How to detect the spacetime curvature without rulers and clocks
A. V. Nenashev, S. D. Baranovskii
TL;DR
This paper shows that it is possible to determine whether a 4-dimensional spacetime is curved or flat solely through a finite set of measurements of causality relations between events, without using rulers or clocks.
Contribution
It proves that non-conformally flat spacetimes can be distinguished from flat ones with only sixteen causal relation measurements, clarifying the finite measurement requirements for detecting curvature.
Findings
Non-conformally flat spacetimes can be identified with 16 measurements.
Causality relations suffice to determine spacetime flatness.
Finite measurement sets can verify spacetime curvature.
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.
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Taxonomy
TopicsRelativity and Gravitational Theory · Noncommutative and Quantum Gravity Theories · Quantum Mechanics and Applications
††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
Department of Physics and Material Sciences Center, Philipps-Universität Marburg, D-35032 Marburg, Germany
S. D. Baranovskii
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 that generally depends on coordinates, i. e. on radius-vector [1, §13.2]. Function (a metric) contains all information about geometry of the space. Then, if the same function 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 eV to eV. Then, it might be no physical objects with lengths between m (the currently probed length scale) and 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 and reflect where is event located with respect to the light cone of event . Event is located on the light cone of event , 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 is located either in the absolute future with respect to (i. e. information from can reach ), of in the absolute past (i. e. information from can reach ), or elsewhere (i. e. it is impossible to send information neither from to , nor from to ). 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?
In the flat Minkowski spacetime, where light travels with a constant velocity , 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 as follows:
[TABLE]
where 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 the situation when event belongs to the light cone of event (or, equivalently, belongs to the light cone of ). In a flat 4-dimensional spacetime (i. e. a Minkowski spacetime), means that
[TABLE]
where , and are Cartesian coordinates, is the time, and is the speed of light. In a curved (pseudo-Riemannian) spacetime, means that events and 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 , , , , , , and , that are all different from each other, from relations , , , , , , , , , , , , , and follows relation .
Table 1 clarifies this set or relations. The relation between and , denoted by the question mark in the table, follows from other 15 relations denoted by symbols .
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 , , , of switching lamps on precede events , , , 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 and ; then the light signals are percepted at events , , , and retranslated further from these four events; finally, retranslated signals reach events and . 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 , then well-stitchedness violates in some vicinity of .
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 that has 20 degrees of freedom. Ten of them are components of the Ricci tensor , and in an empty space they vanish according to Einstein equations. Other ten degrees of freedom are independent components of the Weyl tensor . Moreover, in an empty space, where , 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 simultaneously sends optical signals towards satellites , , and . Satellite does the same in another moment of time, so that each of satellites , , and receives signals from and from simultaneously. As soon as satellite receives the signals, it emits its own signals towards satellites and , and so do satellites , and . Satellite is positioned such that all four signals towards are received simultaneously. Similarly, satellite is set at the point where signals from , and comes at the same time. Then, to reveal the spacetime curvature, one has to detect the difference in time between the moment when signals from , and reach satellite , and the moment when the signal from reaches there.
How large is the time difference that is needed to be detected? One can apply an estimate from Subsection V.7 and find that
[TABLE]
where is a characteristic distance between the satellites, is a suitable component (or a linear combination of components) of the Weyl tensor, and is the speed of light. If the circle lies in the -plane, and satellites , , and are located at equal distances from the center of the circle, then the configuration of events is the same as used in Section V, up to relabeling. In these case, quantity in the latter equation is component , 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 from the center of a massive body as , where is the gravitational radius of the body (see [12, §100] for the Riemann tensor in spherical coordinates). Substituting this expression as into Eq. (3), we obtain
[TABLE]
In the limiting case , which corresponds to a setting shown in Fig. 1b, quantity turns into the Shapiro time delay that has the order of magnitude .
