On the Semiclassical Approach of the Heisenberg Uncertainty Relation in the Strong Gravitational Field of Static Blackhole

Heisenberg Uncertainty and Equivalence Principle are the fundamental aspect respectively in Quantum Mechanic and General Relativity. Combination of these principles can be stated in the expression of Heisenberg uncertainty relation near the strong gravitational field i.e. ∆p∆r ≥ ~/2 1− 2GM/〈r〉 c2 and ∆E∆t ≥ ~/2 1− 2GM/〈r〉 c2 . While for the weak gravitational field, both relations revert to ∆p∆r ∼ ~/2 and ∆E∆t ∼ ~/2 . It means that globally, uncertainty principle does not invariant. This work also shows local stationary observation between two nearby points along the radial direction of blackhole. The result shows that the lower point has larger uncertainty limit than that of the upper point, i.e. ∆pA∆rA = ∆pg∆rg(1− 2gH c2 + ...). Hence locally, uncertainty principle does not invariant also. Through Equivalence Principle, we can see that gravitation can affect Heisenberg Uncertainty relation. This gives the impact to our’s viewpoint about quantum phenomena in the presence of gravitation.


Introduction
In the study of Hawking radiation, the very strong gravitational field near the Schwarezschild radius can generate pair production [1]. Particle move out and antiparticle move in toward the Schwarezschild radius. We have known that the pair production based on quantum field theory is explained by Heisenberg uncertainty [2]. On the other hand, the space and time around massive object like blackhole are affected by gravitation. There has been establish result of Schwarezschild metric that gravitational time dilation and length contraction along the radial coordinate are depend on the ratio of mass with respect to the radius of the blackhole [2]. That reason is intriguing for us to ask about how is the Heisenberg uncertainty in the context of the gravitational time dilation and length contraction?.
On another hand, Weak Equivalence Principle guides us to the understanding that inersial force equivalent with gravitational one [3]. It also valid if two forces are taken in to account relativisticly for the test particle [4]. As the consequence, if an integration is worked Correspondence: ardiantoputraf 2@gmail.com Department of Physics, Universitas Jenderal Soedirman, Purwokerto, Indonesia Full list of author information is available at the end of the article * Equal contributor to both of them along the radial distance r in local region, we obtain potencial energy difference. Then based on the energy conservation, we obtain kinetic energy difference that yields the velocity of particle with respect to the blackhole. This velocity depends on the mass-radius ratio, like the factor which makes gravitational time dilation and length contraction. So, equivalence principle provides explanation that relativistic kinetic energy and momentum depend on the ratio M/r. The question is how the ratio M/r affects the uncertainty of both kinetic energy and momentum such a way that this phenomenon appears in the Heisenberg uncertainty representation?.
The question about whether ratio M/r can affect the uncertainty of time, position, energy, and momentum or not, will bring us to the more fundamental question i.e. Does the Heisenberg uncertainty relation invariant in the presence of gravitation?. If it does, so it will give the impact to our's viewpoint about quantum phenomena in the presence of gravitation.
Based on the paper [5], Kentosh and Mohageg analyzed GPS (Global Positioning System) test to determine the LPI (Local Position Invariance) of Planck constant along the radial distance of the earth. The result shows that the Planck constant is invarian in the such limit which is parameterized by β h < 0.007, with β h explain a violation of LPI.

RETRACTION
It indicates the existence of limited invariance zone of Planck constant. The value of this constant will change along radial direction according to relation . So it is very reasonable to be connected with Heisenberg uncertainty invariance in the gravitational field.
Some considerations above, I propose to reconcile between General Relativity and Quantum Mechanic through both equivalence and uncertainty principle. Because those principles are very fundamental. Based on my hypothesis, properly they are linked uniquely such a way that can be oriented to the invariance aspect of Heisenberg uncertainty relation itself. In this paper, I study how the invariance of Heisenberg uncertainty relation in the gravitational field through Equivalence Principle to look for Heisenberg uncertainty which is consistent with the theory of gravitation.
In this study, I use the semiclasiccal approach by viewing the uncertainty relation in de Broglie wave packet. It is so, because according to the Ehrenfest theorem, the average value of uncertainty connects with the classical domain. So, it can be viewed in the context of Einstein relativity theory.

