ y_{D^{\prime}}-y_{D}\\\
z_{D^{\prime}}-z_{D}\end{pmatrix}+\begin{pmatrix}b_{E}\\\ b_{F}\\\
b_{G}\end{pmatrix}\approx\begin{pmatrix}0\\\ 0\\\ 0\end{pmatrix},$ (121)

where matrix $\hat{B}$ is defined as follows:

$\hat{B}=2\begin{pmatrix}x_{D}-x_{E}&y_{D}-y_{E}&z_{D}-z_{E}\\\
x_{D}-x_{F}&y_{D}-y_{F}&z_{D}-z_{F}\\\
x_{D}-x_{G}&y_{D}-y_{G}&z_{D}-z_{G}\end{pmatrix}=2\epsilon\begin{pmatrix}1&-1&\;\;0\\\
1&\;\;1&\;\;0\\\ 1&\;\;0&-1\\\ \end{pmatrix}$ (122)

and $b_{E}$, $b_{F}$, $b_{G}$ denote the constant terms:

$\displaystyle
b_{E}=2(\vec{r}_{D}-\vec{r}_{E})\cdot(\vec{r}_{E}-\vec{r}_{E^{\prime}})+\delta
s^{2}_{DE}\,,$ (123) $\displaystyle
b_{F}=2(\vec{r}_{D}-\vec{r}_{F})\cdot(\vec{r}_{F}-\vec{r}_{F^{\prime}})+\delta
s^{2}_{DF}\,,$ (124) $\displaystyle
b_{G}=2(\vec{r}_{D}-\vec{r}_{G})\cdot(\vec{r}_{G}-\vec{r}_{G^{\prime}})+\delta
s^{2}_{DG}\,.$ (125)

In the right-hand side of Eq. (123), vector $(\vec{r}_{D}-\vec{r}_{E})$ is
proportional to $\epsilon$, see definitions (50d) and (50e); vector
$(\vec{r}_{E}-\vec{r}_{E^{\prime}})$ is proportional to $\epsilon^{3}$ due to
Eq. (118); and interval $\delta s^{2}_{DE}$ is proportional to $\epsilon^{4}$
according to Eq. (48). Therefore $b_{E}={\cal O}(\epsilon^{4})$, and the same
estimates are true for quantities $b_{F}$ and $b_{G}$.

The solution of system (121) is

$\begin{pmatrix}x_{D^{\prime}}-x_{D}\\\ y_{D^{\prime}}-y_{D}\\\
z_{D^{\prime}}-z_{D}\end{pmatrix}\approx-\hat{B}^{-1}\begin{pmatrix}b_{E}\\\
b_{F}\\\ b_{G}\end{pmatrix}.$ (126)

Here, matrix elements of $\hat{B}^{-1}$ are proportional to $\epsilon^{-1}$,
and terms $b_{E}$, $b_{F}$, $b_{G}$ are proportional to $\epsilon^{4}$.
Consequently the left-hand side of Eq. (126) is proportional to
$\epsilon^{3}$.

Hence, deviations of “shifted” points $A^{\prime}$ – $H^{\prime}$ from
“unshifted” ones $A$ – $H$ do not exceed ${\cal O}(\epsilon^{3})$.

##  Appendix H Calculation of the distance between geodesics in a complete
quadrilateral

Let us consider a figure shown in Fig. 8 (a complete quadrilateral) that
consists of four geodesic lines: $OAB$, $OCD$, $AD$ and $BC$. If the spacetime
were flat, geodesics $AD$ and $BC$ would cross each other at some point
(event) $E_{0}$. In a curved spacetime, however, there may be a gap between
lines $AD$ and $BC$. In this appendix, we address the following questions:

Figure 8: A complete quadrilateral in a curved space. Solid lines $OAB$,
$OCD$, $AE_{1}D$ and $BE_{2}C$ are geodesics. Dashed lines show that geodesics
$AB$ and $CD$ would intersect in some point (event) $E_{0}$ if the spacetime
were flat. Points $E_{1}$ and $E_{2}$ are chosen in a vicinity of point
$E_{0}$. Unit vectors $\vec{b}_{1}$ and $\vec{b}_{2}$ (not shown) are
perpendicular both to $\vec{a}_{1}$ and to $\vec{a}_{2}$.

• How to describe the size of the gap between lines $AD$ and $BC$?

• How the size of this gap is related to the spacetime curvature?

• To which extent the curvature tensor can be restored from the size of these
gap (measured in different configurations of points $O$, $A$, $B$, $C$, $D$)?

