Counterexamples to the maximum force conjecture
CCounterexamples to the maximumforce conjecture
Aden Jowsey and
Matt Visser ID School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand.
E-mail: [email protected] , [email protected] Abstract:
Dimensional analysis shows that the speed of light and Newton’s constant of gravi-tation can be combined to define a quantity F ∗ = c G N with the dimensions of force(equivalently, tension). Then in any physical situation we must have F physical = f F ∗ ,where the quantity f is some dimensionless function of dimensionless parameters. Inmany physical situations explicit calculation yields f = O (1), and quite often f ≤ .This has lead multiple authors to suggest a (weak or strong) maximum force/maximumtension conjecture. Working within the framework of standard general relativity, wewill instead focus on counter-examples to this conjecture, paying particular attentionto the extent to which the counter-examples are physically reasonable. The variouscounter-examples we shall explore strongly suggest that one should not put too muchcredence into any universal maximum force/maximum tension conjecture. Specifically,fluid spheres on the verge of gravitational collapse will generically violate the weak(and strong) maximum force conjectures. If one wishes to retain any general notion of“maximum force” then one will have to very carefully specify precisely which forces areto be allowed within the domain of discourse. Date:
Wednesday 3 February 2021; Tuesday 2 March 2021; L A TEX-ed March 3, 2021
Keywords : maximum force; maximum tension; general relativity. a r X i v : . [ g r- q c ] M a r ontents ρ = ρ s + p . . . . . . . . . . . . . . . . . . . 173.4.1 Radial force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Equatorial force . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.3 DEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Scaling solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.1 Radial force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.2 Equatorial force . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6 TOV equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6.1 Radial force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6.2 Equatorial force . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 – 1 – Introduction
The maximum force/maximum tension conjecture was independently mooted some 20years ago by Gary Gibbons [1] and Christoph Schiller [2]. At its heart one starts bynoting that in (3+1) dimensions the quantity F ∗ = c G N ≈ . × N (1.1)has the dimensions of force (equivalently, tension). Here c is the speed of light invacuum, and G N is Newton’s gravitational constant. Thereby any physical force can always be written in the form F physical = f F ∗ , (1.2)where the quantity f is some dimensionless function of dimensionless parameters. Invery many situations [1–4] explicit calculations yield f ≤ , though sometimes numberssuch as f ≤ also arise [5]. Specifically, Yen Chin Ong [5] formulated strong and weakversions of the conjecture:1. Strong form: f ≤ .2. Weak form: f = O (1).Note that F ∗ = E Planck /L Planck can also be interpreted as the Planck force, though it isnot intrinsically quantum as the various factors of (cid:126) cancel, at least in (3+1) dimensions.Furthermore it is sometimes interesting [6] to note that the Einstein equations G ab = 8 π G N c T ab , (1.3)can be written in terms of F ∗ as T ab = F ∗ π G ab . (1.4)When recast in this manner, maximum forces conjectures have tentatively been relatedto Jacobson’s entropic derivation of the Einstein equations [7].Considerable work has also gone into attempts at pushing various modifications of themaximum force conjecture beyond the framework of standard general relativity [8, 9].Overall, while there is little doubt that the quantity F ∗ is physically important, we feelthat the precise status of the maximum force conjecture is much less certain, and isless than universal. 2e shall investigate these conjectures within the context of standard general relativity,focussing on illustrative counter-examples based on simple physical systems, analyz-ing the internal forces, and checking the extent to which the counter-examples arephysically reasonable. Specifically, we shall consider static spherically symmetric fluidspheres [10–18], and investigate both radial and equatorial forces. We shall also in-clude an analysis of the speed of sound, and the relevant classical energy conditions,specifically the dominant energy condition (DEC), see [19–28]. We shall see that eventhe most elementary static spherically symmetric fluid sphere, Schwarzschild’s constantdensity star, raises significant issues for the maximum force conjecture. Other models,such as the Tolman IV solution and its variants are even worse. Generically, we shallsee that any prefect fluid sphere on the verge of gravitational collapse will violate theweak (and strong) maximum force