Newton's time and space reference system
Under the framework of Newtonian mechanics, time and space are independent of each other. Time passes evenly and space is uniform everywhere which satisfies Euclidean geometry. Under this assumption, the space–time reference system can be simply understood as a Three-Dimensional (3D) space frame and a clock carried by an observer. The former is used as a space reference for direction and distance, and the later as a time reference. A space frame is somewhat alike an electronic total station that can measure distance and direction. Due to the flatness of space, the measurement range of the space frame has no limits theoretically. Therefore, the space–time reference system can be defined by a time scale and a 3D space reference system. A time scale is composed of a starting point and the time unit, which has nothing to do with the choice of space position and coordinates. A space reference system is defined by its origin, orientation of axes and scale or unit of distance. For convenience, it is usually required that the three axes are orthogonal to each other and have the same coordinate unit. There is a special kind of space reference systems in Newtonian mechanics, called inertial reference systems, in which Newton's law of inertia are satisfied. The laws of physics have the same form of expression in all inertial reference systems, which are called Galileo's principle of relativity.
Practically, a complete space reference system needs to define not only a reference frame, but also some geometric parameters and related physical field parameters, such as the reference ellipsoid, the geocentric gravitational constant and gravitational model in the terrestrial reference system, etc..
The concepts of relativistic space–time reference system
Compared with Newtonian mechanics, the concepts of relativistic space–time reference system are much more complicated. First, the physical time and space are not absolutely independent. They are interrelated and cannot be completely separated. In fact, whether the objects in the universe are moving or static, whether the moving speed is large or small, and whether the motion path is a straight or a curve are all determined by the observer or the frame of reference. The space point of one observer may be a line in the eyes of another. What one sees is a straight line may be a curve seen by another. The spatial points, straight lines, curves, even planes and curved surfaces in Euclidean geometry do not have complete objectivity. The distance between two different events in the universe is closely related to the observer or the frame of reference. For different spatial points, the observer's definition of simultaneity is related to distance, and the relativity of distance will inevitably lead to the correlation between time and space. For any two events that occur in the universe, the time intervals and spatial distances measured by observers moving relatively are different, even if they use same clocks and rulers. Therefore, the theory of relativity considers time and space as a whole and supposes that the space-time interval between two events is an invariant quantity that has nothing to do with the observers (Han, 2017). Under this assumption, the coordinate relationship between the two relative moving inertial reference systems no longer satisfies the classic Galileo transformation, but the Lorentz transformation. All the laws of physics remain unchanged under the Lorentz transformation, which is called the principle of special relativity.
Secondly, space–time has non-uniformity or non-Euclidean characteristics. There is a universal interaction among all objects in the universe. The uneven distribution of matter will inevitably lead to the unevenness of the space–time gravitational field, and hence there is no objective straight line in a large-scale space. We know that the basic postulate of Euclidean geometry is straight line, so Euclidean geometry does not hold in a large-scale space. The general theory of relativity considers that space-time including the gravitational field is a curved four-dimensional pseudo-Riemann Space. Therefore, it is impossible to construct a Cartesian coordinate system with a large-scale spatial coverage. Space–time is the basic form of material existence. The vacuum or space without matter is only the result of artificial abstraction, and space–time itself does not have the characteristics of straight or bending. Fundamentally, the curvature of space–time is just the result of gravitation geometrization in general relativity. Therefore, it is easier to be understood if saying that space-time is inhomogeneous rather than that curved (Han, 2017).
The inhomogeneity of space–time also leads to no ideal inertial space in our universe. Both inertia and gravitation are the result of the interaction of substance in the universe. It is impossible to separate them completely. In studying dynamic and kinematic problems, we cannot take all the celestial bodies into account. An effective way is to separate them into two groups, i.e., the near celestial bodies and the far distant ones. The effect of the former is called gravitation, and that of the later is the inertia, which is the so-called Mach principle. Therefore, both gravitational field and inertial space are relative. Inertial space is not only local but also approximate. The spatial scope of application of the inertia depends on remoteness of the celestial bodies that forms the inertial effect, as well as our requirements for the accuracy of space–time measurement.