Near the Earth ( cm), Eq. (4) gives rise to an estimate . 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 , and so on (or, equivalently, to set fifteen time delays , , etc. to zero). For a solar orbit, estimate (4) with the gravitational radius of the Sun km provides the time delay of order of magnitude .
It is important to note that, even in absence of exact positioning, one can get an information about spacetime curvature from sixteen time delays . It can be done by mere replacement of in Eq. (3) with a suitable linear combination of time delays , 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 , one have to measure time delays with the same order 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 , 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 and Desargues configurations. Configurations that have this property express some geometrical theorems [14]. The configuration 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 , 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 , , and that form the so-called complete quadrilateral, shown in Fig. 3. Point , two geodesic lines and passing through , two points , on line , and two points , on line are chosen arbitrarily, with the only restriction that point lies between and , and point — between and . In the flat spacetime, where geodesics are just straight lines, five points , , , and lie in one plane, and therefore lines and must intersect at some point . The same argument is true in a spacetime of a constant curvature. But in a general curved spacetime lines and may not intersect. This occurs, for example, if geodesic line passes near a massive body that deflects this line out of plane . Therefore spacetime curvature can be detected by the observation that lines and do not intersect each other.
We will show in Appendix H that the distance between lines and , 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 , one can measure 19 of 20 degrees of freedom of the Riemann tensor at point : namely, 10 independent components of the Weyl tensor and 9 components of the traceless part of the Ricci tensor (see Appendix H). If in any quadrilateral its lines and intersect, then the above-mentioned 19 degrees of freedom vanish at each point, i. e. the spacetime is both conformally-flat () and Einstein (). 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 and 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 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 -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].
The propagators in the position space, related to lines , , …, and , diverge when the corresponding relations , , …, and 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 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 , due to Theorems 1 and 2. Hence, it is natural to suppose that contribution is a function of . Considering the expansion of 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 . One can however construct of the Weyl tensor a scalar quadratic form, namely . For this reason, one can expect that the first non-vanishing curvature-dependent term in is quadratic in :
[TABLE]
where factor accounts for the invariant volume element, being the determinant of the metric tensor. That is, the curvature-related action that arises from the diagram shown in Fig. 4 may be of the following form:
[TABLE]
where is some constant. In a system of units where , is a dimensionless quantity. The strength of the four-boson interaction is also dimensionless, and it enters into as according to the number of vertices in the diagram. Hence up to a numerical factor. In physical units, the latter relation has a form
[TABLE]
Remarkably, does not depend on the particle’s mass. Of course, a thorough calculation of constant must take renormalization of into account.
Action 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 of general relativity, (where 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 equal to 1. The interval between two points and is defined as
[TABLE]
and relation just means that . The scalar product of two contravariant vectors and is
[TABLE]
Our proof of Theorem 1 is based on three lemmas formulated below.
Lemma 3**.**
For any four points , , , , if relations , , and are fulfilled, then vector is perpendicular to vector (that is, ).
Lemma 4**.**
For any three different points , , , if relations , and are fulfilled, then these points lie on one line.
Lemma 5**.**
For any four points , , , , if relations , and are fulfilled, and points , , lie on one line and are different from each other, then this line is light-like (that is, , and ).
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 , , , , , , , be all different from each other, and 15 relations listed in Definition 1 are fulfilled. First, we consider the generic case, when points , and do not lie on one line, and also points , and do not lie on one line. The opposite case will be discussed later.
Due to Lemma 3, , , , and . Therefore any vector parallel to plane is perpendicular to any vector parallel to plane .
Since , , and , it follows from Lemma 3 that vector is perpendicular to vector . Similarly, as a consequence of relations , , and , vector is perpendicular to . Therefore, vector is perpendicular to plane and hence is parallel to plane . This means that points , , and lie on one plane. Analogous reasoning shows that points , , and lie on one plane.
Hence, all the eight points lie in two planes and , and each vector of the first plane is perpendicular to each vector in the second one.
