Abstract
Solving field equations in the context of higher curvature gravity theories is a formidable task. However in many situations, e.g., in the context of $f(R)$ theories the higher curvature gravity action can be written as Einstein-Hilbert action plus a scalar field action. We show that not only the action but the field equations derived from the action are also equivalent provided the spacetime is regular. We also demonstrate that such equivalence continues to hold even when gravitational field equations are projected on a lower dimensional hypersurface. We have further depicted explicit examples in which the solutions for Einstein-Hilbert and a scalar field system lead to solutions of the equivalent higher curvature theory. The same, but on the lower dimensional hypersurface, has been illustrated in the reverse order as well. We conclude with a brief discussion on this technique of solving higher curvature field equations.
Highlights
The energy scales in particle physics are arranged in a hierarchical manner
In the above two sections we have shown the equivalence of gravitational field equations both in the bulk and in the brane, respectively
The technique essentially hinges on the mathematical equivalence of higher curvature gravity theory, e.g., f (R) theories of gravity with scalar– tensor representation
Summary
The energy scales in particle physics are arranged in a hierarchical manner. While the scale of the weak interaction corresponds to E ∼ 103 GeV, the strong interaction at a scale of E ∼ 1016 GeV exceeds the weak scale by a factor of 1013. We can use it to solve field equations for scalar–tensor theory and obtain the solution corresponding to f (R) action and vice versa. This would be advantageous, since in general solving the field equations for f (R) gravity, where R is not a constant, is difficult1 [22,41,42,43,44,45,46,47]. This is evident, since the conformal factor can change the complete structure of the spacetime This fact was pointed out earlier in [56] by showing that through a conformal transformation one can create matter and as a result, one frame is empty while another has matter, and clearly they are physically non-equivalent. The Latin indices, a, b, . . . runs over the full spacetime indices, while Greek indices, μ, ν, . . . stand for fourdimensional spacetime
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