44
4 Vector-Tensor Gravities and Problem of Lorentz …
and a 4-velocity u α = ±
¯
B α
√
| ¯
B 2 |
, as a result the equations of motion become
G αβ | = ρ m u α u β ,
∂
β F βα [A] = ρ e u α ,
(4.20)
replaying thus the Einstein and Maxwell equations respectively. Effectively we
showed that our background field B μ plays the role of the charged dust. We note
that in principle, the ¯
B α and A α fields can be coupled to usual matter in various
manners being treated either as a usual photon or as a some extra particle.
To conclude, we see that the bumblebee gravity can be treated as a sound candidate, first, to implement the Lorentz symmetry breaking within the gravity context,
second, to display consistency with astronomical observations, due to validity of
most important general relativity solutions. Among other results one can mention
study of dispersion relations in a linearized bumblebee gravity where the constant
bumblebee field triggers deviations from the standard dispersion relations [92]. However, much more aspects of the bumblebee gravity, especially problem of validity
and consistency of many other solutions, are still to be studied. In this context, one
of the most important issues is the study of perturbative aspects of the bumblebee
gravity, and only first steps along this line are done now.
4.4 Conclusions
We discussed vector-tensor gravity models. Just as in the previous chapter, the additional field, in this case the vector one, is treated not as a matter field but as an ingredient of the complete description of the gravity itself. The most important aspect of
these models consists in the fact that some of them, namely those ones involving
potential terms for the vector field, can be extremely useful within the context of the
spontaneous Lorentz symmetry breaking. The known examples of these theories are
the Einstein-aether gravity and the bumblebee gravity.
The Einstein-aether theory has been formulated earlier. Within it, the potential
term generating the spontaneous Lorentz symmetry breaking is implemented through
the constraint with the corresponding Lagrange multiplier field. From one side, this
action is rather simple, but from another side, the presence of the constraint generates
essential difficulties for the perturbative description. Therefore, the bumblebee model
is certainly much more promising. Moreover, the bumblebee approach displays an
advantage in comparison with the naive application of the QFT approach suggesting
to couple dynamical fields with the constant vectors (tensors) which, as we already
noted, cannot be consistently defined in a curved space-time.
The bumblebee approach allows to introduce many Lorentz-breaking vectortensor terms. The term B
μ B
ν R μν from (4.1) is effectively nothing more that the
gravitational aether term proposed in [77]. We note that treating of the B μ as one of
the bumblebee vacua rather than the usual constant vector allows to avoid breaking
4 Vector-Tensor Gravities and Problem of Lorentz …
and a 4-velocity u α = ±
¯
B α
√
| ¯
B 2 |
, as a result the equations of motion become
G αβ | = ρ m u α u β ,
∂
β F βα [A] = ρ e u α ,
(4.20)
replaying thus the Einstein and Maxwell equations respectively. Effectively we
showed that our background field B μ plays the role of the charged dust. We note
that in principle, the ¯
B α and A α fields can be coupled to usual matter in various
manners being treated either as a usual photon or as a some extra particle.
To conclude, we see that the bumblebee gravity can be treated as a sound candidate, first, to implement the Lorentz symmetry breaking within the gravity context,
second, to display consistency with astronomical observations, due to validity of
most important general relativity solutions. Among other results one can mention
study of dispersion relations in a linearized bumblebee gravity where the constant
bumblebee field triggers deviations from the standard dispersion relations [92]. However, much more aspects of the bumblebee gravity, especially problem of validity
and consistency of many other solutions, are still to be studied. In this context, one
of the most important issues is the study of perturbative aspects of the bumblebee
gravity, and only first steps along this line are done now.
4.4 Conclusions
We discussed vector-tensor gravity models. Just as in the previous chapter, the additional field, in this case the vector one, is treated not as a matter field but as an ingredient of the complete description of the gravity itself. The most important aspect of
these models consists in the fact that some of them, namely those ones involving
potential terms for the vector field, can be extremely useful within the context of the
spontaneous Lorentz symmetry breaking. The known examples of these theories are
the Einstein-aether gravity and the bumblebee gravity.
The Einstein-aether theory has been formulated earlier. Within it, the potential
term generating the spontaneous Lorentz symmetry breaking is implemented through
the constraint with the corresponding Lagrange multiplier field. From one side, this
action is rather simple, but from another side, the presence of the constraint generates
essential difficulties for the perturbative description. Therefore, the bumblebee model
is certainly much more promising. Moreover, the bumblebee approach displays an
advantage in comparison with the naive application of the QFT approach suggesting
to couple dynamical fields with the constant vectors (tensors) which, as we already
noted, cannot be consistently defined in a curved space-time.
The bumblebee approach allows to introduce many Lorentz-breaking vectortensor terms. The term B
μ B
ν R μν from (4.1) is effectively nothing more that the
gravitational aether term proposed in [77]. We note that treating of the B μ as one of
the bumblebee vacua rather than the usual constant vector allows to avoid breaking