There are three options: (a) plane contains a time-like vector; (b) plane 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 along a time-like vector lying in plane . Then, any vector in plane is perpendicular to axis . Let us choose axes and along some pair of mutually perpendicular vectors of plane . Axis is then perpendicular to plane and therefore must be parallel to plane . Planes and intersect at some point . Let us choose this point as the origin of the coordinate axes.
With this choice of coordinates, points , , and lie in the plane spanned onto axes and . Similarly, points , , and lie in the plane spanned onto axes and . That is,
[TABLE]
Condition therefore acquires the form
[TABLE]
Let us denote the distance from point to the origin of coordinates as :
[TABLE]
Then, it follows from Eqs. (18) and (19) that
[TABLE]
Similarly, conditions , , along with Eq. (19) lead to
[TABLE]
and conditions , , and Eq. (20) lead to
[TABLE]
Summarizing, we see that points , , and lie on the hyperbola
[TABLE]
and points , , and lie on the circle
[TABLE]
Such an arrangement of points is shown in Fig. 5a.
Finally, Eqs. (23) and (26) can be combined into
[TABLE]
that means . Hence, within option (a), Theorem 1 is proven.
Option (b) can be reduced to option (a) by mere relabeling , , , , 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 is defined as
[TABLE]
and plane is defined as
[TABLE]
with some parameter . In these coordinates, relation , where and , has the form
[TABLE]
Expressing here through by Eq. (30), and through by Eq. (31), one obtains
[TABLE]
Let us denote as the following quantity:
[TABLE]
Using Eq. (33), one can easily deduce from relations and and Eq. (34) that
[TABLE]
Then, it follows from relation and Eq. (35) that
[TABLE]
Points , , , lie on a parabola , and points , , , — on a parabola , see Fig. 5c.
Left-hand sides of Eqs. (37) and (36) are equal to each other:
[TABLE]
According to Eq. (33), this means that relation is fulfilled in option (c). Q.E.D.
We have completed the proof in the generic case, when points , and do not lie on one line, and also points , and also do not lie on one line. What remains is to prove Theorem 1 in the opposite case. Let, for definiteness, points , and lie on the same line . Applying Lemma 5 with , , and , one can ensure that line is light-like, and that relations , and are satisfied. Applying Lemma 4 five times: with , , , with , , , with , , , with , , , and then with , , , one can see that all eight points and lie on the same light-like line (see Fig. 5d). Since points and are connected by a light-like line, then relation 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 . The Weyl tensor is a traceless part of Riemann curvature tensor . By definition, Weyl tensor is equal to [12, §92]
[TABLE]
where is the Ricci tensor, and is the scalar curvature.
For convenience, we set the speed of light 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 . In these coordinates, metric tensor at the origin is the same as that of the special relativity, and the first derivatives of vanish at the same point:
[TABLE]
where
[TABLE]
The second derivatives of are determined by the curvature tensor , such that [56]
[TABLE]
where is the Landau little-o symbol, and .
Additionally, we demand that
[TABLE]
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
In a flat spacetime, relation means that the interval between points and 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 along the geodesic line that connects two given points. Let us define interval between points and as
[TABLE]
where integration is performed over the geodesic line segment between and , is a parameter along this geodesic line, are coordinates of a point on the geodesic line, and two signs are equal to each other and chosen such that the expression under the square root is non-negative.
The function defined by Eq. (45) is nothing else but Synge’s world function [2] multiplied by 2. If the geodesic line that connects and is timelike, than , where is the proper time passed between events and along the geodesic. If this geodesic line is spacelike, than is the squared length of the geodesic line segment between and . And if this geodesic line is lightlike, than .
Hence, if and only if relation is fulfilled.
