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GEOMETRIC ALGEBRA, DIRAC WAVEFUNCTIONS AND
BLACK HOLES
A.N. LASENBY AND C.J.L. DORAN
Astrophysics Group
Cavendish Laboratory
Madingley Road
Cambridge CB3 OHE, U.K.
Abstract. In this contribution we describe some applications of geometric
algebra to the eld of black hole physics. Our main focus is on the proper-
ties of Dirac wavefunctions around black holes. We show the existence of
normalised bound state solutions, with an associated decay rate controlled
by an imaginary contribution to the energy eigenvalue. This is attributable
to the lack of Hermiticity caused by a black hole singularity. We also give
a treatment of the Feynman scattering problem for fermions interacting
with black holes that we believe is new, and produces an analogue of the
Mott scattering formula for the gravitational case. Throughout, the consis-
tent application of geometric algebra simplies the mathematical treatment
and aids understanding by focusing attention on observable quantities. We
nish with a brief review of recent work on the eects of torsion in quadratic
theories of gravity. This work demonstrates that a free torsion eld can play
a signicant role in cosmology.
1. Introduction
This contribution provides a broad survey of a number of recent advances
in the application of geometric algebra to quantum physics and gravita-
tion. The bulk of the presentation deals with the quantum properties of
fermions in a black hole background. We address two related problems.
The rst question is whether a spectrum of bound state solutions exists
around a black hole. The non-relativistic limit of the Dirac equation sug-
gests that this should be the case, and indeed this is borne out by the full
general relativistic theory. The relativistic treatment contains a novel fea-
ture, however, which complicates the search for bound states. The Hamilto-

2
nian representing a Dirac fermion in a Schwarzschild black hole background
is not Hermitian as the singularity acts as a density sink. This means that
the eigenspectrum consists of isolated complex energies. The equations are
too complicated to admit an analysis in terms of known special functions,
so these eigenvalues can only be found by a numerical search. Fortunately,
for a range of masses, the real parts of the energies lie close to their non-
relativistic values, which simplies the search process. The imaginary con-
tribution to the energy is responsible for orbital decay and implies that the
bound states have nite lifetimes. This provides a novel, physical picture
of what happens as a particle falls into a black hole. Some of the questions
raised by this work are discussed here.
The second main application is to the scattering problem for black holes.
This has been discussed by many authors, both in a classical and quantum
context [1]{[6]. Here we tackle the problem in terms of straightforward
Feynman propagator theory. Remarkably, this approach does not appear
to have been tried before. The approach is suggested naturally by the geo-
metric algebra form of the Dirac equation in a Schwarzschild background,
when cast in Newtonian gauge form. The name re ects the fact that many
equations take on a Newtonian form if expressed in this gauge, though of
course all physical predictions are gauge invariant. The result of this ap-
proach is that the eects of the black hole can be described by a single
vertex factor in momentum space. This factor has the unusual property
of vanishing on-shell, so the gravitational cross-section involves a second-
order process (in the Newtonian gauge). The calculation can be reproduced
in other gauges, and in all cases the cross-section, to leading order in the
black hole mass, is
d
d
f
= (GM) 2
4 4 sin 4 (=2)

1 + 2 2 3 2 sin 2 
2 +  4  4 sin 2 
2

; (1)
where  = jpj=E. The cross-section does not depend on the particle mass,
which embodies the equivalence principle at the quantum level. The low-
velocity limit recovers the standard Rutherford formula. The full result is
the gravitational analogue of the Mott formula for the Coulomb scattering
cross-section.
2. Geometric Algebra
Throughout this paper we consistently apply the mathematical language
of geometric algebra. Though this does not aect any of the physical pre-
dictions, it does considerably assist in the understanding and solution of
the mathematics encountered. The foundations of geometric algebra (GA)
were laid in the 19th Century by Hamilton and Grassmann (Fig. 1). Clif-

3
Figure 1. Hermann Gunther Grassmann (1809{1877), born in Stettin, Germany (now
Szczecin, Poland). He worked as a schoolteacher in Germany, and later established a
signicant reputation in linguistics. The mathematical system he introduced, Grassmann
algebra, is now an integral part of dierential geometry and theoretical physics.
ford (Fig. 2) then unied their work by showing how Hamilton's quaternion
algebra could be included in Grassmann's scheme through the introduction
of a new, geometric product. The resulting algebra is known as a Cliord
algebra, though when used in its intended, geometric role most practitioners
prefer Cliord's original name of geometric algebra. The algebra was not
immediately adopted by physicists, as it became tainted by its association
with quaternions. These ran into problems due to their muddled handling
of re ections and rotations, though ironically it is the embedding of quater-
nions within GA which solves all of these diôculties! Cliord algebra was
rediscovered by Pauli and Dirac for use in the quantum theory of spin, and
then developed further by mathematicians in the 50s and 60s. It was rein-
troduced in the guise of geometric algebra to physics in the 70s by David
Hestenes, who is largely responsible for the modern form of the subject.
At its most fundamental level, geometric algebra is a mathematical lan-
guage for directly encoding geometric primitives | points, lines, planes,
volumes, circles, spheres etc. Geometric algebra provides a new product for
vectors which generalizes complex analysis to arbitrary dimensions. The
key to the development of the algebra was Grassmann's introduction of
the outer product in the `Lineale Ausdehnungslehre' of 1844. This product
encodes an oriented plane segment via an antisymmetric `wedge product'
a^b. Antisymmetry in encoded mathematically in the statement
a^b = b^a (2)

4
Figure 2. William Kingdon Cliord 1845{1879. Mathematician and philosopher. He
died tragically young at the age of only 33.
which implies that a^a = 0. For example, suppose that e 1 , e 2 are a pair of
orthonormal basis vectors in two dimensions. With vectors a = a 1 e 1 + a 2 e 2
and b = b 1 e 1 + b 2 e 2 , we nd that
a^b = a 1 b 2 e 1 ^e 2 + a 2 b 1 e 2 ^e 1
= (a 1 b 2 b 2 a 1 )e 1 ^e 2 : (3)
The coeôcient here is seen to represent the area of the parallelogram dened
by the vectors, and the bivector term e 1 ^e 2 denes an orientation. Grass-
mann extended the outer product to arbitrary dimensions and grades. The
latter term refers to the number of vectors in each outer product. The total
antisymmetry ensures that the outer product of a set of linearly-dependent
vectors must vanish. Each step up in grade is therefore a step up in the
spatial dimension of the object of interest.
Cliord's great insight was to realise that Grassmann's outer product
can be combined with the (symmetric) inner product to dene a single
product
ab = a
b + a^b: (4)
Cliord called this the geometric product. It satises the usual axioms of
associativity and distributivity (over addition). Clearly it is not commuta-
tive, and the separate scalar (inner) and outer products can be obtained
from
a
b = 1
2 (ab + ba); a^b = 1
2 (ab ba): (5)

