UNIVERSITÀ DEGLI STUDI DI TORINO. Tesi di Laurea Specialistica. Polytropic stars and no-go theorem in extended theories of gravitation

Transcription

1 UNIVERSITÀ DEGLI STUDI DI TORINO FACOLTÀ DI SCIENZE M.F.N. Corso di Laurea in Matematica Tesi di Laurea Specialistica Polytropic stars and no-go theorem in extended theories of gravitation Candidato: Annalisa Mana Relatore: prof. Lorenzo Fatibene Sessione di Ottobre 2009 Anno Accademico 2008/2009

2 Contents 1 Introduction f R) models How to derive Einstein s field equations How to construct a stellar model Polytropes A no-go theorem for polytropic stars in Palatini fr) gravity Abstract Field equations Stellar model Matching interior and exterior solutions Conclusions Re-examination of polytropic stars in Palatini fr) gravity Abstract Field equations Stellar model Discussion and conclusions Bibliografy 46 i

3 Chapter 1 Introduction It took a long time before it was realized that Newtonian gravity was not a complete theory describing gravity, but a limiting case of Einstein s General Relativity GR) theory, one of the most elegant scientific theories ever developed. GR has been an extremely successful theory of gravity for many years, because it has passed all tests on scales relevant for the solar system. However, it seems that now GR is facing similar difficulties with those that Newton s theory had almost a hundred years ago: scientists have come to the thought that maybe GR is the effective part of a more general theory of gravitation, applicable only on solar system scales. Some of these problems are of theoretical origin, for example the difficulties in finding the quantum counterpart of general relativity and the fact that it has not yet been possible to include it in any grand unification scheme. On the other side, evidence coming from astrophysics and cosmology has revealed a quite unexpected picture of the universe and it seems that GR is unable to explain crucial features of it without introducing very artificial assumptions. In particular, the dynamics of galaxies and clusters of galaxies and the present accelerated expansion of the universe seem to require the introduction of exotic components in the matter content of the cosmos. Actually, the latest datasets coming from different sources, such as the CosmicMicrowave Background Radiation and supernovae surveys, seem to indicate that the energy budget of the universe is the following: 4% of ordinary baryonic matter 22% of dark matter 74% of dark energy. 1

4 The term dark matter refers to an unkown form of matter, which has the same properties of ordinary matter but has not yet been detected in the laboratory. The term dark energy is reserved for an unknown form of energy which hasn t been detected directly, but also doesn t have ordinary properties: this seems to resemble in high detail a cosmological constant. Due to its dominance over matter ordinary and dark) at present times, the expansion of the universe seems to be an accelerated one, contrary to past expectations. Anyway one has to admit that our current picture of the evolution and the matter/energy content of the universe is at least surprising and definitely calls for an explanation. Some people think it could be that our inability to form an acceptable theory of quantum gravity, as well as a simple explanation of the cosmological observations, is due to using an over-restricted theory of gravity. The various problems have so moved many authors to question whether GR is indeed a completely correct theory of gravity on the classical level and to investigate possible alternatives which would not require the inclusion of dark energy: the search for this kind of theories has received a powerful stimulus from current developments in observational cosmology. If one decides that modifying gravity is the way to go, this is not an easy task, because there are numerous ways to deviate from GR. An alternative route is to modify GR by abandoning the simple assumption that the action should be linear in the scalar curvature R. So, instead of using the Einstein-Hilbert Lagrangian, one can use a more general one, which depends on a generic function f R): such theories are called f R) theories of gravity. Unfortunately these theories are not free of problems. 2

5 f R) models First of all they lead to fourth order differential equations which are difficult to attack. Additionally, it is doubtful whether they can pass the known solar system tests and they lead to unavoidable instabilities. There is another modification of gravity that one can consider which does not necessarily involve a modification of the action, but the use of a different variational principle. This variational principle, known as the Palatini formalism, treats the metric and the affine connection as independent geometrical quantities. One has to vary the Lagrangian with respect to both of them to derive the field equations, in contrast with the standard metric variation, where the Lagrangian is varied with respect to the metric alone, and the connection is assumed to be the Christoffel symbols of this metric. It is also known that when the Einstein-Hilbert action is used, the Palatini variational principle leads to the Einstein equations, just like the standard metric variation. This is not true, however, for a more general action. When used together with an f R) Lagrangian, the Palatini formalism leads to second order differential equations instead of the fourth order ones that one gets with the metric variation. At the same time, in vacuum, they straightforwardly reduce to standard General Relativity plus a cosmological constant. This ensures us that, firstly, the theory passes the solar system tests, and secondly, that interesting aspects of GR like static black holes and gravitational waves are still present. Thus there is no real criterion so far about which one of them is better to use. Additionally, the Palatini variation seems to be more general since it yields GR without the need to specify the relation between the metric and the connection. 1.1 f R) models These theories come about by a generalization of the Lagrangian in the Einstein-Hilbert action S EH = 1 2k 2 d 4 x gr 1.1) where k 2 8πG, G is the gravitational constant, g is the determinant of the metric and R is the Ricci scalar c = h = 1), to become a general function of R, f R) S = 1 2k 2 d 4 x g f R) 1.2) 3