In the Minkowski space, where , quantity is the usual special-relativity interval:
[TABLE]
In a curved spacetime that is described by Eq. (43), there must be corrections to Eq. (46). Let us define -neighborhood of point as a set of such points that
[TABLE]
If both points and belong to the -neighborhood of point , then [56]
[TABLE]
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 , on the one side, and the Weyl tensor, on the other side. Let us first express component of the Weyl tensor at point through the Riemann curvature tensor , using Eqs. (39), (40), and the skew symmetry properties :
[TABLE]
Then we choose a spatial scale , small enough such that expansion (48) makes sense, and consider a typical example of eight points (events) – that obey sixteen relations in the Minkowski space:
[TABLE]
where coordinates of the points are listed in order . This set of points is depicted in Fig. 6, and is a particular case of the arrangement shown in Fig. 5a.
Points , , , and lie on hyperbola
[TABLE]
whereas points , , , and lie on circle
[TABLE]
in accordance with Eqs. (27) and (28) of Section IV. Let be one of points , and be one of points . Then, due to Eqs. (51) and (52),
[TABLE]
that is, the first term in the right-hand side of expansion (48) vanishes. Hence, due to Eq. (48), the interval is proportional to the curvature tensor , up to the residual term . This fact can be expressed as follows:
[TABLE]
where tensor is the coefficient at in Eq. (48):
[TABLE]
Let us consider the following combination of sixteen intervals , where , and :
[TABLE]
The choice of signs in this definition enables a remarkable property of stability of under small variations of points – . We will establish this property in subsection V.4.
It is evident from Eqs. (54) and (56) that quantity is also proportional to , up to the residual term:
[TABLE]
where is a combination of tensors :
[TABLE]
(we have omitted tensor superscripts “” in Eq. (58) for brevity).
Tensor 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 , , , , and , i. e. times. We have done this with the aid of a computer (the Matlab code is presented in Appendix F), and obtained the following results:
[TABLE]
and all other components of tensor are equal to zero. Substituting these expressions for the components of into Eq. (57), one can find that
[TABLE]
that can be simplified using the permutation symmetry :
[TABLE]
It is clearly seen from Eq. (49) that the expression in square brackets is equal to . Hence, the linear combination of intervals is expressed through the Weyl tensor component :
[TABLE]
V.4 Stability of combination under small shifts of points –
The aim of this subsection is to find out how quantity changes under small variations of points . Let us choose eight points located near points so that the difference between coordinates of the corresponding points is small with respect to :
[TABLE]
and so on. Then, we replace points with their corresponding points in definition (56) of quantity , and denote the result of this replacement as :
[TABLE]
In this subsection, we will show that the difference between and is small in comparison with .
Let be any of points , and be any of points . Let us find the difference between intervals and 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 . Therefore a change of this term, due to a small shift of points and , must be small with respect to . Then, let us consider the first term in the rhs of Eq. (48). Essentially, this term is a Minkowski-space scalar product . Replacing and with and changes this term by amount of
[TABLE]
where
[TABLE]
According to our choice of points and , each component of vectors and is small compared to , see Eq. (64). For this reason, the last term in Eq. (66) is as small as . Therefore, the difference between intervals and is determined only by the first term in the rhs of Eq. (66), up to :
[TABLE]
The difference between quantities , Eq. (65), and , Eq. (56), is a combination of expressions (68), taken with appropriate signs. After a simple algebra, one can represent this difference as follows:
[TABLE]
Finally, looking at definitions of points – , Eqs. (50a) – (50h), one can see that
[TABLE]
Substitution of these relations into Eq. (69) gives rise to a simple result:
[TABLE]
Remarkably, all terms in the difference cancel each other, except for the residual term . This cancellation became possible as a result of the special choice of signs in the definition of quantity .
Hence, a shift of points – by distances results in as small change of quantity as .
V.5 Making fifteen intervals equal to zero
Generally, all sixteen intervals , that contribute to , 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 – . This will be done in three steps described below. Shifted points are denoted as – .
Step 1. We do not shift points , , and . That is, , , and .
Step 2. For each of points , , and , we leave one of coordinates unchanged, namely , , and . Then we adjust other three coordinates such that three intervals between the given point and points , and vanish. Therefore we obtain shifted points , , and that obey relations
[TABLE]
Step 3. For point , we fix coordinate , and adjust other three coordinates such that three intervals between this point and points , and vanish:
[TABLE]
where is the shifted point .