The benchmark of space–time metric in relativity is essentially the light (Han, 1997). For two determined events, the time intervals measured by observers at different spatial positions are different, even they are relatively static and carry the same ideal atomic clocks which are always consistent with the SI second. Therefore, we believe that the gravitational field will change the frequency or clock speed of an atomic clock. Due to the local flatness of space, the concept of space frame used in Newtonian mechanics is still applicable, but the difference is that it can only be used locally by the observer and cannot extend outward infinitely.
A relativistic space-time reference system consists of a Four-Dimensional (4D) coordinate system and the corresponding metric coefficients. It maps the space-time points, one by one, to the Minkowski Space in which one dimension is imaginary and other three are real. Therefore every event occurred in the universe has a set of clear and unique space–time coordinates, while the trajectory of any object and the measurement characteristics of space–time are determined by the space–time metric. The space–time metric is a second-order symmetric tensor field, which is determined by the matter distribution of space–time and satisfies the Einstein field equation. The metric tensor has 10 independent coordinate components, which are the so called metric coefficients. Obviously, the metric coefficients are the functions of space–time points and closely related to the basis vectors of coordinates. Different basis vectors lead to different metric coefficients. The Einstein field equation has 6 independent nonlinear equations. To solve for 10 metric coefficients, 4 coordinate conditions are required. In the theory of relativity, the coordinate conditions can be arbitrarily selected. Then the space–time coordinates in general relativity have no clear physical or geometric meaning, but arbitrariness and equivalence. Therefore, the coordinates of a space–time point in the gravitational field depend not only on the space–time reference frame located at the coordinate origin, but also on the space–time metric or the coordinate conditions.
Local inertial reference system
According to the principle of equivalence of relativity, for any mass point as an observer that moves freely in space–time, an inertial space reference frame or a local inertial reference system that is applicable to the observer local space–time can be constructed. The local inertial reference system, which may also be called local Lorentz reference frame, meets the following basic conditions:
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The coordinate origin is a mass point freely moving in the space–time.
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The time reference is the reading of the atomic clock coming with the origin.
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The space axes or the coordinate base vectors do not rotate relative to inertial gyroscopes.
Note that the reason why inertial gyroscopes are used here instead of distant celestial bodies to define the non-rotating characteristics of inertial space is that the influences of non-far-distant celestial bodies need to be taken into account for the local inertial space. The spin of a gyro will undergo a so-called de Sitter precession relative to the far-distant celestial bodies, which is also named as geodesic precession.
There is no doubt that the gravitation forces acting on the particles near the origin are almost the same, and free particles move in a uniform form in the eyes of the observer located at the origin. Therefore, we need no gravitation but inertia to describe the particle motions in the local space, in which the tidal forces generated by external celestial bodies can be ignored. However, the local Lorentz reference frame satisfies the Newtonian inertia condition is just a differential approximation in mathematics, and its spatial application range is very limited. Due to the non-uniform nature of the gravitational field, there is no true Newtonian inertial space in the real large-scale space–time.
For a curved space, if the coordinate basis vectors \(\left\{ {e_{\alpha } } \right\}\) of a coordinate system \(\left\{ {x^{\alpha } } \right\}\) are orthogonal to a certain space-time point \(P\left( {x_{A}^{\lambda } } \right)\), and its affine connection coefficients are zero, then the reference frame formed by the coordinate basis vectors of the point is a Lorenz reference frame. Therefore, it forms a local inertial coordinate system nearby. The basic conditions can be expressed as:
$$\left\{ \begin{gathered} g_{\mu \nu } \left( {x_{A}^{\lambda } } \right) = \eta_{\mu \nu } \hfill \\ \varGamma_{\alpha \beta }^{\mu } \left( {x_{A}^{\lambda } } \right) = 0 \hfill \\ \end{gathered} \right.$$
(1)
where \(g_{\mu \nu } \left( {x_{A}^{\lambda } } \right)\) are the common metric coefficients, ημν are the Minkowski ones, \(\varGamma_{\alpha \beta }^{\mu } \left( {x_{A}^{\lambda } } \right)\) are the affine coefficients of connection or expressed in vectors:
$$\left\{ \begin{gathered} {\varvec{e}}_{\mu } \left( {x_{A}^{\lambda } } \right) \cdot {\varvec{e}}_{\nu } \left( {x_{A}^{\lambda } } \right){ = }\eta_{{_{\mu \nu } }} \hfill \\{\varvec{\varGamma}}_{\mu \nu } \left( {x_{A}^{\lambda } } \right) = 0 \hfill \\ \end{gathered} \right.$$
(2)
The first equation of Eq. (1) or Eq. (2) is is the orthogonal normalization condition of the coordinate basis vectors. Although orthogonal normalization is not a necessary condition for the inertial reference system, the Cartesian coordinate system has natural application advantages. Therefore, when establishing a reference system, we always hope that the coordinate bases can meet the condition of orthogonal normalization. The second condition equation is the core of the local inertial system, which requires the coordinate basis vectors to satisfy the characteristics of parallel movement at the origin. In other words, the time axis of the local inertial system is a time-like geodesic, and the space coordinate axes near the origin are space-like geodesic lines.