6 How to derive Einstein s field equations This choice is done because one might imagine replacing the scalar curvature in the Einstein-Hilbert action by some function of it, f R), which could then be expanded in power series, with positive and negative powers of the scalar curvature f R) =... + α 2 R 2 + α 1 R2 2Λ + R + + R3... R β 2 β 3 where the α i and β j coefficients have the appropriate dimensions. Apart from the increased complexity, they lead to fourth order field equations and unfortunately they show problems. In general, most of the research has been focused on the smallest deviations from the linear term in R, that is Lagrangians with an extra 1/R or R 2 dependence. However, for the case of 1/R corrections, proposed to explain the cosmological expansion, they already have an unwanted behaviour and do not seem to pass the solar system tests. These actions are sufficiently general to encapsulate some of the basic characteristics of higher-order gravity, but at the same time they are simple enough to be easily used. They are good tools and it seems that they are able to avoid some problematic singularities. Therefore, fr) gravity is an interesting and relatively simple alternative to GR, from the study of which some remarkable conclusions have been derived already, mostly in order to understand the principles and limitations of modified gravity. However, such an approach is useful, even if it only leads to the conclusion that GR is the only correct theory of gravitation, it will still have helped us to understand it better. 1.2 How to derive Einstein s field equations There are actually two variational principles that one can apply to the Einstein-Hilbert action in order to derive Einstein s equations: the standard metric variation and a less standard variation called Palatini variation. In the first, the metric and the connection are assumed to be dependent variables, whereas in the latter they are considered independent and one has to vary the action with respect to both of them, under the important assumption that the matter action does not depend on the connection. The choice of the variational principle is usually referred to as a formalism, so one can use respectively the terms metric formalism and metric- 4

7 How to derive Einstein s field equations affine formalism, often called Palatini formalism, even though Palatini was not the person who introduced it. However, both variational principles lead to the same field equations for an action whose Lagrangian is linear in R but this is no longer true for a more general action. It obviously follows that there will be two version of f R) gravity, according to which formalism is used: so we have metric f R) gravity and Palatini f R) gravity. In reality, there is even a third version of f R) gravity, which is obtained if one uses the Palatini variation but abandons the assumption that the matter action is independent of the connection: this one is the most general of these theories, but we will not talk about it. Metric formalism Beginning from the action 1.2) and adding a matter term S M, the total action for f R) gravity takes the form S met = 1 2k 2 d 4 x g f R) + S M g µν, ψ) 1.3) where c = h = 1, k 2 8πG, G is the gravitational constant, g is the determinant of the metric g µν, R is the Ricci scalar defined as with the well-known Ricci tensor R = g µν R µν R µν = µ Γ λ λν + λγ λ µν + Γ λ µργ ρ νλ Γλ νργ ρ µλ where Γ λ µν is the Levi-Civita connection of g and ψ denotes the matter fields. Variation with respect to the metric, after some manipulations, gives where, as usual, f R)R µν 1 2 f R)g µν [ µ ν g µν ] f R) = k 2 T µν 1.4) T µν = 2 g δs M δg µν 1.5) R µν is the Ricci tensor, µ is the covariant derivative associated with the Levi-Civita connection of the metric, g µν µ ν and where a prime denotes differentiation with respect to the argument. 5

8 How to derive Einstein s field equations The field equations 1.4) are fourth order partial differential equations in the metric, since R already includes second derivatives of the latter. For an action which is linear in R the fourth order terms the last two on the left-hand side) vanish and the theory reduces to GR. The trace of 1.4) gives the following equation f R)R 2 f R) + 3 f = k 2 T 1.6) where T = g µν T µν, that relates R with T differentially and not algebraically as in GR, where R = k 2 T. This is already an indication that the field equations of f R) theories will admit a larger variety of solutions than Einstein s theory. Equation 1.6) will be very useful in studying various aspects of f R) gravity: for example, we can make some remarks about maximally symmetric solutions, that lead to a constant Ricci scalar. For R =constant and T µν = 0, equation 1.6) reduces to f R)R 2 f R) = 0 1.7) which, for a given f, is an algebraic equation in R. If R = 0 is a root of this equation, then equation 1.4) reduces to R µν = 0 and the maximally symmetric solution is Minkowski spacetime. On the other hand, if the root of equation 1.7) is R = C, where C is a constant, then equation 1.4) reduces to R µν = g µν C 4 and the maximally symmetric solution is de Sitter or anti-de Sitter space, depending on the sign of C, just as in GR with a cosmological constant. Moreover, a calculation reveals that the left hand side of equation 1.4) is divergence-free for the generalized Bianchi identities, implying that µ T µν = 0. In the end, we have to note that it is possible to write the field equations in the form of Einstein equations with an effective stress-energy tensor composed of curvature terms moved to the right hand side, i.e. equation 1.4) can be stated as G µν = R µν 1 2 g µνr = = k2 T µν f R) + g [ f R) R f R)] µν 2 f R) + [ µ ν f R) g µν f R)] f 1.8) R) 6

9 How to derive Einstein s field equations or where T e f f ) µν 1 k G µν = k f T µν + T e f f ) ) µν R) [ [ f R) R f ] R)] g µν + µ ν f R) g µν f R) 2 is an effective stress-energy tensor which does not have the canonical form quadratic in the first derivatives of the field f R), but contains terms that are linear in the second derivatives. This can be put in the form of a perfect fluid energy-momentum tensor, which could be more useful. Palatini formalism We have already mentioned that Palatini formalism consists in taking two independent variations with respect to the metric and the connection separately. The action is formally the same but now the Riemann tensor and the Ricci tensor are constructed with the independent connection. The action now takes the form S pal [g, Γ, ψ] = 1 2k 2 d 4 x g f R) + S M g µν, ψ) 1.9) where, as usual, where c = h = 1, k 2 8πG, G is the gravitational constant, g is the determinant of the metric g µν, ψ denotes the matter fields and the matter action S M is assumed to depend only on the metric and the matter fields and not on the independent connection. Moreover, for clarity of notation, we have denoted the Ricci tensor constructed with the independent connection Γ λ µν as R µν = µ Γ λ λν + λγ λ µν + Γ λ µργ ρ νλ Γλ νργ ρ µλ and the corresponding Ricci scalar turns out to be R = g µν R µν As we know, in order to obtain the field equations, we must vary the action with respect to the various fields present in it. The variation of the action 1.9) can be expressed as δs = 1 2k 2 d 4 x g [ f R) 2 g µνδg µν + δ f R) ] 1.10) 7