conjectures. Consequently, if one wishes to retainany truly universal notion of “maximum force” then one will at the very least have tovery carefully delineate precisely which forces are to be allowed within the domain ofdiscourse. Consider spherically symmetric spacetime, with metric given in Schwarzschild curvaturecoordinates: ds = g tt dt + g rr dt + r ( dθ + sin θ dφ ) . (2.1)We do not yet demand pressure isotropy, and for the time being allow radial andtransverse pressures to differ, that is p r (cid:54) = p t .Pick a spherical surface at some specified value of the radial coordinate r . Define F r ( r ) = (cid:90) p r ( r ) dA = 4 π p r ( r ) r . (2.2)This quantity simultaneously represents the compressive force exerted by outer layersof the system on the core, and the supporting force exerted by the core on the outerlayers of the system.Consider any equatorial slice through the system and define the equatorial force by F eq = (cid:90) p t ( r ) dA = 2 π (cid:90) R s √ g rr p t ( r ) rdr. (2.3)This quantity simultaneously represents the force exerted by the lower hemisphere ofthe system on the upper hemisphere, and the force exerted by the upper hemisphereof the system on the lower hemisphere. Here R s is the location of the surface ofthe object (potentially taken as infinite). As we are investigating with sphericallysymmetric systems, the specific choice of hemisphere is irrelevant.3 Perfect fluid spheres
The perfect fluid condition excludes pressure anisotropy so that radial and transversepressures are set equal: p ( r ) = p r ( r ) = p t ( r ). Once this is done, the radial andequatorial forces simplify F r ( r ) = (cid:90) p ( r ) dA = 4 π p ( r ) r ; (3.1) F eq = (cid:90) p ( r ) dA = 2 π (cid:90) R s √ g rr p ( r ) rdr. (3.2)Additionally, we shall impose the conditions that pressure is positive and decreasesas one moves outwards with zero pressure defining the surface of the object [10–18]. Similarly density is positive and does not increase as one moves outwards, thoughdensity need not be and typically is not zero at the surface [10–18].We note that for the radial force we have by construction F r (0) = 0; F r ( R s ) = 0; and for r ∈ (0 , R s ) : F r ( r ) > . (3.3)In particular in terms of the central pressure p we have the particularly simple bound F r ( r ) < π p R s . (3.4)This suggests that in general an (extremely) weak version of the maximum force conjec-ture might hold for the radial force, at least within the framework outlined above, andas long as the central pressure is finite. Unfortunately without some general relation-ship between central pressure p and radius R s this bound is less useful than one mighthope. For the strong version of the maximum force conjecture no such simple argumentholds for F r , and one must perform a case-by-case analysis. For the equatorial force F eq there is no similar argument of comparable generality, and one must again performa case-by-case analysis.Turning now to the classical energy conditions [19–28], they add extra restrictions toensure various physical properties remain well-behaved. For our perfect fluid solutions,these act as statements relating the pressure p and the density ρ given by the stress- There is a minor technical change in the presence of a cosmological constant, the surface is thendefined by p ( R s ) = p Λ . T ˆ µ ˆ ν . Since, (in view of our fundamental assumptions that pressure anddensity are both positive), the null, weak, and strong energy conditions, (NEC, WEC,SEC) are always automatically satisfied, we will only be interested in the dominantenergy condition (DEC). In the current context the dominant energy condition onlyadds the condition | p | ≤ ρ . But since in the context of perfect fluid spheres, the pressureis always positive, it is more convenient to simply write this as pρ ≤
1; that is p ≤ ρ. (3.5)The best physical interpretation of the DEC is that it guarantees that any timelikeobserver with 4-velocity V a will observe a flux F a = T ab V b that is non-spacelike (eithertimelike or null) [25]. However, it should be pointed out that the DEC, being thestrongest of the classical energy conditions, is also the easiest to violate — indeedthere are several known situations in which the classical DEC is violated by quantumeffects [20–28].The DEC is sometimes [somewhat misleadingly] interpreted in terms of the speed ofsound not being superluminal: naively v s = ∂p/∂ρ ≤