5
These axioms are suôcient to build up an extremely rich algebra. For ex-
ample, in two dimensions the algebra is spanned by the set
1 fe 1 ; e 2 g e 1 ^e 2
1 scalar 2 vectors 1 bivector: (6)
As the basis vectors are orthogonal, the bivector e 1 ^e 2 can be written in
any of the equivalent forms:
e 1 ^e 2 = e 1 e 2 = e 2 ^e 1 = e 2 e 1 : (7)
It follows that the square of the bivector satises
(e 1 ^e 2 ) 2 = (e 1 e 2 )( e 2 e 1 ) = e 1 e 2 e 2 e 1 = 1; (8)
so the unit bivector squares to minus 1! Furthermore, it is responsible for
generating =2 rotations in the plane. One can clearly see that the geometric
algebra of the Euclidean plane naturally incorporates complex analysis,
with the geometric product taking on the role of the complex product.
Similarly, in three dimensions the full algebra is spanned by the set
1 f i g f i ^ j g I =  1  2  3
1 scalar 3 vectors 3 bivectors 1 trivector: (9)
(The reason for the notation will emerge shortly.) Each grade encodes a geo-
metrically signicant object in three dimensions. The highest grade object,
I, is called the pseudoscalar. In three dimensions it has negative square,
I 2 = 1. The pseudoscalar maps vectors to bivectors, and vice-versa. For
the bivectors we can write
 1  2 = I 3 ;  2  3 = I 1 ;  3  1 = I 2 : (10)
It follows that each bivector squares to 1. If we set
i = I 1 ; j = I 2 ; k = I 3 (11)
we nd that i; j; k satisfy
i 2 = j 2 = k 2 = ijk = 1: (12)
This are the dening relations of the quaternion algebra, which is naturally
embedded in the algebra of 3D space. The embedding is important, as it
separates the roles of vectors and quaternions (bivectors). This is essential
for the correct handling of re ections and rotations.
Further information can be found in a number of introductions to ge-
ometric algebra. See for example the papers by Hestenes [7, 8, 9] and

6
Vold [10, 11] and the books [12, 13, 14, 15]. See also [16] for a complete set
of lecture notes and further resources.
3. Spacetime Algebra
The applications in this paper are mainly developed in the geometric alge-
bra of spacetime. This is called the spacetime algebra or STA. We let the set
f  g denote an orthogonal frame of vectors for spacetime, where 2
0 = 1;
and 2
k = 1 for k = 1; 2; 3: The inner product for these is summarised by

 = 1
2 (   +   ) =   = diag(+ ): (13)
These are the dening relations of the Dirac matrix algebra, so the Dirac
matrices form a representation for the STA. We do not use the matrix
representation directly in calculations, though, as it is invariably slower.
For example, one can end up calculating the square of a 4  4 complex
matrix, when all this encodes is the result 2
0 = 1. Also, the matrices are
naturally dened over the complex eld, whereas we are interested in the
real STA.
Repeated multiplication of the basis vector builds up a complete basis
for the STA. This has 2 4 = 16 elements, and is written
1 f  g f k ; I k g fI  g I
scalar vectors bivectors pseudovectors pseudoscalar. (14)
The bivectors  k are dened by
 k = k 0 : (15)
Their algebra is isomorphic to that of Euclidean 3-space, since
 j
 k = 1
2
( j  k +  k  j ) = ô jk : (16)
These dene a set of basis vectors for the rest frame dened by 0 . Dier-
ent timelike vectors dene dierent relative vectors, and hence a dierent
spacetime split of the bivectors in the STA. The spacetime pseudoscalar is
again denoted I, and is given by
I =  1  2  3 = 1 0 2 0 3 0 = 0 1 2 3 : (17)
It follows that I 2 = 1 and, in spacetime, I anticommutes with all vectors.
For a general vector a we write the spacetime split with 0 as
a 0 = a 0 + a (18)

7
where
a 0 = a
0 ; a = a^ 0 : (19)
Relative vectors are denoted in bold, to distinguish them from spacetime
vectors. Of course, in the full STA, relative vectors are treated as bivectors.
For example, the momentum p is split according to
p 0 = p
0 + p^ 0 = E + p: (20)
where E is the energy measured in the 0 frame. We immediately recover
the Lorentz invariant scalar from
p 2 = p 0 0 p = (E + p)(E p) = E 2 p 2 : (21)
(We use natural units throughout, so c =  0 = ~ = 1.) Velocity 4-vectors
are invariably normalised so that v 2 = 1. For these the relative velocity in
the 0 frame is dened by
v = v^ 0
v
0
; (22)
so that we have
v 0 = (v
0 )(1 + v) = (1 + v) (23)
where = (1 v 2 ) 1=2 is the Lorentz factor.
A general multivector in the STA can contain a mixture of grades. It
is useful to have a compact notation to denote the result of projecting out
terms of a given grade. We write this as hAi r , which returns the grade-r
terms in A. For the scalar part (r = 0) we drop the subscript and just write
hAi = hAi 0 : (24)
Multivectors containing terms of only a single grade are called homoge-
neous. For these the inner and outer products extend straightforwardly.
For multivectors A r and B s of grade r and s respectively, we can write
A r
B s = hA r B s i jr sj
A r ^B s = hA r B s i r+s : (25)
In the case where r = 1; so that a = A r is a vector, we have the relation
aB s = a
B s + a^B s : (26)
The symmetry of both the inner and outer product alternate with increasing
grade of B s ,
a
B s = haB s i s 1 = 1
2
(aB s ( 1) s B s a)
a^B s = haB s i s+1 = 1
2 (aB s + ( 1) s B s a): (27)

8
We adopt the convention that, in the absence of brackets, inner and outer
products are performed before geometric products.
An important operation in GA is that of reversion. We write the reverse
of A as e
A; which reverses all the vector products making up the multivector.
This has the property that
(AB)e= e
B e
A: (28)
Given a general multivector
M =  + a +B + Ib + I; (29)
where  and  are scalars, a and b are vectors and B is a bivector, the
reverse satises
f
M =  + a B Ib + I: (30)
Lorentz transformations are spacetime rotations and can be performed
by the use of a rotor. A rotor R is an even-grade element of the STA
satisfying
R e
R = e
RR = 1: (31)
A proper orthochronous Lorentz transformation of a vector a can be written
a 7! a 0 = Ra e
R: (32)
It can be shown that all proper orthochronous Lorentz transformations can
be written in this way. Since R and R encode the same transformation,
rotors form a double cover of the (restricted) Lorentz group. Any rotor can
be written in the form
R =  exp( B=2) (33)
where B is a bivector. The bivectors form the Lie algebra of the Lorentz
group, and directly encode the spacetime plane(s) in which the transfor-
mation is performed. The same double-sided transformation law applies to
multivectors representing geometric objects. For example, the plane a^b
transforms to
a^b 7! (Ra e
R)^(Rb e
R) = 1
2 (Ra e
RRb e
R Rb e
RRa e
R) = R a^b e
R: (34)
The fact that general multivectors transform in this simple manner is a
considerable improvement over matrix-based techniques for carrying out
Lorentz transformations.
The main ingredient in eld theory in the STA is the vector derivative
r. If x  denote the coordinates in each of the  directions, we write
r =  @
@x  = 0
@
@t
+ i @
@x i : (35)

