Quantum Interference Effects in Hořava-Lifshitz Gravity
The relativistic quantum interference effects in the spacetime of slowly rotating object in the Hořava-Lifshitz gravity as the Sagnac effect and phase shift of interfering particle in neutron interferometer are derived. We consider the extension of Kehagias-Sfetsos (KS) solution in the Hořava-Lifshitz gravity for the slowly rotating gravitating object. Using the covariant Klein-Gordon equation in the nonrelativistic approximation, it is shown that the phase shift in the interference of particles includes the gravitational potential term with the KS parameter . It is found that in the case of the Sagnac effect, the influence of the KS parameter is becoming important due to the fact that the angular velocity of the locally non rotating observer is increased in Hořava gravity. From the results of the recent experiments we have obtained lower limit for the coupling KS constant as . Finally, as an example, we apply the obtained results to the calculation of the UCN (ultra-cold neutrons) energy level modification in the gravitational field of slowly rotating gravitating object in the Hořava-Lifshitz gravity.
Keywords: Hořava gravity; Neutron interferometer; Sagnac effect.
Received (Day Month Year)
Revised (Day Month Year)
PACS Nos.: 04.50.-h, 04.40.Dg, 97.60.Gb.
One of the biggest difficulties in attempts toward the theory of quantum gravity is the fact that general relativity is non-renormalizable. This would imply loss of theoretical control and predictability at high energies. In January 2009, Petr Hořava proposed a new theory of quantum gravity with dynamical critical exponent equal to in the UV (Ultra-Violet) in order to evade this difficulty by invoking a Lifshitz-type anisotropic scaling at high energy. This theory, often called Hořava-Lifshitz gravity, is power counting renormalizable and is expected to be renormalizable and unitary .
Having a new candidate theory for quantum gravity, it is important to investigate its astrophysical and cosmological implications. Thus the Hořava theory has received a great deal of attention and since its formulation various properties and characteristics have been extensively analyzed, ranging from formal developments , cosmology , dark energy , dark matter , and spherically symmetric or axial symmetric solutions .
In the paper Ref. ? the possibility of observationally testing Hořava gravity at the scale of the Solar System, by considering the classical tests of general relativity (perihelion precession of the planet Mercury, deflection of light by the Sun and the radar echo delay) for the Kehagias-Sfetsos asymptotically flat black hole solution of Horava-Lifshitz gravity has been considered. The stability of the Einstein static universe by considering linear homogeneous perturbations in the context of an Infra-Red (IR) modification of Hořava gravity has been studied in the paper . In the paper Ref. ? author considered potentially observable properties of black holes in the deformed Hořava-Lifshitz gravity with Minkowski vacuum: the gravitational lensing and quasinormal modes.
The role of the tidal charge in the orbital resonance model of quasiperiodic oscillations in black hole systems and in neutron star binary systems have been studied intensively. The motion of test particles around black hole immersed in uniform magnetic field in Hořava gravity and influence of parameter on radii of innermost stable circular orbit have been studied in papers Ref. ?, ?.
The experiment to test the effect of the gravitational field of the Earth on the phase shift in a neutron interferometer were first proposed by Overhauser and Colella . Then this experiment was successfully performed by Collela, Overhauser and Werner . After that, there were found other effects, related with the phase shift of interfering particles. Among them the effect due to the rotation of the Earth , which is the quantum mechanical analog of the Sagnac effect, and the Lense-Thirring effect which is a general relativistic effect due to the dragging of the reference frames. So we do not consider the neutron spin in this paper.
In the paper Ref. ? a unified way of study of the effects of phase shift in neutron interferometer was proposed. Here we extend this formalism to the case of slowly rotating stationary gravitational fields in the framework of Hořava-Lifshitz gravity in order to derive such phase shift due to either existence or nonexistence of the KS parameter .
The Sagnac effect is well known and thoroughly studied in the literature, see e.g. paper Ref. ?. It presents the fact that between light or matter beam counter-propagating along a closed path in a rotating interferometer a fringe shift arises. This phase shift can be interpreted as a time delay between two beams, as it can be seen below, does not include the mass or energy of particles. That is why we may consider the Sagnac effect as the ”universal” effect of the geometry of space-time, independent of the physical nature of the interfering beams. Here we extend the recent results obtained in the papers Ref. ?, ? where it has been shown a way of calculation of this effect in analogy with the Aharonov-Bohm effect, to the case of slowly rotating compact object in Hořava-Lifshitz gravity.