The coordinates that satisfy the geodesic condition are called Fermi coordinates, so the local inertial system is an orthogonal Fermi coordinate system or Fermi normal coordinates. Due to the in-homogeneity of space–time, the local inertial system of a space–time point is limited not only in space, but also in time. In practice, we often need a local inertial system that is not limited in time, such as a local inertial system centered at a spacecraft. For such a local reference system, the scope of adaptation is not a sphere but a pipeline in the 4D space–time. This unrestricted local inertial system in time is the local Lorentz reference frame for a free observer.
It is very convenient to use the observer's local inertial system to express the events that occur in the space near the observer. But in most cases, a global coordinate system must be used which covers the entire space–time range to describe the movement of substance in a large-scale space. Therefore, it is often necessary to give the transformation relationship between the local inertial system and the global coordinate system.
According to Eqs. (1) or (2), the relationship between the local inertial system \(\{ x^{\prime \alpha } \}\) and the global coordinate system \(\{ x^{\alpha } \}\) can be expressed as:
$$x^{\mu } = x_{A}^{\mu } \left( {t^{\prime } } \right) + e_{j}^{\mu } x^{\prime j} - \frac{1}{2}\varGamma_{\lambda \gamma }^{\mu } \left( {x_{A}^{\kappa } } \right)e_{j}^{\lambda } e_{k}^{\gamma } x^{\prime j} x^{\prime k} + \cdots$$
(3)
or
$$\left\{ \begin{gathered} t = \int {e_{0}^{0} {\text{d}}t^{\prime } } + \frac{1}{c}e_{j}^{0} x^{\prime j} - \frac{1}{2c}\varGamma_{\lambda \gamma }^{0} \left( {x_{A}^{\kappa } } \right)e_{k}^{\lambda } e_{j}^{\gamma } x^{\prime j} x^{\prime k} + \cdots \hfill \\ x^{i} = x_{A}^{i} \left( {t^{\prime } } \right) + e_{j}^{i} x^{\prime j} - \frac{1}{2}\varGamma_{\lambda \gamma }^{i} \left( {x_{A}^{\kappa } } \right)e_{k}^{\lambda } e_{j}^{\gamma } x^{\prime j} x^{\prime k} + \cdots \hfill \\ \end{gathered} \right.$$
(4)
It can be seen from the coordinate relationship Eq. (4) that the relationship is nonlinear between the local inertial system and the global coordinate system. Since the coordinate relationship is developed approximately by using the Taylor series of \(1/c\), the applicable scope of the local inertial system is determined by the convergence of the series.
The barycentric celestial reference system
In astronomy, the mass center of the celestial body or system under study is generally chosen as the origin of the space–time reference system, and its coordinate axes are required to have spatial non-rotating characteristics. The so-called non-rotating has two meanings. One is that the space coordinate axes have no spatial rotation relative to a far-distant celestial body such as the extragalactic radio sources, which is named as the kinematical non-rotating, and the other is that they are relative to gyro or the inertial space and named as the dynamical non-rotating (Han, 1997).
For an isolated celestial system, the kinematic non-rotation and the dynamic non-rotation are equivalent. However, for non-isolated systems, due to the influence of local substance on space–time there will be a slight difference between them. For example, there is a very slow spatial rotation between the geocentric local inertial frame and the geocentric kinematic non-rotating reference frame, which is about 1.92 arc seconds per hundred years and named as geodesic precession. Because of the small dynamical effect on the motion of objects, it can be ignored in the usual cases.