1; whence p ≤ ρ − ρ surface <ρ . But the implication is only one-way, and in addition the argument depends onextra technical assumptions to the effect that the fluid sphere is well-mixed with aunique barotropic equation of state p ( ρ ) holding throughout the interior. To clarifythis point, suppose the equation of state is not barotropic, so that p = p ( ρ, z i ), withthe z i being some collection of intensive variables, (possibly chemical concentrations,entropy density, or temperature). Then we have dpdr = ∂p∂ρ dρdr + (cid:88) i ∂p∂z i dz i dr = v s ( ρ, z i ) dρdr + (cid:88) i ∂p∂z i dz i dr . (3.6)Then, (noting that dρ/dr is non-positive as one moves outwards), enforcing the speedof sound to not be superluminal implies dpdr ≥ dρdr + (cid:88) i ∂p∂z i dz i dr . (3.7)Integrating this from the surface inwards we have p ( r ) ≤ ρ ( r ) − ρ ( R s ) + (cid:88) i (cid:90) R s r ∂p ( ρ, z i ) ∂z i dz i dr dr. (3.8)5onsequently, unless one either makes an explicit barotropic assumption ∂p/∂z i = 0,or otherwise at the very least has some very tight control over the partial derivatives ∂p/∂z i , one simply cannot use an assumed non-superluminal speed of sound to deducethe DEC. Neither can the DEC be used to derive a non-superluminal speed of sound,at least not without many extra and powerful technical assumptions. We have beenrather explicit with this discussion since we have seen considerable confusion on thispoint. Finally we note that there is some disagreement as to whether or not the DECis truly fundamental [21–24]. We shall now consider a classic example of perfect fluid star, Schwarzschild’s constantdensity star [29], (often called the Schwarzschild interior solution), which was discov-ered very shortly after Schwarzschild’s original vacuum solution [30], (often called theSchwarzschild exterior solution).It is commonly argued that Schwarzschild’s constant density star is “unphysical” onthe grounds that it allegedly leads to an infinite speed of sound. But this is a naiveresult predicated on the physically unreasonable hypothesis that the star is well-mixedwith a barotropic equation of state p = p ( ρ ). To be very explicit about this, all realisticstars are physically stratified with non-barotropic equations of state p = p ( ρ, z i ), withthe z i being some collection of intensive variables, (possibly chemical concentrations,entropy density, or temperature). We have already seen that dpdr = ∂p∂ρ dρdr + (cid:88) i ∂p∂z i dz i dr = v s ( ρ, z i ) dρdr + (cid:88) i ∂p∂z i dz i dr . (3.9)Thence for a constant density star, dρ/dr = 0, we simply deduce dpdr = (cid:88) i ∂p∂z i dz i dr . (3.10)This tells us nothing about the speed of sound, one way or the other — it does tellus that there is a fine-tuning between the pressure p and the intensive variables z i ,but that is implied by the definition of being a “constant density star”. We have beenrather explicit with this discussion since we have seen considerable confusion on thispoint. Schwarzschild’s constant density star is not “unphysical”; it may be “fine-tuned”but it is not a priori “unphysical”. 6pecifically, the Schwarzschild interior solution describes the geometry inside a staticspherically symmetric perfect fluid constant density star with radius R s and mass M by the metric: ds = − (cid:32) (cid:114) − MR s − (cid:115) − M r R s (cid:33) dt + (cid:18) − M r R s (cid:19) − dr + r d Ω . (3.11)Here we have adopted geometrodynamic units ( c → G N → T ˆ t ˆ t = ρ = 3 M πR s ; (3.12) T ˆ r ˆ r = T ˆ θ ˆ θ = T ˆ φ ˆ φ = p = ρ (cid:113) − Mr R s − (cid:113) − MR s (cid:113) − MR s − (cid:113) − Mr R s . (3.13)This gives us the relation between density and pressure, as well as demonstrating theperfect fluid condition ( p = p r = p t ), and also verifying that the density is (insidethe star) a position independent constant. In these geometrodynamic units both den-sity and pressure have units 1/(length) , while forces are dimensionless. Note thatthe pressure does in fact go to zero at r → R s , so R s really is the surface of the“star”. Rewriting the relation between pressure and density in terms of the simplifieddimensionless quantities χ = MR s and y = r R s we see p = ρ √ − χy − √ − χ √ − χ − √ − χy . (3.14)Here 0 ≤ y ≤
1, and 0 ≤ χ < . The first of these ranges is obvious from the definitionof y , while the second comes from considering the central pressure at y = 0: p = ρ − √ − χ √ − χ − . (3.15)Demanding that the central pressure be finite requires χ < . (This is actually a rathermore general result of general relativistic stellar dynamics, not restricted to constantdensity, see various discussions of the Buchdahl–Bondi bound [31, 32].)7 .2.1 Radial Force The radial force F r as defined by equation (3.1) can be combined with the pressure-density relation given by equation (3.14), giving: F r = 4 πpr = 4 πρR s y √ − χy − √ − χ √ − χ − √ − χy = 32 χy √ − χy − √ − χ √ − χ − √ − χy . (3.16)As advertised in both abstract and introduction, this quantity is indeed a dimension-less function of dimensionless variables. Furthermore this quantity is defined on thebounded range 0 ≤ y ≤
1, 0 ≤ χ < . To find if F r itself is bounded we analyse themulti-variable derivative for critical points.For ∂ χ F r we find: ∂ χ F r = − y (cid:18) { − χ (3 + y ) }√ − χ √ − χy − { − χ (3 + 5 y − χy ) }√ − χ √ − χy (3 √ − χ − √ − χy ) (cid:19) . (3.17)For ∂ y F r we find: ∂ y F r = − χ (cid:18) { − χ (3 + y ) }√ − χy − { − χy }√ − χ √ − χy (3 √ − χ − √ − χy ) (cid:19) . (3.18)In particular we see that χ∂ χ F r − y∂ y F r = 3 χ y (cid:18) √ − χy ) √ − χ (3 √ − χ − √ − χy ) (cid:19) . (3.19)To have a critical point, ∂ χ F r = ∂ y F r = 0, we certainly require χy = 0. So either χ = 0or y = 0. But for y = 0, and χ ∈ (0 , ) we have ∂ y F r −→ χ (3 χ + 4 √ − χ − √ − χ − > . (3.20)In contrast, for χ = 0, and y ∈ (0 , ∂ χ F r →
