SSix easy roads to the Planck scale
Ronald J. Adler ∗ Hansen Laboratory for Experimental Physics Gravity Probe B Mission,Stanford University, Stanford California 94309
Abstract
We give six arguments that the Planck scale should be viewed as a fundamental minimum orboundary for the classical concept of spacetime, beyond which quantum effects cannot be neglectedand the basic nature of spacetime must be reconsidered. The arguments are elementary, heuristic,and plausible, and as much as possible rely on only general principles of quantum theory andgravity theory. The paper is primarily pedagogical, and its main goal is to give physics students,non-specialists, engineers etc. an awareness and appreciation of the Planck scale and the role itshould play in present and future theories of quantum spacetime and quantum gravity. a r X i v : . [ g r- q c ] J a n . INTRODUCTION Max Planck first noted in 1899 the existence of a system of units based on the threefundamental constants, G = 6 . × − N m /kg (or m /kg s ) (1) c = 3 . × m/sh = 6 . × − J s (or kg m /s )These constants are dimensionally independent in the sense that no combination is dimen-sionless and a length, a time, and a mass, may be constructed from them. Specifically, using (cid:126) ≡ h/ π = 1 . × − J s in preference to h , the Planck scale is l P = (cid:114) (cid:126) Gc = 1 . × − m, T P = (cid:114) (cid:126) Gc = 0 . × − , (2) M P = (cid:114) (cid:126) cG = 2 . × − kg The energy associated with the Planck mass is E P = M P c = 1 . × GeV . The Planck scale is prodigiously far removed from the human scale of about a meter.Indeed we humans are much closer in order of magnitude to the scale of the universe, 10 m ,than to the Planck scale! Present high-energy particle experiments involve energies only oforder 10 GeV , and even the highest energy cosmic rays detected to date, about 10 GeV ,are far below the Planck energy.The presence of (cid:126) in the Planck units in Eq.(2) indicates that the Planck scale is associatedwith quantum effects, c indicates that it is associated with spacetime, and G indicates thatit is associated with gravity. We therefore expect that the scale is characteristic of quantumspacetime or quantum gravity, which is the present conventional wisdom. We wish to illustrate in this paper that the Planck scale is the boundary of validity ofour present standard theories of gravity and quanta. Essentially we travel six roads to theboundary using arguments based on thought experiments. It is beyond our present scopeto cross the boundary and discuss current efforts toward theories of quantum gravity andspacetime; however in section IX we will briefly mention a few such efforts, and we refer thereader to a number of useful references.Observational confirmation of Planck scale effects is highly problematic.
We clearlycannot expect to do accelerator experiments at the Planck energy in the foreseeable future,2ut there are indirect possibilities. One involves the radiation predicted by Hawking to beemitted from black holes if they are small enough to have a “Hawking temperature” abovethat of the cosmic background radiation; in the final stages of black hole evaporation theHawking radiation should have about the Planck energy. Of course Hawking radiation hasnot yet been observed, although many theorists believe it must exist. Another possibility ofinterest is to analyze the radiation from very distant gamma ray bursters, which has beenenroute for about 10 yr . Speculations abound on the effect of Planck scale spacetimegranularity on propagation of such radiation. Finally we mention that in the earliest stagesof cosmological inflation quantum gravity effects might have been large enough to leavean imprint via primordial gravitational radiation on the details of the cosmic microwavebackground radiation. Lacking real experiments we use thought experiments (Gedankenexperiment) in this note.We give plausible heuristic arguments why the Planck length should be a sort of fundamentalminimum - either a minimum physically meaningful length, or the length at which spacetimedisplays inescapable quantum properties i.e. the classical spacetime continuum conceptloses validity. Specifically the six thought experiments involve: (1) viewing a particle witha microscope; (2) measuring a spatial distance with a light pulse; (3) squeezing a systeminto a very small volume; (4) observing the energy in a small volume; (5) measuring theenergy density of the gravitational field; (6) determining the energy at which gravitationalforces become comparable to electromagnetic forces. The analyses require a very minimalknowledge of quantum theory and some basic ideas of general relativity and black holes,which we will discuss in section II. Of course some background in elementary classical physics,including special relativity, is also assumed.We rely as little as possible on present theory, both because we desire mathematical andpedagogical simplicity, and because the general principles we use are most likely to survivethe vagaries of theoretical fashion. We hope that the discussions are thereby made moreaccessible to physics undergraduates and nonspecialists.