In modern astrometry and space geodesy, there are three most important space–time reference systems, i.e., the Barycentric Celestial Reference System (BCRS), the Geocentric Celestial Reference System (GCRS) and the Geocentric Terrestrial Reference System (GTRS). The origin of BCRS is the mass center of the Solar System, which takes into account the distributions of all the masses of the Sun and the planets, and the coordinate axes are required to have no spatial rotation relative to far-distant celestial bodies. It is mainly used to study the orbital motion of celestial bodies of the solar system and the observation modeling of distant celestial bodies. The coordinate origin of GCRS is defined at the center of Earth’s mass and the spatial axes have no spatial rotation relative to BCRS. GCRS is mainly used to study the rotation of the Earth and the orbital motion of artificial Earth satellites. The coordinate origin of GTRS is the same as GCRS, but the space coordinate axes are fixed to the Earth and rotate daily with it, which is mainly used to describe the locations of ground stations and various geophysical phenomena.
Obviously, there exists arbitrariness in the definition and implementation of these reference systems. If there were no standards, the results of observation or research given by different teams could not be compared, communicated or shared. To this end, international organizations, such as the International Astronomical Union (IAU), the International Union of Geodesy and Geophysics (IUGG), and the International Bureau of Weights and Measures (BIPM) have long term commitments in the definition, implementation, and coordination of the recommendations for the space–time reference system and the related physical constants.
According to IAU2000 resolution B1.3, both the BCRS and the GCRS are required to meet the harmonic conditions (Petit, 2000; Soffel et al., 2003). The BCRS space–time metric form can be expressed as:
$$\left\{ \begin{gathered} g_{00} = - \left( {1 - \frac{2w}{{c^{2} }} + \frac{{2w^{2} }}{{c^{4} }}} \right) + O(c^{ - 6} ) \hfill \\ g_{0i} = - \frac{{4{{w}}^{i} }}{{c^{3} }} + O(c^{ - 5} ) \hfill \\ g_{ij} = \delta_{ij} \left( {1 + \frac{2w}{{c^{2} }}} \right) + O(c^{ - 4} ) \hfill \\ \end{gathered} \right.$$
(5)
where δij is Kronecker delta, w and wi are the Newtonian and vector potentials of the gravitational field respectively. Where potential functions
$$\left\{ \begin{gathered} w(t,x^{j} ) = G\int {\frac{{\sigma (t,{\varvec{x}}^{\prime } )}}{{\left| {{\varvec{x}} - {\varvec{x}}^{\prime } } \right|}}} {\text{d}}^{3} x^{\prime } + \frac{G}{{2c^{2} }}\int {\frac{{\partial^{2} }}{{\partial t^{2} }}\sigma (t,{\varvec{x}}^{\prime } )} \left| {{\varvec{x}} - {\varvec{x}}^{\prime } } \right|{\text{d}}^{3} x^{\prime } \hfill \\ {{w}}^{{\mathbf{i}}} (t,x^{j} ) = G\int {\frac{{\sigma^{i} (t,{\varvec{x}}^{\prime } )}}{{\left| {{\varvec{x}} - {\varvec{x}}^{\prime } } \right|}}} {\text{d}}^{3} x^{\prime } \hfill \\ \end{gathered} \right.$$
(6)
here \(t = {\text{TCB}}\), called barycentric coordinate time, and \(\sigma\) and \(\sigma^{i}\) denote mass density and flow density respectively, G is the gravitational constant. Obviously, the potential functions of the metric are zero at infinity (Deng, 2012).
The BCRS can be regarded as a very good inertial reference system in dynamics. The stars outside the solar system are very far away, and the tidal effect generated by them is negligible in the solar system. Therefore, it is not difficult to imagine that if the observer was at the barycenter of the solar system and moved together without rotation with respect to the far-distant celestial bodies, he would have a very flat space, apart from the interaction of the celestial bodies in the solar system, and the time given by the carried atomic clock would be also very uniform. Thus, we can regard the solar system as an isolated system and the BCRS as a space–time reference system with good inertia characteristics and orthogonal coordinates. The interaction among the celestial bodies in the solar system is expressed by the space–time metric determined by the coordinates.