0. So the only critical points lieon one of the boundary segments: ∂ χ F r = ∂ y F r = 0 ⇐⇒ χ = 0 . (3.21)Therefore to find the maxima of F r ( χ, r ) we must inspect all four of the boundarysegments of the viable region. Along three of the boundary segments we can see thatthe three lines corresponding to χ = 0, y = 0, and y = 1 all give F r ( χ, r ) = 0, leavingonly χ → to be investigated. 8e note lim χ → F r ( χ, y ) = 4 y √ − y − − √ − y . (3.22)Inserting this into the partial derivative ∂ y F r reveals:lim χ → ∂ y F r = − (cid:18) √ − y (cid:19) . (3.23)This is a strictly negative function in the range 0 ≤ y ≤ F r ( χ, y ) can be found by taking the limit lim y → giving:( F r ) max = lim y → lim χ → F r = 2 . (3.24)This is therefore bounded, with the radial force approaching its maximum at the centreof a fluid star which is on the verge of collapse. This force violates the strong maximumforce conjecture, though it satisfies the weak maximum force conjecture. This limit caneasily be seen graphically in Figure 1. Figure 1 : Radial force F r ( χ, y ) for the interior Schwarzschild solution.Note F r ( χ, y ) is bounded above by 2 in the region of interest y ∈ [0 , χ ∈ [0 , / .2.2 Equatorial force Using equation (3.2) and the metric defined in equation (3.11), with the relabelling ofthe previous subsection in terms of χ and y gives: F eq ( χ ) = 38 χ (cid:90) √ − χy (cid:18) √ − χy − √ − χ √ − χ − √ − χy (cid:19) dy. (3.25)The integral evaluates to: F eq ( χ ) = 34 (cid:104)(cid:112) − χy + 2 (cid:112) − χ ln (cid:16) (cid:112) − χ − (cid:112) − χy (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) y =1 y =0 . (3.26)Ultimately F eq ( χ ) = 34 (cid:104)(cid:112) − χ (cid:110) − χ ) − (cid:16) (cid:112) − χ − (cid:17)(cid:111) − (cid:105) . (3.27)However, due to the presence of the − ln (cid:0) √ − χ − (cid:1) term in this equation, it canbe seen that as χ → , F eq ( χ ) → + ∞ . Indeed F eq ( χ ) = ln (cid:0) − χ (cid:1) O (1) , (3.28)implying that the equatorial force in this space-time will grow without bound as thestar approaches the critical size, (just prior to gravitational collapse), in violation ofboth the strong and weak maximum force conjectures.So while the interior Schwarzschild solution has provided a nice example of a boundedradial force, F r ( y, χ ), it also clearly provides an explicit counter-example, where theequatorial force F eq ( χ ) between two hemispheres of the fluid star grows without bound. Imposing the DEC (equation 3.5) within the fluid sphere we would require: pρ = √ − χy − √ − χ √ − χ − √ − χy ≤ (cid:112) − χy ≤ (cid:112) − χ, (3.30)whence 1 − χy ≤ − χ ) . (3.31)10pplying the boundary conditions of 0 ≤ χ ≤ , 0 ≤ y ≤
1, we have a solution range: (cid:18) ≤ χ ≤ , ≤ y ≤ (cid:19) (cid:91) (cid:18) < χ ≤ , − χ ≤ y ≤ (cid:19) . (3.32)See figures 2 and 3. Figure 2 : pρ in first range Figure 3 : pρ in second rangeWithin the first region 0 ≤ χ ≤ , ≤ y ≤
1, the radial force is maximised at: χ = 34 , y = 16 (5 − √ ≈ . → F r = 316 ( √ − ≈ . < . (3.33)Under these conditions the strong maximum force conjecture is satisfied. This can beseen visually in figure 4.Within the second region < χ ≤ , − χ ≤ y ≤
1, the radial force is maximised at: χ = 89 , y = 58 ; → F r = 56 > . (3.34)Under these conditions the strong maximum force conjecture is violated, though theweak maximum force conjecture is satisfied. This can be seen visually in figure 5.Turning to the equatorial force, we see that the integrand used to define integral for F eq ( χ ) satisfies the DEC only within the range 0 ≤ χ ≤ . Using the result for F eq ( χ )given above, equation (3.27), we have:( F eq ) max , DEC = F eq ( χ = 3 /
4) = 38 (2 ln 2 − ≈ . < . (3.35)This now satisfies the strong maximum force conjecture.11 igure 4 : Radial force F r ( χ, y ) for the interior Schwarzschild solution in region 1 (cid:0) ≤ χ ≤ , ≤ y ≤ (cid:1) where the DEC is satisfied. Figure 5 : Radial force F r ( χ, y ) for the interior Schwarzschild solution in region 2 (cid:16) < χ ≤ , − χ ≤ y ≤ (cid:17) where the DEC is satisfied.12 .2.4 Summary Only if we enforce the DEC can we then make Schwarzschild’s constant density starsatisfy the weak and strong maximum force conjectures. Without adding the DECSchwarzschild’s constant density star will violate both the weak and strong maximumforce conjectures. Since it is not entirely clear that the DEC represents fundamentalphysics [21–24], it is perhaps a little sobering to see that one of the very simplestidealized stellar models already raises issues for the maximum force conjecture. We shallsoon see that the situation is even worse for the Tolman IV model (and its variants).