As shown in Appendix G, differences between initial points – and their corresponding shifted points – do not exceed 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 . In Subsection V.1, we have chosen a local inertial (at point ) reference frame, in which component of the Weyl tensor is nonzero at . In Subsection V.2, we have introduced such a definition of interval between two points and in a curved spacetime, that if and only if relation is fulfilled. In Subsection V.3, we have defined eight points – within the -neighborhood of point . Then we have found a linear combination of intervals between these points, which is proportional to up to terms . In Subsection V.4, we have shown that this combination remains unchanged under small () variations of coordinates of points – , up to . Finally, we have found in Subsection V.5 such a variation of coordinates that enable to set fifteen intervals to zero.
Collected together, these results consist a proof of Theorem 2. To see this, let us choose a small spatial scale , then define eight points – in the -neighborhood around point according to Subsection V.3, and find shifted points – by the procedure described in Subsection V.5. As a result, all intervals, that define quantity in Eq. (65), vanish, except for . Therefore
[TABLE]
Taking into account stability of under small shifts, Eq. (71), one can see that
[TABLE]
Then, expressing quantity through the Weyl tensor according to Eq. (63), one obtains
[TABLE]
In the chosen reference frame, , see Subsection V.1. When spatial scale is sufficiently small, the term containing exceeds (in absolute value) the residual term in Eq. (76). Therefore, for such value of ,
[TABLE]
The set of points – obeys Eqs. (72) and (73). According to Subsection V.2, this means that fifteen relations
[TABLE]
are fulfilled.
If the spacetime were well-stitched, relation would follow from relations (78). But Eq. (77) means that relation is not fulfilled. Thus, points – violate well-stitchedness.
In conclusion, provided that the Weyl tensor differs from zero at some point , there are eight points – 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 , which is a measure of violation of relation . Inequality means that a light signal emitted at event does not pass through event . We denote as the time delay of this signal, i. e. the difference between the moment of time, when this signal reaches point , and the moment of time of event itself. We are going to find the value of this time delay.
For the sake of generality, let us consider two events and , and denote as the event, in which a light signal from reaches the spatial point of event (see Fig. 7). One can make the reference frame locally-inertial (i. e. set Christoffel symbols to zero) along line by a small distortion of coordinates [12, §85]. In such a frame, the light travels from to with the constant velocity , whence
[TABLE]
where and are differences between and in space and in time, correspondingly. Interval is expressed in this frame just as in special relativity:
[TABLE]
Substituting Eq. (79) into Eq. (80) and neglecting the higher-order infinitesimal , one can express time delay as follows (cf. Ref. [57]):
[TABLE]
Taking from Eq. (76) and from Eq. (50), one can obtain with the help of Eq. (81) that
[TABLE]
This is the time delay, detection of which reveals the spacetime curvature.
Appendix A Proof of Lemma 3
Lemma 3. For any four points , , , , if relations , , and are fulfilled, then vector is perpendicular to vector (that is, ).
Proof. Let , , and be radius-vectors of points , , and , correspondingly. Then, , and . Relations , , and mean that
[TABLE]
Summing up Eqs. (84) and (85), and subtracting Eqs. (83) and (86) from them, and expanding all brackets, one can get
[TABLE]
that is,
[TABLE]
Q.E.D.
Appendix B Proof of Lemma 4
Lemma 4. For any three different points , , , if relations , and are fulfilled, then these points lie on one line.
Proof. Let us choose the coordinate system with the origin at point . It follows from relation that point lies on the light cone with the vertex at :
[TABLE]
Let us turn the axes , , such that point lie in plane . Therefore, according to Eq. (89),
[TABLE]
[Another option is , but in this case we will flip the direction of axis and return to Eq. (90).] Due to relations and , point must lie on both light cones—with vertices at and :
[TABLE]
Subtracting Eq. (91) from Eq. (92) yields
[TABLE]
i. e. . Substituting into Eq. (91), one can find that .