10 How to derive Einstein s field equations where δ f R) represents δ f R) = f R)δ R denoting with f R) the derivative of f with respect to R. From the definition of Ricci scalar, it is easy to see that δ R = δg µν R µν ) = R µν δg µν + g µν δ R µν The next step requires to express δ R µν in terms of δγ λ µν: this can be done using the so called Palatini identity δ R µν = λ δγ λ µν) µ δγ λ λν ) We now manipulate the δ R µν term: that contribution is of the form I = d 4 x gλ µν δ R µν where, in our case, Λ µν f R)g µν. Using the Palatini identity, we get I = d 4 x gλ µν [ λ δγ λ µν) µ δγ λ λν ) ] Using integration by parts and rearranging indices, we find I = { [ g )]} d 4 x λ Λ µν δγ λ µν Λ λν δγ ρ ρν + { [ g } + d 4 x µ Λ µν δβ λ δµ β Λλν)] δγ β λν The first term in brackets is a total derivative and can be discarded, while the second term is the one we need. Putting this back into 1.10) we end up with δs = 1 d 4 x [ g f R) R µν f ] R) 2k 2 g µν d 4 x [ gλ ] g µν µ δβ λ 2k δµ β Λλν ) δγ β λν Knowing that the matter action gives δs M = 1 2 d 4 x gt µν δg µν 8

11 How to derive Einstein s field equations the field equations can be written as follows f R) R µν f R) 2 g µν = k 2 T µν 1.11) µ [ g Λ µν δ λ β δµ β Λλν)] = ) where T µν is the usual stress-energy tensor given by 1.5). Note that the second one is equated to zero because δγ does not appear in δs M. Equation 1.12) can be further simplified if one notices that when λ = β then the equation is identically zero. Taking λ = β then it boils down to β [ gλ λν ] = 0, which is explicitly given by β [ g f R)g µν ] = 0 Finally, variation with respect to metric and connection lead to the following equations f R) R µν 1 2 f R)g µν = k 2 T µν 1.13) [ g ] λ f R)g µν ) = ) So now we can see how the Palatini formalism leads to GR when f R) = R: in this case f R) = 1 and eq. 1.14) becomes the definition of the Levi- Civita connection for the initially independent connection Γ λ µν. Then, R µν = R µν, R = R and eq. 1.13) yields Einstein s equations. It is now clear that generalizing the action to be a general function of R in the Palatini formalism is just as natural as it is to generalize the Einstein-Hilbert action in the metric formalism. However, even if the two formalisms give the same results for linear actions, they lead to different results for more general actions. In the end, it is useful to explain some manipulations of field equations: we will focus on working out a solution for 1.14). Taking the trace of equation 1.13), one gets f R) R 2 f R) = k 2 T 1.15) For a given function f, it is an algebraic equation in R. Given a f, the above equation can be solved as R = RT), that is now a function of the matter and, for this reason, f R and f R are also functions of the matter. 9

12 How to derive Einstein s field equations We can now examine two cases: the one in which T = 0 vacuum case) and the one in which T = 0 with matter). If T = 0, R will be a root of the equation f R) R 2 f R) = ) that is similar to 1.7), and it will be a constant. We will not consider cases for which the previous equation has no roots since it can be shown that the field equations are then inconsistent. Therefore, choices of f that lead to this behaviour should be avoided. Equation 1.16) can also be identically satisfied if f R) = R 2 : this particular choice for f leads to a conformally invariant theory but, as it will be not suitable for a low energy theory of gravity, we will not consider it further. Looking back at the connection equation 1.14), we note that it can be seen as a first order equation for the connection that depends on the metric g µν and the matter [ g ] λ f RT))g µν ) = ) We now discuss how to solve this equation for the connection. In GR, this equation is simply λ [ gg µν ] = 0 and the solution is given by the Christoffel symbols Γ α βγ = gαρ 2 βg ργ + γ g ρβ ρ g βγ ) In order to find a solution for 1.17), we can do the following. We assume that there exists a metric h µν such that the connection that solves 1.17) is the Levi-Civita connection of h µν. This means that the metric h µν satisfies [ hh µν ] = 0 In other words, our ansatz satisfies hh µν = g f R)g µν 1.18) from which it follows that h µν = g µν / f R) and h µν = f R)g µν 1.19) We thus see that the metric h µν that defines our independent connection is conformally related to the spacetime metric g µν. Then, eq. 1.17) 10

13 How to derive Einstein s field equations becomes the definition of the Levi-Civita connection of h µν and can be solved algebraically to give or, equivalently, in terms of g µν Γ λ µν = h λσ µ h νσ + ν h µσ σ h µν ) Γ λ µν = 1 [ ] f R) gλσ µ f R)g νσ ) + ν f R)g µσ ) σ f R)g µν ) Given that equation 1.15) relates R algebraically with T, and since we have an explicit expression for Γ λ µν in terms of R and g µν, we can eliminate the independent connection from the field equations and express them only in terms of the metric and the matter fields. Actually, the fact that we can algebraically express Γ λ µν in terms of the latter two indicates that this connection act as some sort of auxiliary field. Denoting with µ the covariant derivative with respect to the Levi-Civita connection of g µν, we can state that the Ricci tensor will transform under conformal transformations like R µν = R µν f R)) 2 µ f R)) ν f R))+ 1 f µ ν 1 ) R) 2 g µν f R) where = g µν µ ν is the dalambertian. If we contract with g µν we get R = R ) 3 2 f R)) 2 µ f R)) ν f R)) + 3 f R) f R) 1.21) Replacing equations 1.20) and 1.21) in equation 1.13), after some manipulations, one obtains G µν = k2 f T µν 1 2 g µν R ff ) + 1 f µ ν g µν ) f + 3 [ µ 2 f 2 f ) ν f ) 1 2 g µν f ) ] 1.22) 2 Notice that, assuming that we know the root of equation 1.15), we have completely eliminated the independent connection from this equation. Therefore, we have reduced the number of field equations to one and at 11

