Three dimensional Eddington–inspired Born–Infeld gravity: solutions
Three dimensional Eddington-inspired Born–Infeld gravity is studied with the goal of finding new solutions. Beginning with cosmology, we obtain analytical and numerical solutions for the scale factor , in spatially flat () and spatially curved () Friedmann-Roberston-Walker universes with (i) pressureless dust () and (ii) perfect fluid (), as matter sources. When the theory parameter , our cosmological solutions are generically singular (except for the open universe, with a specific condition). On the other hand, for we do find non-singular cosmologies.
We then move on towards finding static, circularly symmetric line elements with matter obeying (i) and (ii) . For , the solution found is nonsingular for with the matter–stress–energy representing inhomogeneous dust. For we obtain nonsingular solutions, for all and discuss some interesting characteristics of these solutions. Finally, we look at the rather simple case where the solutions are either de Sitter or anti-de Sitter or flat spacetime.
Theories of gravity different from General Relativity (GR) have been actively pursued by many, for a variety of reasons. One such reason relates to the possibility of avoiding the singularity problem in GR involving the occurrence of a big bang in cosmology or black holes in astrophysics. In a classical metric theory of gravity one is aware that these singularities are inevitable, as proved through the Hawking–Penrose theorems hawk (), under very general and physical assumptions. However, it is quite possible that in an alternate theory one might obtain non–singular geometries (for example non-singular Friedman–Robertson–Walker type cosmologies) as solutions–a feature which does not exist in the FRW cosmology based on GR. It may be noted that the removal/resolution of a singularity is also expected to be a basic feature of a quantum theory of gravity.
In this article, we look into one such alternate theory. Historically, its origin goes back to Eddington who showed us how de Sitter gravity could be obtained using an action where the Einstein–Hilbert term, ( is the Ricci scalar), is replaced by edd (). Eddington’s formulation allowed the choice of the connection (instead of the metric) as the basic variable –therefore, it is essentially an affine formulation. However, coupling of matter remained a problem in this formulation.
We are also aware of Born–Infeld elecrodynamics born (), which was introduced in order to get rid of the infinity in the field at the location of the charge/current. A gravity theory in the metric formulation inspired by Born-Infeld electrodynamics was suggested by Deser and Gibbons desgib (). Later Vollick vollick () worked on the various aspects of the Deser-Gibbons proposal in a Palatini formulation and also introduced a non-trivial way of coupling matter in such theories. More recently, Banados and Ferreira banados () have come up with a formulation wherein the matter coupling is different and simpler from that introduced in Vollick’s work vollick (). We will focus in the theory proposed in banados () and call it as Eddington-inspired Born–Infeld (EiBI) gravity, for obvious reasons. Note that the EiBI theory has the feature that it reduces to GR, in vacuum.
It may be noted that the theory we consider falls within the class of bimetric theories of gravity (also called bi-gravity). The current bimetric theories have their origin in the seminal work of Isham, Salam and Strathdee salam (). Numerous papers on varied aspects of such bimetric theories have appeared in the last few years. The central feature here is the existence of a physical metric which couples to matter and another auxiliary metric which is not used for matter couplings. One needs to solve for both metrics through the field equations.
Let us now briefly recall Eddington–inspired Born–Infeld gravity. Since we deal with three spacetime dimensions in this article, we prefer to write down the action and ensuing field equations in three dimensional spacetime. The action for the theory developed in banados (), is given as:
where . Variation w.r.t , leads to the metric compatibility condition for the metric. Therefore, the is a valid Riemannian metric and is defined through the relation,
Variation w.r.t gives
In order to obtain solutions, we need to assume a and a with unknown functions, as well as a matter stress energy (). Thereafter, we write down the field equations and obtain solutions using some additional assumptions about the metric functions and the stress energy.
Quite some work on various fronts have been carried out on various aspects of this theory in the last couple of years. Astrophysical aspects have been discussed in the references in eibiastro () while cosmology in those cited in eibicosmo (). Other topics such as a domain wall brane has been analysed in eibibrane (). More recently, generic features of paradigms on matter-gravity couplings have been discussed in eibigen (). However, in eibiprob () a major problem related to surface singularities has been noticed which has put the theory on shaky ground insofar as stellar physics is concerned.