II. COMMENTS ON BASIC IDEAS OF QUANTUM THEORY AND GRAVITY
In this section we review the small amount of quantum theory and gravitational theoryneeded in later sections. Essentially we attempt to compress some root ideas of quantum3heory and general relativity into a few paragraphs! Those familiar with quantum theoryand general relativity may proceed to the first road in section III.The first aspect of quantum theory that we recall is the quantization of light into photons.According to Planck and Einstein light of frequency ν and wavelength λ = c/ν can only beemitted and absorbed in multiples of the energy E = hν = (cid:126) ω (3)where ω is the frequency in rad/s. Thus we may think of light as a rain of photons, eachwith E = hν . According to Einsteins relation for mass-energy equivalence, E = mc , aphoton should interact gravitationally as if it has an effective mass M ef = hν/c (4)Thus, for example, a single photon captured in a mirrored cavity increases the effective massof the cavity according to Eq.(4).Next we recall a standard heuristic derivation of the Heisenberg uncertainty principle. In Fig.1 we show the Heisenberg microscope thought experiment; using the microscope weview a particle with light entering from the bottom of the page. It is well-known from waveoptics (and also rather clear intuitively) that the position of the particle can be determined toan accuracy of about the wavelength λ of the light used; a bit more precisely the uncertaintyin position is given by ∆ x ∼ = λ/sinϑ (5)According to classical physics we could determine the position as accurately as desired byusing very short wavelength light, of arbitrarily low intensity so as not to disturb the particleby the electric field of the light wave. However the quantization of light as photons withenergy hν prevents this since the intensity cannot be made arbitrarily low. A single photonscattering from the particle and into the microscope (at angle less than ϑ ) will impartmomentum of order ∆ p ∼ = psinϑ = ( h/λ ) sinϑ to the particle, so that Eq.(5) implies∆ x ∆ p ∼ = h ≈ (cid:126) (6)which is the Heisenberg uncertainty principle (UP). (In our rough estimates we do notdistinguish between h and (cid:126) , thus taking 2 π ≈
1; this has been referred to as “using4eynman units.”) The UP forces us to consider the position and momentum of the particleto be fundamentally imprecise or “fuzzy” in such a way that the particle occupies a regionof at least (cid:126) in phase space ( x, p ). Thus we cannot speak of the trajectory of a quantummechanical particle but must instead take account of the fuzziness of such a particle in ourdescription, that is in terms of a wave function or probability amplitude.Most quantum mechanics textbooks also give a derivation of the UP from the commu-tation relation for the position and momentum operators, and it is also readily obtainedfrom the fact that the wave functions in position and momentum space are Fourier trans-forms of each other. However neither derivation is as conceptually simple as that using theHeisenberg microscope.There is an energy-time analog of the UP Eq.(6), but it has a somewhat different meaning.Consider a wave of frequency approximately ν and duration T , which thus consists of N ≈ νT cycles. The finite duration of the wave means that its leading and trailing edges will ingeneral be distorted somewhat from sinusoidal, so that N will not be precisely well-definedand measurable, but will have an uncertainty of order ∆ N ≈
1. This uncertainty in N implies in turn an uncertainty in the frequency, given by ∆ νT ≈ ∆ N ≈
1. (This relationis well-known in many fields, such as optics and electrical engineering, wherein it relatesband-width and pulse-width; the time T is subject to a number of somewhat differentinterpretations. The relation is also easy to derive more formally by calculating the Fouriertransform of a finite nearly monochromatic wave train, which is its frequency spectrum.)Since the energy of a photon of light is given by E = hν = (cid:126) ω we see that its energy canbe measured only to an accuracy given by∆ ET ≈ h ≈ (cid:126) (7)where ∆ E is, of course, the absolute value of the energy uncertainty. The same relationholds by similar reasoning for most any quantum system.The expression Eq.(7) formally resembles the UP in Eq.(6), but unlike the position of aparticle, time is not an observable in quantum mechanics, so Eq.(7) has a different meaning: T is the characteristic time of the system (eg. pulse width), and not an uncertainty in atime measurement. Thus, for example the light emitted by an atom is only approximatelymonochromatic, with an energy uncertainty given by ∆ E ≈ (cid:126) /T where T is the lifetime ofthe excited atomic state. 5he quantization relation Eq.