14 How to construct a stellar model the same time both sides of equation 1.22) depend only on the metric and on the matter fields. So the theory has been brought to the form of GR with a modified source. We can now deduce the following: When f R) = R, the theory reduces to GR. For matter fields with T = 0, due to eq. 1.16), R and consequently f R) and f R) are constants and the theory reduces to GR with a cosmological constant. If we denote the value of R when T = 0 as R 0, then the value of the cosmological constant is 1 R 0 f ) R 0 ) = R 0 2 f R 0 ) 4 In the general case T = 0, the modified source on the right hand side of eq. 1.22) includes derivatives of the stress-energy tensor, unlike in GR. These are implicit in the last two terms, since f is a function of T, given that f = f R) and R = RT). 1.3 How to construct a stellar model Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star. These models are based on the solution of a set of differential equations, formulated in terms of stellar parameters that we will see next. As a major simplification, we suppose that all stars studied are in a steady state and, in addiction to this, the matter inside a star is a perfect fluid. Then it follows that the stars considered are spherically simmetric and so the simplest commonly used model of stellar structure is the spherically symmetric quasi-static model. Actually, the evolution of a star may be perceived as a quasi-static process, in which the composition changes slowly allowing the star to maintain local hydrostatic and thermal equilibrium and the temperature is identical for matter and photons. Although this does not strictly hold because the temperature a given shell sees below itself is always hotter than the temperature above, this approximation is normally excellent because the 12

15 How to construct a stellar model photon mean free path is much smaller than the length over which the temperature varies considerably. In forming the stellar structure equations, it is then reasonable to consider perfect-fluid parameters, such as the matter density ρr), the total pressure matter plus radiation) pr), the 4-velocity u µ r), temperature Tr), luminosity lr), energy generation rate per unit mass ɛr) in a spherical shell of a thickness dr at a distance r from the center of the star and the stress-energy tensor of a perfet-fluid T µν, that usually assumes the following form: T µν = ρ + p)u µ u ν + pg µν 1.23) To deduce the gravitational field for a static spherical star, begin with the metric of Special Relativity in the spherically simmetric form given by where ds 2 = dt 2 + dr 2 + r 2 dω ) dω 2 = dθ 2 + sin 2 θdφ ) The simplest way to modify this metric in order to allow a curvature due to the gravitational field produced by the star, while preserving spherical simmetry, is to let those metric components that are already non-zero to assume different values, like ds 2 = e 2Φ dt 2 + e 2Λ dr 2 + r 2 dω ) where Φ = Φr) and Λ = Λr) are the only two unknown functions. This coordinate system and metric have been used in most theoretical models for relativistic stars, since the work of Schwarzschild, Tolman, Oppenheimer and Volkoff. The structure of a star is then given by the set of functions Φr), Λr), ρr) and pr), which are determined in part by the Einstein s field equations G µν = 8πT µν 1.27) and in part by the law of local conservation of energy-momentum in the fluid µ T µν = ) However these are not sufficient to fix the structure uniquely: it is also necessary the functional dependence of the pressure on the density or conversely, that is an equation of state of the matter. Relevant equations of state may have to include the perfect gas law, radiation pressure, pressure produced by degenerate electrons, etc. 13

16 How to construct a stellar model Indicating with a prime the radial derivative, the non-zero components of Einstein tensor are G 00 = e2φ r G rr = e2λ r [ r 1 e 2Λ)] [ 1 e 2Λ] + 2 r Φ G θθ = r 2 e 2Λ [ Φ Φ ) 2 + Φ r Φ Λ Λ r G φφ = G θθ sin 2 θ We examine now first the component G 00 : G 00 = r 2 r 2 e 2Λ r 1 d/dr)e 2Λ ) = r 2 d/dr) ] [ ] r1 e 2Λ ) 1.29) This equation becomes easy to solve as soon as one notices that it is a linear differential equation in e 2Λ. Defining 2mr) r1 e 2Λ ) 1.30) one gets g rr e 2Λ = 1 2mr)/r) 1 and so 1.29) will be G 00 = 2 dmr) r 2 dr while on the other side we have 1.31) 8πT 00 = 8πρ 1.32) Actually the mass increases with radius according to the mass continuity equation: dm dr = 4πr2 ρ 1.33) Integrating the mass continuity equation from the star center r = 0) to the radius of the star r = R) yields the total mass of the star mr) = R 0 4πr 2 ρdr + m0) where m0) is the constant of integration, that is zero if the space geometry is smooth at the origin physically acceptable) or non-zero if the 14

17 How to construct a stellar model geometry has a singularity at the origin physically unacceptable). Turn next to the component G 11 of the field equations, we have: G 11 = r 2 + r 2 e 2Λ + 2r 1 e 2Λ dφ/dr = 8πT 11 = 8πp 1.34) Solving this equation in dφ/dr and replacing e 2Λ by using 1.30), one obtains the source equation dφ dr = m + 4πr3 p rr 2m) 1.35) As regards the conservation of the energy-momentum tensor µ T µν = 0, in most studies of stellar structure, with the help of equation 1.35) one states it as dp dr = p + ρ)m + 4πr3 p) 1.36) rr 2m) that is called the Oppenheimer-Volkoff OV) equation of hydrostatic equilibrium. Its Newtonian limit says that the outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity: dp dr = Gmρ r ) where mr) is the mass inside the shell at r and G is the gravitational constant. Moreover, we have to consider that while the energy is leaving the spherical shell, it satisfies the energy equation: dl dr = 4πr2 ρɛ ɛ ν ) 1.38) where ɛ ν is the luminosity produced in the form of neutrinos which usually escape the star without interacting with ordinary matter) per unit mass. Outside the core of the star, where nuclear reactions occur, no energy is generated, so the luminosity is constant. Actually the energy transport equation takes different forms depending upon the mode of energy transport. Outside a star the density and the pressure vanish and only the metric parameters Φr) and Λr) need to be considered. We also know that the mass mr) stays constant for values of r greater then R i.e. outside the star), so we call mr) = M for r > R 15