Hence, all three points , , belong to the same (light-like) line, which equation is
[TABLE]
Q.E.D.
Appendix C Proof of Lemma 5
Lemma 5. For any four points , , , , if relations , and are fulfilled, and points , , lie on one line and are different from each other, then this line is light-like (that is, , and ).
Proof. The line, that joins points , and , 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 -axis. With this choice of coordinates, relations and obtain the following form:
[TABLE]
Therefore . Taking into account that (since points and are different), one can resolve the latter equality as
[TABLE]
Similarly, one can deduce from relations and that
[TABLE]
It is clearly seen from Eqs. (97) and (98) that . But this contradicts to the proposition that points and are different. Consequently, the line that passes through , and cannot be time-like.
Next, we suppose that this line is space-like. Let us choose this line as -axis. Then, almost the same reasoning, as in the case of a time-like line, leads from relations , and to equations
[TABLE]
and to conclusion that . And again, this contradicts to the proposition that and are different. This means that the line that connects points , , 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 nor plane contains a time-like vector
Let and be 2-dimensional planes in 4-dimensional Minkowski space, and any vector in is orthogonal to any vector in . (In the main text, is plane , and is plane .) We suppose also that neither nor contains a time-like vector. The aim of this Appendix is to show that, in a properly chosen coordinate frame, planes and are defined by Eqs. (30) and (31), respectively.
Let us choose two non-collinear, mutually perpendicular vectors and in plane . 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 and along vectors and , correspondingly. With this choice, plane is parallel to -plane, and consequently any vector of plane is perpendicular to -plane, i. e. parallel to -plane. Therefore plane is parallel to plane . But it contradicts to the supposition that plane does not contain a time-like vector.
Case (ii) contradicts to the choice of vectors and 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 be a light-like vector, and be a space-like one. One can choose the coordinate axes such that vector lie in -plane, and its - and -components be the same:
[TABLE]
It follows from Eq. (101) and from orthogonality of vectors and that . By rotation of -part of the coordinate system, one can put -component of vector to zero, while components of vector remain unchanged. After this, components of vector obeys the following conditions:
[TABLE]
(the latter inequality follows from non-collinearity of vectors and ). Equations (101) and (102) show that vectors and are linear combinations of the following two vectors:
[TABLE]
where the components are listed in order . Therefore, radius-vector of an arbitrary point on plane can be expressed as
[TABLE]
where and are real parameters that vary from point to point, and is a radius-vector of some fixed point on . Similarly, radius-vector of an arbitrary point on plane depends on two real parameters and as follows:
[TABLE]
where is a radius-vector of some fixed point on plane , and each of two non-collinear vectors and is perpendicular to both and . It is convenient to choose vectors and as
[TABLE]
One can easily check that .
Now we will choose points and . First, we take randomly chosen points of planes and as and . Then we shift point along -axis (i. e. along ), until its -coordinate becomes equal to the -coordinate of point . Next, we shift point along -axis (i. e. along ), until its -coordinate becomes equal to that of point . And finally, we shift point along vector , until its -coordinate becomes equal to that of point . In the end of these manipulations, points and differ from each other only by their -coordinates. Let us put the origin of coordinates to point . Therefore
[TABLE]
where is the difference between -coordinates of these points.
One can see from Eqs. (103), (104) and (107) that coordinates of any point on plane are
[TABLE]
where and are real parameters. This means that plane is defined by Eq. (30).
Similarly, Eqs. (105), (106) and (107) express the coordinates of any point on plane through real parameters and :
[TABLE]
that defines plane according to Eq. (31).
Hence, equations (30) and (31) for planes and are established.
Appendix E Possibility of choosing a reference frame, in which
In this Appendix, we consider Weyl tensor at some given point . We suppose that the reference frame is local inertial at this point, and therefore the metric tensor is equal to , i. e. to the metric tensor of special relativity.