In this paper we study quantum interference effects in particular the Sagnac effect and phase shift effect in a neutron interferometer in the Hořava model which is organized as follows. In section 2, we start from the covariant Klein-Gordon equation in the Hořava model and consider terms of the phase difference of the wave function. Recently Granit experiment verified the quantization of the energy level of ultra-cold neutrons (UCN) in the Earth’s gravity field and new, more precise experiments are planned to be performed. Experiments with UCN have high accuracy and that is the reason to look for verification of the gravitational effects in such experiments. In this section as an example we investigate modification of UCN energy levels caused by the existence of KS (Kehagias and Sfetsos) parameter . In section 3 we consider interference in Mach-Zender interferometer and in Section 4 we study the Sagnac effect in the background spacetime of slowly rotating object in Hořava gravity.
Throughout, we use space-like signature , geometrical units system (However, for those expressions with an astrophysical application we have written the speed of light explicitly.). Greek indices are taken to run from 0 to 3 and Latin indices from 1 to 3; covariant derivatives are denoted with a semi-colon and partial derivatives with a comma.
2 The Phase shift
The four-dimensional metric of the spherical-symmetric spacetime written in the ADM formalism has the following form:
where , are the metric functions to be defined.
The IR (Infrared) - modified Horava action is given by (see for more details Ref. ?, ?, ?, ?, ?, ?)
where and are constant parameters, the Cotton tensor is defined as
is the three-dimensional curvature tensor, and the extrinsic curvature is defined as
where dot denotes a derivative with respect to .
Imposing the case , which reduces to the action in IR limit, one can obtain the Kehagias and Sfetsos (KS) asymptotically flat solution for the metric outside the gravitating spherical symmetric object in Horava gravity:
where is the total mass, is the KS parameter and the constant is chosen.
Up to the second derivative terms in the action, one can easily find the known topological rotating solutions given in Ref. ?, ?. This metric in the slow rotation limit has the form:
here is the specific angular momentum of the gravitating object.
Using the Klein-Gordon equation
for particles with mass one cane define the wave function of interfering particles as
where is the nonrelativistic wave function.
In the present situation, both parameters and are sufficiently small and their higher order terms can be neglected. Therefore, to the first order in , and neglecting the terms of , the Klein-Gordon equation in Horava-Lifshitz gravity becomes
where we have used the following notations:
which correspond to the square of the total orbital angular momentum and component of the orbital angular momentum operators of the particle with respect to the center of the Earth, respectively.
After the coordinate transformation , where is the angular velocity of the Earth, we obtain the Schrödinger equation for an observer fixed on the Earth in the following form:
is the Hamiltonian for a freely propagating particle, is the Horava-Lifshitz gravitational potential energy, is concerned to the rotation, is related to the effect of dragging of the inertial frames. The phase shift terms due to and are
where represents the position vector of the instrument from the center of the Earth, , is the area of the interferometer, and is the unit normal vector. If we assume that the Earth is a sphere of radius with uniform density then
if R is perpendicular and parallel to , respectively. Here is the free fall acceleration of Earth.
Now one can easily calculate the phase shift due to the gravitational potential. For the purpose of the present discussion, the quasi-classical approximation is valid and the phase shift
is given by the integration along a classical trajectory. Here is the area of interferometer, is de Broglie wavelength (see the Fig. 1).
Recently published paper Ref. ? describes the precise measurement of the gravitational redshift by the interference of matter waves in the gravitational field of the Earth. Comparing their experimental results with our theoretical predictions one can easily obtain the lower limit on the value of KS parametr .
Astrophysically it is interesting to apply the obtained result for the Hamiltonian of the particle, moving around rotating gravitating object in Hořava gravity, to the calculation of energy level of ultra-cold neutrons (UCN) (as it was done for slowly rotating space-times in the papers Ref. ?, ?). The effect of the angular momentum perturbation of the Hamiltonian on the energy levels of UCN was studied in and subsequent papers. Our purpose is to generalize this correction to the case of the gravitating object (the Earth in particular case) in Hořava model. Denote as the unperturbed non-relativistic stationary state of the 2- spinor (describing UCN) in the field of the rotating gravitating object in Hořava gravity. Then we have
is the Laplacian in the spherical coordinates. By adopting new Cartesian coordinates within and axis being local vertical, when the stationary state is assumed to have the form
one can easily derive from (19) that
where the following notation
has been used.