Equivalence Principle in Ehrenfest Theorem's Viewpoint
Special Relativity is the theory which prevails in the flat spacetime, while General Relativity prevails in the curve one. So Special Relativity is just locally consistent with respect to General Relativity [2]. It can be said that in the tiny region of vaccum space around the massive object, the condition approximates Special Relativity (i.e. the tangen space) which Relativistic Quantum Mechanic i.e. superposition principle can be valid enough. It requires that g µυ ≈ η µυ or specificly g µυ (p) = η µυ (p). This is what Equivalence Principle means. In this condition, locally we can build pseudo-Euclidean (Minkowskian) coordinate by transforming the metric tensor of general coordinate g µυ to the metric tensor of Cartesian (inersial) coordinate η µυ [2].
Equivalence principle shows that according to the non-inersial (accelerated) observers i.e. observers that are placed stationary in the tiny regions along radial distance from the massive object's surface up to the very far place which gravitation is weak, They can view that they are in the inersial reference frames (flat space) with presence of gravitation that is viewed as a type of force [6]. Because we cannot distinguish these frames with non-inersia (accelerated) reference frame where inersia force appears [2,6], so F i = F g i.e. m 0 a = GM m0 r 2 (in the form of non-relativistic). It shows that an object in the tiny region of vacuum space around the massive object, the gravitation field is almost uniform so it cannot be distinguished with the uniform accelerated frame [2,3,4,6].
When we integrate both left and right hand side of m 0 a = GM m0 r 2 with respect to the radial distance, so based on the energy conservation we obtain the relation between v and gravitational potential which has been established in Newtonian picture, i.e. . This equation can be obtained from equivalence principle in the non-relativistic form which v is the escape speed of particle from gravitational bound of a static object with mass M [7], In the context of blackhole, if v = c, so r = R S , but the energy is still in the non-relativistic form. Then by using v 2 = 2GM r , we obtain the relativistic kinetic energy of particle which is written in the following binomial expantion The first term of both left and right hand side are like in Newtonian form with the addition terms giving the relativistic correction. If Eq.(1) is expressed by Lorentz factor, we get that m0c 2 It means that in the relativistic form, equivalence principle prevails, such as in the blackhole case [4]. From the relation above, we can . It shows that the particle velocity is the vector field (the function of r) around the blackhole. It is the consequence of equivalence principle. Based on the Ehrenfest theorem, v is connected with average velocity v i.e. the group velocity v g in the picture of de Broglie wave packet that exhibits correspondence to the classical limit [8,9].
Before we discuss about v , we will explain about acceleration previously. Based on Ehrenfest theorem, acceleration is written in the non-relativistic form [10], with In this condition, the change of potential is slowly with respect to the distance [8,9,10] like in the local laboratory where equivalence principle prevails. We know that F ( r ) and the higher order do not give physical meaning in the context of Newton's second law because there is no term a ( r ) moreover in higher order, although in the form of arbitrary force equation itself such as gravitational force gives the meaning. The existence of a ( r ) does RETRACTION not obey equivalent principle because it is not uniform acceleration. So we must view that F ( r ) = 0. Then because of relativistic case, we take (r − r ) ∼ = λ C , with λ C is the Compton wavelength as the smallest possible uncertainty [11,12]. Automatically it yields r 2 = r 2 . So F (r) = F ( r ). In this condition a(r) = a 1 and a( r ) = a 2 , where a 1 and a 2 are the different constant such a way that acceleration does not depend on radial distance in the local laboratory.
. Suppose that v 0 = 0, such a way that we get the relation between v and r as follow:

Heisenberg Uncertainty in the Gravitational Field
The energy is possed by particle in the gravitational field is consist of relativistic kinetic and potential energy, but we will only take relativistic kinetic energy part instead of potential one in order to get uncertainty which is just contributed by kinetic energy. In special relativity, kinetic energy is K = E − m 0 c 4 . Then, here we use dispersion theory to view the uncertainty [8,10,13], so ∆E = p E c 2 ∆p = 1 v ph c 2 ∆p = v g ∆p. This uncertainty relation can be described naturally by de Broglie wave packet [8,10]. It means that superposition principle can be used here but for local observation only where does Equivalence Principle prevails. Based on the K = E − m 0 c 2 , we get that ∆E = ∆K.
If the stationary observer in such event P measures the test particle which is freely falling near the blackhole, so according to this observer, energy and momentum uncertainty of the test particle are ∆E = γ 3 m 0 v g ∆v and ∆p = γ 3 m 0 ∆v, with ∆v = v 2 − v 2 following standard deviation rule. Then here we define ∆K v = v g ∆p v . As the consequence of Eq. (2), we just replace group velocity inside the Lorentz factor as follow: We can write ∆K v = m 0 v g ∆v ≈ m 0 | −GM r 2 | ∆t as the consequence of Eq. (2). It gives understanding as if energy and momentum respectively is the function of position and time such that contradiction with Heisenberg uncertainty itself. We can understand this by the following explanation.
