Abstract
In this work we present the Karmarkar condition in terms of the structure scalars obtained from the orthogonal decomposition of the Riemann tensor. This new expression becomes an algebraic relation among the physical variables, and not a differential equation between the metric coefficients. By using the Karmarkar scalar condition we implement a method to obtain all possible embedding class I static spherical solutions, provided the energy density profile is given. We also analyse the dynamic adiabatic case and show the incompatibility of the Karmarkar condition with several commonly assumed simplifications to the study of gravitational collapse. Finally, we consider the dissipative dynamic Karmarkar collapse and find a new solution family.
1 Introduction
General Relativity is living unprecedented times, witnessing the transformation of exotic objects – such as black holes – and feeble phenomena – like gravitational waves – from mathematical curiosities to observable physical entities.
There are many notable attempts to explore the properties of physically viable solutions (numeric & analytic) describing either static, stationary, or collapsing relativistic compact objects. All known exact solutions have been obtained by imposing some restrictions, such as symmetry conditions on the metric, the algebraic structure of the Riemann tensor, new coupled field equations, meaningful equations of state for the matter variables, or selecting particular initial and boundary conditions, to mention the most common strategies.
Einstein’s covariant mathematical description of gravitation contrasts with solution obtained which are strongly dependent on the coordinate basis. It is not always easy to understand the qualitative features that these coordinateprone solutions might possess, and the analysis of their general properties could reveal unforeseen features of the theory. Thus it is useful to study the general properties through a coordinate independent formalism.
We have recently implemented a tetrad formalism by an orthogonal splitting of the Riemann tensor. We introduced a full set of equations equivalent to the Einstein system and applied it to the spherical case, showing that it is possible to obtain relevant information from selfgravitating systems [1, 2]. This formalism provides coordinatefree results expressed in terms of structure scalars closely related to the kinematical and physical properties of the fluid.
In this short paper, we shall explore the consequences of imposing the wellknown Karmarkar condition [3], which implies that a curved fourdimensional metric can be embedded into a fivedimensional pseudoEuclidean spacetime. The Karmarkar condition provides a geometric relation between the metric functions and their derivatives; thus, one can choose one of the metric functions and generate the other. In the case of an isotropic static fluid sphere – Pascalian matter distribution – the Karmarkar condition leads to either a Schwarzschild – homogeneous conformallyflat bounded solution – or a Kohler–Chao solution – a nonconformally flat unbounded solution [4]. However, for static anisotropic matter configurations, it provides a geometrical mechanism for implementing equations of state relating the radial and the tangential pressures.
As pointed out by Ivanov [5], the Karmarkar initial embedding motivation changes into a geometical method that generates matter configurations that may represent compact astrophysical objects. As shown in Fig. 1 Karmarkar’s condition has experimented a recent boom, with more than 70 publications in the last 3 years, most of them devoted in describing anisotropic compact objects. There are many interesting models of possible compact objects, depending on the variety of the metric function selected as input: rational functions [6,7,8,9,10,11,12], polynomials [13,14,15,16,17], trigonometric [18,19,20,21] and hyperbolic functions [22,23,24]. Recently, there have been some explorations of the consequences of the Karmarkar conditions on stellar structure models in modified theories of gravity [25, 26].
In the next section, we briefly describe the scalar formalism we use. Following, in Sect. 3, we study the static and dynamic (adiabatic and dissipative scenarios) Karmarkar solutions. For the static case, we implement an algorithm to generate any Karmarkar spherical static anisotropic solution given the energy density profile. Regarding the dynamic adiabatic assumption, we show how restrictive the Karmarkar condition may be, and for the corresponding dissipative environment, we found a new family of dynamical radiating Karmarkar lineelements. Finally, our last section wrapsup some remarks and conclusions.
2 The structure scalar strategy and the general formalism
As we mentioned above, the strategy we shall follow is to formulate two independent sets of equations, – expressed in terms of scalar functions, which contain the same information as the Einstein system.
Let us choose an orthogonal unitary tetrad:
As usual, \(\eta _{(a)(b)}~=~ g_{\alpha \beta } e_{(a)}^\alpha e_{(b)}^\beta \), with \(a=0,\,1,\,2,\,3\), i.e. latin indices label different vectors of the tetrad. Thus, the tetrad satisfies the standard relations:
With the above tetrad (1) we shall also define the corresponding directional derivative operators
The first set can be considered purely geometrical and emerges from the projection of the Riemann tensor along the tetrad [27], i.e.
where \(e^{(a)}_{\alpha ;\beta \gamma }\) are the second covariant derivatives of each tetrad (6) vector indicated with \(a = 0,1,2,3.\)
The second set emerges from the Bianchi identities:
2.1 The tetrad, the source and the kinematical variables
To proceed with the above objective we shall restrict to a spherically symmetric line element given by
where the coordinates are: \(x^0=t\), \(x^1=r\), \(x^2=\theta \).
In this case the tetrad is:
and their covariant derivatives can be written as:
Where: \(J_1\), \(J_2\), \(\sigma _{1}\), \(\sigma _{2}\) and \(a_1\) are expressed in terms of the metric functions and their derivatives as:
with primes and dots representing respectively, radial and time derivatives.