We will consider the size of the whole figure as small, so that geodesic line
segments within the figure are almost straight. We choose a point $E_{1}$ on
geodesic $AD$, and a point $E_{2}$ on geodesic $BC$, both points near $E_{0}$.
Then, the gap between geodesics $AD$ and $BC$ is determined by vector
$\overrightarrow{E_{1}E_{2}}$. To exclude ambiguity of the choice of points
$E_{1}$ and $E_{2}$, this vector should be projected onto the plane
perpendicular to both geodesics $AD$ and $BC$.

We therefore choose two linearly independent unit vectors $\vec{b}_{1}$ and
$\vec{b}_{2}$, both of them perpendicular to geodesics $AD$ and $BC$, and
consider quantities

$\xi_{1}=\vec{b}_{1}\cdot\overrightarrow{E_{1}E_{2}},\quad\xi_{2}=\vec{b}_{2}\cdot\overrightarrow{E_{1}E_{2}}.$
(127)

They completely quantify the distance between the geodesics $AD$ and $BC$.

It is convenient to calculate $\xi_{1}$ and $\xi_{2}$ in the Riemann normal
coordinates [1, §11.6] with the origin at point $O$. The metric tensor
$g_{ik}(\vec{r})$ in these coordinates has the form of Eq. (43), and geodesics
$OAB$ and $OCD$ (that pass through the origin) are simply straight lines. If
points $P$ and $Q$ lie within a small $\epsilon$-neighborhood of point $O$,
then the segment of the geodesic line that joins two points $P$ and $Q$ obeys
the following parametric equation in the normal coordinates [56]:

$r^{i}(\lambda)=(1-\lambda)r_{P}^{i}+\lambda r_{Q}^{i}\\\
+\frac{\lambda(1-\lambda)}{3}R{{}^{i}}_{klm}(r_{Q}^{k}-r_{P}^{k})r_{P}^{l}(r_{Q}^{m}-r_{P}^{m})+{\cal
O}(\epsilon^{4}),$ (128)

where parameter $\lambda$ is equal to 0 at $P$ and 1 at $Q$.

Before applying Eq. (128), we slightly modify it with the aid of skew symmetry
($R{{}^{i}}_{klm}=-R{{}^{i}}_{kml}$) of the Riemann tensor. First, we notice
that $R{{}^{i}}_{klm}r_{P}^{l}r_{P}^{m}=0$ due to this skew symmetry, and
hence the term $(-r_{P}^{m})$ can be omitted in Eq. (128):

$r^{i}(\lambda)=(1-\lambda)r_{P}^{i}+\lambda r_{Q}^{i}\\\
+\frac{\lambda(1-\lambda)}{3}(R{{}^{i}}_{klm}r_{Q}^{k}r_{P}^{l}r_{Q}^{m}-R{{}^{i}}_{klm}r_{P}^{k}r_{P}^{l}r_{Q}^{m})+{\cal
O}(\epsilon^{4}).$ (129)

The term $R{{}^{i}}_{klm}r_{Q}^{k}r_{P}^{l}r_{Q}^{m}$ can be further
transformed to $(-R{{}^{i}}_{klm}r_{Q}^{k}r_{Q}^{l}r_{P}^{m})$ by swapping
indices $l$ and $m$ and employing the skew symmetry:

$r^{i}(\lambda)=(1-\lambda)r_{P}^{i}+\lambda r_{Q}^{i}\\\
-\frac{\lambda(1-\lambda)}{3}R{{}^{i}}_{klm}(r_{Q}^{k}r_{Q}^{l}r_{P}^{m}+r_{P}^{k}r_{P}^{l}r_{Q}^{m})+{\cal
O}(\epsilon^{4}).$ (130)

Later, we assume that the whole figure $OABCD$ belongs to the
$\epsilon$-neighborhood of point $O$ with a small size $\epsilon$, and neglect
the residue term ${\cal O}(\epsilon^{4})$.