The Tolman IV solution is another perfect fluid star space-time, however it does nothave the convenient (albeit fine-tuned) property of constant density like the interiorSchwarzschild solution. The metric can be written in the traditional form [10]: ds = − (cid:18) r A (cid:19) dt + 1 + r A (cid:0) − r R (cid:1) (cid:0) r A (cid:1) dr + r d Ω . (3.36)Here A and R represent some arbitrary scale factors with units of length. This metricyields the orthonormal stress-energy tensor: T ˆ t ˆ t = ρ = 18 π r + (7 A + 2 R ) r + 3 A ( A + R ) R ( A + 2 r ) ; (3.37) T ˆ r ˆ r = T ˆ θ ˆ θ = T ˆ φ ˆ φ = p = 18 π R − A − r R ( A + 2 r ) . (3.38)This demonstrates the non-constancy of the energy-density ρ as well as the perfect fluidconditions. Physically, the surface of a fluid star is defined as the zero pressure point,which now is at: R s = (cid:114) R − A . (3.39)And likewise we can find the surface density ( ρ at R ): ρ s = 34 π A + R R ( A + 2 R ) . (3.40)The central pressure and central density are p = 18 π R − A R A ; ρ = 18 π R + A ) R A . (3.41)13oving forwards, we will likewise calculate the radial and equatorial forces in thisspace-time. Using the previously defined radial force equation, (2.2), we can write the radial forcefor the Tolman IV space-time as: F r ( r, R, A ) = 4 πp r r = r R (cid:18) R − A − r A + 2 r (cid:19) . (3.42)Defining y = r /R s and a = A /R we have y ∈ (0 ,
1) and a ∈ (0 , F r ( a, y ) = (1 − a ) y (1 − y )6 a + 4(1 − a ) y . (3.43)Note this is, as expected, a dimensionless function of dimensionless variables.The multivariable derivatives are: ∂ a F r ( a, y ) = y (1 − y )(1 − a )(2 ay − a − y − ay − a − y ) . (3.44) ∂ y F r ( a, y ) = (1 − a ) (2 ay − ay − y − a )2(2 ay − a − y ) . (3.45)For both derivatives to vanish, (within the physical region), we require a = 1. However a = 1 actually minimises the function with F r ( a, y ) = 0. So we need to look at theboundaries of the physical region. Both y = 0 and y = 1 also minimise the functionwith F r ( a, y ) = 0. We thus consider a = 0: F r (0 , y ) = 14 (1 − y ) , (3.46)where then it is clear that the function is maximised at a = 0, y →
0, which cor-responds ( F r ) max = . This can be seen visually in figure 6. Thus F r ( a, y ) for theTolman IV solution is compatible with the strong maximum force conjecture, but asfor the Schwarzschild constant density star, we shall soon see that the equatorial forcedoes not behave as nicely. 14 igure 6 : F r ( a, y ) for the Tolman IV solution. Using equation (3.2) for this space-time, and combining it with radial surface result ofequation (3.39), we obtain: F eq = 2 π (cid:90) R s √ g rr p ( r ) r dr = 14 (cid:90) (cid:113) R − A (cid:115) r A (cid:0) − r R (cid:1) (cid:0) r A (cid:1) R − A − r R ( A + 2 r ) r dr. (3.47)As an integral this converges, however the resultant function is intractable. Instead, wewill opt for a simpler approach by finding a simple bound. Since the radial coordinateis physically bound by 0 ≤ r ≤ R s = (cid:113) R − A < R , we find that in that range: g rr = 11 − r R r A r A ≥ r A r A ≥ . (3.48)This is actually a much more general result; for any perfect fluid sphere we have g rr = 11 − m ( r ) /r , (3.49)where m ( r ) is the Misner–Sharp quasi-local mass.15o as long as m ( r ) is positive, which is guaranteed by positivity of the density ρ ( r ), wehave g rr >
1, and so in all generality we have F eq > π (cid:90) R s p ( r ) r dr. (3.50)For the specific case of Tolman IV we can write F eq > (cid:90) (cid:113) R − A R − A − r R ( A + 2 r ) rdr. (3.51)Now make the substitutions y = r /R s and a = A /R . We find F eq > πr s (cid:90) p dy = 18 (cid:90) (1 − a ) (1 − y )2 (1 − a ) y + 3 a dy. (3.52)This integral yields F eq > (cid:26) ( a + 2) ln[(2 y − a − y ]32 − (1 − a ) y (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . (3.53)Thence F eq > ( a + 2)[ln(2 + a ) − ln(3 a )32 − (1 − a )16 . (3.54)Under the limit a → − log(3 a ) → + ∞ . So the inequality(3.54) diverges to