9
The spacetime split of r goes as
r 0 = @ t r; (36)
where r =  i @ i is the vector derivative in the relative space picked out
by 0 . The spacetime vector derivative r provides a remarkably compact
encoding of the Maxwell equations. We combine the electric and magnetic
elds into spacetime bivector F = E + IB. The Maxwell equations can
then be written as
rF = J; (37)
where J is the current. This is not merely a cosmetic exercise. The vector
derivative r is directly invertible, which provides a number of new tech-
niques for solving the Maxwell equations.
4. Quantum Theory
The fact that the Pauli and Dirac matrix algebras are representations of
the geometric algebras of real space and spacetime suggests that GA is a
natural tool for studying quantum theory. This is indeed the case. Both
Pauli and Dirac spinors can be handled in the real geometric algebras of
space and spacetime, and this provides a number of insights into their
geometric roles. We start with the case of non-relativistic spinors. A Pauli
column spinor a can be placed in a direct 1 $ 1 correspondence with an
element of the even subalgebra of 3D geometric algebra as follows:
a =
 a 0 + ia 3
a 2 + ia 1

$ = a 0 + a k I k : (38)
The actions of the quantum operators f^ k g, and the unit imaginary i, are
replaced by the operations
^  k j i $  k  3 (k = 1 : : : 3) (39)
ij i $ I 3 : (40)
A feature of this scheme is that the complex structure is taken over entirely
by the bivector I 3 , acting to the right of .
Every calculation that can be performed with the column spinor a
can also be performed with the even element , and in practice the latter
approach is usually easier. One reason for this is the natural decomposition
of into a density term and a rotor:
=  1=2 R; (41)
where
 = e
: (42)

10
For example, the spin vector s has components
s k = h j^ k j i: (43)
In terms of GA these become
s k = h e
 k  3 i =  k
(  3
e
): (44)
It follows that all of the components of the spin vector can be summarised
in the single expression
s =  3
e
= R 3
e
R: (45)
The 3D rotor R is therefore an instruction to rotate the xed  3 vector
onto the observable spin vector. This establishes a natural link with the
description of a rotating rigid body [15, 17].
The normalised observables dene the unit spin vector
^
s =  3
e


= R 3
e
R: (46)
This denes a unit vector which is usually represented as a point on the
Bloch sphere. The quantum density matrix is also easily encoded. For nor-
malised pure states the density matrix is dened by
^
 = j ih j: (47)
This maps directly to the equivalent multivector
^
 = 1
2 (1 +  3 ) e
= 1
2 (1 + ^
s): (48)
For mixed states we simply sum the density matrices and normalise, so that
^
 = 1
2 (1 + P ); P 2  1: (49)
This is suôcient to encode single particle quantum theory, but what we
really need is a version of multiparticle quantum mechanics. Surprisingly,
this requires the full, relativistic theory.
A similar construction for Pauli spinors can be applied to Dirac spinors.
A Dirac spinor has 8 real components, and these are placed in a 1 $ 1
correspondence with a even-grade element of the STA as follows:
j i =

ji
ji

$ =  +  3 : (50)

11
Here ji and ji are two-component spinors, and  and  are their Pauli-
even equivalents. The action of the operators f^  ; ^ 5 ; ig becomes
^
 j i $  0 ( = 0; : : : ; 3)
ij i $ I 3
^
5 j i $  3 :
(51)
With these relations one can immediately write down a form of the Dirac
equation in the STA as
r I 3 eA = m 0 ; (52)
where A is the electromagnetic vector potential and m is the mass. This
equation is entirely equivalent to the standard matrix form of the Dirac
equation, even though it is written entirely in the real STA.
The main observables in Dirac theory are also simply encoded now. For
example, the current is dened by
J  = h j^  j i $ h e
 0 i = 
( 0
e
): (53)
We can therefore reconstitute the vector J = J   to obtain
J = 0
e
: (54)
The remaining observables have equally simple expressions [18]. Further
insight into the role of a spinor is provided by rst writing
e
= e I (55)
so that we can set
=  1=2 e I=2 R: (56)
The object R satises R e
R = 1, so is a spacetime rotor. We now nd that
J = R 0
e
R = v; (57)
so the velocity v is obtained from a boost of the 0 vector onto the observ-
able. The boost is controlled by the rotor R.
Now suppose we wish to extend to describe an n-particle quantum state.
We construct a 4n-dimensional relativistic conguration space, spanned
by the vectors f a
 g, where  = 0 : : : 3 labels the spacetime vector, and
a = 1 : : : n labels the particle space. Vectors from distinct particle spaces
are orthogonal, so we have
a

b
 =   ô ab : (58)

12
With this algebra one can construct a multiparticle Dirac equation, which
provides a simple, geometric encoding of Pauli antisymmetrisation [18]. If
we now apply a spacetime split in one space we obtain the relative vectors
 a
j = a
j a
0 ; j = 1 : : : 3; a = 1 : : : n: (59)
Bivectors from distinct spaces commute, as can be seen from
 a
i  b
j = a
i a
0 b
j b
0
= a
i b
j b
0 a
0
= b
j b
0 a
i a
0 =  b
j  a
i (a 6= b): (60)
It follows that the geometric product of elements of even grade from dif-
ferent spaces is equivalent to the tensor product. This makes it a straight-
forward exercise to construct multiparticle quantum states, and all that
is ever required is the geometric product! This idea is now the basis for
a wide range of applications, including quantum information theory and
NMR [19, 20, 21].
5. Gravitation
The STA is the geometric algebra of ( at) Lorentzian spacetime. This might
appear to pose a problem for its continued application in general relativ-
ity, where spacetime becomes curved. This turns out not to be the case.
The solution is to work with a gauge theory formulation of gravity. The
rst satisfactory gauge treatment of gravity was formulated by Kibble in
1961 [22]. The gauge theory approach leads naturally to an extended ver-
sion of general relativity (GR) known as a spin-torsion theory [23] though,
in the absence of macroscopic spin, the equations reduce to those of GR.
In [24] the gauge theory treatment was reconsidered in the STA framework.
It was shown that gravitation can be developed as a gauge theory built on
arbitrary nite transformations. And, unlike Kibble's earlier theory, the
combination of the gauge treatment and STA produces a theory which is
conceptually and computationally simpler than GR. This theory is known
as gauge theory gravitation, or GTG.
The gauge theory of gravitation requires the introduction of two gauge
elds. The rst is a position-dependent linear function  h(a), or  h(a; x). This
is linear in its vector argument a, and is a general non-linear function of
the position vector x = x   . This gauge eld ensures that the equations
remain covariant under arbitrary, nonlinear displacement. Suppose that the
vector eld J(x) is dened by
J = 
h(r): (61)