We describe figure $OABCD$ by two vectors $\vec{a}_{1}$, $\vec{a}_{2}$ (see
Fig. 8) and two numbers $\mu_{1}$, $\mu_{2}$:

$\vec{a}_{1}=\overrightarrow{OA},\quad\vec{a}_{2}=\overrightarrow{OC},\quad\mu_{1}=\frac{|\overrightarrow{OB}|}{|\overrightarrow{OA}|},\quad\mu_{2}=\frac{|\overrightarrow{OD}|}{|\overrightarrow{OC}|},$
(131)

so that radius-vectors of points $A$, $B$, $C$, $D$ are

$\vec{r}_{A}=\vec{a}_{1},\quad\vec{r}_{B}=\mu_{1}\vec{a}_{1},\quad\vec{r}_{C}=\vec{a}_{2},\quad\vec{r}_{D}=\mu_{2}\vec{a}_{2}.$
(132)

Point $E_{0}$ lies at the intersection of _straight lines_ $AD$ and $BC$.
Hence, its radius-vector expresses as

$\vec{r}_{E_{0}}=(1-\lambda_{1})\vec{r}_{A}+\lambda_{1}\vec{r}_{D}=(1-\lambda_{2})\vec{r}_{C}+\lambda_{2}\vec{r}_{B},$
(133)

where $\lambda_{1}$ and $\lambda_{2}$ are some numbers. Substituting here
$\vec{r}_{A}$, $\vec{r}_{B}$, $\vec{r}_{C}$ and $\vec{r}_{D}$ from Eq. (132),
and collecting the terms with $\vec{a}_{1}$ and $\vec{a}_{2}$ separately, one
can obtain two equations for $\lambda_{1}$ and $\lambda_{2}$:

$1-\lambda_{1}=\lambda_{2}\mu_{1},\qquad\lambda_{1}\mu_{2}=1-\lambda_{2}.$
(134)

Solution of these equations is

$\lambda_{1}=\frac{\mu_{1}-1}{\mu_{1}\mu_{2}-1},\qquad\lambda_{2}=\frac{\mu_{2}-1}{\mu_{1}\mu_{2}-1}.$
(135)

Coordinates of point $E_{1}$ can be obtained by substitution of
$\vec{r}_{A}=\vec{a}_{1}$, $\vec{r}_{D}=\mu_{2}\vec{a}_{2}$ and $\lambda_{1}$
as $\vec{r}_{P}$, $\vec{r}_{Q}$ and $\lambda$ into Eq. (130). It is evident
from comparison with Eq. (133) that the first term in the resulting expression
is equal to $\vec{r}_{E_{0}}$. Hence,

$\vec{r}_{E_{1}}=\vec{r}_{E_{0}}-\frac{\lambda_{1}(1-\lambda_{1})}{3}\mu_{2}R{{}^{i}}_{klm}(\mu_{2}a_{2}^{k}a_{2}^{l}a_{1}^{m}+a_{1}^{k}a_{1}^{l}a_{2}^{m}).$
(136)

Similarly, coordinates of point $E_{2}$ can be obtained by substitution of
$\vec{r}_{C}=\vec{a}_{2}$, $\vec{r}_{B}=\mu_{1}\vec{a}_{1}$ and $\lambda_{2}$
as $\vec{r}_{P}$, $\vec{r}_{Q}$ and $\lambda$ into Eq. (130):

$\vec{r}_{E_{2}}=\vec{r}_{E_{0}}-\frac{\lambda_{2}(1-\lambda_{2})}{3}\mu_{1}R{{}^{i}}_{klm}(\mu_{1}a_{1}^{k}a_{1}^{l}a_{2}^{m}+a_{2}^{k}a_{2}^{l}a_{1}^{m}).$
(137)

The prefactors before $R{{}^{i}}_{klm}$ in equations (136) and (137) are equal
to each other, that can be easily seen by expressing $(1-\lambda_{1})$ and
$(1-\lambda_{2})$ via Eq. (134):

$\frac{\lambda_{1}(1-\lambda_{1})}{3}\mu_{2}=\frac{\lambda_{2}(1-\lambda_{2})}{3}\mu_{1}=\frac{\lambda_{1}\lambda_{2}\mu_{1}\mu_{2}}{3}\,.$
(138)

This fact simplifies calculation of vector
$\overrightarrow{E_{1}E_{2}}\equiv\vec{r}_{E_{2}}-\vec{r}_{E_{1}}$ and
quantities

$\xi_{1}=\vec{b}_{1}\cdot(\vec{r}_{E_{2}}-\vec{r}_{E_{1}})\;\;\text{and}\;\;\xi_{2}=\vec{b}_{2}\cdot(\vec{r}_{E_{2}}-\vec{r}_{E_{1}}).$
(139)

Collecting Eqs. (136) – (139) together, and taking values of $\lambda_{1}$ and
$\lambda_{2}$ from Eq. (135), one can find $\xi_{1}$ and $\xi_{2}$ as