infinity, demonstrating that the equatorial force in the Tolman-IVspace-time can be made to violate the weak maximum force conjecture.Thus, as in the case of the interior Schwarzschild solution, we have shown that theradial force is bounded (and in this case obeys both the weak and strong maximumforce conjectures). However, the equatorial force can be made to diverge to infinity andact as a counter example to both weak and strong conjectures. To see if the DEC is satisfied over the range of integration for the equatorial force, weinquire as to whether or not pρ = ( A + 2 r )( R − A − r )3 A + 7 A r + 6 r + (3 A + 2 r ) R ≤ a = A /R and z = r /R , so that a ∈ (0 , z ∈ (0 , − a ),we can write this as pρ − − a + 6 az + a + 6 z )(7 a + 2) z + 3 a ( a + 1) + 6 z < , (3.56)which is manifestly negative. So adding the DEC does not affect or change our conclu-sions. Indeed, we have already seen that the equatorial force diverges in the limit of a → A →
0. Applying this limit to the ratio p/ρ gives:lim A → pρ = 1 − r r + R = 1 − y y ≤ . (3.57)Again, this is always true within any physical region, so we verify that adding the DECdoes not change our conclusions. For the Tolman IV solution, while the radial force is bounded (and obeys both the weakand strong maximum force conjectures), the equatorial force can be made to diverge toinfinity in certain parts of parameter space ( A →
0) and acts as a counter-example toboth weak and strong maximum force conjectures. For the Tolman IV solution, addingthe DEC does not save the situation, the violation of both weak and strong maximumforce conjectures is robust. ρ = ρ s + p The Buchdahl–Land spacetime is a special case of the Tolman IV spacetime, corre-sponding to the limit A → a → t → At ) can be written: ds = − ( A + r ) dt + 1 + r A (cid:0) − r R (cid:1) (cid:0) r A (cid:1) dr + r d Ω . (3.58)Under the limit A →
0, this becomes: ds = − r dt + 2 R R − r dr + r d Ω . (3.59)17hen the orthonormal stress-energy components are: T ˆ t ˆ t = ρ = 116 π (cid:18) r + 3 R (cid:19) ; T ˆ r ˆ r = T ˆ θ ˆ θ = T ˆ φ ˆ φ = p = 116 π (cid:18) r − R (cid:19) . (3.60)The surface is located at R s = R √ ρ s = 38 πR = 18 πR s . (3.61)At the centre the pressure and density both diverge — more on this point later.We recast the metric as ds = − r dt + 21 − r R s dr + r d Ω . (3.62)This is simply a relabelling of equation (3.59). The orthonormal stress-energy tensoris now relabelled as: T ˆ t ˆ t = ρ = 116 π (cid:18) r + 1 R s (cid:19) ; T ˆ r ˆ r = T ˆ θ ˆ θ = T ˆ φ ˆ φ = p = 116 π (cid:18) r − R s (cid:19) . (3.63)Note that p = ρ − ρ s ; that is ρ = ρ s + p. (3.64)That is, the Buchdahl–Land spacetime represents a “stiff fluid”. This perfect fluidsolution has a naked singularity at r = 0 and a well behaved surface at finite radius.The singularity at r = 0 is not really a problem as one can always excise a small coreregion near r = 0 to regularize the model. Due to the simplicity of the pressure, the radial force can be easily calculated as: F r = 14 (cid:18) − r R s (cid:19) . (3.65)18he radial force is trivially bounded with a maximum of at the centre of the star.This obeys the strong (and so also the weak) maximum force conjecture. The equatorial force is: F eq = 2 π (cid:90) R s (cid:115) − r R s π (cid:18) r − R s (cid:19) rdr. (3.66)This is now simple enough to handle analytically. Using the dimensionless variable y = r /R s , with range y ∈ (0 , F eq = 116 (cid:90) (cid:115) − y (1 − y ) dyy . (3.67)This is manifestly dimensionless, and manifestly diverges to + ∞ . If we excise a smallregion r < r core , (corresponding to y < y core ) to regularize the model, replacing r < r core with some well-behaved fluid ball, then we have the explicit logarithmic divergence F eq = −
116 ln y core + O (1) . (3.68)This violates the weak (and so also the strong) maximum force conjecture. The DEC for this space-time is given by: pρ = ρ − ρ s ρ = 1 − ρ s ρ ≤ . (3.69)which is always true for positive values of r , ρ s . The Buchdahl–Land spacetime is another weak maximum force conjecture counter-example, one which again obeys the classical energy conditions.19 .5 Scaling solution