13
Then, if we replace (x) by  0 (x) = (x 0 ), and transform  h accordingly,
J(x) transforms simply to J 0 (x) = J(x 0 ). By inserting the  h eld at various
points in STA eld equations, we can construct equations which remain
covariant under arbitrary displacements.
The second gauge eld is
denoted
a) =
a; x), and is a bivector-
valued linear function of its argument a. The position-dependence of
a)
is also generally non-linear. The bivector
eld
a) is the connection for
the gauge group of Lorentz transformations. Recall that a Lorentz transfor-
mation can be written as a 7! Ra e
R. Suppose now that all elds are subject
to a local rotation generated by the position dependent rotor R. The gauge
elds transform as:
 h(a) 7!  h
0 (a) = R  h(a)
e
R; (62)
and
a)
7!
0 (a) = R a) e
R 2a
rR e
R: (63)
The gauge elds ensure that equations can remain covariant under local ro-
tations, as well as arbitrary displacements. For example, the Dirac equation
generalises to
 h(
 )D  I 3 = m 0 ; (64)
where
D  = @  + 1
2  )


: (65)
The Dirac wavefunction transforms as 7! R under rotations. Observ-
ables, such as the current 0
e
, then transform as covariant multivectors.
The gravitational eld equations can be expressed in various ways. Sup-
pose we introduce an arbitrary set of coordinates x  in the STA, with as-
sociated coordinate frame fe  g and reciprocal frame fe  g. From these we
dene
g  =  h(e
 ); g  = h 1 (e  ): (66)
(The overbar denotes the adjoint on the linear function h(a)). In terms of
these the metric is dened by
g  = g 
g  : (67)
One can proceed to perform all calculations directly from the metric, but
this is by no means the best way to work. Instead it is preferable to work
with an expanded set of rst-order equations. We dene
L a = a
 h(r) = a
g  @  ; !(a) = a
g

 : (68)
Assuming no torsion is present, the  h(a) and !(a) gauge elds are related
by the bracket identity
[L a ; L b ] = L c ; (69)

14
where
c = L a b + !(a)
b L b a !(b)
a: (70)
The gravitational eld strength is encoded in the Riemann tensor, R(a^b),
which is dened by
R(a^b) = L a !(b) L b !(a) + !(a)!(b) !(c); (71)
with c determined by equation (70). The Ricci and Einstein tensors and
the Ricci scalar are dened by
Ricci Tensor: R(b) = 
R(  ^b) (72)
Ricci Scalar: R = 
R(  ) (73)
Einstein Tensor: G(a) = R(a) 1
2
aR: (74)
6. Spherical Systems
Our analysis of the eects of a spherical black hole starts with the Schwarz-
schild solution in the following form:
ds 2 = dt 2

dr +
 2GM
r
 1=2
dt
! 2
r 2
d
2 : (75)
Here the time coordinate t is the proper time of an observer freely falling
from rest at innity. This simple form of the Schwarzschild solution was rst
given by Painleve and Gullstrand (see [25]). The solution is well-behaved at
the horizon, and has a number of other attractive properties, but it has been
strangely neglected. In terms of the  h function, the solution is generated by
 h(a) = a
p
2M=r a
e r e t : (76)
We call this particular gauge choice the `Newtonian gauge', due to its simple
Newtonian properties (these are discussed further in [24]).
We use this gauge to put the Dirac equation into Hamiltonian form.
The derivation is particularly clear in Cartesian coordinates. The metric is
now
ds 2 =   dx  dx  GM
r dt 2 2
r
 2GM
r
 1=2
a  dt dx  (77)
where a  = (0; x; y; z). The g  vectors are now given by
g 0 = 0 +
 2GM
r
 1=2 x i
r
i ; g i = i (i = 1; 2; 3) (78)

15
and one can check easily that g 
g  = g  . The g  vectors generate a Dirac
equation of the form
r I 3
 2GM
r
 1=2
0 @ r + 3=(4r)

I 3 = m 0 : (79)
The full relativistic wave equation in a black hole background therefore
picks up a single interaction term:
H I =
 2GM
r
 1=2
} (@ r + 3=(4r) ) I 3 : (80)
(The } has been written explicitly here to emphasise the relation with the
standard radial momentum operator i}@ r .) This is impressively simple,
but there is a subtlety connected with the Hermiticity of H I . Writing
H I ( ) = (2M=r) 1=2 r 3=4 @ r r 3=4

I 3 ; (81)
we nd that
Z
d 3 x h y H I ( )i S =
p
2M
Z
d

Z 1
0
r 2 dr r 5=4 h y @ r (r 3=4 )I 3 i S
=
Z
d 3 x h(H I () y i S +
p
2M
Z
d

h
r 3=2 h y I 3 i S
i 1
0
(82)
where h i S denotes the projection onto the `complex' 1 and I 3 terms, and
 y = 0
e
 0 . For all normalised states the nal term in (82) tends to zero as
r !1. But it can be shown that wavefunctions tend to the origin as r 3=4 ,
so the lower limit is nite and H I is therefore not (quite) a Hermitian op-
erator. This immediately rules out the existence of normalisable stationary
states with constant real energy, and gives us an insight into the nature of
wavefunction decay and collapse into a black hole, which will be important
below.
6.1. THE SCHR 
ODINGER LIMIT
To begin understanding some of the physical properties of this interaction
term, we can check that its properties are sensible in the Schrodinger limit.
Dening a reduced radial wavefunction U(r) via
= U
r
(; ) (83)

16
one nds that the standard reduction of the Dirac equation to the Schrodin-
ger equation (here in the presence of the interaction term) yields
d 2 U
dr 2
l(l + 1) U
r 2
2mi
}
 2GM
r
 1=2
r 1=4 d
dr (Ur 1=4 ) = 2mE
} 2
U (84)
Now the spherically symmetric gravitational problem in the Schrodinger
case is formally similar to that of a charged nucleus. In both cases we
have an underlying inverse square force. In fact we can manipulate the
above equation all the way to complete agreement with the Hydrogen atom
problem by carrying out a phase transformation as follows. Dene the grav-
itational Bohr radius
a 0 = } 2
GMm 2
(85)
and set
U = W exp

i(8r=a 0 ) 1=2

: (86)
Then the Schrodinger equation becomes
W 00
 l(l + 1)
r
2
a 0 r
2mE
} 2

W = 0 (87)
which is now identical to the Hydrogen atom case. We can therefore read
o the energy level spectrum:
E = RG
n 2
; (88)
where the gravitational Rydberg is given by
RG = m
2
 GMm
}
 2
: (89)
As we shall see later, these energy levels and associated wavefunctions do
provide good approximations for the relativistic solutions in cases where
the bulk of the probability density lies well outside the black hole horizon.
However, we can see that near the black hole horizon, the Schrodinger
approach becomes inconsistent. If we consider the Schrodinger current, we
nd that it is given by
}
m r
 8r
a 0
 1=2
= 
 2GM
r
 1=2
^ r (90)
where ^ r is a unit vector in the radial direction. This implies a velocity asso-
ciated with the current of (2GM=r) 1=2 | the same as that of an observer

