Compact stellar structures in Weyl geometric gravity

The structure and the properties of the compact astrophysical objects is a central problem in general relativity. This field of study was initiated already in 1916 by Schwarzschild Sch1 , who obtained the interior solution for a sphere of constant density, with vanishing pressure on its vacuum boundary. Despite its theoretical simplicity, and limited interest for realistic stellar objects, the Schwarzschild interior solution did attract a lot of interest, and its properties and extensions have been intensively studied Sch2 ; Sch3 ; Sch4 ; Sch5 ; Sch6 ; Sch7 . An important moment in the development of relativistic astrophysics is related to the work by Tolman T1 ; T2 and of Oppenheimer and Volkoff OV , who obtained the equations of structure of compact general relativistic objects for a static, spherically symmetric geometry. In particular, the equation describing the hydrostatic equilibrium of compact stars was also obtained in these early studies, and it is presently called the Tolman-Oppenheimer-Volkoff (TOV) equation.

The theoretical, as well as the numerical investigations of the TOV equation led to the limiting maximum mass of neutron stars, which was found to be of the order of $3.2M_{\odot}$ RoRu74 . This result was obtained by using the principle of causality, the maximally stiff equation of state $p=\rho c^{2}$, and Le Chatelier’s principle, and it is valid even if the equation of state of matter is unknown in a limited range of densities. On the other hand, Chandrasekhar Ch obtained for the limiting mass of white dwarfs the value $M_{Ch}\approx 1.4M_{\odot}$. Theoretical arguments, as well as observational evidence thus led to the assumption, generally accepted for a long time, that neutron stars must have a mass distribution centered on a value of the order of $1.4M_{\odot}$ Teu . This mass value follows from the result that neutron stars must be supported by the neutron degenerate pressure that becomes dominant after the collapse of the white dwarf. The corresponding radius of a $1.4M_{\odot}$ mass neutron star should be of the order of 10 - 15 km, and its average density is of the order of $6\times 10^{14}\;{\rm g/cm^{3}}$.

However, the standard view on the neutron star masses has changed drastically recently, once more and more exact determinations of the neutron star masses became available Ho . A large number of accurate astronomical observations, as well as the detection of the gravitational waves, has clearly indicated that the masses of neutron stars vary in a much larger range than expected from the simple application of the Chandrasekhar limit. Hence, in MaMe , by using combined electromagnetic and gravitational wave information on the binary neutron star merger GW170817, an upper limit of $M_{max}\leq 2.17M_{\odot}$ for the maximum mass of a neutron star was found. This limit is tighter and less model-dependent than other constraints. From the analysis of the same event in Sh it was pointed out that the neutron matter equation of state has to be sufficiently stiff, implying that the maximum mass of neutron stars has to be much higher than $2M_{\odot}$. This mass value is required so that a long-lived massive neutron star can be formed as the merger remnant for the binary systems of GW170817, for which the initial total mass is greater than $2.73M_{\odot}$. Moreover, since no relativistic optical counterpart was detected, a value of $M_{max}\approx 2.15-2.25M_{\odot}$ can be inferred for the maximum value of the mass of the neutron star. Similar values for the maximum mass of the neutron star have been obtained in Ru and Re , respectively. Other determinations of the masses of neutron stars, by using the Shapiro delay, gave a mass of the order of $1.928\pm 0.017M_{\odot}$ for the pulsar PSR J1614-2230 Fo , and a mass $2.14_{-0.09}^{+0.10}M_{\odot}$ for the Millisecond Pulsar MSP J0740+6620 Cro .

Interestingly enough, more intriguing determinations of neutron star masses came from the gravitational event GW190425, which indicated the total mass of the binary neutron stars to be of the order of $3.4M_{\odot}$ Abb1 , and from GW190408, which did show the possible existence of a neutron star with mass $2.5-2.65M_{\odot}$, merging with a large black hole with mass $26M_{\odot}$ Abb2 . Hence, these recent observations opened new perspectives of the mass distribution of the neutron stars, making the old paradigm of the standard $1.4M_{\odot}$ mass value untenable.

