Scalar Field Static Spherically Symmetric Solutions in Teleparallel F(T) Gravity
Abstract
:1. Introduction
- Quintessence: This form describes a controlled accelerating universe expansion where energy conditions are always satisfied, i.e., [60,61,62,63,64,65,66,67,68,69,70,71,72]. This usual DE form has been significantly studied in the literature in recent decades for the fascination it provokes and the realism of the models.
- Phantom energy : This form can usually describe an uncontrolled universe expansion accelerating toward a Big Rip event (or singularity) [73,74,75,76,77,78,79,80,81]. The energy condition is violated, i.e., . But this DE form is fascinating because we can find new teleparallel solutions and physical models.
- Cosmological constant : This primary DE form is an intermediate limit between the quintessence and phantom DE states, where . A constant scalar field source added by a positive scalar potential will directly lead to this primary DE state. Note that a non-positive scalar potential (i.e., ) will not lead to a positive cosmological constant and/or a DE solution.
- Quintom models: This is a mixture of previous DE types, usually described by some double scalar field models [86,87,88,89,90,91]. This type of model is more complete to study and solve in general. Several types of models are in principle possible and these physical processes need further studies in the future.
2. Summary of Teleparallel Gravity and Field Equations
2.1. Summary of Teleparallel Field Equations
2.2. Static Spherically Symmetric Coframe and Spin-Connection Components
2.3. Static Scalar Field Energy-Momentum Source
2.4. Static Scalar Field Source Field Equations
3. Power-Law Scalar Field Solutions
- Ordinary matter limit ():By substituting Equation (25) into Equation (33), we find as :Equation (35) leads to the ordinary matter (or dust) equivalent EoS under the limit (i.e., for any density of ordinary matter expression). This case describes an ordinary matter and/or a bosonic scalar field. For the current paper, we will consider this case as the ordinary matter limit because Equation (35) under the limit leads to .
- Quintessence: We must consider a value of higher than because of dark energy’s physical limit. Under the consideration of and from Equations (29), (32) and (35)’s respective results, we find the constraints to satisfy for the quintessence process asThe Equation (35) condition for leads to and this last case cannot lead to any DE scalar field. The constraint system of Equations (36)–(38) can yield, in principle, a significant number of possible solutions, because of terms inside each of the equations. Therefore, Equations (36)–(38) provide a minimal value for a characteristic equation solution and Equation (36) will lead to a maximal value for and p.
- Phantom energy: Because of in this type of model, we will find constraints from Equations (29) and (32) to satisfy:Equation (35), under the limit, cannot consistently lead to any phantom DE model. Once again, in the case of Equations (39) and (40), there are a significant number of possible solutions for and p. But the main point is that Equations (39) and (40) provide, respectively, a minimal and a maximal value for .
3.1. Power-Law Ansatz for
3.2. Power-Law Ansatz for
3.2.1. Flat Cosmological Case
- General ():
- : constant. Equation (51) will be a power-law like solution.
- Ordinary matter limit ():
- General ():
- : constant. Equation (54) will be a GR (TEGR-like) solution.
- Ordinary matter limit ():
3.2.2. General Cases
- (a)
- General ():
- (b)
- :
- (c)
- Ordinary matter limit ():
- : Equation (47) will be
3.2.3. Cases
3.2.4. Cases
3.3. Other Ansatzes and Possible Comparisons with the Literature
4. Other Scalar Field Source Solutions
4.1. Exponential Scalar Field Solutions
- By combining Equations (98) and (99), we find in general that for positive and terms. The and classes of solutions found in Section 3.2.3 and Section 3.2.4 as the Section 3.2.1 and Section 3.2.2 are in principle all ideal candidates for describing quintessence processes.
- Phantom Energy: We only require that from Equation (96) asIn this case, we have primarily that the and ideally that term will be dominating for . Only the general solution obtained in Section 3.2.2 for positive values of a can be a candidate for phantom energy models with an exponential scalar field.
- Power-law ansatz with : We find the same solution form as Equation (43) in Section 3.1, but the potential will be Equation (95) with defined by Equation (41). Equation (95) becomes:
- Power-law ansatz with : We find the same solution forms as in Section 3.2, but only the potential expressions change by replacing Equations (28), (31) and (34) by Equation (95) for each subcase treated in this section. Equation (95) will be for the simplest cases:
- (a)
- : The potential in Equation (51) is
- (b)
- and : The expression in Equation (59) is
- (c)
- (d)
- and : The expression in Equation (74) is
- (e)
- and : The expression in Equation (78) is
- (f)
- and : The expression in Equation (84) is
- (g)
- and : The expression in Equation (88) is
The other subcases of Section 3.2 can be computed by the same manner as in the previous simple examples.