All possible realizations of the Weyl tensor consist an irreducible representation of the Lorentz group [58]. Suppose that there are such non-zero realizations of the Weyl tensor, in which its component is equal to zero, and remains zero after any Lorentz transformation. A set of such realizations is also some representation of the Lorentz group. By definition . Also , since a realization with belongs to but not to . But this contradicts to irreducibility of .
It is therefore proven by contradiction, that any non-zero Weyl tensor can be such Lorentz-transformed that inequality will hold after the transformation.
Appendix F MATLAB code for calculation of tensor
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 – and points –
In Subsection V.5, we suggested a method of finding such a set of “shifted” points – that fifteen intervals listed in Eqs. (72) and (73) vanish. Here we discuss how close each “shifted” point is to its “unshifted” counterpart – .
For points , and , this question is trivial because they coincide with , and , correspondingly.
Let us consider point . Its position is defined by a system of three equations
[TABLE]
with three unknowns—namely, three coordinates , and of point . The fourth coordinate is fixed:
[TABLE]
Equations (110) are nonlinear, but can be approximately linearized when the deviation of point from is small in comparison with the spatial scale . Omitting the residual (nonlinear) term in Eq. (68), and taking into account that , one can get the following expressions for intervals , and :
[TABLE]
Substituting Eqs. (111) – (114) into (110), we obtain a system of linearized equations. It is convenient to represent this system in a matrix form:
[TABLE]
where is a matrix of coefficients:
[TABLE]
Matrix is non-degenerate, and therefore system of equations (115) is consistent. Its solution reads:
[TABLE]
The right-hand side of the latter equation is proportional to . Indeed, coefficients of the inverse matrix are proportional to . Intervals , and are proportional to , that can be seen in Eq. (48). Hence,
[TABLE]
Similar considerations for point lead to a linearized system of three equations with three unknowns , and . This system can be written in the same form as Eq. (115), where are replaced with , and matrix is slightly different:
[TABLE]
As a result, one can get the same estimates for coordinate differences between points and as in Eq. (118).
For points and , the analysis is exactly the same as for points and , up to replacements , , , , and . It leads to the conclusion that points , , , differ from their corresponding “unshifted” points , , , by at most along each coordinate.
The last point to consider is . This point is defined by the system of three equations
[TABLE]
with respect to three unknowns—coordinates , and , whereas the fourth coordinate is fixed: . One can express these intervals via Eq. (68), neglecting the nonlinear residual term :
[TABLE]
These expressions provide a linearized form of the system of equations (120):
[TABLE]
where matrix is defined as follows:
[TABLE]
and , , denote the constant terms:
[TABLE]
In the right-hand side of Eq. (123), vector is proportional to , see definitions (50d) and (50e); vector is proportional to due to Eq. (118); and interval is proportional to according to Eq. (48). Therefore , and the same estimates are true for quantities and .
The solution of system (121) is
[TABLE]
Here, matrix elements of are proportional to , and terms , , are proportional to . Consequently the left-hand side of Eq. (126) is proportional to .
Hence, deviations of “shifted” points – from “unshifted” ones – do not exceed .
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: , , and . If the spacetime were flat, geodesics and would cross each other at some point (event) . In a curved spacetime, however, there may be a gap between lines and . In this appendix, we address the following questions:
• How to describe the size of the gap between lines and ?
• 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 , , , , )?
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 on geodesic , and a point on geodesic , both points near . Then, the gap between geodesics and is determined by vector . To exclude ambiguity of the choice of points and , this vector should be projected onto the plane perpendicular to both geodesics and .
We therefore choose two linearly independent unit vectors and , both of them perpendicular to geodesics and , and consider quantities
[TABLE]
They completely quantify the distance between the geodesics and .