As we mentioned before we shall take as our source a bounded, spherically symmetric, locally anisotropic, dissipative, collapsing matter configuration, described by a general energy momentum tensor, written in the “canonical” form, as:
It is immediately seen that the physical variables can be defined – in the Eckart frame where fluid elements are at rest – as:
with \(h_{\mu \nu }=g_{\mu \nu }+V_\nu V_\mu \).
As can be seen from the condition \({\mathcal {F}}^{\mu } V_{\mu }=0\), and the symmetry of the problem, Einstein’s equations imply \(T_{03}=0\), thus:
Clearly \(\rho \) is the energy density (the eigenvalue of \(T_{\alpha \beta }\) for eigenvector \(V^\alpha \)), \({\mathcal {F}}_\alpha \) represents the energy flux four vector; P corresponds to the isotropic pressure, and \(\Pi _{\alpha \beta }\) is the anisotropic tensor, which can be expressed as
with
Finally, we shall express the kinematical variables (the fouracceleration, the expansion scalar and the shear tensor) for a selfgravitating fluid as:
2.2 The splitting of the Riemann tensor and structure scalars
In this section we shall introduce a set of scalar functions – the structure scalars – obtained from the orthogonal splitting of the Riemann tensor (see [28,29,30]) which has proven to be very useful in expressing the Einstein Equations.
Following [28], we can express the splitting of the Riemann tensor as:
with \(\varepsilon _{\mu \nu \gamma } = \eta _{\phi \mu \nu \gamma } V^{\phi }\), and \( \eta _{\phi \mu \nu \gamma }\) the LeviCivita 4tensor. The corresponding Ricci contraction for the above Riemann tensor can also be written as:
where the quantities: \(Y_{\alpha \beta }\), \(X_{\alpha \beta }\) and \(Z_{\alpha \beta }\) can be expressed as
with
and the electric part of the Weyl tensor is written as
2.3 Projections of Riemann tensor
From the above system (3), by using the covariant derivative of equations (7) and the projections of the orthogonal splitting of the Riemann tensor, we obtain the first set of independent equations, for the spherical case, in terms of \(J_1\), \(J_2\), \(\sigma _{1}\), \(\sigma _{2}\), and \(a_1\), (defined in (8)) and their directional derivatives, i.e.
2.4 Equations from Bianchi identities
The second set of equations for the spherical case, emerge from the independent Bianchi identities (4), and can be written as:
3 Karmarkar condition
As it is wellknown, a fourdimensional curved spacetime can be embedded in a fivedimensional pseudoEuclidean space whenever it satisfies the Karmarkar condition which can be stated as [3]:
and provides a geometrical mechanism to implement equations of state relating the radial and the tangential pressures.
3.1 Differential Karmarkar conditions
When considering a line element (5), Karmarkar’s condition (35) leads to
as shown in reference [31], for the particular case of \(A(r,t) = \tilde{A}(r)\), \(B(r,t) = \tilde{B}(r)f(t)\) and \(R(r,t)~=~r \tilde{B}(r)f(t)\).
If we examine a much simpler metric like
the Karmarkar condition (35) can be written as
Which, in the static case, leads to the differential equation
the most common expression for the Karmarkar condition examined in the literature (see references [6] through [24]).
If we provide a particular \(\lambda \)function – listed in Table 1 – we can obtain the other metric coefficient \(\nu \) and then investigate the type of material described by this lineelement. Thus, again, the Karmarkar condition implements a geometrical method to generate anisotropic equations of state, and has boomed a profusion of possible realistic models for compact objects. Unfortunately, (35), the models generated are coordinate dependent, and the general properties obtained are heavily conditioned from this fact.
3.2 Scalar Karmarkar conditions
This coordinate dependence can be overcome in the tetrad framework by projecting the Riemman tensor as
and, from Eq. (21) assuming spherical symmetry, it can be reduced to a simple algebraic scalar relation among several physical variables:
Notice that this scalar relation among the physical variables defined in Eq. (21), despite its simplicity, is valid for any dynamic and dissipative spherical matter distribution described by (5). In the next sections we shall use (41) to study, both the static and dynamic (adiabatic and dissipative) cases.
3.3 The static case
Employing the abovesketeched scalar formalism and assuming the condition (41), we shall find the most general static, spherically symmetric anisotropic Karmarkar solution.
For the line element (5) we can assume, without any loss of generality, \(R=r\) and integrate (24) to obtain:
where \(C_1\) is a constant of integration. Next, from equation (26) it follows at once that:
Clearly, these metric elements (42) and (43) –expressed in terms of the structure scalars \(X_1\) and \(Y_0Y_1\)–, describe any static anisotropic matter distribution [32].
The equivalent Einstein system of Eqs. (23)–(34) can be written, for the static case, as:
and the Karmarkar condition (41) takes the form of
Now, integrating equation (34) we find
On the other hand, by using Eqs. (44)–(46) together with (48)–(49), Eq. (47) can be written as
Now, integrating (51)
and substituting (52) in (42) we get
where again, \(C_2\) is a constant of integration.
Finally, the line element (5) can be rewritten as
which describes any Karmarkar static spherically symmetric anisotropic fluid distribution. Notice, that for this spacetime we have defined
Thus, all metrics will depend on a sole physical parameter: the energy density \(X_0\) and in Table 1 (see the Appendix at the end of the present work), we present the corresponding \(X_0\) for several metrics which appeared in the recent literature.
To illustrate this strategy, let us assume the energy density as
then, from Eq. (55) we obtain that
and the line element (54) can be written as follows
which is the solution given in [33].
3.4 \(Z = 0\), the dynamic adiabatic scenario
Much effort has been dedicated in developing static bounded Karmarkar models, but very little has been done for the dynamic case. In this section, we shall discuss the dynamic adiabatic state, \(Z=0\), and explore the “compatibility” of the Karmarkar condition with other typical restrictions used in studying exact solutions in General Relativity.