(3) and the uncertainty relations Eq.s(6) and (7) are suffi-cient for our analyses in later sections; we need not discuss detailed quantum mechanics orquantum field theory.We next move on to general relativistic gravity and black holes. Recall that the lineelement or metric of special relativity gives the spacetime distance between nearby events.It is usually expressed in Lorentz coordinates ( ct, (cid:126)x ) as ds = ( cdt ) − d(cid:126)x (8)In general relativity gravity is described by allowing spacetime to be warped or distorted, or,more technically correct, curved. Coordinates in general relativity merely label the points inspacetime and do not by themselves give physical distances; for that we need a metric, whichrelates coordinate intervals to physical distance intervals. For a weak gravitational fieldand slowly moving bodies the line element or metric is approximately given by the so-calledNewtonian limit ds = (1 + 2 φ/c )( cdt ) − (1 − φ/c )( d(cid:126)x ) (9)Here φ is the Newtonian potential, with the dimensionless quantity φ/c assumed small, andalso assumed to go to zero asymptotically at large distances from the source; for examplefor a point mass M the potential is φ = − GM/r . What this means is that the proper time,or physical clock time, between 2 events at the same space position and separated only bya time coordinate interval cdt is ds = cdt (cid:112) φ/c (proper time separation) (10a)while the physical meter stick distance between 2 events separated only by a space coordinateinterval dx is dx (cid:112) − φ/c (space separation) (10b)and similarly for the y and z directions.The Newtonian limit Eq.(9) is a quite good approximation in many cases; for examplein the solar system the Newtonian potential is greatest at the surface of the sun, where2 φ/c ≈ − . Thus spacetime in the solar system is extremely close to that of specialrelativity, or flat. Since we are interested only in order of magnitude estimates, we will makefree use of the Newtonian limit Eq.(9) as an approximation. ds = (1 − GM/ c r ) (1 + GM/ c r ) ( cdt ) − (1 + GM/ c r ) d(cid:126)x (11)This expression is correct only in matter-free space outside the body. At large distancesfrom the body it is approximately ds ≈ (1 − GM/c r )( cdt ) − (1 + 2 GM/c r ) d(cid:126)x , (12) GM/c r << , which agrees with the Newtonian limit Eq.(9). If the body is small enough so that r = GM/ c lies outside of it then the coefficient of the time term in Eq.(11) vanishes, whichmeans that the proper time ds is zero, and a clock at that position would appear to a distantobserver to stop. This radius defines the surface of the black hole. Light cannot escape fromthe surface since it undergoes a red shift to zero frequency. A black hole of typical stellarmass is about a km in radius. A minor caveat concerning Eq.(11) is in order. The coordinate r used in Eq.(11) iscalled the isotropic radial coordinate and is not the same as that usually used for theSchwarzschild metric, the Schwarschild radial coordinate. We will not use the Schwarzschildradial coordinate here (or even define it) but only note that it is asymptotically equal to theisotropic coordinate r in Eq.(11) but differs from it at the black hole surface by a factor of 4;that is the black hole surface is at radius 2 GM/c in the Schwarzschild coordinate; 2 GM/c is widely known as the Schwarzschild radius. It is generally believed that if enough mass M is squeezed into a roughly spherical volumeof size about r ≈ GM/c then it must collapse to form a black hole, regardless of internalpressure or other opposing forces; however if the mass is needle or pancake shaped thequestion of collapse is not yet clearly settled. In summary the lesson to take away from the above paragraphs is that spacetime dis-tortion is a measure of the gravitational field. Specifically, since the line element givesthe physical distance between nearby spacetime points or events, such distances are givenroughly by Eq.(10). This corresponds to a fractional distortion given bySpacetime fractional distortion ≈ | φ | c (13a)7lternatively, from Eq.(11), we may say that for a region that is roughly spherical, of size l , and contains a mass M the fractional distortion is of orderSpacetime fractional distortion ≈ GMlc (13b)Even for rather strong gravitational fields this generally holds at least roughly. The aboveresults Eq.(13) will be basic to our arguments in future sections. III. THE GENERALIZED UNCERTAINTY PRINCIPLE
Our first road to the Planck scale is based on the same thought experiment as the Heisen-berg microscope used to obtain the UP in section II and shown in Fig.1, but it includes theeffects of gravity to obtain a generalization of the UP.
According to the UP in Eq.(4)there is no limit on the precision with which we may measure a particles position if we allowa large uncertainty in momentum, as would result from using arbitrarily short wavelengthlight. But this does not take into account the gravitational effects of even a single photon. Asnoted in section II a photon has energy hν and thus an effective mass M ef = hν/c = h/cλ ,which will exert a gravitational force on the particle. This will accelerate the particle, makingthe already fuzzy particle position somewhat fuzzier. Using classical Newtonian mechanicswe estimate the acceleration and position change due to gravity as roughly∆ a g ≈ GM ef /r ef = G ( h/λc ) /r ef , (14)∆ x g ≈ ∆ a g t ef ≈ G ( h/λc )( t ef /r ef )where r ef and t ef denote an effective average distance and time for the interaction. The onlycharacteristic velocity of the system is the photon velocity c, so we naturally take r ef /t ef ≈ c ,and obtain for the gravitational contribution to the uncertainty∆ x g ≈ Gh/λc ≈ ( G (cid:126) /c ) /λ = l p /λ (15)According to the UP Eq.