# Abnormal Quantum Gravity Effect: Experimental Scheme with Superfluid Helium Sphere and Applications to Accelerating Universe

## Abstract

From the general assumption that gravity originates from the coupling and thermal equilibrium between matter and vacuum, after a derivation of Newton’s law of gravitation and an interpretation of the attractive gravity force between two classical objects, we consider the macroscopic quantum gravity effect for particles whose wave packets are delocalized at macroscopic scale. We predict an abnormal repulsive gravity effect in this work. For a sphere full of superfluid helium, it is shown that with a gravimeter placed in this sphere, the sensitivities of the gravity acceleration below could be used to test the abnormal quantum gravity effect, which satisfies the present experimental technique of atom interferometer, free-fall absolute gravimeters and superconducting gravimeters. We further propose a self-consistent field equation including the quantum effect of gravity. As an application of this field equation, we give a simple interpretation of the accelerating universe due to dark energy. Based on the idea that the dark energy originates from the quantum gravity effect of vacuum excitations due to the coupling between matter and vacuum, without any fitting parameter, the ratio between dark energy density and matter density (including dark matter) is calculated as , which agrees quantitatively with the result obtained from various astronomical observations.

###### pacs:

04.60.Bc, 95.36.+x, 98.80.-k## I introduction

Although Newton’s gravity law and Einstein’s general relativity have given marvelous understanding about gravity, it is still the most mysterious problem in the whole field of science Weinberg (). It is widely believed that the unification of gravity and quantum mechanics and the unification of gravity and other three fundamental forces are impossible in the foreseeable future. One of the main reason is that the abnormal effects observed by experiments on earth are highly scarce. To observe possible abnormal effects, most recently, Adler, Mueller and Perl Perl () proposed a terrestrial search for dark contents of the vacuum using atom interferometry, somewhat similarly to the Michelson-Morley experiment.

Recently, Verlinde’s work Verlinde () has renewed enthusiasm add (); add2 () of the possibility that gravity is an entropy force, rather than a fundamental force Jacobson (). Of course, if gravity is not a fundamental force, we should reconsider the unification of quantum mechanics and gravity. Inspired by the thermodynamic origin of gravity for classical objects, we will try to consider several fundamental problems: (i) Why the gravity between two classical objects is attractive? (ii) Does there exist new and observable quantum gravity effect? (iii) Whether there is new clue to solve the mechanism of accelerating universe?

In this paper, the quantum effect of gravity is studied based on the general principle that gravity originates from the coupling and thermal equilibrium between matter and vacuum background. For classical particles, this general principle gives the Newton’s gravity law. For particles described by quantum wave packets, we predict an abnormal quantum effect of gravity. Based on this abnormal quantum gravity effect, we consider the possible origin of the dark energy from the coupling and thermal equilibrium between matter and vacuum background. Quite surprising, the ratio of the dark energy in our simple calculations is , which agrees quantitatively with the result 7/3 obtained from various astronomical observations d1 (); d2 (); d3 (); d4 (); d5 (). Our works also show that, with a sphere full of superfluid helium, there is a feasible experimental scheme to test our idea with a gravimeter placed in the sphere. The sensitivities of below could be used to test our idea, which satisfy the present experimental technique of atom interferometer AI (), free-fall absolute gravimeters Freefalling () and superconducting gravimeters superconductor ().

The paper is organized as follows. First, in Sec. II, we consider the change of entropy for a particle with a displacement in space. For a particle with an acceleration, we give a derivation of the vacuum temperature due to the coupling between matter and vacuum. Based on the consideration of local thermal equilibrium, we give the acceleration of a particle in the presence of a finite vacuum temperature field distribution. In Sec. III, we give a derivation of Newton’s law of gravitation. In particular, we explain the physical mechanism of the attractive gravity between two classical objects. In Sec. IV, we give a brief discussion of the physical mechanism of the equivalence principle based on the thermodynamic origin of gravity. In Sec. V, we consider an abnormal quantum gravity effect for a particle described by the wave function in quantum mechanics. In Sec. VI, we consider an experimental scheme to test this abnormal quantum gravity effect with a superfluid helium sphere. The application of this abnormal quantum gravity effect to test many-world interpretation and de Broglie-Bohm theory is discussed. We also discuss the application of this abnormal quantum gravity effect to condensed matter physics. In Sec. VII, we give the field equation including the quantum gravity effect of vacuum excitations. This gives a possible interpretation of the repulsive gravity effect of dark energy. In Sec. VIII, we calculate the dark energy density from the general principle in this work. In Sec. IX, the general field equation including the classical and quantum gravity effect of matter and radiation is given. We give relevant summary and discussions in the last section.