A possible explanation for the high mass values of the observed compact objects could be obtained by assuming that they contain some exotic components. In Ko1 it was suggested that rotating strange quark stars in the Color-Flavor-Locked (CFL) phase can have masses in the range $3.8-6M_{\odot}$, and thus they can mimic even stellar mass black holes. The low luminosity of the star makes it difficult to detect. On the other hand, Bose-Einstein Condensate stars, consisting of superfluid condensates with particle masses of the order of two neutron masses, forming Cooper pairs, and scattering length of the order of 10 - 20 fm have maximum masses of the order of $2M_{\odot}$. If the particles composing the condensate are kaons, then the mass of the neutron star can be in the range of $2.4-2.6M_{\odot}$ Cha . For a detailed discussion of the astrophysical role of the exotic components, like, for example, quarks and kaon condensates, in the interior of neutron stars see Gle .

Important constraints on the nature of the equation of state of the dense matter in the interior of the neutron stars can be obtained by using multi-messenger astronomy. The observations of the binary neutron star merger GW170817, together with the observation of electromagnetic counterparts across the entire spectrum allowed to obtain some important restrictions on the mass ratio $q\leq 1.38$, and the tidal deformability of the source, $\tilde{\Lambda}\geq 197$ Cou . These constraints rule out sufficiently soft equations of state of the nuclear matter. Compact binary mergers that include neutron stars lead to the creation of the electromagnetic counterpart, the kilonova, which originate from the neutron-rich outflows from the merger Ka . In Cou the analysis of the GW170817 event was performed by using standard general relativity.

Another important avenue for the possible explanation of the high mass values of the neutron stars is modified gravity. Modified gravity theories have been proposed mainly to explain the recent accelerating expansion of the Universe (for reviews of the accelerating Universe, dark energy and modified gravity problems see r1 ; r2 ; r3 ; r4 ).

Modified gravity theories can also open some new windows on the understanding of the structure of compact objects. In modified gravity theories some of the basic stellar structure properties, like the mass-radius relations, maximum masses, or moments or inertia are different as compared with standard general relativity Olmo . It is important to point out that even in the case of non-relativistic stars, like white, red or brown dwarfs, modified gravity effects may play an important role in their internal structure, due to the modification of the gravitational interaction inside these astrophysical objects. These modifications are induced through the modified Poisson equation for the gravitational potential, which leads to modifications in the mass, radius, central density, and luminosity of the stars Olmo . Moreover, the Chandrasekhar mass limit for white dwarfs, and the minimum mass for stable hydrogen burning in high-mass brown dwarfs stars are also influenced. The observational determination of the masses of several neutron stars having values of the order of $2M_{\odot}$ has led to contradictions between the predictions of some realistic equations of state of dense matter, ruling out many of the soft ones (and in particular, the ones including hyperons) Olmo . Modified theories of gravity that go beyond general relativity could help in alleviating or solving these contradictions, through the corrections that appear in the generalized hydrostatic equilibrium equations of extended stellar models Olmo .

There are several extensions of general relativity that could explain the observational properties of the compact astrophysical objects, like neutron stars, and of their mass distribution, which include $f(R)$, $f(R,T)$ and, $f\left(R,L_{m}\right)$ type gravity theories, hybrid-metric Palatini gravity, or theories in which the standard Hilbert-Einstein Lagrangian is extended by a three-form field $A_{\alpha\beta\gamma}$. For a brief review of the astrophysical implications of the above mentioned theories, as well as on the Palatini and metric affine gravity, see Olmo , and references therein. In many modified gravity theories, it is possible to obtain stellar mass values for static, spherically symmetric objects, that are very difficult to be obtained in standard General Relativity, even considering the effects of fast rotation.

