Gravity from the Uncertainty Principle.

Transcription

1 Gravity from the Unertainty Priniple. M.E. MCulloh Otober 29, 2013 Abstrat It is shown here that Newton's gravity law an be derived from the unertainty priniple. The idea is that as the distane between two bodies in mutual orbit dereases, their unertainty of position dereases, so their momentum and hene the fore on them must inrease to satisfy the unertainty priniple. When this result is summed over all the possible interations between the Plank masses in the two bodies, Newton's gravity law is obtained. This model predits that masses less than the Plank mass will be unaeted by gravity and so it may be tested by looking for an abrupt derease in the density of spae dust, for masses above the Plank mass. 1 Introdution The strong, weak and eletromagneti interations are modelled using the quantum exhange of virtual bosons, whereas gravity is modelled using the ontinuum of spaetime whih is urved by matter and energy (Adelberger et al. (2003). The uniation of these two models has been the goal of physis for a entury, but so far it has not been shown that quantum mehanis predits gravity, and although general relativity has been veried lose to the Earth (Ashby, 2003) it may have problems further aeld sine it predits singularities, and also fails to predit observed galaxy rotations and osmi aeleration without the ad ho addition of extra dark mass or energy. It may be that quantum mehanis is the deeper theory, sine its preditions are already known to be very aurate, and also reently Nesvizhevsky et al. (2002) have shown that gravity is quantised. They used neutrons, whih only feel the gravitational fore, and onned them to a gravitational potential well formed by gravity pulling downwards and a mirror reeting them bak up. They showed that the neutrons did not fall smoothly, but jumped vertially - just as eletons jump between quantum levels in an atom. SMSE, Plymouth University, Plymouth, PL4 8AA. mike.mulloh@plymouth.a.uk 1

2 One of the problems in quantising gravity is that the gravitational fore for a single partile in a quantum eld theory at high energy is innite. It is possible to resolve this using a one-dimensional partile instead - a string, but in string theory six additional dimensions of spae are needed and they are undetetable, and string theory itself makes no testable preditions (Rovelli, 2003). Another attempt to quantise gravity is loop quantum gravity (Ashtekar, 1986) in whih spaetime itself is quantised into spin networks that hange in disrete steps. It is onsistent with general relativity without requiring extra dimensions, but this theory also makes no testable preditions as yet (Rovelli, 2003). It is therefore useful to look for alternative ways to quantise gravity. Reently a new quantum mehanial model for inertial mass (quantised inertia or MiHsC) has been proposed by MCulloh (2007). This model assumes that 1) inertia is due to Unruh radiation, and 2) this radiation is subjet to a Hubblesale Casimir eet. The philosophy behind MiHsC is that as the aeleration of an objet redues, the waves of Unruh radiation that it sees, and that are assumed to ause its inertial mass, get longer in relation to the Hubble-sale and so a lesser proportion of them an be observed, so a greater proportion are disallowed by the Hubble-sale Casimir eet, reduing the inertial mass in a new way for low aelerations. MiHsC suessfully predits the observed osmi aelerations without the need for dark energy (MCulloh, 2010) and the anomalous galati rotation urves without dark matter (MCulloh, 2012). Note that MiHsC modies the inertial mass and not the gravitational mass, and the latter is onsidered in this paper. Here, a new quantum mehanial model for gravity is presented. This model assumes that the gravitational interation only ours between whole Plank masses, and it allows Newton's gravity law to be derived from Heisenberg's unertainty priniple. 2 Method Imagine there is a mass m orbiting another of mass M. Now onsider two Plank masses, one within m and one within M. Sine, with Plank masses, we are still, just, in the quantum realm, Heisenberg's unertainty priniple applies to their mutual position (x) and momentum (p), and the total unertainty is twie that for a single partile p x (1) Now we assume that we an write the total average unertainties of all the matter in the two bodies by summing up all the possible interations between the Plank masses in the two bodies. The Plank mass is dened as the mass of two objets whose gravitational potential energy is equal to the energy of a photon whose wavelength is their seperation, ie: /r = /r so that Gm 2 P 2

3 m P = /G = 21.76µg. This is the mass above whih gravity dominates quantum mehanis, so it is the maximum mass to whih we an apply quantum mehanis. So, if m has n Plank masses in it, and M has N we get Sine energy E=p we get p x Ē x N N n (2) n (3) Now we assume that the unertainty of the average position ( x) is the seperation between m and M, whih is the orbital radius r. This gives Ē r (4) Sine work done is fore times distane moved, then E = F x, so we get F x r (5) and x is again the seperation between m and M, r, so we get F r 2 (6) The double summation on the right hand side is equal to the number of Plank masses in mass m (m/m P ) times the number in M (M/m P ), where m P is the redued Plank mass, so F r 2 mm m 2 P (7) Therefore F mm m 2 (8) P r 2 This has the form of Newton's gravity law, and sine m P = /G we get 3

4 F GmM r 2 (9) This is Newton's gravity law, derived from the unertainty priniple. 3 Disussion This derivation an be explained more intuitively as follows: Heisenberg's unertainty priniple states that if the unertainty in position redues, then the unertainty in momentum inreases. So, as the radius of an orbit dereases and so the unertainty in position dereases, then the momentum (dp=fdx/) or fore (F) on the orbiting body must inrease. The derivation shows that when this is assumed, and the result is summed over all the possible interations between the Plank masses in the two bodies, the Newtonian gravity law is obtained. In the above derivation, the orret value for the gravitational onstant G is only obtained when it is assumed that the gravitational interation ours between whole multiples of the Plank mass. The derivation is being presented here, not beause it is laimed to be a omplete theory, but merely to point out that Newton's gravity law an be derived in this way, with some intuitive reasoning behind it. One predition of this model is that gravitational attration is only possible for masses greater than the Plank mass, whih is about the size of a dust partile, or m P = kg. This means that spae should show an abundane of dust partiles of less than this mass, sine they are not gravitationally attrated to large bodies, and there should be an abrupt drop o in their number for sizes over the Plank mass. 4 Conlusion The unertainty priniple states that if the unertainty in position redues, then the unertainty in momentum inreases. So, as the radius of an orbit dereases and the unertainty in position dereases, then the momentum, or fore, on the orbiting body must inrease. When this is summed over all the possible interations between all the Plank masses in the two bodies, Newton's gravity law is derived. This model an be tested by looking at the size distribution of spae dust: it predits a derease in their number above the Plank mass. Aknowledgements Thanks to B. Kim for enouragement. 4

5 Referenes Ashby, N., Relativity in the global positioning system. Living Rev. Relativity, 6, 1. Ashtekar, A., New variables for lassial and quantum gravity. Phys. Rev. Letters., 57, 18: Adelberger, E.G., B.R. Hekel, A.E. Nelson, Tests of the gravitational inverse-square law. MCulloh, M.E., Modelling the Pioneer anomaly as modied inertia. MNRAS, 376, MCulloh, M.E., Minimum aelerations from quantised inertia. EPL, 90, MCulloh, M.E., Testing quantised inertia on galati sales. Astrophysis and Spae Siene, Vol. 342, No. 2, Nesvizhevsky, V.V. et al., Quantum states of neutrons in the Earth's graviational eld. Nature, 415, Rovelli, C., A dialog on quantum gravity. Int. J. Mod. Phys., D12,