18 How to construct a stellar model By integrating the equation in dφ/dr, with p = 0 and m = M, and by imposing the boundary condition Φr = ) = 0, one finds Φr) = 1 2 ln 1 2M ) for r > R 1.39) r Consequently, outside the star the spacetime geometry becomes ds 2 = 1 2M ) dt 2 dr + 2 r 1 2M/r) + r2 dθ 2 + sin 2 θdφ 2) 1.40) that is called Schwarzschild line element, because Karl Schwarzschild discovered it as an exact solution to Einstein s field equation a few months after Einstein had formulated General Relativity theory. Summing up, the most efficient procedure to construct a stellar model is to evaluate the four equations of 1.28), then evaluate enough field equations six eq.) to obtain a complete set ten eq.)and in the end evaluate the remaining four field equations as checks of the results of the previous points. Combined with a set of boundary conditions, a solution of these equations completely describes the behavior of the star. Unfortunately, for realistic equations of state, these equations cannot be integrated analytically and numerica integration is necessary; however we will consider idealized equation of state for which analytic solutions exist. Stellar structure equations associated with boundary conditions, all gathered together with the line element, result as follows: line element ds 2 = 1 2M r ) dt 2 + dr2 1 2M/r) + r2 dθ 2 + sin 2 θdφ 2) for r > R mass equation mr) = R 0 4πr2 ρdr with mr = 0) = 0 there is no mass inside the center of the star, as required if the mass density remains finite) and mr = R) = M the total mass of the star is the star s mass) hydrostatic equilibrium dp dr = p+ρ)m+4πr3 p rr 2m) with pr = R) = 0 the pressure at the surface of the star is zero) and pr = 0) = p c central pressure) source equation for Φ dφ with Φr = R) = 1 2 ln 1 2M R dr = m+4πr3 p rr 2m) ) and Φr = 0) = Φ0 equation of state with TR) = T e f f the temperature at the surface is the effective temperature of the star). 16

19 Polytropes Recalling that Birkhoff s theorem guarantees that the exterior spacetime will be the Schwarzschild one, we easily deduce that the metric functions will be given by g rr = r 4πr3 3 0) ρ for r R interior) ) 1 2M 1 r for r > RSchwarzschild) ) e Φ 3 = 2 1 2M 1/2 ) 1/2 R Mr2 for r R interior) R 3 1 2M R for r > RSchwarzschild) where M is the gravitational mass of the star given by 1.4 Polytropes M = R 0 4πr 2 ρ 0 dr = 4π 3 R3 ρ 0 A fundamental principle that enables a simple solution of the structure equations is finding a property that changes moderately enough from the stellar centre to the surface to allow us to consider it as uniform independent on the radius r). At fisrt sight, this appear strange, keeping in mind that for example the pressure is expected to change throughout a star by more then 14 orders of magnitude. However there are properties, such as the matter composition, that do not change significantly with radial distance. Looking for a simple model based on the principle of a uniform property, we will examine polytropic models, that will be very useful in the following chapters. In astrophysics, a polytrope refers to a solution of the Lane-Emden equation in which the pressure depends upon the density in the form p = Kρ n+1)/n 1.41) where p is the pressure, ρ is the density, n is a constant known as the polytropic index and K is another constant. Given an ideal gas and given a polytropic index, the constant K is ) NA k K = B T ρ 1/n µ 17

20 Polytropes and the expression on the right hand side is assumed to be constant throughout the solution. Here N A is Avogadro s number, k B is Boltzmann s constant, T is absolute temperature and µ is mean molecular weight, i.e. the atomic weight of an average gas particle. We also have to know that the higher the polytropic index is, the more condensed at the centre is the density distribution. According to this we can state some physical examples of models with a specific polytropic index: 0.5 < n 1: neutron stars n = 1.5 γ = 5/3): degenerate electron gas in star cores like those of red giants), white dwarfs, brown dwarfs, giant gaseous planets like Jupiter), in the non-relativistic case n = 3 γ = 4/3): main sequence stars like our Sun Eddington standard model of stellar structure), in the extreme relativistic limit n = 5 with an infinite radius: self-consistent stellar system first studied by A. Schuster in 1883) n = : isothermal sphere, that is an isothermal self-gravitating sphere of gas, whose structure is identical with the structure of a collisionless system of stars like a globular cluster. Actually, what we are going to do later on is to refer to an equation of state that looks similar to the thermodynamic relation above, that we call polytropic equation of state. However we will use a more useful expression for our calculations, that is p ) 1/γ p ρp) = ) K 1 γ Looking back at the stellar stucture equations, we notice that if the pressure were only a function of the density and composition, of course), the first two equations in dp/dr and dm/dr would be independent and could be solved separately: analytic solutions of this form are more than a century old. Multiplying equation 1.37) by r 2 /ρ and differentiating with respect to r one gets d r 2 dr ρ ) dp = G dm dr dr 18

21 Polytropes Substituting equation 1.33) on the right-hand side, we obtain 1 d r 2 r 2 dr ρ ) dp = 4πGρ 1.43) dr We now consider a polytropic equation of state of the form p = Kρ γ 1.44) where K and γ are costants, and where it is customary to define the corresponding polytropic index, denoted by n, as γ = n 1.45) Substituting equations 1.44) and 1.45) into equation 1.43), we obtain a second-order differential equation n + 1)K 4πGn 1 d r 2 dr r 2 ρ n 1 n ) dρ = ρ 1.46) dr The solution ρr) for 0 r R, called a polytrope, requires two boundary conditions: ρ = 0 at the surface r = R) and dρ/dr = 0 at the centre r = 0). Hence a polytrope is uniquely defined by three parameters, K,R and n, and it enables the calculation of additional quantities as functions of radius, such as pressure and mass. 19