Following to the papers Ref. ?, ? one can compute ”KS parameter ” modification of the energy level as the first-order perturbation:
Assume (where is the latitude angle) and to be equal to 1, that is . Assuming now one can extend (24) as
We remember that is the average value of for the stationary state . For the further calculation we use formula for from
Now one can easily estimate the relative ”KS parameter ” modification of the energy level of the neutrons as
We numerically estimate the obtained modification using the typical parameters for the Earth: , , , , and ,
From the obtained result (28) one can see, that the in influence of parameter will be stronger in the vicinity of compact gravitating objects with small . Recent experiments on measuring energy levels of UCN has an error , which does not allow to obtain the influence of parameter on energy levels of UCN. Further improvements of the experiments would give either exact value or lower limit for the above mentioned parameter.
3 The interference in a Mach-Zehnder-type interferometer
The components of the tetrad frame for the proper observer for metric (6) are
and the acceleration of the Killing trajectories is
and we obtain for the nonvanishing component of the acceleration:
The nonvanishing orthonormal (”hatted”) components of rotation tensor of the stationary congruence in the slowly rotating Hořava-Lifshitz gravity are given by
The simple form of the vector potential of the electromagnetic field in the Lorentz gauge in the spacetime (6) is . Here the integration constant , where gravitational source is immersed in the uniform magnetic field being parallel to its axis of rotation (properties of black hole immersed in external magnetic field have been studied, for example, in Ref. ?, ?, ?, ?, ?, ?), and the other integration constant can be calculated from the asymptotic properties of spacetime (6) at the infinity:
One can write the total energy of the particle in the weak field approximation in the following form:
where is electric charge of the particle. This is interpreted as total conserved energy consisting of gravitationally modified kinetic and rest energy , a modified electrostatic energy .
For the further use note the measured components of the electromagnetic field, which are the electric and magnetic fields:
where is the field tensor, is the pseudo-tensorial expression for the Levi-Civita symbol , .
Now one can obtain the total phase shift as
where , is the angle of the baseline with respect to and is the tilt angle. Therefore one can independently vary the angles and , and extract from the phase shift measurements the following combinations of terms:
Using above obtained results one can estimate lower limit for KS parameter . Using the results of the Earth based atom interferometry experiments would give us an estimate .
4 The Sagnac effect in the Horava gravity
It is well known that the Sagnac effect for counter-propagating beams of particles on a round trip in an interferometer rotating in a flat space-time may be obtained by a formal analogy with the Aharonov-Bohm effect. Here we study the interference process of matter or light beams in the spacetime of slowly rotating compact gravitating object in braneworld in terms of the Aharonov-Bohm effect . The phase shift
is detected at uniformly rotating interferometer and the time difference between the propagation times of the co-rotating and counter-rotating beams is equal to
and is the unit four-velocity of particles:
From (6) and coordinate transformation , where one can see that the unit vector field along the trajectories will be
where we have used the following notation
Now inserting the components of into the equation (47) one can obtain
where is the angular velocity of Lense-Thirring.
Following to the paper Ref. ? one can find a critical angular velocity
which corresponds to zero time delay . is the angular velocity of zero angular momentum observers (ZAMO).
We have studied quantum interference effects including e.g. the phase shift and time delay in Sagnac effect in the spacetime of rotating gravitational objects in Hořava gravity and found that they can be affected by the KS parameter . Then, we have derived an additional term for the phase shift in a neutron interferometer due to the presence of KS parameter and studied the feasibility of its detection with the help of ”figure-eight” interferometer. We have also investigated the application of the obtained results to the calculation of energy levels of UCN and found modifications to be rather small for the Earth but more relevant for the compact astrophysical objects. The result shows that the phase shift for a Mach-Zehnder interferometer in spacetime of gravitational object in Hořava gravity is influenced by the KS parameter . Obtained results can be further used in laboratory experiments to detect the interference effects related to the phenomena of Hořava gravity. Recently authors of the paper Ref. ? from Solar system tests obtained values for parameter as follow: (from perihelion precession of the Mercury), (light deflection by the Sun), (radar echo delay). Here we have estimated lower limit for parameter as using the experimental results of the recent paper Ref. ? on the precise measurement of the gravitational redshift by the interference of matter waves.
The work was supported by the UzFFR (projects 5-08 and 29-08) and projects FA-F2-F079 and FA-F2-F061 of the UzAS. This work is partially supported by the ICTP through the OEA-PRJ-29 project. Authors gratefully thank Viktoriya Morozova for useful discussions. AB acknowledges the TWAS for associateship grant. AA and AB thank the IUCAA for the hospitality where the research has been completed.
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