Salecker-Wigner-Karolyhazy Gedankenexperiment in light of the self-gravity
aa r X i v : . [ g r- q c ] A ug Salecker-Wigner-Karolyhazy Gedankenexperimentin light of the self-gravity
Michael Maziashvili ∗ School of Natural Sciences and Medicine, Ilia State University,3/5 Cholokashvili Ave., Tbilisi 0162, Georgia
Abstract
In Gedankenexperiment mentioned in the title, the imprecision in space-time measurement isrelated to the spreading of clock’s wave-function with the passage of time required for the measure-ment. Special relativity puts a bound on the measurement time, it cannot be reduced arbitrarilyas the signal used for the measurement cannot propagate with speed greater than that of light. Inview of this reasoning, one is led to conclude that the clock should be heavy enough to slow downits wave-function from spreading with time. However, the general relativity puts an upper boundon clock’s mass, since its size must remain greater then the Schwarzschild radius associated to it.This way one reaches a limit in length measurement. However, as is discussed below, an additionalinsight into the question comes by taking into account self-gravitational effects. As a result, theuncertainty in length measurement is reduced to the Planck length. ∗ [email protected] . PREFACE One of the characteristic features of quantum mechanical system is the presence of zero-point-fluctuations. That is, even in the ground state, physical quantities are characterizedapart from their average values with the fluctuations, which are usually estimated by themean square deviations. It is enough to mention that the vacuum fluctuations of the elec-tromagnetic field is responsible for a number of well known phenomena. For instance, itstimulates a spontaneous emission of atom [1], its another manifestation is Casimir force [2]and the Lamb shift also can be explained by means of it [3]. In general, it is hard to estimatethe rate of zero-point-fluctuations in quantum field theory, as it turns out to be a divergentquantity. Alternatively, one could try to use various Gedankenexperiments for estimatingorder of magnitude of the fluctuations of a given physical quantity. Such Gedankenmes-sungen usually account for the unavoidable disturbances caused by the interaction duringthe measuring process. One may recall a well known example of this sort of discussionconcerning the electromagnetic field [4, 5]. In contrast to other fields, the metric that de-scribes the gravitational field - determines at the same time the background space-time.Thus, one may consider the measurement of gravitational field by means of the motion oftest particles [6–9] (as is the case with electromagnetic field [4, 5]) or one may discuss themeasurement of space-time characteristics like curvature [10–12] and space-time intervals[13, 14]. While there are no objections that the quantum fluctuations prevent one frommeasuring position with greater accuracy than the Planck length, l P ≈ − cm, [15], thereis still controversy about the rate of length fluctuations [16–23]. Karolyhazy supplementedthe discussion of Salecker and Wigner [13, 14] by noting that the minimum size of a clock isset by its Schwarzschild radius and found that the length l cannot be measured with greateraccuracy than δl & l / P l / [16]. This result was criticized by devising new Gedankenexper-iments [17, 19, 20, 22] and supported again in a series of papers [18, 21, 23]. We are notgoing to discuss the counterexamples and their refutations but instead we shall argue thatthe bound δl & l P , which is considered by some authors to be the proper one, can readilybe achieved by taking into account the effect of self-gravity in Salecker-Wigner-KarolyhazyGedankenexperiment. 2 I. SALECKER, WIGNER, KAROLYHAZY