(4) the position uncertainty neglecting gravity is about ∆ x ≈ (cid:126) / ∆ p ; we add the gravitational contribution in Eq.(15) to obtain a generalized uncertaintyprinciple or GUP, ∆ x ≈ (cid:16) (cid:126) ∆ p (cid:17) + l p (cid:16) ∆ p (cid:126) (cid:17) (16)8his same expression for the GUP has been obtained in numerous ways, ranging in sophis-tication from our very naive Newtonian approach to several versions of string theory; itwould thus appear to be a rather general result of combining quantum theory with gravityand may indeed be correct. Almost needless to say the GUP should be considered a rough estimate for positionuncertainty, with the coefficient of ∆ p/ (cid:126) in the second term only being of order of thesquare of the Planck length, and the whole expression being valid in order of magnitude aswe approach the Planck scale from above. For example, we could just as well have factors of2 π or 10 or 1 /α = 137 etc . multiplying l P in Eq.(16). Moreover we add the uncertainties dueto the standard UP and the gravitational interaction linearly; we could equally well havetaken the root mean square; this makes little difference in our conclusions, as the readermay verify.From the GUP in Eq.(16) we see that the position uncertainty of a particle has a minimumat (cid:126) / ∆ p = l P and is about ∆ x min ≈ l p (17)as shown in Fig.2. This minimum position uncertainty corresponds to a photon of wavelengthabout l P and energy E P .Since we cannot measure a particle position more accurately than the Planck length, theabove result suggests that from an operational perspective the Planck length may representa minimum physically meaningful distance. As such one may plausibly question whethertheories based on arbitrarily short distances, such as string theory, really make sense froman operational point of view; spacetime at the Planck scale and below might not be a usefulconcept. IV. LIGHT RANGING
The next argument uses a thought experiment that is particularly simple conceptu-ally, and has the virtue that length is defined via light travel time, which is the actualpresent practice: the definition of a meter is “the distance traveled by light in free spacein 1/299,792,458 of a second.”Fig.3 shows the experimental arrangement, which we refer to by the generic name light9anging, in analogy with laser ranging. We send a pulse of light with wavelength λ froma position labeled A to one labeled B, where it is reflected back to A, and measure thetravel time with a macroscopic clock visible from A. Since light is a wave we cannot askthat the pulse front can be much more accurately determinable than about λ , so there is anuncertainty of at least about ∆ l w ≈ λ in our measurement of the length l .If nature were actually classical we could use light of arbitrarily short wavelength andarbitrarily low energy so as not to disturb the system, and thereby measure the distanceto arbitrarily high accuracy. However in the real world we cannot use arbitrarily short λ since this would put large amounts of energy and effective mass in the measurement region,even if we use only a single photon. According to our comments in section II the energy ofthe light will distort the spacetime geometry and thus change the length l by a fractionalamount | φ | /c according to Eq.(13). We may estimate the Newtonian potential due to thephoton, which is somewhere in the interval l , to be about φ ≈ GM ef l ≈ G ( E/c ) l ≈ Ghνc l ≈ G (cid:126) clλ (18)so the spatial distortion is about∆ l g ≈ l ( φ/c ) ≈ ( G (cid:126) /c ) /λ = l p /λ (19)In Fig.3 the letter B labels a point in space, but we could instead take it to be a small bodyin free fall, which would move during the measurement (actually only during the return tripof the light pulse), and this would also affect the measurement. We have already estimatedjust such motion in section III; it is given in Eq.(15) by∆ x g ≈ l p /λ (20)That is the space distortion in Eq.(19) and the motion in Eq.(20) are comparable, and toour desired accuracy we simply write for either effect ∆ l g ≈ l P /λ .Since we only know the photon position to be somewhere in l we interpret this as anadditional uncertainty due to gravity. We add it to the uncertainty due to the wave natureof the light to obtain ∆ l ≈ ∆ l w + ∆ l g ≈ λ + l p /λ (21)This expression Eq.(21) for the total uncertainty has a minimum at λ = l P , where it isequal to ∆ l ≈ l P , so we conclude that the best we can do in measuring a distance using10ight ranging is about the Planck length. The energy and gravitational field of the photonprevents us from doing better.A final note is in order. We have here assumed implicitly that we have access to a perfectclassical clock for timing the light pulse. However if the clock is assumed to be a smallquantum object then there will be a further contribution to the uncertainty due to thespread of the position wave function of the clock during the travel of the light pulse. Someauthors suggest that such a quantum clock should be used in the thought experiment, andarrive at a larger uncertainty estimate for light ranging, one involving the size of the system l . However other authors point out that such a quantum clock may not be appropriatesince it could suffer decoherence and behave classically. Predictions of this nature may betestable with laser interferometers constructed as gravitational wave detectors. V. SHRINKING A VOLUME