Though IAU2000 Resolution B1.3 has given the form of space–time metric for BCRS, the orientation of the spatial coordinate axes are not given. For this reason, IAU2006 Resolution B2 further clarifies that for all practical applications, unless otherwise stated, the BCRS is assumed to be oriented according to the ICRS axes.
ICRS is the International Celestial Reference System, which is a realization of BCRS, including the International Celestial Reference Frame (ICRF) and related standards, constants, and models. ICRF realizes an ideal reference system by precise equatorial coordinates of extragalactic radio sources observed with Very Long Baseline Interferometry (VLBI). It is established and maintained by the International Earth Rotation and Reference Systems Service (IERS). IERS was jointly established by IAU and IUGG in 1987. Its basic mission is to provide Earth rotation, space reference systems and related data and standard services for astronomy, geodesy, and geophysics. The establishment and maintenance of the time scale is the responsibility of BIPM.
The geocentric celestial reference system
Due to the orbital motion of the Earth's center of mass relative to the solar barycenter and the influence of tidal forces caused by other celestial bodies of the solar system, the Earth cannot be regarded as an isolated body. The geocentric reference system is very complicated in conception and definition. Simply, the coordinate origin of GCRS is defined at the mass center of the Earth, and the coordinate axes near the Earth are orthogonal to each other with no spatial rotation relative to the coordinate axes of BCRS. The coordinate axes of GCRS are essentially defined by the coordinate relationship between GCRS and BCRS, which are only locally straight and orthogonal. Therefore, if BCRS were viewed as straight line coordinates, GCRS would be curvilinear coordinates.
According to IAU2000 Resolution B1.3, the spatial coordinate axes of GCRS are consistent with the spatial orientation of BCRS. The metric of GCRS is required to take the same form as the barycentric one:
$$\left\{ \begin{gathered} G_{00} = - \left( {1 - \frac{2W}{{c^{2} }} + \frac{{2W^{2} }}{{c^{4} }}} \right) + O(c^{ - 6} ) \hfill \\ G_{0i} = - \frac{{4{{W}}^{i} }}{{c^{3} }} + O\left( {c^{ - 5} } \right) \hfill \\ G_{ij} = \delta_{ij} \left( {1 + \frac{2W}{{c^{2} }}} \right) + O\left( {c^{ - 4} } \right) \hfill \\ \end{gathered} \right.$$
(7)
W = W0 is the scalar potential, which is the sum of the earth’s gravitational potential and the tide forces of the Sun and other external celestial bodies. Where the potential functions:
$$W^{\mu } {(}T,{{X}}^{k} {) = }W_{{\text{E}}}^{\mu } {(}T,{{X}}^{k} {) + }W_{{{\text{ext}}}}^{\mu } {(}T,{{X}}^{k} {)}$$
(8)
\(W_{\text{E}}^{\mu}, W_{\text{ext}}^{\mu}\) represent the geocentric potentials and the external potentials. In the outer space of the Earth, the geocentric potentials can be expressed as:
$$\left\{ \begin{gathered} W_{{\text{E}}} {(}T,{{X}}^{k} {) = }\frac{{GM_{{\text{E}}} }}{R}\left\{ {1 + \sum\limits_{l = 2}^{\infty } {\sum\limits_{m = 0}^{l} {\left( {\frac{{a_{{\text{E}}} }}{R}} \right)^{l} P_{lm} (\text{cos}\theta )} } \left[ {C_{lm} \text{cos}(m\phi ) + S_{lm} \text{sin}(m\phi )} \right]} \right\} \hfill \\ W_{{\text{E}}}^{i} {(}T,X^{k} {) = }\frac{G}{{2R^{3} }}\varepsilon_{ijk} S_{{\text{E}}}^{j} {{X}}^{k} \hfill \\ \end{gathered} \right.$$
(9)
Here \(T \equiv {\text{TCG}}\), called Geocentric Coordinate Time (TCG), \(M_{{\text{E}}}\) denotes the mass of the Earth, R is the radius of the earth, Plm(cosθ) is Legendre expansion, \(S_{{\text{E}}}^{i}\) the angular momentum of Earth rotation, \(a_{{\text{E}}}\) the semi-major axis of the Earth's equator, and \(\left( {C_{lm} ,S_{lm} } \right)\) the geocentric gravitational potential coefficients.