$\xi_{1}=\frac{\mu_{1}\mu_{2}(1-\mu_{1})(1-\mu_{2})}{3(\mu_{1}\mu_{2}-1)^{2}}R_{iklm}\,b_{1}^{i}\\\
\times\left[(\mu_{2}-1)a_{2}^{k}a_{2}^{l}a_{1}^{m}-(\mu_{1}-1)a_{1}^{k}a_{1}^{l}a_{2}^{m}\right]+{\cal
O}(\epsilon^{4})$ (140)

and

$\xi_{2}=\frac{\mu_{1}\mu_{2}(1-\mu_{1})(1-\mu_{2})}{3(\mu_{1}\mu_{2}-1)^{2}}R_{iklm}\,b_{2}^{i}\\\
\times\left[(\mu_{2}-1)a_{2}^{k}a_{2}^{l}a_{1}^{m}-(\mu_{1}-1)a_{1}^{k}a_{1}^{l}a_{2}^{m}\right]+{\cal
O}(\epsilon^{4}).$ (141)

Equations (140) and (141) solve the problem of calculating the gap between
geodesics $AD$ and $BC$ (described by $\xi_{1}$ and $\xi_{2}$) expressed
through parameters $\vec{a}_{1}$, $\vec{a}_{2}$, $\mu_{1}$, $\mu_{2}$ of
figure $OABCD$, unit vectors $\vec{b}_{1}$, $\vec{b}_{2}$ perpendicular to the
figure, and Riemann curvature tensor $R_{iklm}$ at point $O$. Though we have
obtained these equations with a special choice of coordinates, they are
written in an invariant form, and therefore are valid in any coordinate frame.

It is evident from Eqs. (140) and (141) that the quantities $\xi_{1}$ and
$\xi_{2}$, that characterize the distance between geodesics $AD$ and $BC$, are
proportional to some linear combinations of components of curvature tensor
$R_{iklm}$. Measuring the gaps between geodesics $AD$ and $BC$ in different
figures $OABCD$, one can thus restore some information about tensor
$R_{iklm}$. Let us now figure out, how much information about $R_{iklm}$ can
be obtained in this way.

For this goal, it is enough to fix the values of $\mu_{1}$ and $\mu_{2}$ at

$\mu_{1}=\mu_{2}=2,$ (142)

i. e. to restrict our attention to those figures, in which
$|\overrightarrow{OA}|=|\overrightarrow{AB}|$ and
$|\overrightarrow{OC}|=|\overrightarrow{CD}|$. In this case, Eqs. (140) and
(141) are simplified to

$\xi_{1}=\frac{4}{27}R_{iklm}\,b_{1}^{i}\left(a_{2}^{k}a_{2}^{l}a_{1}^{m}-a_{1}^{k}a_{1}^{l}a_{2}^{m}\right)$
(143)

and

$\xi_{2}=\frac{4}{27}R_{iklm}\,b_{2}^{i}\left(a_{2}^{k}a_{2}^{l}a_{1}^{m}-a_{1}^{k}a_{1}^{l}a_{2}^{m}\right)$
(144)

(we have also omitted the residue term ${\cal O}(\epsilon^{4})$). In
particular, choosing a small length scale $\epsilon$ and setting

$\vec{a}_{1}=\epsilon\vec{e}_{x},\quad\vec{a}_{2}=\epsilon\vec{e}_{y},\quad\vec{b}_{1}=\vec{e}_{z},\quad\vec{b}_{2}=\vec{e}_{t},$
(145)

where $\vec{e}_{x}$, $\vec{e}_{y}$, $\vec{e}_{z}$ and $\vec{e}_{t}$ are unit
vectors along the coordinate axes, one obtains

$\xi_{1}=\frac{4\epsilon^{3}}{27}\left(R_{zyyx}-R_{zxxy}\right),\quad\xi_{2}=\frac{4\epsilon^{3}}{27}\left(R_{tyyx}-R_{txxy}\right).\quad$
(146)

Similarly, setting

$\vec{a}_{1}=\epsilon\vec{e}_{x},\quad\vec{a}_{2}=-\epsilon\vec{e}_{y},\quad\vec{b}_{1}=\vec{e}_{z},\quad\vec{b}_{2}=\vec{e}_{t},$
(147)

one can get

$\xi_{1}=\frac{4\epsilon^{3}}{27}\left(R_{zyyx}+R_{zxxy}\right),\quad\xi_{2}=\frac{4\epsilon^{3}}{27}\left(R_{tyyx}+R_{txxy}\right).\quad$
(148)