## Ii Entropy, vacuum temperature and inertia law

Compared with electromagnetism, weak interaction and strong interaction, the gravitation has several different features. (i) The gravitation is universal. (ii) The gravitation “charge” is in a sense the energy-momentum tensor. Hence, the gravitation “charge” is not quantized. (iii) The coupling between the energy-momentum tensor and spacetime leads to the gravity force for another particle. (iv) The gravity law closely resembles the laws of thermodynamics and hydrodynamics Bekenstein (); Bardeen (); Hawking (); Davies (); Unruh (); Pand (). These features strongly suggest that the gravitation deserves studies with completely different idea, compared with other forces. It is well known that the force in classical and quantum gases can be understood in a natural way with statistical mechanics. Following the intensive pioneering works to eliminate the gravitation from a fundamental force, we will study the thermodynamic origin of gravitation, and in particular the quantum gravity effect when both quantum mechanics and thermodynamics are considered.

Although the theoretical predication about the abnormal quantum gravity effect addresses a lot of subtle problems, our starting points are the following two formulas about the change of entropy for a displacement of a particle, and the vacuum temperature because of the vacuum excitations induced by an accelerating object.

(1) The formula about the change of entropy after a displacement for a particle with mass .

(1) |

In the original work by Verlinde Verlinde (), this postulation motivated by Bekenstein’s work Bekenstein () about black holes and entropy, plays a key role in deriving the Newton’s law of gravitation. The above formula means that after a displacement for a particle with mass , there is an entropy increase of for the whole system. To understand this formula, we emphasize three aspects about this formula. (i) The whole system about the entropy includes the vacuum background. (ii) There is a strong coupling between the particle and vacuum background. This can be understood after little thought. Without a strong coupling, it is nonsense to define the location and time for a particle existing in the spacetime ( the vacuum background). (iii) As a medium for various matters, the zero-point (or ground-state) energy density of the vacuum background is extremely large with the standard quantum field theory. The vacuum zero-point energy density is of the order of if the energy cutoff is the Planck energy. This also gives the reason why there is a strong coupling between matter and vacuum.

In a sense, the motion of a particle in the vacuum background is a little similar to a speedboat moving in the sea. The speedboat left behind a navigation path in the sea. After a navigation of distance , the speedboat stopped. After waiting sufficient time, we cannot know the navigation path in the sea. If the location resolution in the navigation path in the sea is , about bits of information are lost. In this situation, there is an entropy increase of . For a matter (similar to the speedboat) moving in the vacuum background (similar to the sea), it is similar to understand the relation with being the location resolution in the sight of the vacuum background.

Here we consider a method to calculate the location resolution . From special relativity, the energy of the particle is . Together with quantum mechanics, the eigenfrequency is . It is understandable that various gauge fields in the vacuum have the propagation velocity of light velocity . Hence, the relative velocity between matter and the gauge field in the vacuum is . In this situation, we get the coherence length of the particle in the sight of the vacuum background. With the standard quantum mechanics method, we have . This coherence length can also be regarded as the location resolution in the sight of the vacuum background. More detailed discussions about this coherence length and entropy are given in the Appendix.

(2) Vacuum temperature induced by a uniformly accelerating object in the vacuum background.

Considering a particle with acceleration , we have

(2) |

Here . In this paper, all bold symbols represent vectors. From Eq. (1), we have

(3) |

In addition, from , we have

(4) |

Using the fundamental thermodynamic relation for the whole system including the vacuum background, we have

(5) |

In this process, there is no entropy increase for the particle itself. The entropy increase comes from the vacuum. Hence, refers to the vacuum temperature at the location of the particle.

Because temperature is a concept of statistical average, rigorously speaking, the right-hand side of the above expression should be written as the form of statistical average,

(6) |

If the fluctuations of the acceleration can be omitted, we have

(7) |

This shows that the acceleration of a particle will induce vacuum excitations, and thus lead to finite vacuum temperature. Although the above formula is the same as the Unruh temperature Unruh (), the physical meaning is in a sense different from that of the Unruh effect. In the present work, the temperature denotes the vacuum temperature due to the vacuum excitations. Because the Unruh effect itself addresses a lot of subtle problems, in this paper, we will not discuss the detailed difference between and Unruh temperature. We will apply the above expression with our understanding of in this work.

The meaning of the above equation is further shown in Fig. 1. For a particle at location B with acceleration (shown by the dashed red arrow), the coupling between the particle and the vacuum establishes a temperature field distribution shown by the dashed blue line with peak value . In a sense, the strong coupling between the particle and the vacuum leads to a “dressed” state which includes the local vacuum excitations and the particle itself. If the particle has no size, the width of the local vacuum excitations is of the order of the Planck length, with the same consideration of the derivation of the Newton’s law of gravitation in the following section.

In Fig. 1, there is a temperature field distribution shown by the red line. At location B, there is a particle denoted by red sphere. To establish local thermal equilibrium, the red sphere will accelerate so that the peak temperature of the dressed state is equal to the temperature of the vacuum temperature field at the location B, . In this situation, we have

(8) |

However, the acceleration is a vector, while the temperature is a scalar. Therefore, the above formula is not clear about the direction of the acceleration. We further consider the free energy of the whole system which is defined as . Here is the overall energy of the system, which is a conserved quantity. Because the system evolution has the tendency to decrease the free energy with the most effective way, we get the following formula to determine the magnitude and direction of the acceleration in the presence of a vacuum temperature field.