Let's notice again Eq.(2). If we take uncertainty of this equation we get that v g ∆v ≈ | −GM r 2 |∆r with v 2 − v 2 = ∆v 2 and r 2 − r 2 = ∆r 2 . The maximum fixed value of v 2 = c, but v can change from zero at infinity distance up to c at the horizon. While, the maximum fixed value of r 2 = ∞ up to Schwareszchild radius r S . When the particle rests at infinity distance of blackhole, we can set ∆v ≈ 0 and ∆r ≈ ∞ correspond to the condition in which v ≈ 0 and r ≈ ∞. This condition is suitable with v g ∆v ≈ | −GM r 2 |∆r. Then if the particle closes to the horizon, we also take ∆v ≈ 0 and ∆v ≈ ∞ correspond to the condition that v ≈ c and r ≈ r S . So it gives that v g ∆v | −GM r 2 |∆r. So it gives that v g ∆v ≈ | −GM r 2 |∆r. It will not be consistent with equivalence principle in the classical correspondence. The problem is how to keep the uncertainty form of Eq. (2) is still consistent for ∆v ≈ 0 and ∆r ≈ ∞. It is impossible. The way out of this problem is viewing Lorentz Fitz-Gerald contraction effect to position uncertainty ∆r. This effect will be explained evidently in the next section after we formulate position uncertainty in Eq. (5). Because of Lorentz Fitz-Gerald contraction to ∆r, it shows that ∆r ≈ 0 not ∆r ≈ ∞ at a point close to horizon. In this condition, of course that a(r) ≈ a( r ). Hence the uncertainty form of Eq.(2) is valid in all condition. But we must take the consequence that Heisenberg uncertainty seems to be violated because of ∆r ≈ 0 for ∆v ≈ 0. It is no problem because it is just position uncertainty which is according to the observation. The position uncertainty which will be infinite is the proper position uncertainty i.e. ∆r p (see Eq. (5)), such that does not commute with velocity and not violates Heisenberg uncertainty principle. This point will has been clear later, in the last formulation of relativistic Heisenberg uncertainty.Then from relation v g ∆v ≈ | −GM r 2 |∆r, we can cancel v g so that ∆v ≈ | −GM r 2 |∆t. When ∆v ≈ c and ∆r ≈ λ C correspond to the condition that v ≈ 0 and r ≈ ∞, so ∆t ≈ 0 following Lorentz-Fitzgerald contraction of ∆r.
Further from the g µv ≈ η µv , locally we can define Cartesian coordinate system as follow [2] ds 2 ≈ c 2 dt 2 − dr 2 − r 2 dθ 2 − r 2 sin 2 θdϕ 2 , but we will write in the sense of uncertainty i.e. ∆s 2 = c 2 ∆t 2 − ∆r 2 − r 2 ∆θ 2 − r 2 sin 2 θ∆ϕ 2 . It is good enough in describing local flat space time with ∆θ = ∆ϕ = 0, because particle just radially moves. The ∆r can be stated as ∆r = v g ∆t. It is the contracted width of wave packet RETRACTION during the time ∆t. So, we obtain uncertainty relation between position r and time t which can form the relation ∆E∆t = ∆p∆r. So there is analogy between energy-time and momentum-position uncertainty.
In quantum field theory, r and t are treated as a parameter, hence both of them are not properties of particle [14,15]. The ∆t and ∆r respectively show how long and how far the state changes correspond to the arbitrary observable Q [8,16]. These process are describe as ∆t = 1 dQ dt ∆Q and ∆r = vg dQ dt ∆Q.
Nevertheless, in the context of Heisenberg uncertainty, we can view both ∆r and ∆t as if an uncertainty such a way that can be connected with ∆p and ∆E.