Hence, from measurement of quantities $\xi_{1}$ and $\xi_{2}$ for the two
different quadrilaterals, one can restore four components of the curvature
tensor:

$R_{zyyx},\quad R_{zxxy},\quad R_{tyyx},\quad\text{and}\;\;\;R_{txxy}.$

In the same manner, measuring other quadrilaterals, with geodesics $OAB$ and
$OCD$ directing along other coordinate axes, one can obtain all curvature
tensor components of form $R_{ikkl}$, where $i\neq k$, $i\neq l$ and $k\neq
l$. Some of these components are equal to each other due to symmetries of the
Riemann tensor, namely $R_{ikkl}=R_{lkki}$. There are 12 algebraically
independent components of this form:

$\displaystyle R_{xtty},\quad R_{xttz},\quad R_{yttz},\quad R_{txxy},\quad
R_{txxz},\quad R_{yxxz},$ $\displaystyle R_{tyyx},\quad R_{tyyz},\quad
R_{xyyz},\quad R_{tzzx},\quad R_{tzzy},\quad R_{xzzy}.$

Hence, _at least 12 independent components of curvature tensor $R_{iklm}$ can
be obtained from measuring the gaps between geodesics $OAB$ and $OCD$ in
different quadrilaterals $OABCD$._

But what about other 8 components? One can study more orientations of
quadrilaterals on the subject of extracting more information about the
curvature tensor. Instead, we will address much simpler considerations based
on group theory.

All possible values of the Riemann curvature tensor $R_{iklm}$ consist a
20-dimensional representation of the Lorentz group. It can be decomposed into
three _irreducible_ representations: a 10-dimensional one that describes
possible values of the Weyl tensor, a 9-dimensional one related to the
traceless part of the Ricci tensor, and a one-dimensional representation for
the scalar curvature [58]. Correspondingly, all 20 degrees of freedom of the
Riemann tensor consist of $10+9+1$ degrees of freedom related to the Weyl
tensor, the traceless part of the Ricci tensor, and the scalar curvature. Each
of these three pieces either can be completely restored from measurements on
different quadrilaterals, or cannot be restored at all in such a way.

Suppose that the Weyl tensor cannot be restored from measuring the gaps
between geodesics in different quadrilaterals. Then, no more than 10 degrees
of freedom of the Riemann tensor $R_{iklm}$ can be restored—namely, all 20
degrees of freedom of $R_{iklm}$ minus 10 degrees of freedom of the Weyl
tensor _at most_. But this contradicts to the above-mentioned statement that
_at least_ 12 independent components of $R_{iklm}$ can be obtained in such a
way. Hence, the argument by contradiction leads us to the conclusion that _the
Weyl tensor can be restored from measuring the gaps between geodesics in
different quadrilaterals $OABCD$._

The same argument can be applied to the traceless part of the Ricci tensor,
that leads to the following conclusion: _the traceless part of the Ricci
tensor also can be restored from measuring the gaps between geodesics in
quadrilaterals._

The only remaining question is whether the scalar curvature can be restored in
a similar way. If we suppose that it can, then the whole Riemann tensor
$R_{iklm}$ could be obtained from measuring the gaps in quadrilaterals.
Therefore we could distinguish between the flat spacetime, where $R_{iklm}=0$
and a spacetime of a constant nonzero sectional curvature (de Sitter or anti-
de Sitter), where $R_{iklm}\neq 0$, just by measuring the gaps in
quadrilaterals. But it is impossible, since there is _no gap_ between
geodesics $AD$ and $BC$, as in the flat spacetime, as in the de Sitter/anti-de
Sitter one (see Section III). Again, we come to a contradiction, and thus
prove that _the scalar curvature cannot be restored from measuring the gaps
between geodesics in quadrilaterals._

To summarize, the gap between geodesics $AD$ and $BC$ in the configuration
$OABCD$ shown in Fig. 8 (a complete quadrilateral) is characterized by two
quantities $\xi_{1}=\vec{b}_{1}\cdot\overrightarrow{E_{1}E_{2}}$ and
$\xi_{2}=\vec{b}_{2}\cdot\overrightarrow{E_{1}E_{2}}$. Equations (140) and
(141) express these quantities via parameters of the quadrilateral and the
local Riemann tensor $R_{iklm}$. By measuring $\xi_{1}$ and $\xi_{2}$ for
different quadrilaterals around point $O$, one can get 19 of 20 parameters of
the Riemann tensor at $O$: namely, 10 independent components of the Weyl
tensor and 9 independent components of the traceless part of the Ricci tensor.

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