Does a physical field have to share the symmetries of the ambient spacetime?
Open a typical textbook on classical electrodynamics and you will find numerous examples of wrinkled and twisted electromagnetic fields, in a sheer contrast with the maximally symmetric Minkowski spacetime they inhabit. These, however, are the weak fields which do not “bend” the spacetime “fabric”. Once we allow the field to interact with the spacetime geometry via gravitational field equations, the symmetry constraints become much more stringent. When the answer to the opening question is affirmative we say that the field inherits the spacetime symmetries. Symmetry inheritance is not only used as a convenient assumption in a choice of the ansatz, but is also an important ingredient of the various gravitational uniqueness theorems. Its breaking may point to some novel, interesting physical phenomena.
The symmetry inheritance of the scalar fields was recently thoroughly analysed in the paper 2015 Class. Quantum Grav. 32 145010. The upshot is that the noninheriting real scalar fields are rare and divergent. An example is the Wyman’s solution of the Einstein-Klein-Gordon’s equations, a stationary spacetime in which the real scalar field grows linearly in time. On the other hand, noninheriting complex scalar fields are more diverse and may even provide a black hole hair, such as the one recently found by Herdeiro and Radu. It remains to be seen whether such configurations might occur in a realistic astrophysical settings.
A series of older results reduce the noninheriting 4-dimensional electromagnetic fields to a small number of well known, “pathological” cases which are usually excluded in the presence of the black hole horizon. What about the electromagnetic fields in everybody’s favorite toy model, the 3-dimensional one? In our last paper, published in CQG, we have proven that the 3-dimensional electromagnetic fields necessarily inherit the spacetime symmetries! The beauty of the argument is that it works for any continuous symmetry (described by some Killing vector field), wide range of gravitational theories and even in the presence of the gauge Chern-Simons terms.
So, is 4-dimensional case an exception? The question about the symmetry inheritance properties of the higher dimensional electromagnetic fields remains wide open.
Read the full article for free* in Classical and Quantum Gravity:
Does three dimensional electromagnetic field inherit the spacetime symmetries?
Ivica Smolić, Maro Cvitan and Predrag Dominis Prester 2016 Class. Quantum Grav. 33 077001
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