The scaling solution is ds = − r w w dt + (cid:18) w + 6 w + 1(1 + w ) (cid:19) dr + r d Ω . (3.70)This produces the following stress energy tensor: T ˆ t ˆ t = ρ = w π ( w + 6 w + 1) r ; T ˆ r ˆ r = T ˆ θ ˆ θ = T ˆ φ ˆ φ = p = w π ( w + 6 w + 1) r . (3.71)This perfect fluid solution has a naked singularity at r = 0 and does not have a finitesurface — it requires r → ∞ for the pressure to vanish. Nevertheless, apart from asmall region near r = 0 and small fringe region near the surface r = R s , this is a goodapproximation to the bulk geometry of a star that is on the verge of collapse [34, 35].To regularize the model excise two small regions, a core region at r ∈ (0 , r core ), and anouter shell at r ∈ ( r fringe , R s ), replacing them by segments of well-behaved fluid spheres.Note that for r ∈ ( r core , r fringe ) we have p/ρ = w , (and since ρ > w > w ∈ (0 , Using equation (2.2), we find that the radial force is very simply given by: F r = 2 w w + 6 w + 1 . (3.72)This is independent of r and attains a maximum value of when w = 1, giving abounded force obeying the strong maximum force conjecture. Now, using equation (3.2), the equatorial force can be calculated as: F eq = (cid:90) r fringe r core (cid:115) w + 6 w + 1(1 + w ) (cid:18) w ( w + 6 w + 1) r (cid:19) dr + O (1) . (3.73)20hat is F eq = w (1 + w ) √ w + 6 w + 1 ln( r fringe /r core ) + O (1) , (3.74)which trivially diverges logarithmically as either r core → r fringe → ∞ , providing acounter-example to weak maximum force conjecture. Again we have an explicit model where the radial force F r is well-behaved, but theequatorial force F eq provides an explicit counter-example to weak maximum force con-jecture. This counter-example is compatible with the DEC. Let us now see how far we can push this sort of argument using only the TOV equationfor the pressure profile in perfect fluid spheres — we will (as far as possible) try toavoid making specific assumptions on the metric components and stress-energy. TheTOV equation is dp ( r ) dr = − { ρ ( r ) + p ( r ) }{ m ( r ) + 4 πp ( r ) r } r { − m ( r ) /r } . (3.75) From the definition of radial force F r = 4 πpr , we see that at the maximum of F r wemust have (2 pr + r p (cid:48) ) (cid:12)(cid:12) r max = 0 . (3.76)Thence, at the maximum( F r ) max = (4 πpr ) max = − π ( r p (cid:48) ) (cid:12)(cid:12) r max . (3.77)In particular, now using the TOV at the location r max of the maximum of F r :( F r ) max = 2 π (cid:20) ( ρ + p ) r ( m + 4 πpr )(1 − m/r ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r max . (3.78)Let us define the two parameters χ = (cid:20) m ( r ) r (cid:21) r max = 2 m ( r max ) r max , and w = (cid:20) p ( r ) ρ ( r ) (cid:21) r max = p ( r max ) ρ ( r max ) . (3.79)21hen ( F r ) max = 12 (cid:18) πp (1 + w ) r ( χ/ πpr )1 − χ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r max . (3.80)Simplifying, we see: ( F r ) max = 12 ( F r ) max [1 + 1 /w ] [( F r ) max + χ/ − χ (3.81)Discarding the unphysical solution ( F r ) max = 0, we find( F r ) max = 4 w − χ − wχ w ) = 2 w w − χ w w . (3.82)The physical region corresponds to 0 ≤ χ <
1, while w >
0. Furthermore we have( F r ) max >
0, whence 4 w − χ − wχ >
0, implying χ < w/ (1 + 5 w ) < /
5. That is, atthe location r max of the maximum of F r we have (cid:20) m ( r ) r (cid:21) r max = 2 m ( r max ) r max < . (3.83)This is not the Buchdahl–Bondi bound, it is instead a bound on the compactness ofthe fluid sphere at the internal location r max where F r is maximized.Observe that ( F r ) max is maximized when χ = 0 and w = ∞ , when ( F r ) max → w ≤
1, and ( F r ) max is maximized when χ = 0and w = 1, when ( F r ) max →