One of the important extensions of the Riemann geometry is represented by the Weyl geometry Weyl ; Weyl1 . In Weyl geometry the metric condition is abandoned, and the covariant divergence of the metric tensor in non-zero, $\nabla_{\alpha}g_{\mu\nu}=Q_{\alpha\mu\nu}\neq 0$, where $Q_{\alpha\mu\nu}$ is the non-metricity. In its initial formulation by Weyl, for the non-metricity the particular form $Q_{\alpha\mu\nu}=\omega_{\alpha}g_{\mu\nu}$ was assumed, where $\omega_{\alpha}$ is the Weyl vector. For a presentation of the Weyl geometry, and its historical aspects, see Scholz . Another important idea introduced by Weyl is the idea of the conformal invariance of physical laws, and the necessity of reformulating Einstein’s gravity as a conformally invariant theory. Recently, conformally invariant theories of gravity, and of elementary particle physics, were proposed and discussed in P1 ; P1a ; P2 ; P3 ; P4 ; G1a ; G2 .

A conformally invariant theory of gravity, based on the action $S=-\alpha_{g}\int{C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa}\sqrt{-g}d^{4}x}=-2\alpha_{g}\int{\left(R_{\mu\nu}R^{\mu\nu}-R^{2}/3\right)\sqrt{-g}d^{4}x}$, where $C_{\lambda\mu\nu\kappa}$ is the (conformally invariant) Weyl tensor, was introduced, and extensively investigated, in Ma0 ; Ma1 ; Ma2 ; Ma3 ; Ma4 ; Ma5 . In particular, the conformally invariant vacuum field equations do admit an exact solution of the form $ds^{2}=e^{\nu}dt^{2}-e^{-\nu}dr^{2}-r^{2}d\Omega$, where $e^{\nu}=1-2m/r+\gamma r-\Lambda r^{2}$, where $m$, $\gamma$, $\Lambda$ are constants Ma0 .

Weyl geometry, and the corresponding gravitational theory, have found a large number of applications in theoretical physics, astrophysics, and cosmology. Dirac Dirac1 ; Dirac2 has proposed an extension of Weyl’s theory, which is based on three geometric quantities, the symmetric metric tensor $g_{\mu\nu}$, the Weyl connection vector $\omega_{\mu}$, and the Dirac gauge function $\beta$. The Weyl-Dirac theory was extensively discussed, and applied to cosmology Rosen ; Isrcosm , and can give a systematic description of the matter creation in the Universe, as well as for its late acceleration. Weyl geometry is the theoretical foundation of the $f(Q)$ type modified gravity theories Q1 ; Q2 ; Q4 , and of its generalizations Q10 ; Q17 ; Q20 ; Q23 ; Q24 . In the $f(Q)$ theory, the basic quantity describing the gravitational field is the (metric dependent) non-metricity $Q$. The action of this theory is given by $S=\int{f(Q)\sqrt{-g}d^{4}x}$, where $f(Q)$ is an arbitrary function of the non-metricity. An observational investigation of several modified $f(Q)$ models using the redshift approach was performed in Laz , by using a variety of observational probes (Type Ia Supernovae, Quasars, Gamma Ray Bursts, Baryon Acoustic Oscillations and Cosmic Microwave Background data). It turns out that $f(Q)$ gravity provides a credible alternative model for the explanation of the late time acceleration of the Universe. $f(Q)$ type theories have been extended to include the effects of the coupling between non-metricity and matter, described by the matter Lagrangian $L_{m}$ in C0a . This approach was extended to theories with action described by an arbitrary function of the non-metricity and the trace of the energy-momentum tensor $T$, in C1a ; C2a ; C3a , leading to the so-called $f(Q,T)$ theory of gravity. The role of the torsion in the couplings between matter and the basic geometric quantities was investigated in C4a . A cosmological model based on $f\left({\cal R}\right)$ gravity, with $f\left({\cal R}\right)=\alpha{\cal R}-\beta{\cal R}^{2}/2-\gamma/3{\cal R}$ was investigated in Pinto , by using the Palatini approach. The model was interpreted in a Weylian geometric framework, and the definitions of the luminosity distance, proper distance, and redshift were extended to Weyl type geometries.