(9) |

Here denotes a three-dimensional spatial vector. In Fig. 1, for the particle at location B, the direction of the acceleration is determined based on the consideration of the free energy. We will show in due course that the above equation will interpret why the gravity force is attractive between two spatially separated objects.

Note that Eq. (9) is invalid for uniformly distributed vacuum temperature field. For uniformly distributed vacuum temperature field such as location A in Fig. 1, from the consideration of the free energy, the direction of the acceleration of a particle is completely random. This means that the direction of the acceleration is highly fluctuating. Although for this case, if the uniformly distributed vacuum temperature is nonzero. By using Eq. (6), we have .

From and , we have . Using Eqs. (2)-(5), we have

(10) |

From the above equation, we get the following inertia law (Newton’s second law)

(11) |

Of course, from we can directly get this inertia law. The above derivation of the inertia law shows that it is self-consistent to consider the origin of classic force from statistical mechanics and the coupling between matter and vacuum background.

If , we have . This is Newton’s first law. The coupling between the particle and vacuum background, and the consistency with Newton’s first law suggest that the vacuum background is in a sense a superfluid. In this superfluid, the propagator velocity of various gauge fields is the light velocity . If is regarded as the sound velocity of the vacuum background, the critical velocity to break the superfluidity is .

## Iii A derivation of Newton’s law of gravitation

As discussed in previous section, the coupling between particle and vacuum background leads to an entropy increase for a displacement. The physical mechanism of this entropy increase is that there is a strong coupling between matter and vacuum. In this section, we give a derivation of Newton’s law of gravitation physically originating from these vacuum excitations.

When there are vacuum excitations due to the coupling between matter and space, we assume as the correlation length of the vacuum excitations. At least for the situations considered in the present work, the vacuum excitation energy density is much smaller than the vacuum zero-point energy density. We will show in due course that is of the order of the Planck length with being the gravitational constant, which is the length scale at which the structure of spacetime becomes dominated by quantum gravity effect. In Fig. 2, we consider the three-dimensional space with showing the structure of the space due to quantum gravity effect. Although the microscopic mechanism of is not completely clear, it is not unreasonable to assume the existence of in the structure of the space. These lowest space structures are a little similar to atoms in a solid. We assume the energy of a particle shown by red sphere in Fig. 2 is . We further assume the freedom of the lowest space structure is . From the local thermal equilibrium, at the location of the particle, the temperature of the vacuum is determined by

(12) |

Here denotes a dimensionless coupling strength between matter and space. From the ordinary statistical mechanics, should be of the order of . We have

(13) |

When , . Hence, one expects the temperature field distribution shown in Fig. 3. For another particle with mass in this temperature field distribution, from Eq. (9), the acceleration field distribution is then

(14) |

Here the radial unit vector . To get the above expression, the spherical symmetry of the system for is also used. This explains the attractive gravity force between two classical objects. It is worthy to point out that both in Newton’s gravitation law and Einstein’s general relativity, this attractive gravity force is imposed from the observation results, rather than microscopic mechanism. One of the merit of thermodynamics lies in that even we do not know the exact collision property such as scattering length between atoms, the macroscopic force such as pressure can be derived. When the thermodynamic origin of gravity is adopted, there is a request that one gets the correct result of the direction of gravity force.

From the continuous property (Gauss’s flux theorem) of the force , we have

(15) |

Note that the above expression holds for , so that the spherical symmetry approximation can be used.

Combined with Eq. (13), we have . Although we do not know the exact value of , it is understandable that is of the order of . In this situation, we have

(16) |

To get the above expression, we have used . Using Eq. (14), we have

(17) |

Assuming

we get the standard result of Newton’s gravitation law

(18) |

We see that the correlation length of the vacuum excitation is of the order of the Planck length. The temperature field distribution for becomes

(19) |

We consider above the situation of a particle whose size is of the order of the Planck length. If the size of the particle is larger than the Planck length, by using the Gauss’s flux theorem, for being much larger than the size of the particle, the above result still holds.

It is worthy to further consider the meaning of the vacuum excitations due to the coupling between matter and space. The above derivations give the temperature field distribution. Assuming the vacuum zero-point energy density is , we consider the situation that a particle with mass suddenly emerges at the location . This will lead to the establishment of the vacuum temperature field distribution (19). However, one should note that in the establishment process of the temperature field distribution, the whole energy should be conserved. Hence, assuming that is the vacuum energy density in the presence of , we have

(20) |

(21) |

This physical picture is further shown in Fig. 4.

For an assembly of classical fundamental particles (here “classical” means that the quantum wave packet effect is negligible) shown in Fig. 5, we assume the temperature field distribution due to the th particle is . Because there is no quantum interference effect between different classical particles, the force (a measurement result in quantum mechanics) on the object with mass is

(22) |

The above expression is based on the assumption of the linear superposition of gravity force . In the Newton’s law of gravitation for an assembly of classical particles, this implicit assumption is also used. We stress here that, rigorously speaking, the summation in the above expression is about all fundamental particles.