17
freely falling from rest at innity. The problem, of course, is that this be-
comes equal to c at the horizon, meaning that the Schrodinger approach
becomes inconsistent there. Instead, therefore, we must work with the full
Dirac theory.
6.2. RELATIVISTIC BOUND STATES
We start by assuming
(x) = (x)(t) (91)
in equation (79). As usual, the solution of the t-equation is
(t) = exp( I 3 Et); (92)
where E is the separation constant. The non-Hermiticity of H I means that
E cannot be purely real if is normalisable. The imaginary part of E is
determined by equation (82) and, for suitably normalised states, we nd
that
Im(E) = lim
r!0
2
p
2M h y ir 3=2 : (93)
This equation shows that the imaginary part of E is necessarily negative, so
the wavefunction decays with time. This is consistent with the fact that the
streamlines generated by the conserved current 0
e
are timelike curves
and, once inside the horizon, must ultimately terminate on the origin. How-
ever, we can still ask the question of whether a discrete set of complex
energy eigenvalues exist. This would then give us a set of spectral levels
similar to that in a Hydrogen atom, except with the novelty that each
state decays with time, making it a type of resonance.
We now show this does in fact happen, but a crucial question before
continuing is whether the energy eigenvalues so obtained are physically
meaningful. In the GTG approach, this will be the case if we can show that
the values obtained are gauge invariant. In [24], we show that the radial
coordinate r is physically well-dened (e.g. it can be experimentally deter-
mined by tidal forces) and hence that the full set of gauge transformations
compatible with our setup in the spherically symmetric case are:
t 7! t + f(r); position gauge change
7! R ; rotation gauge change
with the rotor R being a function of x only (R = R(x)). Suppose that we
have a solution in some gauge
(x) = (x) exp( I 3 Et); (94)

18
where E = E r + I 3 E i . We now carry out a gauge transformation of the
above form, obtaining
R (x 0 ) = R(x) exp( I 3 E(t + f(r)))
= R(x)(x) exp( I 3 Et) exp( I 3 Ef(r))
Thus
@ t R (x 0 )I 3 = R (x 0 )(E r + I 3 E i ) (95)
i.e. the new wavefunction is still an eigenfunction of the evolution operator,
with the same eigenvalue. This means E r and E i are gauge invariant, and
hence potentially physically observable.
Proceeding with a solution in the Newtonian gauge, we carry out an
angular separation via writing
(x; ) =
(
m
l u(r) + ^ r m
l v(r)I 3  = l + 1
^ r m
l u(r) 3 + m
l iv(r)  = (l + 1);
(96)
where  is a non-zero integer and u(r) and v(r) are complex functions of r
(sums of a scalar and an I 3 term). Here the m
l are spherical monogenics
(see e.g. [18]), which carry the angular dependency of the wavefunction.
Specically, the unnormalised monogenic m
l is dened by
m
l = [(l +m+ 1)P m
l (cos) P m+1
l (cos)i  ]e mI3 ; (97)
where l  0, (l + 1)  m  l, and P m
l are the associated Legendre
polynomials.
Substituting (96) into the Dirac equation with the time dependence
separated out, and using the properties of the spherical monogenics, we
arrive at the coupled radial equations

1 (2M=r) 1=2
(2M=r) 1=2 1

u 0
1
u 0
2

= A

u 1
u 2

(98)
where
A =

=r j(E +m) (2M=r) 1=2 (4r) 1
j(E m) (2M=r) 1=2 (4r) 1 =r

;
(99)
u 1 and u 2 are the reduced functions dened by
u 1 = ru u 2 = jrv; (100)
and the primes denote dierentiation with respect to r. (We employ the
abbreviation j for right-sided multiplication by I 3 .)

19
To analyse (98) we rst rewrite it in the equivalent form
1 2M=r

 u 0
1
u 0
2

=
 1 (2M=r) 1=2
(2M=r) 1=2 1

A
 u 1
u 2

: (101)
This makes it clear that the equations have regular singular points at the
origin and horizon (r = 2M ), as well as an irregular singular point at
r = 1. To our knowledge, the special function theory required to deal with
such equations has not been developed. In the massless case the equations
can be manipulated into a second order equation of Heun type [26], but
this does not appear to help with the present problem. In the absence of a
suitable mathemtical theory, we must either attempt a numerical solution,
or look for power series with a limited radius of convergence. We start by
considering the latter approach, and look for power-series solutions around
the horizon. To this end we introduce the series
u 1 =  s
1
X
k=0
a k  k ; u 2 =  s
1
X
k=0
b k  k ; (102)
where  = r 2M . The index s controls the radial dependence of at
the horizon, so represents a physical quantity. To nd the values that s
can take, we substitute (102) into (101) and set  = 0. This results in the
equation
s
2M
 a 0
b 0

=
 1 1
1 1
 =(2M) j(E +m) (8M) 1
j(E m) (8M) 1 =(2M)
 a 0
b 0

(103)
Rewriting this in terms of a single matrix and setting its determinant to
zero yields the two indicial roots
s = 0 and s = 1
2
+ 4jME: (104)
The s = 0 solution is entirely sensible | the power series is analytic, and
nothing peculiar happens at the horizon. If one calculates the conserved
current 0
e
associated with this solution, one nds it is nite and inward-
pointing at the horizon, as one would expect. The second root leads to
solutions which are ill-dened at the horizon, and have a discontinuity in
the current there. This raises some delicate issues as regards the physical
meaning of these solutions, some of which are discussed in Section 8 of
[24]. If one disregards these problems then, remarkably, it turns out that
the discontinuity can be interpreted as the creation of a net outward ux
of particles at the horizon, described by a Fermi-Dirac distribution at a
temperature given by the Hawking temperature [27]
T = 1
8MkB : (105)