In order to demonstrate principal limitations on space-time measurement due to quantumand gravitational effects Salecker and Wigner proposed the following Gedankenexperiment[13, 14]. The clocks are placed at the points the distance between which is being measured(the clock can be viewed as a spherical mirror inside which light is bouncing), and bymeasuring the time a light signal takes from one clock to another we estimate the distancebetween those points. Clock is characterized with some mass m and radius r c . Because ofclock’s size, the points are marked with the precision ≃ r c . In addition clocks are subject toquantum fluctuations, δp ≃ /r c , that give for fluctuation velocity: δv ≃ /mr c . Thus, thetotal uncertainty in measuring the length l = t (we use ~ = c = 1 system of units) takes theform δl & r c + lδv ≃ r c + lmr c . Minimizing this equation with respect to r c , one gets r c ≃ r lm , δl ≃ r lm . (1)It seems that at the expense of mass we can always minimize the δl as much as we want.But, as it was noticed by Karolyhazy, gravity brings new insight into the problem [16, 18].Namely, the clock is characterized by the Schwarzschild radius r g ≃ l P m and to avoid itsgravitational collapse, the size of clock should be greater than its Schwarzschild radius l P m . r lm . It gives an upper limit on m m . l / l − / P , and puts a lower bound on δl δl min ≃ l / l / P . (2)Let us note that the above discussion has been carried out without paying any attention tothe self-gravitational effects. However, one has to draw attention to the fact that the optimalmeasurement in Salecker-Wigner-Karolyhazy Gedankenexperiment is done by a clock whosecharacteristics are very close to that of a black hole [24]. If we bear in mind that it means thewave-function describing the clock to be shrunk to its Schwarzschild radius, we are driven to3he conclusion that the gravitational attraction becomes very strong and it may drasticallyaffect the wave-packet expansion. We discuss this matter in the next section. III. SUPPRESION OF WELLENPAKET EXPANSION DUE TO SELF-GRAVITY
In the above discussion the clock (as a whole) is treated as a free quantum mechanicalobject/body described by the Gaußsche Wellenpaket ψ ( t, r ) = e − r / a (2 π ) / (cid:20) r c (cid:18) it mr c (cid:19)(cid:21) − / , where a = r c (cid:18) it mr c (cid:19) . From this wave-packet one finds δl ( t ) ≃ s r c + t m r c & r c + t mr c . (3)Taking now into account the self-gravity of the Gaußian wave-packet - its dynamics getsmodified. For gravity prevents expansion, on general grounds one concludes that the valueof δl should be smaller than the expression (3). To get a qualitative picture, let us denoteby r wp the radius of the wave-packet. Without gravity r wp ( t ) ≃ s r c + t m r c , r wp (0) = r c , ˙ r wp (0) = 0 . (4)The quantum mechanical acceleration responsible for this expansion has the form¨ r wp ( t ) = 14 m (cid:16) r c + t m r c (cid:17) / = 14 m r wp . (5)One can derive the results (1, 2) immediately from Eq.(4). Minimizing the r wp ( t ) withrespect to r c one gets r c = r t m . After substituting it into Eq.(4) one finds r wp ( t ) = r tm .
4n the other hand, the gravitational acceleration that prevents expansion of the wave-packetlooks like [25] a g = l P mr wp . So that, the net acceleration takes the form a = 14 m r wp − l P mr wp . Thus, we have to solve the equation¨ r wp = 14 m r wp − l P mr wp , ⇒ ˙ r wp m r wp − l P mr wp = const. ≡ A . (6)As r wp (0) = r c , ˙ r wp (0) = 0, one finds A = 18 m r c − l P mr c . The solution can be written in the form r wp Z r c dx q A + l P mx − m x = t . (7)A typical form of the potential governing the dynamics of r wp is shown in Fig.1. It has aminimum at r c = 14 l P m , (8)corresponding to the state of stable equilibrium. Gaußian wave-packet having this radius inthe initial state neither contracts nor expands in course of time. From Eq.(8) one sees thatthe larger the mass - the smaller the clock size. However, there is an upper bound on themass set by the Schwarzschild radius, m max ≃ r c l P , which together with Eq.(8) yields r c ≃ l P , ⇒ δl ≃ l P . wp / m r w p − l P m / r w p FIG. 1. The potential: 1 / m r wp − l P m/r wp . It seems likely that one will arrive at the same result by solving the Schr¨odinger-Newtonsystem [25–27] i∂ t ψ = − m △ ψ − mϕψ , △ ϕ = 4 πl P m | ψ | , (9)with the initial state given by the Gaussian wave-packet ψ ( t = 0 , r ) = e − r / r c (2 πr c ) / . It is worth noting that, apart from the above discussed effect, the self-gravity implies alsothe reduction of clock’s mass. As this observation is significant for all discussions concerningthe space-time measurements, let us confine our attention to this problem now.