One may still consider the following possibility of calculating the acceleration field

(23) |

Obviously, the above expression contradicts with the Newton’s law of gravitation. Considering the astrodynamics addressing three bodies, this method to calculate the gravity force should be ruled out. The reason of the error in the above calculations lies in the method to get . Even for the ordinary situation, the above method to get is also not correct. We consider the presence of thermal sources at different location . The temperature increase for an observer due to these thermal sources is . If only one thermal source exists, we assume that the temperature increase for the observer is . It is obvious that when thermal sources coexist, the temperature increase for the observer is not .

## Iv the equivalence principle

It is well known that the equivalence principle is based on the assumption that the gravitational mass is equal to the inertial mass . It is still a mystery why because it seems that the gravitational mass and inertial mass are different physical concept, if the gravity force is regarded as a fundamental force, similarly to other fundamental forces. For example, for electromagnetism, the inertial mass is completely different from the electric charge leading to electromagnetism. Previous studies clearly show that if the thermodynamic origin of gravity force is adopted, the gravitational mass and the inertial mass are the same mass in the mass-energy relation in special relativity. In all our derivations, we do not need especially introduce the gravitational mass at all. In this situation, the gravitational mass and inertial mass are in fact the same physical concept. Hence, it is not surprising that .

As discussed previously, the velocity of light is in a sense the sound velocity of the vacuum. If the vacuum is regarded as a superfluid, there is a breakdown of the superfluidity if the velocity of an object is larger than . This gives a possible physical mechanism why is the speed limit for all objects. Hence, the validity of special relativity is in a sense determined by the vacuum background where there is a finite and extremely large zero-point energy density. In other words, it’s the vacuum background which leads to the theory of special relativity. This is the reason why there is an internal relation between and vacuum zero-point energy density.

In a sufficiently small region of a freely falling system, the temperature due to the acceleration of this system is equal to the vacuum temperature field. In this small region, the object will not experience the finite vacuum temperature effect. This suggests the validity of the strong principle of equivalence. Hence, when the quantum effect is included in the field equation of gravitation, we will adopt the strong principle of equivalence, which is also adopted by Einstein in deriving his field equation for gravitation.

From and , we see that there is another definition of the mass. For different fundamental particles, there are different correlation lengths in the coupling between particles and vacuum. The mass is then . Considering previous studies that Newton’s first law, second law and Newton’s gravitation law can be derived from , this gives the definition of mass with the view of information. For the particle with inertial mass , we consider the possible change of due to a large mass near us, such as the Milky Way galaxy. At the location of this particle, assuming the presence of the Milky Way galaxy leads to a change of the correlation length , we have . Assuming at the location of the particle, the vacuum temperature field due to the Milky Way galaxy is , because comes from the coupling between the vacuum and particle, to the first order, would be of the order of with being the Planck energy. In this situation, we have . We see that, to observe the possible change of the inertial mass, extremely large would be needed. In most situations we can imagine, we cannot distinguish the gravity effect between Einstein’s general relativity and Mach’s principle. It is not clear, whether at the singular point of black hole or during the Planck epoch et al., there would be obvious effect predicted by Mach’s principle.

## V abnormal quantum effect of gravity

In the thermodynamic origin of gravity, for an object with mass , it establishes a temperature field of . Using the formula about the relation between acceleration and temperature, we get the Newton’s gravity law between two classical objects. In particular, the attractive gravity force between two classical objects is explained. In this section, we will consider the quantum gravity effect by including quantum mechanics in the thermodynamic origin of gravity.

To give a general study on the quantum gravity effect, we consider the following wave function for a fundamental particle (such as electron) with mass ,

(24) |

The average density distribution is shown by the black quantum sphere in Fig. 6(a).

For , similarly to the consideration of a classical particle, it is easy to get

(25) |

At , from the consideration of the spherical symmetry, . Hence, we have . It is clear that the temperature field distribution is that shown in Fig. 7. For , using again the spherical symmetry, we have

(26) |

For the quantum wave packet shown in Fig. 6(a), from the relation between acceleration field and temperature field distribution given by Eq. (9), we have

(27) |

This means a remarkable predication that in the interior of the quantum sphere, the gravity force is repulsive! This abnormal effect is further shown in Fig. 6(b). It is clear that this abnormal quantum gravity effect physically originates from the quantum wave packet effect for the particle in the black sphere. At , the average acceleration field is zero. However, because of the finite vacuum temperature, the direction of the acceleration is highly fluctuating, and . It’s this highly fluctuating acceleration field leads to different value of in the interior and exterior of the quantum sphere.

If there are particles in the same quantum state given by Eq. (24), we have

(28) |

For this abnormal quantum gravity effect, one should be very careful. Without experimental verification, it is only a theoretical predication. Although we will show that it is possible that this abnormal quantum gravity effect just gives an interpretation to the accelerating universe due to dark energy, it is necessary to consider whether this theoretical predication is self-consistent.