Our work here is reasonably modest. In the two subsequent sections, we discuss cosmological and circularly symmetric solutions in three spacetime dimensions, successively. In the final section, we briefly summarize and conclude. Some of our solutions are analytical and simple. They also maintain some of the generic features noted in the original work of Banados and Ferreira banados ().
Let us assume a homogeneous and isotropic Friedmann-Robertson-Walker (FRW) line element in 2+1 dimensions, given as:
where for closed, flat and open universe respectively. The energy–momentum tensor is taken to be that of a fluid with . The conservation of energy–momentum leads to
which implies (for ) and (for ). Further, we assume the auxiliary line element to be of the form
Using the auxiliary metric (), physical metric () and stress-energy tensor () in the field equation obtained by varying w.r.t. , we get two equations, given by,
where, , , and . We define two quantities and , given as,
Let us now examine two special cases: () pressureless dust () filled flat universe () with the cosmological constant, and () radiation dominated () flat universe with .
where, . Note that from the conservation law is a constant. Using this in Eq. (10), we obtain the solution for the scale factor for as,
where the constant . The solution clearly demonstrates that for the EiBI theory cannot avoid the initial singularity ( can become zero at a finite and hence we have infinite curvature). However, for , the Eq. (10) becomes,
From Eq. (12), we note that there exists a maximum density () or, equivalently, a minimum value for the scale factor (). The solution for the scale factor for is given by,
where is a constant. It is easy to see that the scale factor is never zero and thus, there is no curvature singularity. Both these solutions are plotted in the top row of Fig. 1.
For case (),(i.e radiation dominated universe), the Friedmann equation becomes:
For the flat universe (), the last term in the square-bracket of the right hand side of the Eq. (14) does not contribute and the equation becomes,
In this case, for , for an arbitrary, but physically justifiable value of (i.e for ). Thus, here also, a curvature singularity appears in the solution. Using the equation of state :, where is a constant, we can solve numerically the Eq. (15) for the time evolution of the scale-factor , which is shown in Fig. 1. However, for , the Eq. (15) is rewritten as,
The presence of the factor in the Eq. (16) leads to being negative when . Therefore, in this case, for , there exists a maximum density or, equivalently, a minimum, non–zero value for the scale factor.
Let us now define two dimensionless variables (),. Using these we recast the equations as:
Numerically solving the above equation, we plot and note the time evolution of the scale factor using the Eqs. (17). In the bottom row of Fig. 1 we note a non-singular scale factor for –a result showing the existence of a bounce, similar to that obtained in dimensions. For , the solution appears to be singular.
The above solutions (especially, the ones for the case) are instructive because they are, as far as we know, the only known analytical solutions in EiBI cosmology.
Introducing curvature in the spatial slices (i.e. ) does not yield anything drastically new in the dust () case primarily because of the fact that .The solutions for and are given by,
where is a constant ( for ). For a solution there is a lower bound on () whereas, for , is an arbitrary positive, real constant. When , we recover the earlier results (Eq. 11 and Eq. 13 ). It is easy to see that there is nothing new in the solutions.
However, in the radiation dominated () case we do get some interesting results, though the main conclusion regarding the appearance of a singularity or otherwise, is almost the same. The introduction of spatial curvature results in an additional term as shown in the square-bracket of the R.H.S. of Eq. (14) and this leads to all the differences. For a closed universe (), instead of a bounce we get an oscillation of the universe for . For there is an additional feature implying a maximum value of the scale-factor, along with the singularity. These are shown in Fig.2. In an open universe (), we do not see any characteristic novelties for , but for along with the singularity, we also get, under certain circumstances, a non-singular loitering phase of the early universe. This is shown through the plots in Fig. 3.
