Is nonrelativistic gravity possible?
Abstract
We study nonrelativistic gravity using the Hamiltonian formalism. For the dynamics of general relativity (relativistic gravity) the formalism is well known and called the ArnowittDeserMisner (ADM) formalism. We show that if the lapse function is constrained correctly, then nonrelativistic gravity is described by a consistent Hamiltonian system. Surprisingly, nonrelativistic gravity can have solutions identical to relativistic gravity ones. In particular, (anti)de Sitter black holes of Einstein gravity and IR limit of Hořava gravity are locally identical.
pacs:
04.20.q, 04.20.Cv, 04.20.Fy, 04.70.BwI Introduction
We use the Hamiltonian formalism Ref:ADM , Ref:FiMa79a , Ref:FiMa79b , Ref:AAK for the dynamics of nonrelativistic gravity in WheelerDeWitt superspace Ref:DeWitt . The formalism leads naturally to the study of consistency of the nonrelativistic gravity. The equations of the rate of change of energy and momentum are computed. As is well known, the relativistic theory is characterised by identically zero energy rather than just the total integrated energy being zero Ref:Misner , Ref:FiMa72 . A question arises: Can one generalise nonrelativistic theories and recover an identically zero energy condition? In other words: Can one generalise the lapse function from being a function of time only to a function of space and time? We show that the answer is negative, unless a very strong consistency condition is satisfied. Thus, generically, the lapse function of consistent nonrelativistic theories must be time dependent only.
Ii Nonrelativistic gravity
ii.1 Superspace
Let be an oriented without boundary smooth ddimensional manifold. Let denote the space of all smooth symmetric two tensors on and let be the manifold of positive definite Riemannian metrics on . The tangent bundle of is
Let be the space of all symmetric two contravariant tensor densities on . The cotangent bundle of is
We have a natural pairing between and given by
where , , , , and .
The DeWitt metric on is given by Ref:DeWitt
where is a constant, , , and . The metric has an inverse metric given by
where
ii.2 Hamiltonian formalism
We investigate a dynamical system on given by an invariant action
(1) 
where
(shift vector field) is a time dependant vector field on , (lapse function) is a function of only, i.e. is a constant function in the space of realvalued functions , is the Lie derivative, and the potential is a scalar density.
The canonical momenta conjugate to are
and the Hamiltonian is
(2) 
where
Hamiltonian equations have the following form Ref:FiMa79a , Ref:FiMa79b :
(3) 
where
and its adjoint map
are defined by
Here we follow Ref:FiMa79a and consider the potential as a function of the undifferentiated metric coefficients that do not appear in the Christoffel symbols , and of the Christoffel symbols, and we write .
ii.3 Constraints
The invariance of the Hamiltonian with respect to the spatial diffeomorphisms implies the following Ref:FiMa79a :
for an arbitrary vector field . Therefore, we have the following conservation law (constraint)
(4) 
Then from (2) we get
(5) 
but not necessarily a much stronger constraint
(6) 
as in relativistic gravity. As is well known Ref:Misner , Ref:FiMa72 , in any topologically invariant theory (6) holds rather than just (5).
But, is it possible to impose (6) on nonrelativistic gravity? In order to answer this question let us compute the rate of change of and along a solution of (3) for general and . It is straightforward to show that (cf. Ref:FiMa79a )
(7)  
where  
(8) 
Incidentally, (7) is equivalent to the Dirac canonical commutation relations (cf. Ref:Dirac , Ref:FiMa79a , Ref:FiMa79b ).
Let us define Ref:FiMa79b
If , then we have for all for which the solution exists, but for all if and only if the restriction of to vanishes, i.e. the following condition holds for all
(9) 
If one assumes that is a function of and for a nonrelativistic theory, then the theory will be consistent if and only if (9) holds. This is a very strong condition. By definition we have
However, (9) does not hold for all and a general potential . We know one theory (possibly the only one if ), that of general relativity satisfying the condition. If (9) does not hold, then the Hamiltonian system is not consistent. Hence, (6) cannot be imposed and one has to consider as a function of only. In that case (7) can be written in the following form:
(10)  
where  
(11) 
Thus, it is obvious that nonrelativistic gravity is possible, provided one considers a time only dependant lapse function, a projectable function (see Ref:H2 ). If one generalises the lapse function, then the only meaningful, consistent theory is Einstein gravity.
However, if (9) does not hold for all solutions it can hold for specific solutions. Indeed, there could exist solutions with , then and . These types of solutions would mimic relativistic ones. They will be called Lorentz symmetry recovering (LSR) solutions.
ii.4 Examples
Let us consider some important (non)relativistic theories.
Einstein gravity. For the relativistic potential
with arbitrary we have
and  
(12) 
where , and . Thus, we see that and are critical values as noted in Ref:H1 , Ref:H2 . Theories with are very different from Einstein gravity, because of the last term in (12). The DeWitt metric’s dependence on is crucial too. If , then and full relativistic gravity is recovered. Therefore, one is free to choose a space and time dependent lapse function.
Hořava gravity Ref:H1 , Ref:H2 . We consider a more general potential
For simplicity, we assume that and the spatial metric is flat , and then it is trivial to show that all solutions are LSR ones. Moreover, there is a bijection between solutions of Hořava and Einstein gravity. In particular, for a spherically symmetric metric, all solutions are locally equivalent to the SchwarzschildKottler solution in Lemaître coordinates Ref:Lem . For example, for and , we have
where
Thus, there is no “new” (A)dS black hole solutions in Hořava gravity. One will find new solutions if one considers a space and time dependent lapse function, but then the theory becomes inconsistent. However, nonflat geometries are not necessarily LSR solutions.
Iii Conclusions
The Hamiltonian formalism is used to study nonrelativistic gravity. The evolution (7) for and is derived and a consistency condition (9) is proposed. It is shown that if one considers a time only dependant lapse function, then nonrelativistic gravity is possible and described by a consistent Hamiltonian system. A typical nonrelativistic gravity will be an inconsistent theory if we assume a space and time dependant lapse function. One could conjecture that only Einstein gravity is consistent with a space and time dependant lapse function if . The other possibility is Hořava gravity if (see Ref:H1 , Ref:H2 ).
The results of the paper can be extended to include field theories coupled to gravity. One is tempted to extend the approach and investigate the nonrelativistic WheelerDeWitt equation Ref:DeWitt
All of these directions will be investigated in further study and hopefully a more important question, “Is physically meaningful nonrelativistic gravity possible?” will be answered.
Note added.–While this work was being prepared for submission, we became aware of Ref:SM where similar questions are addressed.
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