20
The value of this temperature comes directly from the imaginary part of
the second index s in (104). The surprising feature here is that a Fermi-
Dirac distribution is obtained without any of the apparatus of quantum eld
theory. It turns out that repeating this exercise with a eld of integer spin
(for example a scalar eld, or the electromagnetic eld) yields the Hawking
temperature again, with a ux described by a Bose-Einstein distribution.
This is obviously an interesting and important area to explore more
fully, but here we wish to concentrate instead on the question of the ex-
istence of bound state solutions. By analogy with the rejection of un-
normalisable solutions in the Hydrogen atom problem, we here reject the
solution which is singular at the horizon, and ask whether bound state solu-
tions exist composed wholly of the regular solution, with index s = 0. Since
the wavefunction we are using is described by two complex functions of r
(the u(r) and v(r) in equation (96)), there are four real degrees of freedom.
Two of these are absorbed when we discard the singular solution, and the
remaining two correspond to the amplitude and phase of the regular solu-
tion at the horizon. The equations are linear so, without loss of generality,
we can x the parameters to arbitrary values. This means we have no de-
grees of freedom left in the wavefunction, and can proceed to try to nd the
(complex) values of energy which lead to normalisable solutions | these
will be the eigenfunctions we desire. In the absence of the necessary special
function theory, this has been carried out numerically. We hope that one
day a more analytic approach may be possible.
The numerical method chosen involved picking trial values for the real
and imaginary components of the energy, and then integrating outwards
from the horizon as far as possible. What sets the limits to such an integra-
tion is that either numerical instability sets in, or the wavefunction density
starts diverging exponentially, due to an incorrect (non-eigenvalue) value
of E being chosen. The search method involving minimizing the density
as a function of the imaginary component of E automatically for each E r
over a grid of values of E r . Some typical results are shown in Fig. 3. These
are for l = 0, a black hole mass M = 1, and particle mass m = 0:1. The
y-axis is log 10 of the wavefunction density, so a dynamic range in selecting
eigenvalues of about 10 4 is displayed here.
The search for eigenvalues is aided by searching near regions in E r
where the non-relativistic theory would predict a bound state. We would
expect the non-relativistic energy found in equation (88) to act as a small
correction to the rest mass energy mc 2 , giving
E n  mc 2 GMm
2a 0 n 2
(106)
where a 0 is the gravitational Bohr radius (85), and n is the quantum number
appropriate in the non-relativistic analysis. For the ground state with m =

21
Figure 3. A plot of wavefunction density at a xed large distance from the black hole,
for various values of the real part of the particle energy Er , with automatic minimization
carried out in E i . The particle mass is 0:1, the angular quantum number l = 0 and the
black hole mass is 1:0. The y-axis shows log 10
of the wavefunction density at a radius
(800 GM=c 2 ).
0:1, M = 1, this predicts E r = 0:0995, indeed very close to where the rst
dip is seen in Fig. 3. n = 2 would yield E r = 0:099875, close to the second
dip. The wavefunction density corresponding to this rst excited state is
shown in Fig. 4.
To establish formally that we have genuine eigenvalues requires integrat-
ing out to innite radial coordinate, and demonstrating that the density
integral remains nite. This can be achieved by integrating inwards from
innity and simultaneously outwards from the horizon, and matching at
an intermediate point. This process is complicated by the fact that the
wavefunction has two types of essential singularity at innity, which have
to be allowed for before series solutions at innity can be found. One is the
exp(
p
m 2 E 2 r) behaviour which one expects by analogy with the Dirac
solution solution for the Hydrogen atom. The second is an essential singu-
larity in the phase, given by a dependence of the form exp(j2E
p
2Mr).
A term of this kind could be expected from what we found above for the
gravitational Schrodinger equation.

22
Figure 4. A plot of r 2
 wavefunction density for the rst excited state of a fermion
bound to a black hole. As in Fig. 3 the particle mass is 0:1, the angular quantum number
l = 0 and the black hole mass is 1:0.
TABLE 1. Groundstate energies calculated
for two values of m and M = 1.
m Er E i
0.1e0 0.099468827746 -2.7870281e-6
0.2e0 0.194834514694 -.000751079389
We have conrmed that this process works, which means we can be
condent that our eigenvalues are correct. A factor reinforcing condence
is that all the computations have been repeated in a dierent gauge (the
advanced Eddington-Finkelstein gauge | see below) and exactly the same
numerical values were found here as well. Furthermore, one can compare
the numerically obtained imaginary component of E with that found by
integrating the wavefunction to as large a radius as possible and normalising
it, and then using equation (93) at the origin. This also checks out precisely.
Some ground state eigenvalues are as shown in Table 1 and for the rst
excited state in Table 2.
What physical regimes might these solutions be interesting in? We can

23
TABLE 2. First excited state energies calcu-
lated for two values of m and M = 1.
m Er E i
0.1 0.099870235768 -.356282508e-6
0.2 0.19880214433 -.00009960784655
get some idea of the importance of the decay by dening a dimensionless
quantity
 = gravitational Bohr radius
Schwarzschild radius/2 (107)
We can re-write this relation in the form
Mm = 1
p

(2:2  10 8 kg) 2 : (108)
So, for example, if we take m equal to the electron mass, m = m e =
9:1  10 31 kg, then M = (1=
p
) 5:3  10 14 kg. Primordial black holes of
size 10 12 to 10 14 kg say, then lie in an interesting range as regards quantum
eects.
The size of  is inversely related to the importance of decay eects.
A crude estimate, obtained by working with non-relativistic approxima-
tions to the ground state wavefunctions, suggests that (in the near non-
relativistic regime) we should nd
E i  8 5=2 mc 2 (109)
which is roughly borne out by the numerical results. For electrons coupled
to primordial black holes, this is therefore of order the `zitterbewegung'
time }=(m e c 2 ), (approximately 10 21 seconds) which is obviously very fast,
and it is not clear how such a decay would manifest itself. Typical questions
which should now be faced include:
1. What does the detailed energy spectrum look like?
2. What is the signicance of the antiparticle solutions, which have op-
posite sign of the real part of their energy?
3. Do the energy dierences between shells correspond to something ob-
servable, given the rapid decay rates?
4. If they do, what is the mechanism of radiation, and what quantum
jumps are involved?
5. Are there bound states, for which the expectation value of r lies wholly
inside the horizon?

24
6. What is the extension to the Reissner-Nordstrom and Kerr cases?
7. Can we incorporate multiparticle eects?
The last question is important since a multiparticle approach is neces-
sary to provide a proper link to the Hawking radiation. Equally, in the same
way that for the calculation of the Lamb shift it is necessary to evaluate a
sum involving bound state energies of the Hydrogen atom, so presumably
in the black hole case, vacuum uctuation eects should be calculated tak-
ing proper account of the existence and spectrum of bound states. We are
not sure that so far this has been carried out.
7. Propagators and black hole scattering cross-sections
We have considered bound states, so an obvious next topic is fermion scat-
tering by black holes. What we wish to look at is again in the spirit of
seeing how far we can get in applying conventional quantum mechanics to
a gravitational context, using the added facilities of geometric algebra.
In the Coulomb scattering of an electron by a point charge, the rst
quantum corrections to the Rutherford formula are embodied in the Mott
scattering cross-section. A convenient way to arrive at this is via using the
Feynman propagator to nd the rst order S-matrix linking initial and -
nal states. In a geometric algebra approach, one nds that the S-matrix is
replaced by a scaled rotor, which as well as giving the quantum mechanical
amplitude, also explicitly embodies the spin transformation properties of
the process. In simple cases, this means that the need for spin-sum calcu-
lations is obviated, leading to a considerable reduction in the length of the
calculations as compared to the conventional approach (see e.g. [18, 28]).
Specically, let i (x) be an initial `input' wavefunction and f (x) the
nal `output' function and Consider the Coulomb scattering problem to
rst order. What one nds in the GA approach (see [18] for details) is
f (x) = S fi i (x)I 3
(2) 2
E f
ô(E f E i ) (110)
where E i and E f are the initial and nal energies and S fi is the scaled rotor
just referred to. Clearly it rotates from initial to nal momentum and spin
states, while its magnitude determines the cross-section via
d
d
f
= S fi
e
S fi : (111)
In the case of Coulomb scattering from a nucleus with charge Z one nds
explicitly, to rst order, that
S fi = Z
q 2
(2E + q): (112)