To further understand the abnormal quantum gravity effect, we consider a classical sphere with the following density distribution

(29) |

This density distribution for a classical sphere is shown in Fig. 6(c). At first sight, one may get the same acceleration field distribution, compared with the situation of the quantum sphere. This is not correct. For a classical sphere, assuming there are particles, it is very clear that the wave packets of all particles are highly localized. Hence, for a particle at location , the temperature field distribution due to this particle is . From , we get the result given by Newton’s law of gravitation, shown in Fig. 6(d). It is easy to show that the quantum states of the quantum sphere and classical sphere are completely different. In the quantum sphere, the many-body wave function is

(30) |

Here is given by Eq. (24). For the classical sphere, the many-body wave function is

(31) |

Here , , are highly localized wave functions. denotes all the permutations for the particles in different single-particle state. is a normalization factor. We consider here the situation of bosons. It is similar for Fermi system. It’s this essential difference of the many-body wave function that leads to different gravity effect.

Assuming there are fundamental particles whose wave functions are , we give here the formulas to calculate the acceleration field due to these particles. Previous studies directly lead to the following two formulas.

(32) |

and

(33) |

Here is the mass of the th fundamental particle. The integral in the right-hand side of Eq. (33) is due to the quantum wave packet of the th fundamental particle, while the norm of the vector after calculating this integral is due to the fact that is a scalar field larger than zero and is an observable quantity based on Eq. (32). It is easy to show that if all these particles are highly localized classical particles, we get the Newton’s gravity law , with being the location of the th particle.

We assume that a fundamental particle with mass is uniformly distributed in the sphere with radius . For the vacuum temperature field due to this particle, the maximum temperature is

(34) |

From the mass-energy relation, even all the energy is transferred into temperature with only one freedom, we get a limit temperature

(35) |

Obvious, there is a request of

(36) |

This means that

(37) |

This result shows that any object having strong coupling with the vacuum background cannot be distributed within a sphere of radius , irrelevant to its mass. This provides a physical mechanism why in our calculations of , the volume of the smallest space unit is of the order of .

## Vi experimental scheme and potential applications of quantum gravity effect

Now we turn to consider a feasible experimental scheme to test further the quantum gravity effect with superfluid He, shown in Fig. 8. For brevity’s sake, we consider a sphere full of superfluid He. There is a hole in this sphere. In this situation, from Eqs. (32) and (33), the gravity acceleration in the sphere due to superfluid He can be approximated as

(38) |

Here the liquid helium density is kg/m. From this, the anomalous acceleration is . The gradient of this anomalous acceleration is . Even only the condensate component of superfluid He is considered, for a superfluid He sphere whose radius is , the maximum anomalous acceleration is about . Quite interesting, this value is well within the present experimental technique of atom interferometer AI () to measure the gravity acceleration. Nevertheless, this is a very weak observable effect. Thus, it is unlikely to find an evidence to verify or falsify this anomalous acceleration without future experiments. Apart from atom interferometer, the measurement of gravity acceleration with Bloch oscillation Bloch () for cold atoms in optical lattices, superfluid helium interferometry HeliumAI (), free-fall absolute gravimeters Freefalling () and superconducting gravimeters superconductor () provide other methods to test the abnormal quantum gravity effect. In particular, the standard deviation of free-fall absolute gravimeters in the present technique is about Freefalling (), while the superconducting gravimeters have achieved the sensitivities of one thousandth of one billionth () of the Earth surface gravity.

Because this abnormal quantum gravity effect could be tested by contrast experiment with superfluid helium and normal helium, and the prediction that this abnormal quantum gravity effect is location-dependent, the sensitivities of a gravimeter could be used to test the abnormal quantum gravity effect. This makes it very promising to test the abnormal quantum gravity effect in future experiments.

Because highly localized helium atoms will not lead to quantum effect of gravity, the measurement of acceleration will give us new chance to measure the fraction of highly localized helium atoms, which is still an important and challenging topic in condensed matter physics. In our theory, the quantum effect of gravity does not rely on superfluid behavior. It depends on whether the wave packet of a particle is localized. It is well known that whether there is wave packet localization is a central topic in condensed matter physics and material physics, such as the long-range order problem at a phase transition. Considering the remarkable advances in various gravimeters, it is promising that the quantum gravity effect would have potential applications in our understanding of condensed matter physics and material physics, etc.

A possible risk in the decisive test of the abnormal quantum gravity effect with superfluid helium sphere lies in our understanding of the superfluid behavior of liquid helium. In the ordinary understanding of superfluid helium, the superfluid fraction can achieve almost while the condensate fraction is about Penrose-1 (). Because of the strong interaction between helium atoms, the liquid helium is a very complex strongly correlated system. Considering the fact that there are a lot of open questions in strongly correlated systems, we cannot absolutely exclude the possibility that the wave packets of all helium atoms are localized, although the whole system still exhibits superfluid behavior. This significantly differs from the Bose-Einstein condensate in dilute gases, where the wave function of the atoms in the condensate is certainly delocalized in the whole condensate.