In this thought experiment we shrink a volume containing a mass M as much as possi-ble, until we are prevented from continuing. We assume the volume is intrinsically three-dimensional, about l in all of its spatial dimensions, as shown in Fig.4. A difficulty occursdue to gravity when the size approaches the Schwarzschild radius, l ≈ GM/c (22)The system may then collapse to form a black hole as discussed in section II and cannot bemade smaller. There is of course no lower limit to this size if we choose an arbitrarily smallmass M .A different difficulty occurs due to quantum effects. From the UP the uncertainty in themomentum of the material in the volume is at least of order ∆ p ≈ (cid:126) /l . Since the energy inthe volume is given by E = M c + p c the uncertainty in the energy is roughly∆ E ≈ c ∆ p ≈ (cid:126) c/l (23)If l is made so small that this energy uncertainty increases to about 2 M c then pairs ofparticles can be created and appear in the region around the mass M, as shown in Fig.4. The localization is thereby ruined and the volume cannot shrink further. This limit happens11hen the energy and size are about
M c ≈ ∆ E ≈ (cid:126) c/l, l ≈ (cid:126) /M c (24)The quantity (cid:126) /M c is known as the Compton radius or Compton wavelength of mass M .Our inability to localize a single particle to better than its Compton radius is wellknownin particle physics. Indeed one fundamental reason that quantum field theory is used inparticle physics is that it can describe the creation and annihilation of particles whereas asingle particle wave function cannot. We now have two complementary minimum sizes for the volume containing a mass M :the Schwarzschild radius dictated by gravity is proportional to M , and the Compton radiusdictated by quantum mechanics is inversely proportional M . The overall minimum occurswhen the two are equal, which happens for l ≈ (cid:126) /M c ≈ GM/c , M ≈ (cid:126) c/G ≡ M p , (25) l ≈ (cid:126) /M p c ≈ (cid:112) (cid:126) G/c = l p Thus the combination of gravity and quantum effects creates insurmountable difficulties ifwe attempt to shrink a volume to smaller than Planck size.A minor caveat should be repeated here: we have assumed that the volume is effectively 3dimensional in that all of its dimensions are roughly comparable; if one of the dimensions ismuch larger or much smaller than the others the region is effectively one or two dimensionaland the question of gravitational collapse is less clear, as noted in section II.
VI. MEASURING PROPERTIES OF A SMALL VOLUME
For this rather generic thought experiment we use a quantum probe, such as a light pulse,to measure the size, energy content or other physical properties of a volume of characteristicsize l , as shown in Fig.5. Such properties may in general fluctuate significantly in the timeit takes light to cross the volume, so we would naturally want to do the measurement withinthat time, T ≈ l/c . For this we need a probe with frequency greater than c/l and energygreater than about E ≈ (cid:126) ( c/l ). 12ut there is a limit to how much probe energy E or effective mass M ef ≈ E/c = (cid:126) /lc can be packed into a region of size l , as we discussed in section II. According to Eq.(13) thefractional distance uncertainty in the volume containing such an effective mass is about∆ ll ≈ | φ | c ≈ c (cid:16) GM ef l (cid:17) = Gc l (cid:16) (cid:126) lc (cid:17) = G (cid:126) c l = l p l (26)If l is made so small that this approaches 1 then the geometry becomes greatly distortedand the measurement fails, which happens at l ≈ l P .Another way to see the limit effect is to note that the effective mass M ef = h/cl injectedinto the region by the probe can induce gravitational collapse to form a black hole (asalready noted in section V) when the region size approaches the Schwarzschild radius ofabout GM ef /c ; this happens for l ≈ GM ef c ≈ G (cid:126) c l , l ≈ (cid:112) G (cid:126) /c ≈ l p (27)We thus conclude that any attempt to measure physical properties in a region of aboutthe Planck size involves so much energy that large fluctuations in the geometry must occur,including the formation of black holes and probably more exotic objects such as wormholes. Such wild variations in geometry were first dubbed spacetime foam by J. A. Wheeler; thephrase has become quite popular to express vividly the supposed chaotic nature of geometryat the Planck scale. VII. ENERGY DENSITY OF GRAVITATIONAL FIELD