25
Here  is the ne structure constant, E = E i = E f is the electron energy
and q = p f p i is the change in relative momentum. (Recall the relative
momentum is dened by p 0 = p
0 + p^ 0 = E + p.). This leads to the
Mott cross-section
d
d
f
= Z 2  2
q 4
(4E 2 q 2 ) = Z 2  2
4p 2  2 sin 4 (=2)
1  2 sin 2 (=2)

(113)
where
q 2 = (p f p i ) 2 = 2p 2 (1 cos ) and  = jpj=E: (114)
Now the interaction Hamiltonian for the Coulomb potential is the simple
scalar operator
HC = Ze 2
4 0 r (115)
while for an electron interacting with a black hole using the Newtonian
gauge we have seen the interaction Hamiltonian has the form
H I =
 2GM
r
 1=2
} (@ r + 3=(4r) ) I 3 (116)
The essential part of this is another scalar operator, but this time with a
derivative in r present in it. The simplicity of this form suggests we ask
whether we can employ the same Feynman propagator techniques we used
in the Coulomb calculation to obtain an analogue of the Mott scattering
formula in the gravitational case. Along the way we would nd the gravi-
tational analogue of the `rotor' S fi , which would again have the benet of
obviating the need for spin sums.
The key part of the derivation will be nding the momentum space
representation of H I . We may write schematically
H I (p 2 ; p 1 ) =
Z
d 3 xe jp 2

x H I e jp 1

x (117)
which yields
H I (p 2 ; p 1 ) = j3 3=2 (GM) 1=2
jp 2 p 1 j 7=2
(p 2
2 p 2
1 ): (118)
On multiplying this by 0 we obtain the desrired vertex factor. This has
the unusual property of vanishing on-shell, when jp 1 j = jp 2 j. (Energy is
conserved throughout since the interaction Hamiltonian is independent of
time). In retrospect, this is as we might expect, since otherwise we would
have the matrix element going as
p
M in leading order, rather than as M ,
which is what what we would expect on the basis of the analogy with Z in
the Coulomb case.

26
While this makes sense, it means we have to go to second order in
our calculations, which is a perhaps unfortunate feature of the Newtonian
gauge. It is thus of interest to nd an alternative gauge in which the desired
scattering result can emerge at rst order. If this answer agrees with that
from the second order Newtonian gauge answer, this would be a powerful
indication that both are right and that the result we are calculating is
physically meaningful (gauge invariant).
The suitable gauge in which to attempt this is the advanced Eddington
Finkelstein (AEF) gauge. This has as its h-function
 h(a) = a + M
r
a
n n where n = 0 ^ r (119)
and ^ r is a unit spacelike vector in the radial direction, so that n is null.
This h-function has a simple form, and the gravitational eects of the black
hole enter linearly in M , as we want, rather than proportional to
p
M as in
the Newtonian gauge case. However, unlike the latter case, the interaction
Hamiltonian is no longer a simple scalar operator on , but has a multivec-
tor structure, not analogous to the Coulomb case. Specically, the Dirac
equation in this gauge is
r I 3 + GM
r
( 0 ^
r)
 @
@t
@
@r
1
2r

I 3 = m 0 (120)
We can still apply the Feynman techniques even in this case, however, by
using the following route. For the general Dirac equation in a gravitational
eld,
D I 3 m 0 = 0 (121)
let us rewrite it instead as
r i 3 m 0 = r i 3 D i 3 (122)
This appears to be a trivial rewriting, but means that we can view the right
hand side as an interaction term which can be solved for using the free-
particle Feynman propagator appropriate to the left hand side. This is quite
a powerful technique, and will apply wherever there are asymptotically free
in and out states. Some details of the evaluation of the matrix element in
this case are discussed in [6], but here we simply state the result:
S fi = GM
q 2
E(2E + q) + p 2 + p f p i


: (123)
This is the scaled rotor which transforms between initial and nal states
for a fermion scattering from a black hole, to rst order in M , calculated

27
using the AEF gauge. This is to be compared and contrasted with the
equivalent Coulomb matrix element (112). It is similar in the rst term,
though with an extra factor of E, and then contains two extra terms. We
note immediately that the presence of the extra E in the rst term is a
manifestation of the equivalence principle! Without it, the cross-section
would depend explicitly on the particle mass m. With it, the cross-section
is a function of the particle velocity only.
Before continuing to discuss the cross-section coming from this matrix
element, we have to satisfy ourselves that it is physically meaningful. Here
we have the advantage that we can verify that we obtain the same result
using the Newtonian gauge to second order. The details are given in [6] but
it turns out that indeed the result is identical to (123), which helps verify
gauge invariance. A further argument in favour of gauge invariance is that
the similar computation in the Coulomb case is denitely gauge invariant
| if we replace the electromagnetic potential A with A r(r), so that
A 7! A + ^ r d=dr, one nds that (on-shell) the matrix element S fi , and
therefore the cross-section, is unaected. A nal piece of evidence is that
we can be certain that in the AEF gauge we will not obtain an eect at
order M 3=2 , since the perturbation series will involve whole powers of M .
This predicts that the third order eect in the Newtonian gauge should
vanish. This can be shown to be indeed the case by explicit computation
in the Newtonian gauge. We therefore take it that our result is physically
meaningful, although an explicit proof of gauge invariance (ideally to all
orders in M) would of course be desirable.
The cross-section following from (112) is
d
d
f
= (GM) 2
4 4 sin 4 (=2)

1 + 2 2 3 2 sin 2 
2 +  4  4 sin 2 
2

(124)
and is plotted for a particular case in Figure 5. It is possible to check
our result against the classical limit for black hole scattering by comparing
with the result of Collins, Delbourgo and Williams in [1]. They were able to
obtain an explicit series solution for the classical cross-section in the limit
of small . The rst term in their series is of order M 2 and is (translating
to our notation)
d
d

=
 2GM(2 2 1)
 2 ( 2 1)
 2
(125)
where is the Lorentz factor (1  2 ) 1=2 . This agrees precisely with the
rst term in the small angle expansion of our result, equation (124). Their
second term
3(GM) 2 (5 2 1)
4 3 ( 2 1)
(126)