In Special Relativities viewpoint, we know that position-time uncertainty (∆r − ∆t) according to the stationary observer is expressed as ∆r = γ −1 ∆t p . Then, equivalence principle shows that with ∆r p are the proper length but ∆t p is not the proper time duration. I state like that because the truly definition of proper time is ∆τ = ∆x p /c, not ∆t p = ∆x p /v g . Then because of treating ∆r as the length, it is oriented to Lorentz Fitz-Gerald contraction and by cancelling v g , the time is shorter also. So ∆r as the width of wave packet of freely falling particle when it is moving with v g close to the speed of light and ∆t as the duration for the moving wave packet to spend it's width (not time dilation), which are measured by the stationary observer. If we want to view that the time runs slower, we have to use the Lagrangian in expressing the energy, consistent with the rule that . It means that according to the stationary observer, the time is elapsed by the freely falling test particle near the blackhole runs slower. As the consequence, the spreading of the wave packet is slower also. So we can state that Heisenberg uncertainty relation of the freely falling test particle according to the stationary observers in every local laboratory along the radial distance obey the relation Then, from the Lagrangian form and time dilation, we get ∆L∆t = . Equation (7) and (8) are analogue each other but not for Lagrangian-time dilation form. So we will not use this form later. If we notice, the right hand side of Eq. (7) and (8) to be the minimum limit of Heisenberg uncertainty in relativistic domain with /2 = ∆p v ∆r p for Eq. (7) and /2 = ∆K v ∆t p for Eq. (8). We can give the terminology that the left hand side of Eq.(7) until Eq.(8) as the coordinate Heisenberg uncertainty while the right hand side, i.e. ∆p v ∆r p and ∆K v ∆t p whose value are /2 are the proper Heisenberg uncertainty. It is like the concept of proper length/time and coordinate length/time. It means that globally, Heisenberg uncertainty minimum limit does not invariant. We can state that ∆K v , ∆p v , ∆t p , and ∆r p are the forms which correspond to the non-relativistic uncertainty. Those forms can be understood as uncertainty of particle if the particle is far away from gravitational influence such a way that ∆p∆r ∼ ∆p v ∆r p and ∆E∆t ∼ ∆K v ∆t p . So, the weak gravitational field will be not significant to increase the Lorentz factor in the uncertainty relation of particle. Equation (7) and (8) are consistent in the each tangen space where special theory of relativity prevails, i.e. the tiny region of space along the radial vacuum space around the blackhole. Notice that ∆K v according to freely falling observer S' is not equal to ∆K v according to stationary observer S which correspond to the relation ∆E = γ 3 ∆K v , because in this relation, ∆K v contains v g whose value close to c, while v g which is convenient to freely falling observer is 0, so it seems contradictive that (∆K v ) S > (∆K v ) S . As the consequence (∆t p ) S < (∆t p ) S (because of relation /2 = ∆K v ∆t p ). It shows the difference duration of ∆t p between which is observed by S and S'. Nevertheless it is no problem, because (∆K v ∆t p ) S = (∆K v ∆t p ) S = /2. Hence we can still take ∆K v ∆t p according to stationary S observer in to account.

Local Observation between Nearby Points
A local laboratory where is equivalence principle prevails, requires that gravitational field must be uniform. Suppose that stationary observer has a local laboratory that it's height is H, connecting the upper point A and the lower point B. Freely RETRACTION falling particle moves from A to B as it is observed by two stationary observers. The form of relativistic momentum uncertainty at A is (∆p) A and at B is (∆p) B . Then the relativistic form of energy uncertainty at A is (∆E) A and at B is (∆E) B . According to two local stationary observers at A and B, they relatively see relativistic energy-momentum uncertainty increases when the particle is freely falling from A to B. The observers them self who stay at two points would feel the difference of their energy-momentum uncertainty if we view they as a particle which are identic with the freely falling particle. While for the non-relativistic momentum uncertainty (∆p v ) A > (∆p v ) B because v g at B is greater than that at A, while the v 2 = c. So we can insert multiplication factor ξ for ∆p v between A and B to be ( We can see that when v g at A is smaller, so the ∆p v is greater. It is conversely at B. Hence we get that the Then, by using binomial expansion for the Lorentz factor, we obtain Further, because of (∆p v ) A = ξ(∆p v ) B , it gives the consequence that (∆r p ) A will be not equal to (∆r p ) B and because of (∆K v ) A =(∆K v ) B , the (∆t p ) A will be equal to (∆t p ) B . Hence (∆r p ) A = ξ −1 (∆r p ) B , so (γ∆r) A = ξ −1 (γ∆t) B . The ∆r − ∆t is the space-time uncertainty of freely falling particle according to observer at two points and ∆r p − ∆t p is the non-relativistic form of spacetime uncertainty . By using the same procedure i.e. binomial expansion, we get It shows that in the local laboratory, the Stationary observer will relatively see the difference of width and moving time of wave packet between upper point A and lower point B. Wave packet at B is shorter than that at A like in Lorentz-Fitzgerald contraction according to stationary observer and also for it's elapsing time. Stationary observers also feel that their radial length contracts. Lorentz contraction happen globally along radial distance correspond to Eq. (5). Every wave packet shows local observation region where gravitation is uniform. If we zoom in this region, the width of wave packet changes with respect to the change of radial velocity correspond to Eq.(11) because of uniform gravitational field g. It means that locally, the state of particle is changed by g. This condition is shown in Figure 1.