This argument is based on the uncertainty in the energy density of the gravitationalfield, and field fluctuations that correspond to the uncertainty. Algebraically it resemblessomewhat the argument of section VI, but has a different conceptual basis.We first obtain an expression for the energy density of the gravitational field in Newtoniantheory. Consider assembling a spherical shell of radius R and mass M by moving smallmasses from infinity to the surface, as shown in Fig.6. From Newtonian theory the energydone moving a small mass dM to the surface is dE = − GM dM/R (28)and the total energy for the assembly is the integral of this energy over the mass, which is E = − GM / R (29)13his binding energy may be viewed as energy in the gravitational field, and is negativebecause gravity is attractive. To obtain a general expression for the energy density weassume it is proportional to the square of the gravitational field (cid:126) g = −∇ φ . That is we set ρ g = λ ( ∇ φ ) (30)This is in direct analogy with the energy density of the electric field — except of coursefor the opposite sign! The proportionality constant λ can be determined by integrating ρ gin Eq.(30) over the volume between R and infinity in Fig.6 to get the total field energy.Equating this energy expression with the total binding energy given by Eq.(29) we obtain λ (4 πG M /R ) = − GM / R, λ = − / πG (31)Thus the energy density of the Newtonian gravitational field is written as ρ g = − ( ∇ φ ) / πG (32)In general relativity the problem of gravitational field energy is notoriously more subtleand complex. This is due to the nonlinearity of the field equations, which in turn is relatedto the fact that gravity carries energy and is thus a source of more gravity. In this sensegravity differs fundamentally from the electric field, which does not carry charge and thus isnot the source of more electric field. For our present purpose we will content ourselves withthe rough estimate given by Eq.(32).We now consider a space region of size l that is nominally empty and free of gravity,except for fluctuations allowed by the energy-time uncertainty relation Eq.(7). As in sectionVI we attempt to measure the gravitational energy in the region in time l/c , with accuracylimited to ∆ E ≈ (cid:126) c/l . We thus cannot verify that the region is truly free of gravity, butonly that the gravitational energy in the region is no more than about ∆ E ≈ (cid:126) c/l . FromEq.(32), this limit implies the following limiting relation for the Newtonian potential field l ( ∇ φ ) / πG ≈ (cid:126) c/l (33)As a rough estimate ( ∇ φ ) ≈ (∆ φ/l ) , where ∆ φ is the uncertainty or fluctuation in thenominally zero Newtonian potential field, so from Eq.(33) we obtain∆ φ ≈ √ (cid:126) cG/l (34)14his fluctuation corresponds roughly to a fractional spacetime distortion given by Eq.(13),∆ l/l ≈ ∆ φ/c ≈ (cid:112) (cid:126) G/c /l, ∆ l ≈ (cid:112) (cid:126) G/c = l p (35)That is, the allowed nonzero value of the energy density of the gravitational field correspondsto Newtonian potential fluctuations and thus metric and distance fluctuations; the distancefluctuations are, once again, about the Planck length. VIII. EQUALITY OF GRAVITY AND ELECTRIC FORCES
Our final argument characterizes the Planck scale in terms of the mass or energy atwhich gravitational effects become comparable to electromagnetic effects and thus cannotbe ignored in particle theory. The argument is simple to remember and provides a goodmnemonic for quickly deriving the Planck mass.In most situations we encounter gravity as an extremely weak force; for example thegravitational force between electron and proton in a hydrogen atom is roughly 40 orders ofmagnitude less than the electric force, and can safely be ignored. However if we insteadconsider two objects of very large mass M (or rest energy) with the electron charge e, thenthe gravitational and electric forces become equal when GM r ∼ = e r , M ∼ = e G (36)The dimensionless fine structure constant is defined by α ≡ e / (cid:126) c ∼ = 1 / M ∼ = α (cid:126) cG = αM p , M ∼ = √ αM p ∼ = M p
12 (37)That is equality occurs within a few orders of magnitude of the Planck mass, at least interms of Newtonian gravity. We may plausibly infer that such equality also occurs whencharged massive particles scatter at near the Planck energy. Quantum electrodynamics (QED), describing the electromagnetic interactions of electronsand other charged particles, ignores gravitational effects. Clearly this is not reasonable forenergies near the Planck scale. Thus virtual processes described by loop integrals are clearlynot handled correctly since they involve arbitrarily high energies, and indeed most of themdiverge.
We may therefore hope that a more comprehensive theory that includes gravitymight be free of such divergences. 15espite the simplistic nature of this section it does hint at the germ of deep ideas. Inthe standard model of particle physics the electromagnetic and weak forces are unified intothe so-called electroweak force, which has been very successful in predicting experimentalresults; at low interaction energies the weak and electromagnetic forces differ greatly, butat an energy above a few thousand GeV they become comparable.
They may be viewedas different aspects of a single force rather than as fundamentally different forces. Similarlythe strong force is widely believed to become comparable and similarly unified with theelectroweak force in some grand unified theory or GUT at energies of about 10 GeV , onlya few orders of magnitude below the Planck energy. Quite roughly speaking then, all thefundamental forces of nature are believed to become comparable near the Planck scale.
