## Abstract

Consider a rotating, gravitating system whose mass centre and intrinsic spin define a natural axis of symmetry. A pair of quantum–mechanical polar coordinates, a continuous radial coordinate and an angular coordinate with discrete eigenvalues, are tied to the system's geometry. The gravitational scalar potential generated by the system is a quantum operator that is a function of the two polar coordinates. Exact expressions for the potential's gradient and Laplacian are derived, which involve forward, backward and central differences. The system's binding energy, an integral over Euclidean 3-space, comprises a radial part and an integral over the unit 2-sphere. The latter is shown to amount to a summation over the angular eigenvalues. The result is applied to the Newtonian analogues of Kerr black holes, a class of gravitational potentials with classical, ring singularities. It is shown that any fermionic, *Kerr*–*Newton particle* has a finite binding energy. This is remarkable because the classical binding energy is divergent. Furthermore, the binding energy tends to a constant in the limit of large spin.

## 1. Introduction

This paper is devoted to some extraordinary properties of spinning objects, an object's ‘spin’ being its total angular momentum about its centre of mass. Our Universe abounds with spinning objects from elementary particles to planets, from stars to galaxies. Many of these possess axes of symmetry that coincide with their spin vectors.

Owing to their rotations, the objects will take the shapes of oblate spheroids and the gravitational fields that they produce will display quadrupole and higher multipole structures as well as the usual monopole. If standard polar coordinates are tied naturally to the geometry of a given axisymmetric object, the gravitational potential *Φ* will depend only on two of them.

If *O* is the object's centre of mass and ** J** is its spin vector, the axis of symmetry will pass through

*O*with direction

**. Consider now a sensor**

*J**X*located at a distance

*r*from

*O*and whose displacement

*OX*defines a unit vector

**. The polar angle**

*n**θ*is defined via the component

*Q*of the spin in the direction

**,(1.1)**

*n*The potential *Φ* will therefore be a function of ** J**. In non-relativistic quantum mechanics,

*r*and

**are quantum operators. It follows that the Newtonian gravitational potential is a quantum operator.**

*J*The argument is expanded in §2. *r* and *θ* are commuting operators, the radial parameter *r* having continuous eigenvalues with the polar angle *θ* having discrete eigenvalues. In fact, the potential *Φ* is expressed as a function of *r* and *Q*.

By way of mathematical preliminary, §3 derives expressions for the gradients of functions of *Q*. These expressions are exact, but they resemble numerical approximations. By contrast, §4 considers the process of integrating functions of *Q* over the unit sphere. It is found that such integrals may be replaced by finite sums over the eigenvalues of *Q*. Again, this is exact but resembles a numerical approximation. The results of §§3 and 4 are applied in §5, expressions being derived for the gradient of the potential, for the effective mass density of the system and for its gravitational binding energy.

Over a decade ago (Bramson 1992), a study was made into *Kerr*–*Newton particles*, a term coined by a creative referee for certain real solutions to Laplace's equation with ring singularities. Kerr–Newton particles are the Newtonian analogues of Kerr's solution to Einstein's field equations in general relativity. The diameter and orientation of each ring is determined by the mass and spin of the corresponding particle. Within the framework of non-relativistic quantum mechanics, it was shown that the singularities disappear for fermionic spins, i.e. whenever the eigenvalue specifying the magnitude of the spin is half odd integral.

The process of singularity removal is intimately related to the unusual geometry provided by the observer as part of the measurement process. For example, the reason why the polar angle has discrete eigenvalues is related to the fact that the cross product of the spin vector with itself vanishes in classical mechanics but not in quantum theory.

Kerr–Newton particles are reviewed in §6. The results of §5 are then applied in §7 to their gravitational binding energies. It is shown that the binding energy of any fermionic, Kerr–Newton particle is finite, tending to a constant value as the spin tends to infinity. This result is significant because the classical binding energy diverges for all spins.

Throughout, units are chosen for which Planck's constant, Newton's gravitational constant and the speed of light are all unity.

## 2. Quantum coordinates and the gravitational potential

Before embarking on a quantum discussion, recall the standard approach to modelling *classical*, axisymmetric, spinning systems in Euclidean 3-space. Consider such a system with centre of mass *O* and intrinsic spin vector ** J** (its angular momentum about

*O*). Its axis of symmetry passes through

*O*with direction

**, where**

*k***is the unit vector,(2.1)**

*k*Let *X* be an observer with displacement ** x** from

*O*and whose distance

*r*from

*O*is defined by(2.2)

As long as *X*≠*O*, the outgoing radial direction from *O* to *X* is given by a unit vector ** n**, such that(2.3)

The vectors ** k** and

**yield a natural polar angle**

*n**θ*, the angle between

**and**

*k***, according to(2.4)**

*n*Within this framework, suppose that the system produces a gravitational field specified by a potential *Φ* that reflects the symmetry of the system. The value of *Φ* at *X* will therefore depend only on *r* and ** k**.