1. This still violates the strong maximum force conjecturebut not the weak maximum force conjecture. Consequently the weak conjecture for F r generically holds for any prefect fluid sphere satisfying the TOV. As we have by now come to expect, dealing with the equatorial force will be considerablytrickier. In view of the non-negativity of the Misner–Sharp quasi-local mass we have: F eq = 2 π (cid:90) R s √ g rr p rdr = 2 π (cid:90) R s (cid:112) − m ( r ) /r p rdr > π (cid:90) R s p r dr. (3.84)To make the integral (cid:82) R s p r dr converge it is sufficient to demand p ( r ) = o (1 /r ).However, for stars on the verge of gravitational collapse it is known that p ( r ) ∼ K/r ,see for instance [34, 35]. More specifically, there is some core region r ∈ (0 , r core )22esigned to keep the central pressure finite but arbitrarily large, a large scaling region r ∈ ( r core , r fringe ) where p ∼ K/r , and an outer fringe r ∈ ( r fringe , R s ) where one has p ( r ) → p ( R s ) = 0. Then we have the identity (cid:90) R s p r dr = (cid:90) r core p r dr + (cid:90) r fringe r core p r dr + (cid:90) R s r fringe p r dr. (3.85)But under the assumed conditions this implies (cid:90) R s p r dr = O (1) + (cid:20)(cid:90) r fringe r core Kr dr + O (1) (cid:21) + O (1) . (3.86)Thence (cid:90) R s p r dr = K ln ( r fringe /r core ) + O (1) . (3.87)Finally F eq > πK ln ( r fringe /r core ) + O (1) . (3.88)This can be made arbitrarily large for a star on the verge of gravitational collapse, sothe weak and strong maximum force conjectures are both violated.Note that technical aspects of the argument are very similar to what we saw for theexact scaling solution to the Einstein equations, but the physical context is now muchmore general. We see that the weak maximum force conjecture generically holds for the radial force F r when considering perfect fluid spheres satisfying the TOV. In contrast we see thatthe weak maximum force conjecture fails for the equatorial force F eq when consideringperfect fluid spheres satisfying the TOV that are close to gravitational collapse. With the notion a natural unit of force F ∗ = F Planck = c /G N in hand, one can similarlydefine a natural unit of power [36–40] P ∗ = P Planck = c G N = 1 Dyson ≈ . × W , (4.1)23 natural unit of mass-loss-rate( ˙ m ) ∗ = ( ˙ m ) Planck = c G N ≈ . × kg/s , (4.2)and even a natural unit of mass-per-unit-length( m (cid:48) ) ∗ = ( m (cid:48) ) Planck = c G N ≈ . × kg/m . (4.3)Despite being Planck units, all these concepts are purely classical (the various factorsof (cid:126) cancel, at least in (3+1) dimensions).Indeed, consider the classical Stoney units which pre-date Planck units by some 20years [41–43], and use G N , c , and Coulomb’s constant e π(cid:15) , instead of G N , c , andPlanck’s constant (cid:126) . Then we have F ∗ = F Planck = F Stoney . Similarly we have P ∗ = P Planck = P Stoney , ( ˙ m ) ∗ = ( ˙ m ) Planck = ( ˙ m ) Stoney , and ( m (cid:48) ) ∗ = ( m (cid:48) ) Planck = ( m (cid:48) ) Stoney .Based ultimately on dimensional analysis, any one of these quantities might be used toadvocate for a maximality conjecture: maximum luminosity [36–40], maximum mass-loss-rate, or maximum mass-per-unit-length. The specific counter-examples to the max-imum force conjecture that we have discussed above suggest that it might also be worthlooking for specific counter-examples to these other conjectures [39].
Through the analysis of radial and equatorial forces within perfect fluid spheres ingeneral relativity, we have produced a number of counter-examples to both the strongand weak forms of the maximum force conjecture. These counter-examples highlightsignificant issues with the current phrasing and understanding of this conjecture, asmerely specifying that forces are bounded within the framework of general relativity ismanifestly a falsehood. As such, should one wish some version of the maximum forceconjecture to be considered viable as a potential physical principle, it must be veryclearly specified as to what types of forces they pertain to.
Acknowledgments
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