It is convenient to calculate and in the Riemann normal coordinates [1, §11.6] with the origin at point . The metric tensor in these coordinates has the form of Eq. (43), and geodesics and (that pass through the origin) are simply straight lines. If points and lie within a small -neighborhood of point , then the segment of the geodesic line that joins two points and obeys the following parametric equation in the normal coordinates [56]:
[TABLE]
where parameter is equal to 0 at and 1 at .
Before applying Eq. (128), we slightly modify it with the aid of skew symmetry () of the Riemann tensor. First, we notice that due to this skew symmetry, and hence the term can be omitted in Eq. (128):
[TABLE]
The term can be further transformed to by swapping indices and and employing the skew symmetry:
[TABLE]
Later, we assume that the whole figure belongs to the -neighborhood of point with a small size , and neglect the residue term .
We describe figure by two vectors , (see Fig. 8) and two numbers , :
[TABLE]
so that radius-vectors of points , , , are
[TABLE]
Point lies at the intersection of straight lines and . Hence, its radius-vector expresses as
[TABLE]
where and are some numbers. Substituting here , , and from Eq. (132), and collecting the terms with and separately, one can obtain two equations for and :
[TABLE]
Solution of these equations is
[TABLE]
Coordinates of point can be obtained by substitution of , and as , and into Eq. (130). It is evident from comparison with Eq. (133) that the first term in the resulting expression is equal to . Hence,
[TABLE]
Similarly, coordinates of point can be obtained by substitution of , and as , and into Eq. (130):
[TABLE]
The prefactors before in equations (136) and (137) are equal to each other, that can be easily seen by expressing and via Eq. (134):
[TABLE]
This fact simplifies calculation of vector and quantities
[TABLE]
Collecting Eqs. (136) – (139) together, and taking values of and from Eq. (135), one can find and as
[TABLE]
and
[TABLE]
Equations (140) and (141) solve the problem of calculating the gap between geodesics and (described by and ) expressed through parameters , , , of figure , unit vectors , perpendicular to the figure, and Riemann curvature tensor at point . 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 and , that characterize the distance between geodesics and , are proportional to some linear combinations of components of curvature tensor . Measuring the gaps between geodesics and in different figures , one can thus restore some information about tensor . Let us now figure out, how much information about can be obtained in this way.
For this goal, it is enough to fix the values of and at
[TABLE]
i. e. to restrict our attention to those figures, in which and . In this case, Eqs. (140) and (141) are simplified to
[TABLE]
and
[TABLE]
(we have also omitted the residue term ). In particular, choosing a small length scale and setting
[TABLE]
where , , and are unit vectors along the coordinate axes, one obtains
[TABLE]
Similarly, setting
[TABLE]
one can get
[TABLE]
Hence, from measurement of quantities and for the two different quadrilaterals, one can restore four components of the curvature tensor:
[TABLE]
In the same manner, measuring other quadrilaterals, with geodesics and directing along other coordinate axes, one can obtain all curvature tensor components of form , where , and . Some of these components are equal to each other due to symmetries of the Riemann tensor, namely . There are 12 algebraically independent components of this form:
[TABLE]
Hence, at least 12 independent components of curvature tensor can be obtained from measuring the gaps between geodesics and in different quadrilaterals .
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 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 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 can be restored—namely, all 20 degrees of freedom of 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 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 .
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 could be obtained from measuring the gaps in quadrilaterals. Therefore we could distinguish between the flat spacetime, where and a spacetime of a constant nonzero sectional curvature (de Sitter or anti-de Sitter), where , just by measuring the gaps in quadrilaterals. But it is impossible, since there is no gap between geodesics and , 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 and in the configuration shown in Fig. 8 (a complete quadrilateral) is characterized by two quantities and . Equations (140) and (141) express these quantities via parameters of the quadrilateral and the local Riemann tensor . By measuring and for different quadrilaterals around point , one can get 19 of 20 parameters of the Riemann tensor at : namely, 10 independent components of the Weyl tensor and 9 independent components of the traceless part of the Ricci tensor.
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