The change of the width of wave packet is not caused by localizing process, but that is the Lorentz contraction effect. In this case, we do not localize particle. with uncertainty terms follow Eq. (7) and (8). It means that even locally, Heisenberg uncertainty minimum limit does not invariant also (Compare this with Eq. (7) and (8) which show globally not invariant). Equation (13) tells that in the local zone Stationary observer B see relatively that Heisenberg uncertainty of freely falling particle at him is larger than that at observer A.
It is because (∆p v ∆r p ) B is viewed as (∆p∆r) B at the lower point, while (∆p v ∆r p ) A is viewed as (∆p∆r) A at the upper one whose value is smaller than that at the lower. At the same moment, Freely falling particle will relatively see the Heisenberg uncertainty of the lower observer increases than that of the upper if the RETRACTION observers are viewed as identical particles like freely falling one. Observer B can feel that his uncertainty is larger than that of A, because they are in the inersial frame with the presence of gravitational effect, while freely falling particle cannot feel that it's uncertainty increases, because it is in the inersial frame in the absence of gravitational effect. According to paper [5], relation This setting is equivalent with one we have done from Eq.(9) until Eq. (12). If we use viewpoint of this setting for Local Observation between two nearby Point, so it means that we have taken /2 for ∆p v ∆r p and ∆K v ∆t p . Consequently, we can view that ∆p v ∆r p and ∆K v ∆t p are not as /2 again. Both ∆p v ∆r p and ∆K v ∆t p can decrease even until zero at the horizon. Hence uncertainty relation between two stationary observers will become can be used to give the reason from theoretical viewpoint in explaining the LPI violation of the Planck constant if we set that to the non-relativistic limit, because LPI violation data is analyzed for the weak gravitational field, i.e. the gravitation of earth. The analysis of LPI gives A B = (1 + β h ∆U/c 2 ), while theoretically I get A B = (1 + 2∆U/c 2 ) from Eq. (11).
Nevertheless, we must understand that although the set of Eq. (14) is equivalent with Eq.(9) until (12), but we cannot use this set. The reason is that when we use the set of Eq. (14), so the value of Eq (7) and (8) will be /2 in the left hand side, while the /2 in the right one will decrease up to zero at the horizon. It means that the minimum limit of Heisenberg uncertainty can than /2. Whereas, commonly in quantum mechanics, the minimum limit of Heisenberg uncertainty must be /2 as the proper value. So I decide to use the previous manner in the sense that Heisenberg uncertainty proper limit is /2, and it will be larger in the strong gravitational field. In this condition, the states of the particle is constrained by larger phase space p − r than that at a point which far away from the blackhole. At the horizon, Heisenberg uncertainty will be infinite. For the weak gravitational field, we get ∆p A ∆r A /∆p B ∆r B = (1 − 2∆U/c 2 ). It is consistent with quantum mechanic minimum limit /2 but contradict with A / B = (1 + 2∆U/c 2 ). It can be the subject for the next study. Heisenberg uncertainty in this study is actually more appropriate for the real particles which fall to the blackhole, while particles in Hawking radiation are virtual. However we can use these equations also practically to describe vacuum fluctuation, because principally, that phenomenon connected with uncertainty. The relation between ratio M/r and Heisenberg uncertainty in this study can unify universal constants G, , and c. The discussion which implicates the Planck scale and the concept of quantum gravity [17,18,19] was explained by generalized uncertainty relation ∆p∆x ≥ /2 1 + β(∆p) 2 + ... . This relation also shows the unification of G, , and c. The discussion about relation between that form and the result in this paper is beyond of this paper. Principally, both of those forms show that the minimum limit of Heisenberg uncertainty will increase even blowing up in a such situation.

Conclusion
Heisenberg uncertainty relation in the strong gravitational field is ∆p∆r ≥ . While the forms revert to ∆p∆x ∼ /2 and or ∆E∆t ∼ /2 if gravitational field is weak where is the non-Relativistic Quantum Mechanic prevails. For the local observation between two nearby points, the uncertainty is ∆p A ∆r A = ∆p B ∆r B (1 − 2gH c 2 + ...). Unification between general relativity and quantum domain based on my understanding gives the Heisenberg uncertainty relation as the function of gravitational RETRACTION field with expression of three fundamental constants. This uncertainty relation is the result from combining equivalence principle and uncertainty principle. Through equivalence principle, gravitation can affect Heisenberg uncertainty relation.