\(X_1=0\), homogeneous energy density. The uniform density spherical matter configuration is the standard entry point in all textbooks of General Relativity and Relativistic Astrophysics [34,35,36,37]. We have recently shown [2] that, despite its simplicity and pedagogical interest, this widespread assumption is very restricted. Any dynamic homogeneous density profile satisfying the Karmarkar condition will lead to the Schwarzschild solution. It can be easily obtained from (34) assuming \(X_0=X_0(t)\) and establishing regularity conditions at the origin we found \(X_1=0\). Next, substituting this result into (41), it leads to \(Y_1=0\), i.e., conformally flat perfect fluid solution with homogeneous energy density: the Schwarzschild solution.

\(Y_1=0\) vanishing complexity factor. Recently, L. Herrera introduced a new concept of complexity for selfgravitating systems [38]. This concept includes the influences from energy density inhomogeneities and local anisotropy of the pressures on the active gravitational (Tolman) mass. Assuming the vanishing complexity condition, \(Y_1=0\), in Eq. (41) we obtain \(X_1=0\), and because \(Y_0\ne 0\), we reobtain only the Schwarzschild solution.

\({\mathcal {E}}=0\), conformally flat case. If \({\mathcal {E}}=0\), then \(X_1=Y_1\), and from (41) we obtain \(X_1=Y_1=0\), due to \(X_0+Y_0\ne 0\) because the regularity at the origin.

\(\Pi _1=0\), Pascalian isotropic fluids. Karmarkar condition (41) and the relation \(X_1+Y_1=8\pi \Pi _1\) lead to
$$\begin{aligned} X_1=\frac{(Y_0X_08\pi \Pi _1)\sqrt{(Y_0X_08\pi \Pi _1)^232\pi \Pi _1 X_0}}{2}\nonumber \\ \end{aligned}$$(59)and
$$\begin{aligned} Y_1=\frac{(Y_0X_0+8\pi \Pi _1)+\sqrt{(Y_0X_0+8\pi \Pi _1)^232\pi \Pi _1 Y_0}}{2}\nonumber \\ \end{aligned}$$(60)Now, from (59) and (60) it is clear that isotropy, \(\Pi _1=0\) leads to \(X_1=Y_1=0\), which is again the Schwarzschild solution.