The abnormal quantum gravity effect gives a possible experimental scheme to test many-world interpretation Many (). In many-world interpretation, there is no “true” wave packet collapse process. For a particle described by a wave packet, the measurement result of the particle at a location does not mean that the wave packet of this particle at other locations disappears. It suggests that the wave function of the whole universe evolves into a series of orthogonal many-body wave functions due to the interaction between the measurement apparatus and the particle. The observed result of the particle at a location corresponds to one of the orthogonal many-body wave functions. Considering again the superfluid helium sphere, if we increase the temperature so that it becomes normal liquid, based on the many-world interpretation, the wave packets of helium atoms (at least the fraction of the helium atoms initially in the condensate) are still delocalized in the whole sphere. In this situation, if the many-world interpretation is correct, it is possible that one may also get the abnormal quantum gravity effect. This would mean a gravity effect dependent on the history of a system. At least, it seems that all previous experiments or astronomical observations do not overrule this possibility. The present work clearly shows that it’s time to consider more seriously the new view of gravity, in particularly by future experiments.

In the above studies of the quantum gravity effect, we adopt in a sense the ordinary understanding of quantum mechanics. It is worthy to consider the quantum gravity effect based on other understanding of quantum mechanics. Here we give a brief discussion with the de Broglie–Bohm theory. We consider particles with the same wave function shown in Fig. 9. In this pilot-wave theory, the particle is still a highly localized particle with a well-defined trajectory guided by the wave function. We can not predict the exact location of a particle because the initial position of the particle is not controllable by the experimenter, and the motion of the particle guided by the wave function is determined by the initial position of the particle. If we assume further that the energy is mainly carried by the particle, rather than the wave, for the situation shown in Fig. 9, the gravity effect due to these particles would be the same as that of classical particles whose density is . Hence, this experimental scheme also gives us chance to test the de Broglie-Bohm theory. If the abnormal quantum gravity effect is verified, the de Broglie-Bohm theory would be excluded in a sense.

## Vii field equation including quantum gravity effect of vacuum excitations

As shown in Sec. IV, the gravitational mass and inertial mass are the same mass in Einstein’s mass-energy relation, when gravity force is regarded as a thermodynamic effect. This explains in a natural way the equivalence principle. For the existence of vacuum excitations at location , a local reference system with acceleration given by Eq. (32) will not experience these vacuum excitations. Therefore, in this reference system, the physical law of special relativity would hold. This strongly suggests that Einstein’s general relativity should be included to consider the quantum gravity effect. Because different locations can have different accelerations, different locations have different reference systems satisfying the physical law of special relativity. To construct the connection between the reference systems at different locations, Riemannian geometry is the most convenient mathematical tool. Therefore, although we try to argue here that thermodynamics and the coupling between matter and vacuum are the essential mechanisms of gravity force, Riemannian geometry is still needed to construct a systematic theory, because of the same reason in constructing Einstein’s general relativity.

To clearly introduce the new field equation including the quantum effect of gravity, we first give a brief introduction of Einstein’s field equation for classical object (see Ref. Weinberg ()). In weak field approximation, we have with being the gravitational potential. From and , we have

(39) |

If is due to classical particles, from the attractive gravity force between classical objects originating from the consideration of the free energy, the negative sign in the right-hand side of the above equation is included. We stress again that, in Einstein’s derivation of his field equation, the negative sign in the right-hand side of the above equation is due to the observed phenomena that the gravity force between two classical objects is attractive, rather than a property derived from fundamental principle. The development of the above equation to relativistic case gives

(40) |

With general considerations of Riemannian geometry and general covariance, we get the following Einstein’s field equation

(41) |

Note that we have adopted the unit with and the following Minkowski space-time

(42) |

The presence of matter and radiation in the universe will establish various temperature field distributions. These temperature field distributions lead to the force for the matter and radiation in the universe. As shown previously, the physical mechanism of these temperature field distributions is due to the vacuum excitations. In higher-order calculations, the gravity effect of the vacuum excitations themselves should also be considered. Although this would be a complex nonlinear coupling process, general conditions (such as the principle of general covariance) could be imposed to attack this challenging problem. In a reasonable field equation including the gravitation effect of vacuum excitations themselves , one should introduce an extra energy-momentum tensor , apart from for ordinary matter and radiation. Because the vacuum excitations are virtual processes from the view of quantum field theory, these vacuum excitations are different from the ordinary matter. Because of this, these vacuum excitations cannot be described by the form of the energy-momentum tensor for the ordinary matter or radiation. In a sense, these vacuum excitations are integrated into the space time. Fortunately, in principle, one can add an extra term in Eq. (41) without the violation of Riemannian geometry and principle of general covariance. The sole form of the field equation is Weinberg ()

(43) |

Here

(44) |

This general consideration leads to a request that the energy density in is uniform. Assuming is the ground-state energy density of the vacuum, is the energy density of the vacuum in the presence of matter and radiation, we have . Here is determined by Eq. (7). The role of this spatially-dependent and has been included in Einstein field equation (41). The energy density of the vacuum excitations is . There are two equivalent ways to understand the average considered here: the average about the time interval which is much larger than the response time of the vacuum, and the average about the space scale which is much larger than the Planck length. We see that although is spatially dependent, in principle, could be spatially independent. Of course, this is only a statement that there would be no contradiction between spatially dependent and spatially independent . The spatial independence of is due to the general principle of covariance for spacetime itself. The role of is included through in Eq. (43).