An interesting approach to Weyl gravity, and its applications, was initiated, and extensively developed, by using a perspective suggested by elementary particle physics, in Gh1 ; Gh2 ; Gh3 ; Gh4 ; Gh5 ; Gh6 ; Gh7 ; Gh8 . The basic idea is to linearize in the action of the Weyl quadratic gravity the (geometric) term $\tilde{R}^{2}$ with the help of an auxiliary scalar field. The linearized Weyl quadratic gravity undergoes a spontaneous breaking of $D(1)$ by a geometric, Stueckelberg type, mechanism, with the Weyl gauge field acquiring mass from the spin-zero mode of the $\tilde{R}^{2}$ term in the action. The Stueckelberg mechanism is implemented by setting the scalar field $\phi$ to a constant value, taken as its vacuum expectation value, so that $\phi\rightarrow<\phi>$. Then, the Weyl vector field becomes massive, and it absorbs the dynamical scalar field $\phi$, which disappears from the initial scalar-vector-tensor theory. In this way the Einstein-Proca action is recovered from the Weyl action, by eliminating the auxiliary scalar field $\phi$, and thus returning to a vector-tensor theory, as the initial Weyl theory is.

This mode also generates the Planck scale, and the cosmological constant, with the Einstein-Proca action emerging in the broken phase. Moreover, all mass scales, including the Planck scale, and the cosmological constant, have a geometric origin Gh8 . Moreover, the Higgs field of the standard model of elementary particle physics has a similar origin, generated by Weyl boson fusion in the early Universe. The coupling of matter and geometry in Weyl gravity was considered in C1 ; C2 ; C3 ; C4 , with the Palatini formulation of the theory in the presence of geometry-matter coupling was investigated in C2 . The quadratic Weyl gravity $\tilde{R}^{2}+R_{\mu\nu}^{2}$ in the Palatini formalism was studied in Gh5 , by assuming that the Weyl connection and the metric are independent. The theory has a spontaneous breaking of gauged scale symmetry, and mass is generated as a purely geometric effect. The theory leads to a successful inflationary scenario, by predicting a tensor-to-scalar ratio $0.007\leq r\leq 0.01$ for the spectral index $n_{s}$ (at 95 % CL) and $N=60$ efolds. A comparative study of inflation in the Weyl quadratic gravity and in the theory obtained in the Palatini approach to the considered action was performed in Gh6 . These theories have different vectorial non-metricities, induced by the gauge field, and thus lead to different physical predictions. Both theories have a small tensor-to-scalar ratio, $r\sim 10^{-3}$, which is somewhat larger in the Palatini case. The metric Weyl theory gives a dependence $r\left(n_{s}\right)$ similar to that in the Starobinsky inflation.

It is the goal of the present paper to consider static, spherically symmetric stellar models in the simplest model of Weyl geometric gravity. Our starting point is a gravitational action consisting of the sum of the square $\tilde{R}^{2}$ of the Weyl scalar, and of the field strength $F_{\mu\nu}^{2}$ of the Weyl vector. The action can be linearized in the Weyl scalar by introducing an auxiliary scalar field, and then can be reformulated as an effective scalar-vector-tensor theory in Riemann geometry, with the action containing effective couplings between the scalar field and the Ricci scalar, and the Weyl vector field. These terms are conformally invariant by construction. Moreover, a matter term is also added to the total action. The field equations corresponding to this action are obtained in the metric formalism, by varying the action respect to the metric tensor, Weyl vector and the scalar field.

An important question in conformally invariant gravitational actions is how to implement the conformal invariance of the matter terms. In this work we implement the requirement of the conformal invariance of the matter action by imposing trace condition on the effective matter action $\mathcal{L}_{m}$, constructed with the help of the ordinary matter action $L_{m}$, and the square of the Weyl vector $\omega^{2}$, so that $\mathcal{L}_{m}=\mathcal{L}_{m}\left(L_{m},\omega^{2}\right)$. Once this condition is satisfied, the corresponding gravitational field equations, and their solutions, are conformally invariant.

