Abstract
We have derived effective gravitational field equations on a lower dimensional hypersurface (known as a brane), placed in a higher dimensional bulk spacetime for both Einstein and $f(\mathcal{R})$ gravity theories. We have started our analysis on $n$-dimensional bulk from which the effective field equations on a $(n-1)$-dimensional brane has been obtained by imposing $Z_{2}$ symmetry. Subsequently, we have arrived at the effective equations in $(n-2)$-dimensions starting from the effective equations for $(n-1)$ dimensional brane. This analysis has been carried forward and is used to obtain the effective field equations in $(n-m)$-dimensional brane, embedded in a $n$-dimensional bulk. Having obtained the effective field equations in Einstein gravity, we have subsequently generalized the effective field equation in $(n-m)$-dimensional brane which is embedded the $n$-dimensional bulk spacetime endowed with $f(\mathcal{R})$ gravity. We have also presented applications of our results in the context of Einstein and $f(\mathcal{R})$ gravity. In both the cases we have discussed vacuum static spherically symmetric solutions as well as solutions in cosmological context. Implications are also discussed.
Highlights
A simplistic model which captures most of the key notions is a 5-dimensional model in which matter fields are confined to 4-dimensional spacetime while gravity exists in all the five spacetime dimensions
Due to the presence of an extra spacetime dimension it is expected that there would be deviation of various physical results from Einstein gravity, which would become more significant in the high energy limit
As already mentioned we will start with 6-dimensional bulk spacetime and obtain cosmological solutions from the effective field equation on a 4-dimensional brane
Summary
A simplistic model which captures most of the key notions is a 5-dimensional model in which matter fields are confined to 4-dimensional spacetime while gravity exists in all the five spacetime dimensions. There exists an elegant way in which an effective field equation for gravity on a lower dimensional spacetime can be obtained This involves the use of geometric quantities like the metric and the curvature induced on the brane from the bulk. We generalize the above setup by working in an n-dimensional bulk and obtaining effective equation on an (n − m)-dimensional brane where m can take any values This seems natural from a string theory view point, where the universe is 11-dimensional, while our universe is a three-brane with four spacetime dimensions. Following this argument we have derived effective gravitational field equation on a (n − m)-dimensional brane embedded in a ndimensional bulk. Capitalized Latin letters A, B, . . . run over the (n − 2) spacetime indices
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