22 Chapter 2 A no-go theorem for polytropic stars in Palatini fr) gravity 2.1 Abstract First of all we have examined from the mathematical point of view the work of Enrico Barausse, Thomas P.Sotiriou and John C.Miller, called A no-go theorem for polytropic spheres in Palatini f R) gravity, published on the Classical Quantum Gravity the 4th of March In this work non-vacuum static spherically symmetric solutions in Palatini f R) gravity are examined. Briefly it is shown that for generic choices of f R), there are commonly used equations of state for which no satisfactory physical solution of the field equations can be found, apart from in the special case of GR, casting doubt on whether Palatini f R) gravity can be considered as an alternative to GR. As mentioned before, we examine the Palatini formalism of f R) gravity and some of its properties seen in the previous chapter. Recalling how this class of theories comes about as a generalization of GR, we have to keep in mind that the most important thing in this case is that Einstein s equations will be derived from the Einstein-Hilbert action by making independent variations with respect to the metric and the connection. As the article was rather short with plenty of omitted calculations, we prefer deriving again all the equations, step by step, with the help of Maple too, in order to avoid mistakes and organize a complete mathematical exposition of the subject. 20

23 Field equations 2.2 Field equations As we consider separately the metric g µν and the connection Γ λ µν, for clarity of notation we always denote the quantities constructed with the independent connection with tilde. Note one more that in order to derive the fields equations we have to assume that the matter action does not depend on the independent connection. Considering the well-known action for Palatini theories S[g, Γ, ψ] = 1 2k 2 d 4 x g f R) + S M [ gµν, ψ ] 2.1) where, as usual, where we are using relativistic units c = h = G = 1, k 2 8πG, G is the gravitational constant, g is the determinant of the metric g µν, R = g µν R µν is the Ricci scalar constructed with the independent connection, ψ denotes collectively the matter fields and S M is the matter action. In addiction to this, f R) is a function of the scalar curvature. As we have already seen, independent variations with respect to the metric and the connection give f R) R µν 1 2 f R)g µν = k 2 T µν 2.2) λ [ g f R)g µν] = 0 2.3) that are perfectly analogous to 1.13) and 1.14), where f R) = f / R, T µν is the usual stress-energy tensor of the matter given by 1.5) and µ is the covariant derivative with respect to the independent connection Γ λ µν. Setting f R) = R, they leads to the standard GR Einstein s equations, while 2.3) in this case incorporates the definition of the Levi-Civita connection. Many studies have been made about Palatini s f R) theory and its cosmological aspects. However, the question of obtaining solutions describing stars and compact objects has received only a small attention so far. We now focus on the problem of finding consistent solutions for static spherically symmetric stellar models when f R) = R. We note that fortunately Palatini f R) gravity retains a fundamental characteristic of GR: the exterior spherically symmetric solution is unique Birkhoff s theorem). As we have seen before, we need to take the trace of 2.2): f R) R 2 f R) = k 2 T 2.4) 21

24 Stellar model that is exactly equation 1.15), where T g µν T µν. For a given function of the scalar curvature, this is an algebraic equation in R and therefore it can be solved to give R as a function of T and for us this equation always has roots. Since equation 2.4) also implies that if T = 0, i.e. in vacuum, R must be constant, denoting the value of R when T = 0 by R 0 and insert it 2.3), this equation reduces to the covariant conservation of g µν, where the connection is the Levi-Civita one. Thus, in vacuum 2.2) reduces to R µν Λ R 0 )g µν = 0 2.5) where R µν is now the Ricci tensor of the metric and Λ R 0 ) = R 0 /4. According to whether R 0 is zero or not, which of course depends on the choice of f R), the theory reduces in vacuum to GR without or with a cosmological constant. So we know that the vacuum spherically symmetric solution is unique in both the cases, being respectively Schwarzschildde-Sitter or Schwarzschild-anti-de-Sitter solution. Having determined an exterior solution, we then need to find an interior solution and perform a matching between the two. Following our previous calculation, one has to solve 2.3) for Γ λ µν, insert this into 2.2) and express the resulting equation in terms only of metric quantities. Then the field equations 2.2) and 2.3) can be rewritten as a single one, just like 1.22): G µν = k2 R ff ) + 1 f µ ν g µν ) f + f T µν 1 2 g µν 3 2 f 2 [ µ f ) ν f ) 1 2 g µν f ) ] 2.6) 2 where µ is the covariant derivative with respect to the Levi-Civita connection of g µν and = g µν µ ν is the dalambertian. This equation will be the starting point in order to derive the structure equations for a star. 2.3 Stellar model Stucture equations Now we use the static spherically symmetric ansatz ds 2 = e Ar) dt 2 + e Br) dr 2 + r 2 dω 2 2.7) 22

25 Stellar model with dω 2 = dθ 2 + sin θ 2 dφ 2 It is well-known that the solution to Einstein s field equation in vacuum is the static Schwarzschild-de-Sitter metric or Schwarzschild-antide-Sitter one) in the form e A = Ct) e B = 1 2GM Λ ) r 3 r2 1 2GM Λ ) 1 r 3 r2 where obviously A = Ar) and B = Br), Ct) and M are integration constants and Λ is the cosmological constant. If we focus now on the interior solution, we have to write equations 2.6). The non-zero components of Einstein tensor on the left-hand side are G tt = ea B r r + e B 1) r 2 e B G rr = A rr e B + 1 r 2 G θθ = 1 r2a r 2B r + 2rA rr + ra 2 r ra r B r ) 4 e B G φφ = 1 r2a r 2B r + 2rA rr + ra 2 r ra r B r ) 4 e B sin 2 θ where A r = dar)/dr, B r = dbr)/dr and A rr = d 2 Ar)/dr 2. Looking at the right-hand side, we first need to consider perfect-fluid matter with T µν = ρ + p)u µ u ν + pg µν 2.8) where ρ = ρr) is the energy density, p = pr) is the pressure and u µ = [α, 0, 0, 0] is the fluid 4-velocity. Inserting the ansatz 2.7), it becomes ρe A T µν = 0 pe B pr pr 2 sin 2 θ Then, calculating all the terms, one gets the following non-zero components of the right-hand side that we call collectively T µν e A T tt = 1 4 re B f ) 2 4k2 f re B ρ + 2re B R f ) 2 2 f re B f f r f rr 2 f r f rb r + 8 f f r 3 f r) 2 ) 23

