Abstract
We derive conservation laws for collisions of self-gravitating n-branes (or n-dimensional shells) in an (n+2) dimensional spacetime including induced gravity on the brane. Previous work has shown how geometrical identities in general relativity enforce conservation of energy-momentum at collisions. The inclusion of induced gravity terms introduces a gravitational self-energy on the brane which permits energy-momentum conservation of matter fields on the brane to be broken, so long as the total energy-momentum, including induced gravity terms, is conserved. We give simple examples with two branes (one ingoing and one outgoing) and three branes.
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Notes
We here dropped the normalisation factor to simplify notation. For our purpose, this would not play any role because this normalisation factor would be identical for every branes and thus would always simplify.
We require that the curvature of the n-space must be the same for two regions separated by a brane.
As the careful reader might have noticed, we have here chosen “angles” between two branes and not only between a bulk and a brane. This is possible by using the fact that α a|b =α a|I +α I|b with I the region between the branes “a” and “b”.
Note that (32) necessarily lead to identical spacetime on both side of the brane. Indeed for nil-branes, it is understandable that “both sides” are identical because a nil-brane is the vacuum.
Just as a remind, due to the \(\mathbb{Z}_{2}\) symmetry of branes “a” and “c”, α a|c =0.
We did not study a system of more that three \(\mathbb {Z}_{2}\) branes because one dimension of spacetime would disappear at the collision.
Note that “time” can be reversed and the normal incoming brane can become an outgoing brane. The instant decay would become an instant creation of matter.
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Acknowledgements
I would like to thank D. Wands for his supervision and comments, M. Krämer for suggestions, the ICG of Portsmouth for its hospitality and J. Emery & T. Kidani for fruitful discussions. Diagrams have been drawn using JaxoDraw (Binosi and Theussl 2004).
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Pellen, M. Conservation laws for colliding branes with induced gravity. Astrophys Space Sci 357, 24 (2015). https://doi.org/10.1007/s10509-015-2296-5
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DOI: https://doi.org/10.1007/s10509-015-2296-5