**; it will be independent of azimuth. A common practice involves expanding**

*n**Φ*in terms of Legendre polynomials,(2.5)

Now proceed to the quantum theory. ** x** and

**become quantum operators that commute with each other.**

*J***commutes with itself while the commutation relations for**

*x***are summarized by(2.6)**

*J*Thus, the cross product of ** J** with itself does not vanish.

The possible values of *r* are the non-negative reals. The possible values of *J*^{2}, for non-zero ** J**, are given by(2.7)

Henceforth, fix *J*^{2} to be one of these. Thus, consider systems that are eigenstates of *J*^{2}. For fixed, non-zero |** J**|, it turns out expedient to define(2.8)and to regard

*Φ*as a function of

*r*and

*Q*,(2.9)

There is no factor ordering ambiguity on quantizing equations (2.1), (2.3), (2.4), (2.8) and (2.9). In particular, the gravitational potential *Φ* is now a quantum operator. Throughout this paper, *Φ* will be constrained to be non-singular in the sense that all its eigenvalues will be well defined. Note that the cosine of the polar angle operator now becomes(2.10)which means that the angle *θ* between ** J** and

**takes (2**

*n**j*+1) possible values in the range 0…

*π*,(2.11)

To end this section, two remarks are worth making. First, the objections raised by Peierls (1980) do not apply here. This is because the angle *θ* defined by Peierls via *plane* polar coordinates has azimuthal properties, while the angle *θ* defined in equation (2.11) has polar properties and is constrained to the range 0 … *π*. Secondly, *θ* cannot take the values 0 or *π*. Thus, ** n** can never lie parallel or anti-parallel to

**, except in the limit of large**

*J**j*. Furthermore, for fermionic systems,

**.**

*J***can never vanish,**

*n**θ*can never take the value

*π*/2 and

**can never be orthogonal to**

*n***. Thus, although an Euclidean geometry was postulated**

*J**a priori*, the geometry perceived in terms of the eigenvalues of observables is quite different.

## 3. Taking gradients orthogonal to the radial direction

For fixed *j*, consider a non-singular, quantum operator *f*(*Q*), where *Q* is defined in equation (2.8). Here, ‘non-singular’ means that the eigenvalues of *f*(*Q*) are all well defined. Note that, for fixed *j*, *f*(*Q*) must be a polynomial in *Q* of degree at most 2*j*. To see this, let the eigenvalues of *Q* be *q*_{1}, *q*_{2} … *q*_{2j+1}. Define (2*j*+1) polynomials of degrees 2*j*,(3.1)

Then, by examining its application to each eigenstate of *Q*, *f*(*Q*) may be written as(3.2)

To take the gradient of *f*(*Q*), it will therefore be sufficient to consider(3.3)

Now(3.4)where ** j** is that part of

**orthogonal to the unit radial vector**

*J***,(3.5)**

*n*It follows that, to make progress, it will be useful to derive expressions for the action of ** j**, by left or right multiplication, on functions of

*Q*.

*With Q defined in equation* *(2.8)*, *j**defined in equation* *(3.5)* *and f(Q) a non-singular operator,*(3.6)*where*(3.7)

The statement of lemma 3.1 involves the decomposition of ** j** into the sum of two complex null vectors(3.8)

If , ** m** and

**form a right-handed orthonormal triad, it is easy to relate the vectors**

*n*

*j*_{±}to the standard operators

*J*

_{±}that raise and lower the eigenvalues of

*Q*,(3.9)

Indeed, note that *j*_{±} satisfies(3.10)

Next, the commutation relations(3.11)where ** a** and

**are arbitrary vectors that commute with each other and with**

*b***, imply that(3.12)**

*J*It follows that(3.13)whence, for any positive integer *n*,(3.14)

Noting the decomposition of ** j** presented in equation (3.8), it follows that(3.15)the second of these being the Hermitian conjugate of the first. It follows that, for any non-singular

*f*(

*Q*), by virtue of the expansion depicted in equation (3.2), pre- and post-multiplication by

**yield equations (3.6). This proves lemma 3.1. ▪**

*j*Lemma 3.1 is now used to derive an expression for the gradient of *f*(*Q*).

