GR-GSG Hybrid Gravity
arXiv:1606.08980 [gr-qc]
Nahomi Kan, Takuya Maki and Kiyoshi Shiraishi
We propose a model of gravity in which a General
Relativity metric tensor ( g ) and an effective metric ( q)
generated from a single scalar formulated in Geometric
Scalar Gravity are mixed. We show that the model yields
the exact Schwarzschild solution, along with accelerating
behavior of scale factors in cosmological solutions.
§1. Brief review of GSG
§2. Construction of hybrid model
§3. Static Spherical Solutions
§4. Cosmology with two metrics
§5. Cosmology with composite metric
§6. Summary and Prospects
§1. Brief review of GSG
M. Novello, E. Bittencourt, U. Moschella, E. Goulart, J. M. Salim and J. D. Toniato,
JCAP 1306 (2013) 014; JCAP 1401 (2014) 01, E01. arXiv:1212.0770 [gr-qc].
E. Bittencourt, U. Moschella, M. Novello and J. D. Toniato,
Phys. Rev. D90 (2014) 123540.arXiv:1412.4227 [gr-qc].
E. Bittencourt, M. Novello, U. Moschella, E. Goulart, J. M. Salim and J. D. Toniato,
Nonlinear Phenomena in Complex Systems 17 (2014) 349.
J. D. Toniato, "A teoria geometrica-escalar da gravitacao e sua aplicacao a cosmologia", Tese de Doutorado (Rio de Janeiro, 2014).
E. Bittencourt, U. Moschella, M. Novello, and J. D. Toniato, Phys. Rev. D93 (2016) 124023.
J. D. Toniato and M. Novello, arXiv:1607.01037 [gr-qc].
I. C. Jardim and R. R. Landim, arXiv:1508.02665 [gr-qc]
Effective metric:
where , η
μν
: flat metric, V(Φ) : scalar potential
action:
variation:
where
If we choose ,
GSG yields the Schwarzschild solution exactly as
a vacuum solution.
§2. Construction of hybrid model
Einstein-Hilbert action:
mixing of g and q ('minimal mass term')：
variation：
Total action of GR-GSG Hybrid Gravity
(not including matter coupling):
Equations of motion:
where
with
§3. Static Spherical Solutions
non-dynamical flat metric:
ansatz: Φ＝Φ(R)
Changing the variable yields
with
For the metric g ,
equation of motion (1)
equation of motion (2)
equation of motion (3)
equation of motion (4)
where
Solution for Weak gravity
where
Hence
,
, and upto this order.
§4. Cosmology with two metrics
matter action:
equations of motion:
where
where
metric ansatz
For the metric q:
where ,
For the metric g:
matter: perfect fluid
Cosmological equations of motion
where ,
conservation:
From Bianchi identity， . This leads to an algebraic equation:
also if we require →
, , ,
time evolution of scale factors
,
deceleration→acceleration→deceleration
20
,
deceleration→acceleration→deceleration
21
acceleration if
§5. Cosmology with composite metric
Composite metric:
equations of motion
Bianchi: ,
where and
,
a new cosmological time T :
Composite metric
, ， ↑
deceleration→acceleration→deceleration
Composite metric
,
27
Composite metric
, ， , ↑
deceleration→acceleration→deceleration
Composite metric
,
29
§6. SUMMARY AND PROSPECTS
We have presented a GR-GSG hybrid model of gravity.
In this model, we have found:
＊spherical solution given asymptotically
＊accelerating phase of the universe
+qualitative analyses, because many tunable parameters and
a possible mixing of g and q in S mix (g,q) other than the minimal
choice exist. → should be followed in future
Important issues:
＊Incorporating a mechanism for inflation
＊Initial fluctuations
＊Anisotropic solutions
＊Quantum cosmology
・the initial conditions？・the singularity problem？
＊Compact objects