IV. REDUCTION OF MASS DUE TO SELF-GRAVITY
According to the papers [28–32], we can safely say that self-gravity affects the clock’smass. The conclusion reached in the papers [28–30] implies the modification of the clockmass in the following way m = m c + l P m c r c ⇒ m c = l − P (cid:18)q r c + 2 l P r c m − r c (cid:19) , (10)where m is to be identified with the mass in absence of gravity: l P →
0. It is plain to seethat m c is always positive. Duff, in his expository paper [32], points out that it is not aproper conclusion and suggests the correct version in the form6 c = m (cid:18) − l P m r c (cid:19) . (11)The source of this mistake is well explained in [32], however, we will not dwell on the details.Instead we point out that the Eq.(11) itself is very suggestive for the speculation (see [28])that leads to the Eq.(10). Namely, one can interpret the Eq.(11) as the correction to themass due to self-gravity in the framework of Newtonian gravity. However, one may claimthat in general relativity it is the total mass that interacts gravitationally and not just themass m . This way one arrives at Eq.(10). We shall consider both expressions separately.Let us assume that the reader has no objections with regard to the Eq.(10) and pose thequestion - how to operate with these two masses in the above discussed Gedankenexperi-ment? Before proceeding further, we have to make a few remarks to clarify the Eq.(10). m c is the mass that enters the exterior Schwarzschild solution. Hence, this mass determines theSchwarzschild radius. In addition, one has to require r c > l P m/ r c > l P m/ r c > l P m c .To carry the idea further, let us note that in Salecker-Wigner-Karolyhazy Gedanken-experiment the clock is described by the wave-function whose breadth is given by Eq.(4).Therefore, r wp plays the role of the radius of clock-mass distribution and, accordingly, onehas to replace r c in Eqs.(10, 11) by this expression (recall that r wp (0) = r c ) m c = l − P (cid:16)q r wp + 2 l P r wp m − r wp (cid:17) , (12) m c = m (cid:18) − l P m r wp (cid:19) . (13) m c is the mass determining the gravitational field that affects the dynamics of the wave-packet. The Eq.(6) gets modified as˙ r wp m r wp − l P m c r wp = const . . (14)In view of Eq.(12), the one-dimensional potential governing the time evolution of r wp inEq.(14) takes the form 78 m r wp − s l P mr wp + 1 . It has the same qualitative behavior as the potential depicted in Fig.1. It has a minimumat the point determined by the equation r c = 14 l P m s l P mr c . Now r c is greater than the solution (8). From this equation it readily follows that for m ≃ l − P the minimum occurs at r c ≃ l P .Now let us turn to the Eq.(13). In this case the potential governing the dynamics of r wp reads (cid:18) m + l P m (cid:19) r wp − l P mr . Hence r c = 14 l P m + l P m , and again r c ≃ l P for m ≃ l − P . V. CONCLUDING REMARKS
The results can be summarized as follows. Salecker and Wigner found that one can alwayschoose the size of the clock in such way that the total uncertainty in length measurementis minimized to δl ≃ p l/m . One can read this result also in the following way. If thereis a clock of size r c and mass m , then the maximum distance which can be measured bythis clock with accuracy r c is r c ≃ p l max /m [24] (see Eq.(1)). Their discussion uses thefiniteness of the speed of light and the quantum mechanical expansion of the wave packetdescribing the clock - no mention of the effects of general relativity. Further insight intothis Gedankenexperiment was obtained by Karolyhazy, who noted that minimum size of8he clock is set by the Schwarzschild radius and thus one can not measure the length withgreater accuracy than δl ≃ l / P l / . This rate of length fluctuations is certainly much lagerthan δl ≃ l P lending thus extra interest to the issue from the standpoint of experimentalsignatures. It should be noted, however, that such clock is very close to the black hole andone naturally expects strong gravitational effects that will essentially affect the wave packetdynamics. We have seen that self-gravity prevents the expansion of the wave-packet and thusreduces the uncertainty in length measurement to δl ≃ l P . One more point of importancerelated to self-gravity is the mass reduction. In view of the discussion presented in sectionIV, we see that it does not change our conclusion made in the previous section but, in anycase, it would be desirable if one could provide a numerical study of the Schr¨odinger-Newtonequation by taking into account the effect of the mass reduction due to self-gravity. For thispurpose one could use basic idea underlying the Schr¨odinger-Newton equation (9) as a guide.This system makes use of the Schr¨odinger equation in the background gravitational field,which in its turn is created by the mass distribution m | ψ ( t, r ) | . But the self-gravitationalmass reduction implies that the gravitational field for an external observer, r & r wp , issourced by the reduced mass m c . Hence, one has to make the following replacement inEq.(9) m | ψ ( t, r ) | → m c | ψ ( t, r ) | . From Eqs.(12, 13) it is obvious that as far as r wp ≫ l P m - the corrections are negligiblysmall.In closing this section, we wanted to draw attention to the fact that the modificationof Schr¨odinger-Newton system by replacing m with the gravitating mass, see Eqs.(12, 13),implies the dependence of the equation on the wave-packet breadth. The modification ofSchr¨odinger equation due to quantum fluctuations of the background space suggested in[33, 34] is of similar nature. To stress once more our point of view, physically meaningfulincorporation of l P into quantum mechanics should be expressed by some function of theratio l P /r wp rather than by a function of l P h p i , where h p i stands for average momentum.Otherwise one may obtain evidently misleading results [33].9 CKNOWLEDGMENTS
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