*With Q defined in equation* *(2.8)*, *j**defined in equation* *(3.5)*, *j*_{±} *defined in equations* *(3.7)* *and f(Q) a non-singular operator,*(3.16)*where*(3.17)

The proof begins by expanding *f*(*Q*) as in equation (3.2) and recognizing that for a typical term *Q*^{n}, equations (3.3) and (3.4) imply that(3.18)

Using equations (3.15) and summing a pair of geometric progressions yields the equivalent expressions(3.19)(3.20)

By virtue of the expansion (3.2), it follows that the gradient of any non-singular *f*(*Q*) has the equivalent forms presented in equations (3.16). This proves theorem 3.2. ▪

The expressions (3.17) resemble the forward and backward differences used in numerical approximations to derivatives (Morton 1964). However, the gradients of *f*(*Q*) presented in equation (3.16) are exact.

## 4. Integrating over the unit sphere

Consider a unit 2-sphere *S* embedded in Euclidean 3-space. Let ** n** be the outward-pointing, unit vector joining the centre of

*S*to points on

*S*. Let

**be a quantum–mechanical spin vector and write**

*J**Q*=

**.**

*J***. Assume that**

*n***commutes with itself and with**

*n***. This section seeks expressions for integrals of a non-singular**

*J**f*(

*Q*) over

*S*. By way of introduction, consider integrals of the form(4.1)

with *n* a non-negative integer. (Integrals of *Q*^{2n+1} vanish.) In classical mechanics, the answer is *J*^{2n}/(2*n*+1), where *J*^{2n} means (*J*^{2})^{n}. Quantum mechanically, things are different because ** J** does not commute with itself. Formally, the answer is(4.2)where round brackets indicate symmetrization over the indices. For example,(4.3)

(4.4)

(4.5)

Thus, the classical and quantum expressions agree for *I*_{0} and *I*_{2} but not for *I*_{4}. It turns out that the integral displayed in equation (4.1) may be computed by averaging *Q*^{2n} over the eigenvalues of *Q*.

*Consider a unit 2-sphere S embedded in Euclidean 3-space. Let* *n**be the outward-pointing, unit vector joining the centre of S to points on S. Let* *J**be a quantum–mechanical spin vector and write Q=**J**.**n**. Assume that* *n**commutes with itself and with* *J**. Then, if the eigenvalues of f(Q) are non-singular,*(4.6)

The proof is by induction on *j*. First note that the theorem is true for *j*=0, both sides of equation (4.6) amounting to *f*(0). Now suppose the theorem to be true for some *j* (but for all smooth *f*), i.e. suppose that, at the quantum level,(4.7)

A consequence of this supposition is that, again at the quantum level,(4.8)

(The left-hand side must be a vector. The only candidate has the form *λ*** J**.

*λ*is determined by taking the inner product with

**.)**

*J*Now, inspired by a technique employed in Penrose (1971), attach a spin system to the original system in such a way that the total spin increases by . Do this by writing(4.9)where ** J** and

**commute with each other, and by demanding that(4.10)**

*S*The necessary and sufficient condition for *j*′ indeed to be is(4.11)

To evaluate(4.12)use the dependency on ** S**.

**, whose eigenvalues are ±, to make the expansion(4.13)**

*n*(For example, when *f*(*Q*)=*Q*^{2}, both sides equal .) Then *I*′ involves the sum of two terms which, by the inductive hypothesis, take the forms(4.14)

Using condition (4.11) yields(4.15)

Now deal separately with and . In the former, putting yields(4.16)the range of *q*′ having been extended to include −*j*′, because the argument of summation actually vanishes there. Similarly,(4.17)

Putting the two halves together gives(4.18)

Thus, if theorem 4.1 is true for *j*, then it is also true for , which completes the proof by induction. ▪

Note that, classically, if |** J**|=

*j*,(4.19)

Equation (4.6) resembles a numerical approximation to equation (4.19) (though worse than both Simpson's approximation and the trapezoidal rule), but it is exact.