28
Black Hole scattering cross section (beta=0.99)
--2
0
2
4
6
8
10
log10(cross­section)
20 40 60 80 100 120 140 160
theta (deg.)
Figure 5. Plot of the black hole scattering cross-section formula (124) for a high speed
particle ( = 0:99), and with M = 1.
is also of order M 2 , and therefore might be expected to agree with us also.
However, our second term is
(GM) 2 4 4 7 2 + 1


3 2 ( 2 1) 2
(127)
which as well as being quite dierent numerically, is also dierent as re-
gards its order in , going as  2 rather than  3 . We cannot explain this
discrepancy, and indeed it seems strange that an odd power of  could be
obtained at all, even in the classical result.
The massless limit m 7! 0 is also well-dened and leads to the simple
formula
d
d
= (GM) 2 cos 2 (=2)
sin 4 (=2) : (128)
Again, the low-angle limit recovers the classical formula for the bending of
light. This result also predicts zero amplitude in the backward direction,  =
. Null geodesics produce a signicant ux in the backward direction, and
the fact that zero is predicted here is a diraction eect for neutrinos which

29
goes beyond the predictions of geometric optics. A similar prediction of zero
back-scattering for neutrinos was made in [3]. A more detailed analysis
of the cross section in the backward direction also reveals a large `glory'
scattering [3, 5]. In the geometric optics limit this is attributable to multiple
orbits, and in the quantum description the glory scattering is described by
higher-order terms in GM . To describe these eects in the present scheme
requires extending to higher order in perturbation theory. This is currently
under investigation.
Extending to higher orders also raises the question of the convergence
of the iterative scheme proposed here. This is not a straightforward issue
to address as there is no dimensionless coupling constant in the problem.
Also, it is not clear whether higher-order quantum terms should still be
expected to obey the equivalence principle. One can easily formulate de-
sirable criteria for convergence, such as GME < 1 or GMEv < 1, but
these are too restrictive, given that the low angle formula we arrive at is
expected to be valid for all masses and velocities. It would appear that the
only way to investigate convergence is to compute the next order terms in
the perturbation series directly.
This work should also have claried the importance of working consis-
tently to the correct order in M . This is particularly clear in the Schwarz-
schild gauge, where the interaction term contains factors which go as 1
(1 2GM=r) 1=2 . An iterative scheme based on this gauge choice should ex-
pand out the vertex factor as a series in M , and then keep all of the terms
up to the desired order. Such a scheme is workable, but has the disadvan-
tage of introducing new vertex terms at each order in the series solution. It
is straightforward to conrm that such a scheme reproduces our result for
the fermion cross section, to lowest order.
The next steps include the following:
1. Extension of the above results to second order in the AEF gauge (and
perhaps 4th order in Newtonian gauge, to verify gauge invariance at
order M 2 ). Potential new features which may emerge include the `glory'
back scattering eect, mentioned above ([5]).
2. Extension to the Riessner-Nordstrom and Kerr cases. The interaction
with the angular momentum in the Kerr case will be particularly in-
teresting, but the computations currently look diôcult.
3. Another vertex which can be tried is interaction with a photon, in the
presence of the background gravitational eld, in order to calculate
the gravitational equivalent of bremsstrahlung. This could shed some
light on the long-standing problem of the radiation caused by a freely
falling electron.

30
8. Riemann-squared theory and torsion eects in the early uni-
verse
We would like nally to discuss some progress in applying GA techniques
to higher-order Lagrangian eld theories. Our group has already looked at
the eects of torsion in gauge theory gravity, and found a new solution
for the Dirac eld coupled self-consistently to gravity in which torsion was
important [29, 30]. We have also considered a GA approach to quadratic
gravitational Lagrangians and to topological invariants, in [31]. Here, we
wish to highlight some recent work which may be important in application
to the early universe.
The higher-order Lagrangian theory which is perhaps best motivated in
analogy with other gauge theories is one where the Lagrangian is given by
the same expression as would be used in any Yang-Mills theory | the eld
strength tensor squared. Written in the GA approach this is a term in
L 2 = R(e  ^ e  )
R(e  ^ e  ): (129)
This has been explored several times and is known to lead to spherically
symmetric metrics of Schwarzschild-like form but with

1 2M
r
+ r 2

in place of

1 2M
r

(130)
If one demands asymptotic atness, then  = 0 and one is back with the
standard Schwarzschild metric, which means that the theory passes all the
standard solar system tests.
A key feature of this type of theory, which makes it particularly attrac-
tive, is that it satises scale covariance as well as position and rotation
gauge covariance. It thus encompasses all the symmetries one might expect
to be present. A further feature is that it includes torsion. This is the case
even if there is no source of quantum spin, which is a signicant dierence
from theories using the using a Ricci scalar Lagrangian for the gravitational
eld. For these the torsion is generated entirely by matter. In cosmology,
for example, we can consider coupling self-consistently with a Dirac eld,
which is the form of torsion considered in [29] and [30].
Working with the Lagrangian L 2 we can derive the eld equations ap-
propriate for cosmology. In what follows we have assumed spatial atness.
The equations yields a remarkable result. Writing
A = 2H 2 + _
H 1
4 Q 2 and B = 3
2 QH + 1
2
_
Q (131)
where H is the Hubble parameter and Q is a measure of the `free-space' tor-
sion (both real), we nd that it is possible to write the evolution equations

31
Hubble parameter
Torsion
Legend
Variation of torsion and Hubble parameter
--40
--30
--20
--10
0
10
20
1 2 3 4
time
Figure 6. Illustrative plot of variation of Hubble parameter (H) and and free-space
torsion (Q) with Q starting at small values. In this case, the matter density is zero at all
times, so it is only the torsion eld providing the dynamics.
together jointly in a complex form:
d
dt
(A + iB) = iQ(A + iB) (132)
Thus the modulus of the quantity (A + iB) is constant, but its phase gets
driven round in a potentially chaotic fashion by the torsion, Q. These are
the full equations of the cosmology. They reduce to equations people have
studied before in Riemann squared cosmology for Q = 0, but it is not clear
whether the above behaviour is known for the torsion case.
If one computes numerically some typical results, one nds that the
regular behaviour of H is disturbed by the torsion Q which, provided it
is not precisely zero initially, builds up and then forces a rapid change in
H. Q then declines in value again, before another episode of building up.
A typical example is shown in Figure 6. This is actually computed for
zero matter density, and shows that the addition of torsion by itself is quite
suôcient to give some interesting dynamics. A problem for the matter sector
of any scale invariant theory, is that the matter stress-energy tensor has to

32
have zero trace. This precludes using normal matter, and means that such
cosmologies are probably unrealistic. One of the main current cosmological
problems, however, is how to generate a period of in ation in the early
universe, and then a hugely smaller eective cosmological constant today.
It is certainly interesting that torsion can give somewhat chaotic behaviour
of the type illustrated, with very large swings of the Hubble parameter,
unrelated to the underlying matter eld.
Acknowledgements
ANL would like to thank Venzo de Sabbata for the opportunity of attending
the meeting in Erice and contributing to this volume. CJLD is supported
by the EPSRC.
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