The main differences, in the context of cosmology, between the results in dimensions and those in the dimensional version of EiBI theory are the following. In dimensions, for , the cosmological solutions are singular for matter satisfying and . In contrast, the dimensional cosmological solutions are non-singular for both and . Further, in , for , we find a bounce solution for , whereas in , as obtained by I. Cho et. al.eibicosmo ()(see Fig.2 there), the early universe is de Sitter spacetime and is constant at early times. In eibicosmo (), an analytical expression for the scale factor in the dimensional case, has indeed been obtained but the expression involves non-invertible functions. In , with , we find analytical scale factors which involve simple functions. The loitering phase of the early universe for in dimension is absent in , except for a specific situation with an open universe (). For a radiation dominated closed universe (), we obtain a big crunch solution for (see Fig. 2), whereas, in dimensions, the solution under a similar situation has a non-singular loitering phase. Thus, the lower dimensional toy model cosmologies have expected similarities and differences when compared with their higher dimensional counterparts.
Iii Circularly symmetric, static solutions
Let us now turn to a completely different class of line elements –i.e. those which are circularly symmetric and static. We consider a simple ansatz for the physical line element ,
where , are non–negative functions and extends from minus infinity to plus infinity. represents the radius of a circle at each value of . is the so-called redshift function. The energy-momentum tensor is assumed as, =diag.. Let us further assume the auxiliary line element to be of the form
where and are non–negative functions of . The field equation obtained from variation yields,
The other field equation obtained from variation yield the following equations,
Consistency of the last two equations (since the L. H. S. of both are the same) leads to the simple relation
where is a constant. Further, the conservation law implies
Using the expressions for and given earlier, we arrive at
which, using the expression becomes
We can now look for possible solutions of the above equations.
The first of these involves assuming which, from the equations imply . Equivalently, , i.e. and are both constants. With , Eqn (22) is vacuous and the remaining single equation for is
To solve this equation we need another condition. With , we note that and hence there is no scope of assuming an equation of state. In other words the solution we are looking for is produced by inhomogeneous dust in the presence of a . To find one such solution, we assume a relationship between and . Let us take
where is a constant. Thus, we can transform the second order differential equation (Eq. 29) into an easily solvable first order ordinary differential equation for . Other choices for the R. H. S. of Eq. 30 (may not be a constant) could give other mathematical solutions for which we need to verify whether the energy-density distributions () are physically acceptable.
We now demonstrate one solution assuming the condition in Eq. (30) and . Let us also assume . The solution we obtain is specified by the and given below:
The energy density is given as
and is positive as long as and . The corresponding Ricci scalar for the physical metric is,
which is clearly non-singular and has asymptotically constant negative curvature. The Kretschmann scalar is
and is non-singular as well. The physical line element, for this case can be written as,
One can also rewrite it as,
and . Thus, and a metric singularity occurs in the transformed metric function only at . We plot the transformed metric function and have shown it in Fig. 4. Note that this solution is not asymptotically flat (i.e. does not tend to zero as ). However, there is a minimum radius and the geometry is symmetric as . Also . Thus, the features are similar to that of a Lorentzian wormhole wormhole () though the geometry is not asymptotically flat (in fact it is asymptotically anti-de Sitter).
In Fig. 5 we have shown the and metric functions, the Ricci scalar (), and the energy-density () for the solution quoted above.
We note that the energy-density is asymmetric as , even though the physical metric is symmetric in . This happens because the metric is asymmetric and the field equation which contains the energy–momentum tensor, has contributions from the and metrics. The asymmetry is also evident from the expression for which clearly depends only on .
If , assuming the same relation (Eq. 30) between and , we get a singular solution. In this case, becomes restricted and at the boundary of , both the Ricci scalar and the energy density () diverge.
In this case, the equations we need to solve are Eqns. (21), (22), (23) or (24), (25) and (28) which are six equations in the six variables , , , , and . However, we now show that Eqns (22) and (24) (or (23)) are not independent equations. To this end, we define
Using these definitions and the other equations, one can show that Eqn. (22) reduces to
Further, one may rewrite Eqn (24) using the new variables defined above. as obtained from the reduced version of (22) matches with the as obtained from (24). Hence these two equations are not independent. We exploit this feature of the system of equations to impose an equation of state restriction. This is chosen to be .