\(\sigma _1=\sigma _2\), shearfree and \(a_1=0\) geodesic fluids. Finally, considering shearfree and geodesic conditions in Eqs. (23) y (24), we again obtain \(Y_1=0\)
3.5 The dissipative case: \(Z\ne 0\)
A recent paper [31] develops a model of a radiating relativistic sphere that satisfies the Karmarkar condition. It is the first dynamic dissipative model obtained.
For the present case, we shall consider a shearfree fluid, i.e.
From these two assumptions (Karmarkar and shearfree) we can find two new families of dissipative solutions.
If we assume that \(X_1=0\), we can see immediately, from (27) and (30) that
and from (30) that
Now, subtracting Eq. (33) from Eq. (32), we obtain
Next, by using the Karmarkar condition (41) and taking into account (25), Eq. (64) can be written as:
where \(\tilde{x}_0(t)\) is a constant of integration.
Thus, we can identify two possible cases:

1.
For the case \(a(t)=0\), from (26) we get \(Z=a_1\sigma \) and by integrating (65) we find
$$\begin{aligned} A=C_2(t)\sqrt{r^2+C_1(t)},\quad R=b(t)r,\quad B=b(t) \end{aligned}$$(66)which is the family of solutions shown in [31].

2.
For \(a(t)=\frac{\tilde{a}}{b^2(t)}\), \(x_0=\frac{2 b^2}{\dot{b}^2}\) and integrating (65) we find
$$\begin{aligned}&A(t,r)=\nonumber \\&\quad \frac{\dot{b} \sqrt{C(2+\tilde{a} C)+\tilde{a}r^4(1+\tilde{a}^2 C^2)+2r^2(1\tilde{a}C+\tilde{a}^2 C^2)}}{\sqrt{2}(1+\tilde{a} r^2)}\nonumber \\ \end{aligned}$$(67)and
$$\begin{aligned} R(t,r)=\frac{b(t) r}{1+\tilde{a}r^2},\qquad B(t,r)=\frac{b(t) }{1+\tilde{a}r^2} \end{aligned}$$(68)with \(C=\frac{4\tilde{c}(t)}{\dot{b(t)}}\), where \(\tilde{c}(t)\) is a constant of integration.
The matter variables for the solution (67) and (68) are
with
4 Final remarks
It surprises the number of works published by slight variations in the metric functions. Then, after the integration of the Karmarkar condition (39), dozens of models (see Table 1) are obtained with negligible or no discussion in their interrelations, remaining most of these efforts, in very descriptive stages.
In this short article, we tried to explore some general consequences derived from the Karmarkar condition (35). By using a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor, we have presented it in terms of the structure scalars. Thus the new expression (41) becomes an algebraic relation among the physical variables, and not a differential equation between the metric coefficients shown in (36) and (38). Taking advantage of its simplicity, we have studied the static, dynamic adiabatic, and dynamic dissipative Karmarkar solutions.
For the static case, we developed a method to obtain any spherically, static, anisotropic Karmarkar solution, parameterized by the energy density profile. We think it opens the possibility to explore new anisotropic matter configurations, starting from realistic isotropic nuclear equations of state.
Much effort has been made on static bounded Karmarkar models, but very little considering the dynamic scenario. The simplicity of the scalar Karmarkar condition allows us to study the adiabatic and radiant cases efficiently. Regarding the adiabatic dynamic matter configuration, we have shown that combining the Karmarkar condition with several other common simplifying assumptions, we inexorably obtain the homogeneous Schwarzschild solution.
This raises a possible conjecture that for the spherical case, the Karmarkar dynamic adiabatic condition is incompatible with any other simplifying assumption. If they are combined, we necessarily obtain the homogeneous Schwarzschild solution. This possible conjecture should be further, and carefully explored in the future.
Finally, for the dynamic dissipative case, we recovered a known previous solution [31] and found a new shearfree Karmarkar radiating solution.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental or computational data were generated or analysed during the development of this work.]
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Acknowledgements
J.O. acknowledge financial support from Ministerio de Ciencia, Innovación y Universidades, Spain (Grant PGC2018096038B100) and Junta de Castilla y León, Spain (Grant SA083P17). J.O acknowledges hospitality of School of Physics of the Industrial University of Santander, Bucaramanga Colombia. L.A.N. gratefully acknowledge the financial support of the Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander and the financial support provided by COLCIENCIAS, Departamento Administrativo de Ciencia, TecnologTecnología e e Innovación under Grant no. 8863.
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Ospino, J., Núñez, L.A. Karmarkar scalar condition. Eur. Phys. J. C 80, 166 (2020). https://doi.org/10.1140/epjc/s1005202077388
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DOI: https://doi.org/10.1140/epjc/s1005202077388