From , we have

(45) |

For a flat universe, assuming the scale factor as in the Fiedmann-Robertson-Walker (FRW) metric at the present time of our universe, we have

(46) |

Considering the fact that various fields in the vacuum have the propagation velocity of , it is not unreasonable to assume that various fields in the vacuum background have delocalized wave packets in the whole observable universe. Another reason is that when the big-bang model of the universe is adopted, various fields just have sufficient time to propagate in the whole observable universe. This could lead to delocalized wave packets for the vacuum excitation fields at least within the observable universe. The third reason is that when the coupling between matter and vacuum is considered, the vacuum has the characteristic of superfluidity. These analyses suggest a repulsive gravity effect by nonzero and positive . This requests that one should take the negative sign in the above expression of . We finally get

(47) |

Here . We see that vacuum excitations as a whole have positive energy and abnormal negative pressure. In our theory, it is clearly shown that the abnormal negative pressure physically originates from the quantum characteristic of vacuum excitations.

## Viii dark energy density

In this section, we will try to calculate based on the whole thermal equilibrium in the coupling between matter and vacuum. If the gravity effect is regarded as the thermodynamics and the coupling between matters and vacuum, it is a natural request that the whole vacuum excitations (dark energy) should be determined based on the thermodynamic origin of gravity. Now we turn to consider the whole average effect of these vacuum excitations for large scale of the universe. We will try to calculate the whole average vacuum excitations, the dark energy density. In calculating the dark energy density, we will use the assumption of isotropy and homogeneity on the large scale of the universe. Another assumption is the spatially independent discussed in previous section.

Using spherical polar coordinates for three-dimensional space, we have

(48) |

Here . , , are time-independent co-moving coordinates. is the scale factor. , , for spherical, hyperspherical and Euclidean cases. In the present work, we consider the case of verified by various astronomical observations.

If the evolution of the universe is considered, the radial coordinate of a source that is observed now with redshift is Cosmology ()

(49) |

Here , , . , and are the present values of the dark energy density, average cold matter ( dust) energy density and hot matter ( radiation) energy density. and are the present values of the scale factor and Hubble constant. Because the gravity field ( the vacuum excitations) has the propagation velocity of , for the observer at , the above expression of about has the merit that, is the radial coordinate of a source whose gravity field is observed now with redshift . This gravity field at earlier time is that the observer could experience at the present time. This is the same reason why the above expression is very useful in calculating the luminosity of distant stars.

Various astronomical observations have given precision measurement of and . Assuming , we have

(50) |

From this equation, we can also get . Note that the radial coordinate for co-moving sources is time-independent. We will use this sort of radial coordinate to calculate the dark energy density.

For an observer at of the present time, the overall energy of cold matter and radiation the observer experiences is then

(51) |

Here for cold matter is given by

(52) |

is the cold matter density in the co-moving radial coordinate , without considering the expansion of the universe. When this radial coordinate is adopted, the cold matter density in this coordinate system should not depend on the scale factor. Hence, when the present value of the scale factor is takes as , . The existence of the average relative velocity between the observer and the matter at makes the cold matter energy density for the observer becomes . This is the physical origin of the factor in the above expression. Because denotes the redshift, we have .

The factor in the right-hand side of the above equation originates from two physical effects. When the thermodynamic origin of the gravity is considered, there are various sounds in the temperature field of the vacuum due to the presence of matter. In this situation, for the observer, the rate of arrival of the individual sounds in the gravitation field (temperature field) is reduced by the redshift factor . On the other hand, the energy of the individual sounds experienced by the observer is also reduced by the redshift factor . Hence, the effective energy density is reduced by the factor in the above equation.

There is another way to understand the factor in the above equation. For the matter with the radial coordinate , we consider the gravity field emitted at appropriate time . When this gravity field emitted at time just arrives at the observer, the effective distance between the matter that we consider and the observer becomes because of the expansion of the universe. For the observer, this is equivalent to the case that the energy the observer experiences for the matter at is reduced by the multiplication of the factor .

As for , because the radiation field propagates at a speed equal to , we have

(53) |

It is understandable that the vacuum excitations leading to the gravity effect have also the propagation velocity of . In this situation, it is similar to get for the vacuum excitations (dark energy), which is given by

(54) |

Assuming that there is a thermal equilibrium between matter and vacuum on the large scale of the universe, we have

(55) |

The above equation is also due to the reason that the smallest space unit () is the same for cold matter, hot matter and vacuum excitation, so that the overall freedom ( with being the volume of the universe) is the same for cold matter, hot matter and vacuum excitation.

It is straightforward to get

(56) |

Combined with Eq. (50) to get , one can get the ratio . The numerical result is , agrees quantitatively with the result in astronomical observations d1 (); d2 (); d3 (); d4 (); d5 ().