## 5. The gravitational field, mass density and binding energy

In Newton's classical theory of gravitation, the gravitational field ** F** is defined as (minus) the gradient of the Newtonian potential

*Φ*, the mass density

*ρ*is related via Poisson's equation to the Laplacian of the potential and the binding energy is found by integrating the square of the field over Euclidean 3-space,(5.1)(5.2)(5.3)

For the quantum system envisaged in this paper, it seems natural to mimic Newton's approach, but starting with the scalar quantum operator *Φ*(*r*,*Q*). (Only non-singular operators are considered here.) Using(5.4)with ** n** defined implicitly in equation (2.3), together with equations (5.1) and (3.16), it follows that(5.5)(5.6)

Here,(5.7)(5.8)

According to equation (5.2), the system's mass density requires the Laplacian of the potential. Using the relations(5.9)together with the definitions (3.7), it follows that the divergences of *j*_{±} are given by(5.10)

The Laplacian of the potential then takes the form(5.11)where, for any function *f*(*Q*) of *Q*,(5.12)

Further simplification yields(5.13)

The application of to *f*(*Q*) involves the double application of a forward (or backward) difference and a central difference (Morton 1964). Both feature in numerical approximations respectively to the second and first derivatives of *f*. Thus, equation (5.13) resembles a numerical approximation. Nevertheless, it is exact. Indeed, using equation (2.10) for large *j*, putting(5.14)the expression for is approximately that found in Legendre's equation. Specifically,(5.15)

Finally, the system's binding energy, defined in equation (5.3), requires the evaluation of the energy density −*F*^{2}/8*π*. This is obtained from the scalar product ** F**.

**, with the first**

*F***obtained from equation (5.5) and the second from equation (5.6),(5.16)**

*F*(There are no cross terms because ** n**.

*j*_{±},

*j*_{±}.

**, and all vanish.) The binding energy is given by(5.17)which, on using theorem 4.1, becomes(5.18)**

*n*The expression for *F*^{2}, presented in equation (5.16), when used in equation (5.18), implies summing over contributions from both and . In fact, the two summations are identical; this may be seen on putting *q*′=*q*−1 in the second case and noting that [*j*(*j*+1)−*q*′(*q*′+1)] vanishes when *q*′=−*j*−1 and *q*′=*j*. The upshot is(5.19)

## 6. Kerr–Newton particles

The gravitational potential *Φ* generated by a Kerr–Newton particle of mass *m* and spin ** J** is given by the real part of a complex potential

*V*,(6.1)where the parameter

*b*, having the dimensions of length, is taken to be

*m*

^{−1}, the Compton wavelength of the particle (Bramson 1992). Here,(6.2)is defined by its action on the simultaneous eigenstates |

*r*′

*jq*〉 of

*r*,

*J*^{2}and

*Q*,

*r*′ being an eigenvalue of

*r*,(6.3)with the root chosen, so that the result behaves like

*r*′ for large

*r*′ (see also §5 of Bramson 1992).

Classically, |** x**−i

**/**

*J**m*| vanishes when

*r*

^{2}=

*J*^{2}/

*m*

^{2}and

**.**

*J***=0, and this yields a ring singularity for**

*x**V*. However, quantization removes the singularity for Fermions because

*q*can never vanish.

It proves useful to make the rescalings(6.4)where(6.5)

Note that(6.6)(6.7)

For future use, write(6.8)

To proceed, write(6.9)where *X* is the positive real solution to(6.10)(6.11)

For *q*>0, *α* decreases monotonically from *π* to 0, as *x* increases from 0 to ∞. For *q*<0, *α* increases monotonically from −*π* to 0, as *x* increases from 0 to ∞. The Kerr–Newton potential is given by(6.12)and this is illustrated in figure 1 for three values of *u*.

For large *x*,(6.13)

The *x*^{−3} term arises from the quadrupole moment of the source, but note that, for spin , it vanishes. For small *x*,(6.14)so that the field is actually repulsive at short range, i.e. for *r* small compared with . In fact, the magnitude of the potential has a minimum |*ϕ*|_{min} occurring at *x*=*x*_{min}. For example, writing ,(6.15)

(6.16)

## 7. The binding energies of fermionic, Kerr–Newton particles

The binding energy of a classical Kerr–Newton particle diverges, and this is shown in appendix A. The question therefore arises as to whether quantum mechanics, which for Fermions removes the singularity, will make the binding energy finite.