IX. SUMMARY AND FURTHER COMMENTS
We have tried to show that the Planck scale represents a boundary when we attempt toapply our present ideas of quantum theory, gravity, and spacetime on a small scale. To gobeyond that boundary, new ideas are clearly needed.There is much speculation by theorists on such new ideas. Here we will only mentionthree (of many) such efforts very superficially, and refer the reader to references 4 to 7. Thefirst effort involves perturbative quantum gravity, studied for many years by many authors,notably by Feynman and Weinberg; in perturbative quantum gravity the flat space of specialrelativity is taken to be a close approximation to the correct geometry, and deviationsfrom it are treated in the same way as more ordinary fields such as electromagnetism.Just as in quantum electrodynamics Feynman diagrams may be derived to describe theinteractions between particles and the quanta of the gravitational field, called gravitons.The theory has the serious technical drawback that it does not renormalize in the sameway as quantum electrodynamics, and in fact contains an infinite number of parameters andgraviton interactions. Even more importantly it does not truly address the quantum natureof spacetime. The second and best-known effort involves super-string theory, or simplystring theory, in which the point particles assumed in quantum field theories are replacedby one-dimensional strings of about Planck size. String theory purports to describe allparticles and interactions, and has been studied intensively for decades, and is consistent withgravitational theory since it accommodates a particle with the properties of the graviton, that16s zero mass and spin 2. As yet however there is no experimental or observational evidencethat its basic premise is correct. The third effort, which we may call affine loop gravity,recasts the mathematics of general relativity in such a way that the fundamental object isnot the metric but a mathematical object called an affine connection, which is analogous tothe gauge potentials describing other non-gravitational fields, such as the vector potentialof the electromagnetic field. In affine loop gravity areas and volumes are indeed quantized,and the theory has other attractive features.Various authors visualize spacetime as a boiling quantum foam of strange geometriessuch as virtual black holes and wormholes, or as a dense bundle of 6 dimensional Calabi-Yaumanifords, or as a subspace of a more fundamental 10 or 11 dimensional space, or as a lowerdimensional holographic projection, or as the eigenvalue space of quantum operators, or as aspin network, or as a woven quantum fabric etc. etc. But despite intense effort over decadesnone of the many speculative ideas and theories has yet reached a high level of success orgeneral acceptance, and we remain free to consider many possibilities.Perhaps the oddest possibility is that spacetime at the Planck scale is not truly observableand may thus be an extraneous and sterile concept, much as the luminous ether of thenineteenth century proved to be extraneous after the advent of relativity and spacetime —thus obviating decades of theoretical speculation. At present it is certainly not clear whatmight replace our present concept of spacetime at the Planck scale.
ACKNOWLEDGEMENTS
This work was partially supported by NASA grant 8-39225 to Gravity Probe B. Thanksgo to Robert Wagoner, Francis Everitt, and Alex Silbergleit and members of the GravityProbe B theory group for useful discussions and to Frederick Martin for patient reading andcomments on the manuscript. ∗ Electronic address: Electronic mail: [email protected] Plancks comments on his unit system preceded his discovery of the correct black body radi-ation law: M. Planck, “Uber irreversible Strahlungsvorgange,” Sitzungsberichte der KoniglichPreussischen Akademie der Wissenschaften zu Berlin 5, 440 V 480 (1899). J. A. Wheeler clearly identified the Planck scale as appropriate to quantum gravity, bringing itto the wider attention of theorists: J. A. Wheeler, “Geons,” Phys. Rev. 97, no. 2, 511-536 (1955).Reprinted in: J.A. Wheeler, Geometrodynamics (Academic Press, New York and London 1962). Ronald J. Adler, “Gravity,” in The New Physics for the Twenty-first Century, edited by GordonFraser, (Cambridge University Press, Cambridge UK, 2006). R. P. Feynman, “Lectures on Gravitation,” notes taken by F. Morinigo and W. G. Wagner(California Institute of Technology bookstore 1971). An up-to-date technical reference is: Daniele Oriti, “Approaches to Quantum Gravity,” (Cam-bridge Press, Cambridge UK 2009), in particular see the chapter by C. Rovelli, “The unfinishedrevolution,” in which he points out that the subject of quantum gravity dates back at least to:M. P. Bronstein, “Quantentheories schwacher Gravitationsfelder,” Physikalische ZeitschriftderSowietunion 9, 140 - ? (1936). A list of quantum gravity theories online: en.wikipedia.org/wiki/Quantum gravity G. Amelino-Camelia, “Planck-scale Lorentz-symmetry Test Theories,” arXiv:astroarXiv:astro-ph/0410076 (2004). G. Amelino-Camelia, “Gravity-wave interferometers as quantum-gravity detectors,” Na-ture , 216 - 218 (1999). G. Amelino-Camelia, ‘Quantum Gravity Phenomenology,”arXiv:0806.0339v1[gr-qc] Ronald