26 T rr = 1 4r f ) 2 4k2 f rpe B + 2re B R f ) 2 2 f re B f + T θθ = 1 4 T φφ = f f r A r r + 8 f f r + 3 f r) 2 r) r e B f ) 2 4k2 f rpe B + 2re B R f ) 2 2 f re B f f f r + 2 f f r A r r + 4 f r f rr 2 f r f rb r 3 f r) 2 r) r e B f ) 2 4k2 f rpe B + 2re B R f ) 2 2 f re B f + Stellar model + 4 f f r + 2 f f r A r r + 4 f r f rr 2 f r f rb r 3 f r) 2 r) sin 2 θ where f d f /dr, f r d f /dr, f rr d 2 f /dr 2. Thus we can write the equations G µν = T µν in an explicit form. The combinations G tt ± G rr of the latter, written in a compact form, are the following A r = 1 1 e B eb 1 + σ r f k2 rp + α ) 2.9) r where B r = σ α r e B r f ) 2 r f + 2 f r r f + eb 2 β r 2 f rr f 3 2 eb f k2 rρ + α + β r f ) ) 2 r f ) 2.10) R f f ) ) 2.11) 2.12) σ r f r 2 f 2.13) The conservation law for energy-momentum tensor, i.e. µ T µν = 0, can be expressed as follows p r = A r ρ + p) 2.14) 2 and if we substitute equation 2.9) in this one, we get p r = σ ρ + p) rr 2Mr)) Mr) + k2 r 3 p 2 f α r 2Mr)) 2 ) 2.15) 24

27 Stellar model where we have defined Mr) r1 e B )/2 2.16) Unlike in GR, it is in general not possible to obtain an explicit solution for Br), or equivalently, for Mr). However, we can use 2.10) to obtain a differential equation for Mr), that is M r = 1 k 2 r 2 ρ 1 + σ 2 f + α + β 2 Mr) ) α + β σ) ) where Br) is completely eliminated in favour of Mr). Given an equation of state p = pρ) by which we consider that matter can be described, equations 2.15) and 2.17) will completely determine pr), ρr) and Mr), and also the metric via definition 2.16) and the source equation 2.9). So, when the equation of state is specified, one can in principle solve the above equations and derive an interior solution. However, this is hard to do in practice because the equations are implicit: if we pay attention, we see that their right-hand sides include through f r and f rr both first and second derivatives of the pressure. We need therefore to put them in an explicit form, which allows us not only to solve them numerically, but also to study their behaviour at the stellar surface. Multiplying 2.15) by d f /dp and using the definitions of α and σ, we get a quadratic equation in f r, whose solution is f r = 4r f C f )r 2M) + D 2 r 2 3C 4 f )r 2M) 2.18) where M = Mr) given by 2.16), D = ±1 and where we have defined C = d f d f dρ ρ + p) = p + ρ) 2.19) dp dρ dp [ ] = f r 2 r 2M) 8 f C f ) 2 r 2M) [ ] f r 2 r 2M) C4 f 3C)2k 2 p f R + f )r f M) 25

28 Politropic equation of state We will now focus on polytropic equation of state given by Stellar model p = Kρ γ 2.20) where as usual p is the pression, ρ is the mass density and K and γ are constants. As we have seen in the previous chapter talking about polytropes, this equation can be rewritten as p ) 1/γ p ρp) = + K 1 γ giving a direct link between p and ρ. Recalling now equation 2.19) C = d f d f dρ ρ + p) = p + ρ) dp dρ dp 2.21) we note that have written C in terms of d f /dρ because we can prove that it is finite at the stellar surface, i.e. r = r out where p = ρ = 0. In fact, d f dρ = d f d R d R dt dt dρ = d f d R d R dt 3 dp ) dρ ) where d f /d R and d R/dT are in general finite even when the trace of the energy-momentum tensor given by T = 3p ρ goes to zero. Let us examine now the term dp/dρ: from equation 2.20) we easily get p ) 1/γ ρ = 2.23) K Deriving equation 2.20) with respect to ρ and susbstituting 2.23), we find [ dp p ) ] 1/γ γ 1 p ) γ 1 dρ = Kγργ 1 γ = Kγ = Kγ 2.24) K K where it is evident that dp dρ 0 for p 0 26

29 Matching interior and exterior solutions This allow us to say that d f /dρ is finite approaching the stellar surface. Deriving instead equation 2.23) with respect to p, we have where it is clear that dρ dp = γ p γ 2.25) γ K1/γ dρ dp for p 0 However, if we compute the product p + ρ)dρ/dp where in this case ρ + p) dρ dp ρ + p) dρ dp γ p γ 2.26) γ K2/γ 0 for p 0 if 0 < γ < 2 To sum up, we have seen that when p 0, while dρ/dp diverges, the term p + ρ)dρ/dp of C goes to zero if γ < 2. Therefore, for a polytrope with γ < 2, in according with equation 2.19), having d f /dρ finite and p + ρ)dρ/dp 0, we find that C = 0 at the surface. 2.4 Matching interior and exterior solutions We now try to match the interior solution of the field equations to the exterior one. For what we said before, the general solution to 2.5) is e Br) = le Ar) = 1 2m r R 0 r ) where l and m are integration constants which have to be fixed by requiring continuity of the metric coefficients across the surface and R 0 is again the vacuum value of R. Using the definition 2.16) of Mr) and eq. 2.27), this gives in the exterior, Mr) = r 2 1 e Br) ) = r 2m 2 r R ) 0 r 2 = m + r 3 R )