4.2. Logarithmic Scalar Field Solutions
- By satisfying the Equation (112) criterion, we automatically satisfy Equation (113)’s requirement. To guarantee a -valued expression, it requires that in Equations (112) and (113) for a quintessence process. For the subcase, Equation (112)’s constraint simplifies as . Any Section 3.2 may lead in principle to quintessence solution.
- Phantom Energy: Equation (111) with requirement leads to the following constraint:For any positive and solution (or -valued), we need to satisfy the criterion. In a such case, only the Section 3.2.2 solutions may lead to the phantom energy models and any Section 3.2.3 and Section 3.2.4 cannot lead to this type of models (due to the negative values of a).
- Power-law ansatz with : As in Section 4.1, the solution is under the same form as Equation (43) in Section 3.1. The potential will be Equation (110) with defined by Equation (41) as
- Power-law ansatz with : As in Section 4.1, we find the same solution forms than Section 3.2, but only the potential expressions change for the Equation (110) form for each subcases treated in this section. Equation (110) will be for the simplest cases:
- (a)
- : The potential in Equation (51) is
- (b)
- and : The expression in Equation (59) is
- (c)
- (d)
- and : The expression in Equation (74) is
- (e)
- and : The expression in Equation (78) is
- (f)
- and : The expression in Equation (84) is
- (g)
- and : The expression in Equation (88) is
As in Section 4.1, the other subcases of Section 3.2 can be computed in the same manner as in the current section examples.
4.3. Graphical Comparisons and Summary of Main Results
Summary of Main Analytical Solutions
5. Concluding Remarks
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
AL | Alexandre Landry |
DE | Dark Energy |
EoS | Equation of State |
GR | General Relativity |
KV | Killing Vector |
NS | Neutron star |
TRW | Teleparallel Robertson–Walker |
BH | Black Hole |
DoF | Degree of Freedom |
FE | Field Equation |
KS | Kantowski–Sachs |
NGR | New General Relativity |
TEGR | Teleparallel Equivalent of General Relativity |
WD | White Dwarf |
Notation | |
coordinate indices | |
tangent space indices | |
spacetime coordinates | |
, , or | coframe expressions (tetrad for orthonormal frames) |
, | spin-connection |
gauge metric | |
spacetime metric | |
curvature tensor | |
torsion tensor | |
T | torsion scalar |
superpotential | |
teleparallel theory function of T | |
, | derivatives with respect to (w.r.t) T |
Einstein tensor | |
, | conserved energy-momentum tensor |
covariant derivative | |
Lorentz Transformation | |
hypermomentum | |
scalar field | |
radial coordinate derivative |
Appendix A. Field Equation Components
Appendix A.1. General Components
Appendix A.2. A 3 =c 0 = Constant Power-Law Components
Appendix A.3. A 3 =r Power-Law Components
Appendix B. Tables of Section 3.2.3 and Section 3.2.4 Special Functions and Solutions
b | ||
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N.A. | ||
1 | N.A. | |
N.A. | ||
2 | ||
N.A. | ||
3 | ||
b | |
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1 | |
2 | |
3 |
b | ||
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1 | ||
N.A. | ||
2 | N.A. | |
N.A. | ||
4 | ||
N.A. | ||
6 | ||
b | |
---|---|
1 | |
2 | |
4 | |
6 | |
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Landry, A. Scalar Field Static Spherically Symmetric Solutions in Teleparallel F(T) Gravity. Mathematics 2025, 13, 1003. https://doi.org/10.3390/math13061003
Landry A. Scalar Field Static Spherically Symmetric Solutions in Teleparallel F(T) Gravity. Mathematics. 2025; 13(6):1003. https://doi.org/10.3390/math13061003
Chicago/Turabian StyleLandry, Alexandre. 2025. "Scalar Field Static Spherically Symmetric Solutions in Teleparallel F(T) Gravity" Mathematics 13, no. 6: 1003. https://doi.org/10.3390/math13061003
APA StyleLandry, A. (2025). Scalar Field Static Spherically Symmetric Solutions in Teleparallel F(T) Gravity. Mathematics, 13(6), 1003. https://doi.org/10.3390/math13061003