After obtaining the gravitational field equations of the Weyl geometric gravity and the consistency condition in static spherical symmetry, we consider a number of specific stellar models, whose general relativistic counterparts have been intensively investigated. Thus, we investigate constant density, stiff fluid, radiation fluid, quark and Bose-Einstein Condensate stars, by numerically solving the Weyl geometric gravity field equations. In each case a detailed analysis of the astrophysical properties of the stars (mass and density profiles, mass-radius relation, Weyl vector and scalar field behavior) is performed, and the dependence of the stellar structures on the Weyl geometric gravity theory parameters is presented. As a general conclusion of our study it turns out that a larger variety of stellar type objects can be constructed in Weyl geometric gravity as compared to standard general relativity.

The present paper is organized as follows. In Section II we review the basic of Weyl geometry, we introduce the quadratic Weyl geometric gravitational action, and we discuss its linearization in the Weyl scalar. The gravitational field equations are obtained by varying the action with respect to the metric tensor, to the Weyl vector and to the scalar field, together with the consistency condition on the Weyl current. A specific form of the effective matter Lagrangian $\mathcal{L}$ is also adopted, and the corresponding field equations are written down. In static spherical symmetry the field equations of the Weyl geometric theory are obtained in Section III, where a dimensionless representation of the geometric and physical variables is also introduced. Several stellar models are constructed, by numerical solving the gravitational field equations, in Section IV. Finally, we discuss and conclude our results in Section V.

In the present Section we first briefly review the geometrical foundations of the conformally invariant Weyl geometric gravity. The action of the theory is also written down, and linearized in the Weyl, and Ricci curvatures by introducing an auxiliary scalar field. The gravitational field equations are obtained in a general form by varying the action with respect to the metric, and the scalar and Weyl vector field.

In the present Section we present the static, spherically symmetric gravitational field equations, describing the interior structure of compact Weyl geometric type stars. We assume that the interior line element in coordinates $\left(t,r,\theta,\phi\right)$ is given by the standard expression,

where $f(r)$ and $g(r)$ are arbitrary functions of the radial coordinate $r$, and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. Moreover, we represent $g(r)$ as,

where $m(r)$ has the physical interpretation as the total (effective) mass of the star. For the Weyl vector we assume the form,

where $h(r)$ is a function to be determined from the field equations. With be above assumptions for the metric and the vector field $\omega_{\mu}$, the gauge condition, $\nabla_{\mu}\omega^{\mu}=0$, is automatically satisfied.

For the Lagrangian of the ordinary matter we adopt the expression

where $\rho$ is the energy density of the baryonic matter. The energy momentum tensor of the ordinary matter is given by

where by $p$ we have denoted the thermodynamic pressure of the matter, while $u_{\mu}$ is the matter four-velocity.

The non-zero component of the Weyl vector field equation is obtained as

The $(00)$ component of the metric field equation (II.2) is given by,

The spatial components of the metric field equation (II.2) are obtained as,

and

respectively. The non-conservation (balance) equation of the energy momentum tensor is given by

The scalar field equation (36), and the trace constraint equation (37) can be written as,

and

respectively. The trace condition allows us to obtain immediately the initial conditions at the center of the star for the Weyl vector $h$, which are given by

In the next Sections, in order to obtain the numerical solutions of the field equations, we use a set of dimensionless parameters and variables for the geometrical and physical quantities, which are defined as

In the calculations we set $\rho_{c}=2.45\times 10^{15}\,{\rm g/cm}^{3}$.

In the present Section we will consider several specific stellar models in conformally invariant Weyl geometric gravity. In particular, we will consider constant density stars, stiff fluid stars, described by the Zeldovich equation of state, photon stars, quark stars, and Bose-Einstein Condensate stars, respectively. In all cases the static spherically symmetric field equations of the Weyl geometric gravity are integrated numerically, with the consistency condition of the Weyl current fully satisfied.