30 Matching interior and exterior solutions Besides continuity of the metric, one has to impose some junction conditions for A r : this is what we are going to do now. The exterior solution evaluated at the surface, i.e. r = r out, gives A r r = r out ) = 2r 3 out R 0 12m) r out R 0 r 3 out 12r out + 24m) 2.29) The value of A r r out for the interior solution can be calculated with 2.9) and for this we need to express f rr = r out ) evaluating equation 2.16) at the surface, because this term appears in σ that itself appears in 2.9). As we have seen before, when r = r out, p = ρ = 0 and also C = 0, whereas R, f and f take respectively their constant vacuum values R = R 0, f = f 0 and f = f 0 = 1 2 f 0 R 0 Thus from 2.18) evaluated at the surface, we get f rr = r out ) = 4r out f 0 f 0 )r out + D 2 r 2 out 4 f 0 )r out = 4r2 out f D) 4 f 0 r3 out = 4r2 out f Dr out 2 f f 0 r3 out = 1 + D) f 0 r out 2.30) where D = Dsignr out 2Mr)). More precisely, we have to remark that, unlike in GR, one cannot prove that r out > 2Mr) because p r is not necessarily positive, although one may expect this in sensible solutions. Choosing D = 1 implies σ = 1 at the surface: in fact σ = r f r 2 f = r out 2 f 0 2 f 0 ) = 1 r out This gives, looking back at equation 2.9), that whereas according to 2.29) A r for r rout from the interior) A r < for r r out + from the exterior) Because G µν involves A rr, this infinite discontinuity leads to the presence of Dirac deltas in the field equations. These Dirac deltas cannot be = 28

31 Matching interior and exterior solutions cancelled by the derivatives of f r on the right-hand side, because the discontinuity of f r is only a finite one, and one should invoke an infinite surface density at r = r out. Since this is unreasonable, we focus only on the choice D = 1, for which f rr = r out ) = 0 when r r out making both f r and A r continuous across the surface. In order to examine now the behaviour of Mr) at the surface, we need first to derive an explicit expression for f rr. If we take the derivative of 2.18), f rr appears on the left-hand side and also on the right-hand side through M r, calculated from 2.17) and the definition of β 2.12), giving a linear equation in f rr. The solution of this, evaluated at the surface, i.e. r = r out, is ) R 0 r f rrr out 3 = r out ) = 8Mr) C 2.31) 8r out r out 2Mr)) where C is the derivative of C with respect to r, that can be easily calculated from definition 2.31), using the chain rule, equation 2.14) and the fact that the trace of the stress-energy tensor for a perfect fluid is T = 3p ρ as mentioned before. So we get C = dc [ d dp p r = 2 f dp 2 ρ + p)2 + d f 1 + dρ ) ] Ar ρ + p) dp dp 2 = [ = A r C + d f ) dr dt dρ 2 ρ + p)] + 2 dr dt dρ dp [ A r {ρ + p) 2 d f ) dr d2 ρ 2 dr dt dp 2 + d f d 2 R dr dt 2 3 dρ ) ]} 2 + dp [ A r {ρ + p) 2 d 2 f ) dr 2 2 dr 2 3 dρ ) ]} 2 dt dp 2.32) Evaluating α, β and σ at the surface using f rr out ) = 0 and f rr given by 2.31), and inserting all into 2.17), one gets M r r = r out ) = 2 f 0 R ) 0 rout 2 + R 0 rout 3 8Mr) C 16 f ) where C is still defined by equation 2.32). By the way, we have to observe now that for equation 2.14), or equivalently 2.15), C turns out to 29

32 Matching interior and exterior solutions be proportional to ρ + p), i.e. C = dc dp p r dc ρ + p) 2.34) dp and if we use the definition of C stated in 2.19) in the latter, we get dc d d f ) ρ + p) = ρ + p) ρ + p) = dp dp dp [ d = 2 f d f ρ + p) dρ )] 2.35) ρ + p) dp2 dp dp Defining dc ρ + p) = dp f p d f dp and f pp d2 f we are able to re-express the last equation as [ f ppρ + p) + f p dp dρ dp )] ρ + p) 2.36) Inserting now the polytropic equation of state, after some manipulations one gets [ C f p p ) p ) ] 1/γ p2 γ)/γ + γ K γ K2/γ [ + f pp p ) ] 2/γ 2.37) p K Since pγ+1)/γ K1/γ f p = d f dp = d f dρ dρ dp and remembering equation 2.25), i.e. as well as its square dρ dp = γ p γ γ K1/γ we can rewrite f p as ) dρ 2 = γ) dp γ 2 p γ K2/γ f p = d f dρ dρ dp = d f 1 dρ γ 1 1 γ p γ K1/γ 30

33 Conclusions Collecting all together, we are finally able to express the following term of 2.37) ) F f p γ p γ = d f ) ) γ p γ γ p γ 2.38) γ K2/γ dρ γ K1/γ γ K2/γ and it is now evident that F p 3 2γ γ 2.39) We can conclude that this term, and consequently C of which it is part of, goes to zero at the surface for γ < 3/2 so that expression 2.33) is finite and it even gives continuity of M r across the surface. However, for γ > 3/2, this term, and consequently C, diverges as the surface is approached, i.e. when p 0. So, while M keeps finite, the divergence of C drives to infinity M r and of f rr as evident from equations 2.33) and 2.31) respectively. Looking then back to the definition of β in 2.12), we find out that also β diverges, as it contains f rr. Moreover, because of the fact that β appears in B r as equation 2.10) reveals, B r itself diverges too. If we calculate the Ricci scalar of the metric given by 2.7) at the beginning, we find it to be R = 1 2 [ rrar + 4)A r B r ) + 2r 2 A rr 4e B ] + 4 r 2 e B 2.40) As we have just proved that B r diverges, it follows that the Ricci scalar or scalar curvature), which is the simplest curvature invariant, diverges too: in this case, we can say that the singularity is true. 2.5 Conclusions To sum up this part, we have tried to find static spherically symmetric solutions of Einstein s equations in Palatini f R) gravity, for a generic f R). Looking for the exterior and interior solutions and trying to match them, we find out the presence of an unavoidable singularity, as a curvature invariant diverges, if one considers a polytropic equation of state in the range γ > 3/2. This obviously implies that the matching on the surface of polytropic stars isn t possible and that such a type of solution doesn t exist in this case. 31

34 Conclusions The same problem was reconsidered later by Kainulainen, who argued that the singularities had more to do with the peculiarities of the equation of state used than with the theory of Palatini f R) gravity: he found that the singularity appears only when the use of a polytropic equation on state would be unrealistic. However, Barausse replied successfully to this. 32