J. Adler, Ilya Nemenman, James Overduin, David Santiago, “On the detectability ofquantum spacetime foam with gravitational wave interferometers,” Physics Letters B , 424-428 (2000). Ronald J. Adler, Pisin Chen, and David Santiago, “The Generalized Uncertainty Principle andBlack Hole Remnants,” General Relativity and Gravitation 33, 2101 V 2108 (2001). ∼ ejb/faq.html One interesting example of many speculations on spacetime foam: John Ellis, N. E. Mavromatos,D. V. Nanopoulos, “Derivation of a Vacuum Refractive Index in a Stringy Space-Time FoamModel,” Phys. Lett. B , 412 - 417 (2008). Joseph Silk, “The Cosmic Microwave Background.” arXiv:astro-ph/0212305v1 For history of quantum mechanics: Hans C. Ohanian, “Principles of Quantum Mechanics,”(Prentice-Hall, Englewood Cliffs, N. J. 1990), see section 1.4 on the Heisenberg microscope. Aclassic in quantum theory is: D. Bohm, “Quantum Theory,” (Prentice-Hall, Englewood Cliffs, N.J. 1951), see chapter 5. An online reference is: en.wikipedia.org/wiki/Introduction to quantummechanics. For quantum theory in general: R. Shankar, “Principles of Quantum Mechanics,” second edition(Plenum Press, New York and London 1994), see chapter 3 on history. M. Born and E. Wolf, “Principles of Optics,” seventh edition (1999) (Cambridge Press, Cam-bridge UK 1959), see chapter 10. Also see section 5.3 of Ohanian in 15 E. R. Taylor and J. A. Wheeler, “Spacetime Physics,” (W. H. Freeman, San Francisco 1963),see chapter 1. R. J. Adler, M. Bazin, and M. Schiffer, “Introduction to General Relativity,” (McGraw Hill, N.Y. 1965, second edition 1975), chapter 9. C. W. Misner, K. S. Thorne, and J. A. Wheeler, “Gravitation,” (W. H. Freeman, San Francisco1970), chapter 39. B. F. Schutz, “A first course in general relativity,” (Cambridge University Press, CambridgeUK 1985) chapter 1 on special relativity and chapter 8 on Newtonian limit. C. Will, “Theory and experiment in gravitational physics,” (Cambridge University Press, Cam-bridge UK 1981, revised edition 1993), chapter 4. See section 6.2 of 20 See chapter 14 of 20 See chapters 33 and 34 of 11 Ronald J. Adler and David I. Santiago, “On gravity and the uncertainty principle,” Mod. Phys.Lett. A , 1371 V 1381 (1999), F. Scardigli, Generalized uncertainty principle in quantumgravity from micro black hole gedanken experiment, Phys. Lett. B , 39 - 44 (1993). Also see11. E. Witten, “Reflections on the fate of spacetime,” Physics Today, April 1996, 24 V 30 (1996), See the string theory references in 5, and for cogent criticisms see: Lee Smolin, “The TroubleWith Physics,” (Houghton Mifflin, Boston and New York 2006), in particular parts II and III. B. N. Taylor and A. Thompson, “Guide for the Use of the International System of Units,”Special Publication 811 (National Institute of Standards and Technology, Gaithersburg MD2008), and online discussion at: en.wikipedia.org/wiki/Metre J. Y. Ng and H. van Dam, “On Wigner’s clock and the detectability of spacetime foam withgravitational-wave interferometers,” Phys. Lett. B , 429 - 435 (2000); J. Y. Ng and H. vanDam, “Measuring the foaminess of space-time with gravity-wave interferometers,” Found.Phys. , 795-805 (2000); W.A. Christiansen, J. Y. Ng, and H. van Dam, “Probing spacetime foamwith extragalactic sources,” Phys. Rev.Lett. , 051301 (2006) James D. Bjorken and S. D. Drell, “Relativistic Quantum Mechanics,” (McGraw Hill, New York1964), see chapter 3. James D. Bjorken and S. D. Drell, “Relativistic Quantum Fields,” (McGraw Hill, New York1964), see chapter 11. Anthony Zee, “Quantum Field Theory in a Nutshell,” (Princeton Uni-versity Press, Princeton N. J. 2003), see chapter 1. See chapter 44 of ref. 21, and chapter 15 of ref. 20. Wheeler coined many descriptive terms in use today, such as quantum foam and black hole.See: C. W. Misner, Kip S. Thorne, and W. H. Zurek, “John Wheeler, relativity, and quantuminformation,” in “Physics Today,” April 2009. See chapter 11 of ref. 20. Chris Quigg, “Particles and the standard model,” in The New Physics for the Twenty-firstCentury, edited by Gordon Fraser, (Cambridge University Press, Cambridge UK, 2006). See chapters 7 and 8 of ref. 34. IG. 1: Fig.1. Left: a particle is illuminated from below by light of wavelength λ , which scattersinto the microscope whose objective lens subtends an angle of 2 ϕ . Right: the particle nature ofthe scattering is emphasized, with an effective separation r ef shown.FIG. 2: The position uncertainty due to the standard UP and the additional gravitational effectembodied in the GUP. The minimum uncertainty occurs at about twice the Planck length.FIG. 3: A light pulse is sent from A and reflected back from B. Its energy causes a distortion ofthe spacetime between A and B and hence affects the length l . IG. 4: Shrinking of a volume containing mass M is limited by gravitational collapse to a blackhole in the top figure, and by the creation of particle anti-particle pairs in the lower figure.FIG. 5: A region of space of size l to be measured in time l/c . As the size approaches the Plancklength there can occur wild variations in the geometry, including such things as black holes andwormholes.FIG. 6: Potential energy of many small dM elements reappears as energy density ρ g of the gravi-tational field.of the gravi-tational field.