Using the above equation of state, it is straightforward to write down all the unknown functions as functions of . These are given below as,
where, we have assumed , defined and introduced a non-zero positive constant . Further, we have, from the field equations, the following equation for ,
Due to the presence of the factor in the denominator of the second term in the L.H.S, the Eqn. (44) can be solved only for for and for . Though the range of is restricted but can approach its limiting values and be as close as required. One can obtain qualitative analytical information about the nature of the solutions by using approximations. For instance, when , we can show, using a Taylor series about , that the solution will behave as . On the other hand, when , the solution behaves generically as (, being constants). The equation for can be easily solved numerically and the solutions for and are shown in Figs 6,7. Note that the plot for as a function of matches qualitatively with the approximate solutions found in the neighborhoods of and .
We also note that the Ricci scalar is regular everywhere though the does become zero at . We will now try to see analytically, why the solution is regular everywhere. The expression for the Ricci scalar as a function of turns out to be
From the differential equation for we can show, by taking another derivative w.r.t. , that is finite everywhere including the location of the maximum density at where and . The remaining terms are all finite everywhere and thus the Ricci scalar is finite. One can also check that the Kretschmann scalar
is finite everywhere and therefore is also finite. Thus at the location of a maximum density the geometry remains nonsingular though the radius becomes zero. This implies the vanishing of the circumference of the circle at constant and . The vanishing of at would have implied a singularity (as in cosmology, where implies a singularity) in GR. However, in EIBI theory it does not imply a curvature singularity essentially due to presence of a non–zero . In the cosmological context, such regular solutions with vanishing volume spatial hypersurfaces have been noted earlier in the literature van (). The form of as shown in Figs. 6,7 indicates the absence of any horizon ( is never zero) in the geometry. The energy density and the pressure are both finite and positive definite everywhere as is clear from the graphs in Figs. 6,7. We also note that the spacetime is asymptotically flat and tends to one of the vacuum () solutions, const., and ( and being constants), at infinity.
In this simple case we solve a similar set of five equations, assuming the equation of state , with . From the Eqn. (26), we have ( is a positive constant). Using this in Eqn. (21), we get and . Further, using these results in the remaining equations, we finally obtain the second order linear homogeneous differential equation for , which is
Solving the Eqn. (47) and using other relations, we get trivial but non-singular solutions. For , the solutions for , and are
The resulting physical line element is de-Sitter spacetime. Similarly, for , we get de-Sitter spacetime (if but ). For and , we get,
This solution represents anti-de Sitter spacetime. For and , we find,
which is just flat spacetime.
The main purpose behind this article was to find simple solutions in three dimensional EiBI gravity. We have focused here on two types of solutions –cosmological and static, circularly symmetric. We summarise our results pointwise below.
We have found analytical, spatially flat cosmologies for pressureless dust () and numerical solutions for . We have also explored the cases when spatial curvature is present. The cosmological solutions presented here seem to point to a generic feature that singularities are not present if whereas singularities do arise if .
In the circularly symmetric, static case we have found an analytical solution for inhomogeneous dust (). We have also obtained static, circularly symmetric, numerical solutions for the equation of state and have briefly analysed the case. In the class of circularly symmetric, static solutions found we note that they are nonsingular for all , except in the case for . The solution exhibits the curious feature of being a regular (non-singular) solution though the radius of the circle vanishes at , which usually would imply a singularity. This feature is exclusively due to the structure of the modified theory (EiBI).
We are aware that in General Relativity extensive work has been done on cosmological solutions (see for example barrow ()). It may be possible to use some of these ideas in EiBI gravity too. Further, our results for the circularly symmetric, static case can easily be generalised to equations of state such as or the well-known polytropic one.
In conclusion, we raise a few relevant questions. For instance, one may ask– are there black hole solutions in the dimensional EiBI theory? The answer is surely hidden in the field equations and is worth exploring. In the context of the cosmological solutions, it is important to know about the behaviour of fluctuations about a given background solution. A study of such fluctuations can surely be carried out using the exact analytical solution found here (for a spatially flat FRW line element with ) as the background. For the more general cases (say ) the numerically obtained spacetimes may be used as backgrounds to investigate fluctuations, numerically.
Finally, it is possible that our work on solutions in this toy three dimensional theory may provide viable pointers towards finding new analytical or numerical geometries in the real four dimensional world.
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