The Kerr–Newton potential presented in equation (6.12) is an even non-negative function of *q*. Consequently, the binding energy, given by equation (5.19), may be simplified by employing only positive values of *q*. Recalling that *b*=*m*^{−1},(7.1)where *q* is related to *u* by equation (6.4) and the rescaling(7.2)implies that(7.3)

Note that the second term of the integrand in equation (7.1) may be written as(7.4)which, on applying the mean value theorem (Phillips 1962), becomes(7.5)for some *δ*(*u*, *x*) between 0 and 1. Thus, an equivalent expression for the binding energy is(7.6)

It turns out that equation (7.1) may be evaluated numerically for after coordinate transformations that deal separately with the *ϕ*_{x} and *ϕ*_{+} parts. In the *ϕ*_{x} part, *x* is replaced by a function of *α* consistent with equations (6.11), while, in the *ϕ*_{+} part, *x* is replaced by tan *ψ* (i.e. independent of *u*), the variables *α* and *ψ* having finite range. The details are set out in appendix B. In this way, the expression for the binding energy has been evaluated numerically for a range of fermionic spins using Simpson's approximation with 10 000 steps. A sample of values to four decimal places is presented in table 1; this reveals little variation with spin. In fact, the gravitational binding energy for spin , namely –0.4195*m*^{3}, may be evaluated analytically from tables of elliptic integrals.

As *j*→∞, appears to tend to a limit of around −0.4213*m*^{3}. The existence of a limit is supported by the second expression for the binding energy presented in equation (7.6). In appendix C, bounds are produced for this expression involving Riemann's zeta function. The limit of large spin yields(7.7)

To four decimal places, the bounds are (0.3340, 0.9253)*m*^{3}.

## 8. Conclusion and outlook

The purpose of this paper has been to present a framework for Newtonian quantum gravity, given axisymmetric, spinning sources. Of course, this is quantum gravity without gravitons. There are no gravitational waves in Newton's theory. The quantization is driven solely by the spin of a given source and by the spatial discreteness with regard to the natural polar angle that this implies. A side effect has been to generalize the analysis concerning Kerr–Newton particles, as discussed previously (Bramson 1992).

The potential, field, energy density and total energy of a general, spinning, axisymmetric system have all been examined. These quantum–mechanical operators on Euclidean 3-space depend on two naturally defined arguments, a radial coordinate with continuous eigenvalues and an angular coordinate, the polar angle, with discrete eigenvalues. The expressions for the gradient and Laplacian of the potential involve forward, backward and central differences. While resembling numerical approximations, they are, in fact, exact. By analogy, integrating the energy density involves a discrete summation. This strange relationship between quantum–mechanical discreteness and the discreteness related to numerical approximations holds for *any* function respecting the symmetry of the system.

The results are applied to Kerr–Newton particles, the Newtonian analogues of Kerr's solution in general relativity and whose classical binding energies are divergent. Remarkably, it is shown that the binding energies of fermionic, quantum–mechanical Kerr–Newton particles are finite. Of course, this is related to the fact that quantization has removed the ring singularity. Specifically, the magnitude of the cosine of the polar angle has smallest eigenvalue [*j*(*j*+1)]^{−1/2}, and this provides a natural cut-off to the integral that defines the binding energy. For a spin particle of mass *m*, the gravitational binding energy may be evaluated analytically as −0.4195*m*^{3} to four decimal places. For very large spins, numerical approximation suggests a limit of around −0.4213*m*^{3}. That a limit exists is supported by the analytic calculation of rigorous bounds −0.3340*m*^{3} and −0.9253*m*^{3}.

Two avenues for future research should be considered. First, the constraint of axisymmetry may be relaxed via the superposition of two or more Kerr–Newton particles. For two such particles, subtracting the individual binding energies from the total system binding energy should yield their energy of interaction. Second, if quantization can remove singularities and yield finite binding energies in a non-relativistic context, what might be expected of a relativistic theory? An early investigation (Bramson 1993) revealed that spin , Kerr–Minkowski particles, the generalizations of spin Kerr–Newton particles to Minkowski space-time, were singularity-free. Furthermore, the electromagnetic field produced by a charged version of such a particle had finite self-energy. However, one aspect of the relativistic analysis departed radically from its non-relativistic counterpart; the operator specifying the distance from sensor to particle had eigenvalues that were bounded below by roughly the particle's Compton wavelength. Thus, the space-time defined by the relationship between observer and observed appeared to be Minkowski space-time with a region removed. A generalization to other spins is currently underway.

## Acknowledgments

It is a great pleasure to thank John Jefferson for reviewing this paper. His comments have added some clarity of purpose that would otherwise have been absent. I also thank Jonathan Pritchard and David Haworth for much encouragement.

## Footnotes

- Received May 4, 2006.
- Accepted September 5, 2006.

- © 2006 The Royal Society