IAU2006 Resolution B2 clearly stated that GCRS orientation is derived from ICRS-oriented BCRS. According to the Eq. (4), if the geodesic precession is ignored, under the post-Newtonian approximation, the coordinate relationship between GCRS and BCRS can be expressed as follows:
$$\left\{ \begin{gathered} t \equiv T + \int {\left( {\gamma_{{\text{E}}} - 1} \right)} {\text{d}}T + e_{j}^{0} X^{j} /c \hfill \\ x^{j} \equiv x_{{\text{E}}}^{i} \left( T \right) + e_{j}^{i} {{X}}^{j} + {{\left[ {\frac{1}{2}a_{{\text{E}}}^{i} {{X}}^{k} X^{k} - a_{{\text{E}}}^{k} {{X}}^{i} X^{k} } \right]} \mathord{\left/ {\vphantom {{\left[ {\frac{1}{2}a_{{\text{E}}}^{i} {{X}}^{k} {{X}}^{k} - a_{{\text{E}}}^{k} {{X}}^{i} {{X}}^{k} } \right]} {c^{2} }}} \right. \kern-\nulldelimiterspace} {c^{2} }} \hfill \\ t \equiv T + \int {\left( {\gamma_{{\text{E}}} - 1} \right)} {\text{d}}T + e \hfill \\ \end{gathered} \right.$$
(10)
where
$$\left\{ \begin{gathered} \gamma_{{\text{E}}} \equiv e_{0}^{0} = \left[ {1 - \frac{1}{{c^{2} }}(2\overline{w}_{{\text{E}}} + v_{{\text{E}}}^{2} ) + \frac{1}{{c^{4} }}\left( {2\overline{w}^{2}_{{\text{E}}} + 8\overline{w}^{j}_{{\text{E}}} v_{{\text{E}}}^{j} - 2\overline{w}_{{\text{E}}} v_{{\text{E}}}^{2} } \right)} \right]^{\frac{1}{2}} \\ { = }1 + \frac{1}{{c^{2} }}\left( {\overline{w}_{{\text{E}}} + \frac{1}{2}v_{{\text{E}}}^{2} } \right) + \frac{1}{{c^{4} }}\left( {\frac{1}{2}\overline{w}^{2}_{{\text{E}}} + \frac{5}{2}\overline{w}^{{}}_{{\text{E}}} v_{{\text{E}}}^{2} + \frac{3}{8}v_{{\text{E}}}^{4} - 4\overline{w}^{j}_{{\text{E}}} v_{{\text{E}}}^{j} } \right) + O\left( {c^{ - 6} } \right) \\ e_{0}^{i} = \gamma_{{\text{E}}} v_{{\text{E}}}^{j} /c \\ e_{j}^{0} = \gamma_{{\text{E}}} v_{{\text{E}}}^{j} /c + \frac{1}{{c^{3} }}\left( {2\overline{w}^{{}}_{{\text{E}}} v_{{\text{E}}}^{j} - 4\overline{w}^{j}_{{\text{E}}} } \right) + O\left( {c^{ - 5} } \right) \\ e_{j}^{i} = \delta_{ij} \left( {1 - \frac{1}{{c^{2} }}\overline{w}_{{\text{E}}} } \right) + \frac{1}{{2c^{2} }}v_{{\text{E}}}^{i} v_{{\text{E}}}^{j} + O\left( {c^{ - 4} } \right) \\ \end{gathered} \right.$$
(11)
and \(x_{{\text{E}}}^{i} ,v_{{\text{E}}}^{i} ,a_{{\text{E}}}^{i}\) are respectively the position, velocity, and acceleration of the center of the Earth in BCRS, \(\overline{w}_{{\text{E}}}^{{}} ,\overline{w}_{{\text{E}}}^{i}\) the scalar potential and the vector potential at the center of the Earth. It can be seen from the coordinate relationships Eqs. (4) and (10) that if the coordinates in BCRS are considered as Euclidean linear coordinates, the coordinates of GCRS are curvilinear coordinates. Moreover, the scope of application of GCRS is limited to the local space near the Earth, which is much smaller than the space range of the Earth-Moon system. Since the tidal potentials of the space–time metric in GCRS are from the BCRS, the definition of GCRS is conceptually derived from BCRS.