In the co-moving coordinate, the dark energy density is then

(57) |

Because the co-moving radial coordinate is time-independent, and can be also regarded as the cold matter density and radiation energy density in the co-moving coordinate. For an observer at other times, in the co-moving coordinate, and are also the cold matter density and radiation energy density for this observer. It is easy to show that the dark energy density is still given by the above equation. Together with this time-independent dark energy density in the co-moving coordinate, we see that is a universal value once the following conditions are satisfied.

(i) There is a big-bang origin of the universe.

(ii) The universe and its evolution are isotropic and homogeneous.

(iii) In the co-moving coordinate, the radiation energy density is much smaller than the cold matter energy density.

(iv) The evolution of the universe is a quasi-equilibrium process. This condition suggests that for cosmic inflation process, the calculation of deserves further studies.

One may consider the problem that why we do not introduce the similar concept of luminosity distance , so that the temperature field due to matter and radiation at the location of the observer becomes

(58) |

With the similar idea, the temperature field at the location of the observer due to vacuum excitations is

(59) |

From , we get

(60) |

This method to calculate the dark energy density is wrong. In this method, at the location of the observer, the sum of the vacuum temperature due to the matter and radiation at different locations is adopted. As shown already previously, this sort of sum of the vacuum temperature due to the matter and radiation is nonsense. With the above equation, the numerical result of is about . As expected, it does not agree with the astronomical observations.

In Eq. (57), the dark energy density is given in the time-independent co-moving coordinate. If the evolution of the scale factor is considered, , and in the ordinary coordinate (where the proper distance is adopted) need further studies. As shown previously, if the energy-momentum tensor given by Eq. (47) is adopted for the dark energy, the whole energy-momentum tensor can be written as

(61) |

Here and are the energy-momentum tensors for cold matter and radiation. In the evolution of the universe, when the Friedmann-Robertson-Walker metric is used, the energy conservation law leads to

(62) |

From this we have . If for dark energy as discussed in previous section, this requests that the dark energy density is constant in the evolution of the universe. It is similar to get the well-known results of and .

For coexisting cold matter, radiation and dark energy, when there is no interchange of energy between different components, these evolutions for different components always hold. It is consistent with the result of the dark energy density in previous calculations. The above energy conservation law clearly shows a further condition in previous calculations of the dark energy density, the time-independent dark energy density relies on the condition that there is no energy interchange (or material conversion) between cold matter and hot matter including radiation. If this energy interchange happens, it is possible that the dark energy density becomes time-dependent. This deserves further theoretic studies for the early non-equilibrium universe, to find observable effect.

Most quantum field theories predict a huge value for the quantum vacuum. It is generally agreed that this huge value should be decreased times to satisfy the observation result. Because there are no “true” wave functions for these “quantum zero-point states”, based on our theory, even there are huge vacuum energy due to “quantum zero-point states”, the gravity effect should be multiplied by zero! Another reason is that, the temperature of the vacuum including only “quantum zero-point states” is zero. The finite temperature characteristic of the vacuum is due to the excitations from the vacuum, which influence the motion of the matters. Based on our theory, the cosmological constant problem is not a “true” problem at all.

In this work, the vacuum energy calculated by us is due to the coupling and thermal equilibrium between matter and vacuum background. The coupling and thermal equilibrium lead to various “true” excitations from the vacuum. What we calculated in this work is in fact aims at these excitations which can be described by sophisticated and delocalized wave functions. In Fig. 10, we give a summary of the role of different forms of vacuum energy.

For the sake of completeness, now we consider the evolution of our universe in a brief way. for flat universe is

(63) |

We assume the present value of the scale factor as . If the matter (including dark matter) and radiation are regarded as classical when the evolution of the whole universe is studied, in Eq. (43) takes the ordinary form. From Eqs. (43) and (63), we have Peeble ()

(64) |

and

(65) |

## Ix field equation including quantum gravity effect of matter

The quantum gravity effect of vacuum excitations has been included in Einstein’s field equation, which explains in a simple way the remarkable astronomical observations about dark energy. The basic clues of the combination of quantum gravity effect of matter and general relativity are:

(1) The direction of the acceleration field due to gravity force should be determined by the tendency of decreasing the free energy.

(2) The field equation should satisfy the principle of general covariance.

We first consider further the abnormal quantum gravity effect based on Eqs. (32) and (33). For a particle with mass and wave function , Eqs. (32) and (33) have another equivalent form as follows.

(66) |

(67) |

Here is the acceleration field without considering quantum effect of gravity. is the acceleration field when quantum effect of gravity is considered. For a classical particle, . Note that . From Eq. (66), we have . In this situation, we get a simple relation between and , which is given by

(68) |

Here . The sign at a location is determined by the rule that the direction of the acceleration should point to the increasing of in the neighboring region including . In classical case, this rule physically originating from the property of the free energy has explained why the gravity force between two classical objects is attractive. Obviously, it is natural to generalize this rule to quantum wave packet, because classical mechanics has been replaced by quantum mechanics, when fundamental physical law is addressed.

In reality, the wave function of a lot of particles may be very complex. For brevity’s sake, we consider a system of identical bosons which can be directly generalized to more complex case. The many-body wave function is assumed as

(69) |

The single-particle density matrix is given by