Approximate solutions in General Relativity via deformation of embeddings

0.27 MB·English

by  Richard Kerner

Approximate solutions in General Relativity
via deformation of embeddings
⋆ ⋆
Richard Kerner and Salvatore Vitale
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0 February 3, 2008
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n
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J Abstract

1
3 Asystematic studyof deformations of four-dimensionalEinsteinian space-times
embedded in a pseudo-Euclidean space EN of higher dimension is presented. In-
]
c finitesimal deformations, seen as vector fields in EN, can be divided in two parts,
q
tangent to the embedded hypersurface and orthogonal to it; only the second ones
-
r
are relevant, the tangent ones being equivalent to coordinate transformations in
g
[ the embedded manifold.

1 The geometrical quantities can be then expressed in terms of embedding func-
v tions zA and their infinitesimal deformations vAzA z˜A = zA +εvA. The de-
8 →
formations are called Einsteinian if they keep Einstein equations satisfied up to a
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8 given order in ε. The system so obtained is then analyzed in particular in the case
4 of the Schwarzschild metric taken as the starting point, and some solutions of the
.
1 first-order deformation of Einstein’s equations are found.
0
We discuss also second and third order deformations leading to wave-like so-
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lutions and to the departure from spherical symmetry towards an axial one (the
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: approximate Kerr solution)
v
i
X
1
r
a
1 ⋆ Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,
Universit´e Pierre-et-Marie-Curie - CNRS UMR 7600
Tour 22, 4-`eme ´etage, Boˆıte 142,
4, Place Jussieu, 75005 Paris, France
1
1 Introduction
The study of the two-body problem in General Relativity must include possible emission
of gravitational waves. In this respect General Relativity runs into difficulties akin to
those of classical electromagnetism, where the full treatment of charged particle motion
plus the radiative field can be performed only via approximation techniques. Great
progress was achieved since the first papers by Ll. Bel and N. Deruelle, [1], followed
by the throughout analysis of post-Newtonian and post-post-Newtonian approximations
including gravitational radiation. The most important contributions have been made by
Th. Damour, L. Blanchet, G. Shaeffer and many others, [2], [3], [4].
Their approach was based on a particular choice of the initial approximation. In
physical situations where the gravitational field is not too strong and when the relative
velocities of bodies under consideration are very low as compared with the speed of light
c, the Newtonian theory provides us with exact (although not explicit) solutions. One
takes one of these solutions as a starting point, adding successive corrections resulting
from the inclusion of the relativistic effects: the finite propagation of gravitational field,
the curvature of space, and so forth. The corrections are consequently made to the
trajectory, to the law of motion, and to the gravitation potential (identified with the
corrections to the space-time metric tensor).
Ina series of paperspublished afew years ago[6], [7], [8], [9]) analternative methodof
determining relativistic motion of test particles in a spherically symmetric gravitational
field has been proposed, without need to use the Newtonian limit of General Relativity.
The idea was to take as a starting point the very special explicit solution of geodesic
equation in Schwartzschild metric background: a test particle moving with constant
speed along the circle around a spherically symmetric mass. Let us denote this particular
trajectory by xµ(s), with s denoting the proper spacetime length of the curve. Then one
can consider a deviation from this worldline, x˜µ(s), which can be expanded in a power
series of some small parameter ε:
x˜µ(s) = xµ(s)+εnµ(s)+ε2bµ(s)+... (1)
ThegeodesicequationinSchwarzschild backgroundcanbealsoexpandedinaseriesof
equations whose solutions in terms of unknown deviations nµ(s), bµ(s), etc., will provide
us with successive approximations to the exact solution, up to any order required. The
parameter ε is roughly proportional to the eccentricity of the new trajectory. Already
the first approximation, linear in ε, predicts the perihelion advance for orbits with very
small eccentricity.
The shortcoming of this method, which was entirely focused on the trajectories, was
the total lack of any variation of the gravitational field, i.e. the Schwarzschild metric
which was maintained invariable for all orders of geodesic deviation. This fact reduced
the validity of the method only to the case of test particles with mass m negligible
when compared with the mass M of the central body appearing in the Schwarzschild
1
background metric. More precisely, the successive approximations of planet’s trajectory
remain valid as long as the dimensionless parameter (m/M) can be considered as negli-
gibly small, i.e. (m/M) << 1. When the mass m is no more a negligible quantity, its
presence must inevitably alter the geometry of the initial Schwarzschild metric, and its
influence can be therefore represented by a power series in the small parameter (m/M).
In this article we shall describe the departure from the initial Schwarzschild metric in
terms of the embedding functions. Embeddings of the exterior Schwarzschild geometry
in pseudo-Euclidean flat spaces are known since a long time ([13], [14], [12]), and once
such an embedding is given, all intrinsic geometric quantities of the embedded manifold
can be expressed in terms of derivatives of the embedding functions which depend on
four “internal” parameters which are the space-time coordinates. A general analysis
of deformations of the embedded Einstein spaces was given in [10]; nevertheless, only
the theoretical setup was considered, without any concrete solution describing Ricci-flat
deformations of known exact Einstein spaces, and in the first place, Schwarzschild and
Kerr metrics.
The present article is intended to explore not only the first linear approximation, but
also the effects of second and third order in the expansion of deformations in powers of
small parameter ǫ, including the corrections describing gravitational waves. The depar-
ture from spherical symmetry (Schwarzschild’s metric) towards axial symmetry (Kerr’s
metric) as Einsteinian (Ricci-flat) deformation of the corresponding embedded manifold
is also discussed.
2 Isometric embeddings and their properties
2.1 The embedding functions and the induced metric
Consider the embedding of a four-dimensional Riemannian space parametrized by local
coordinates (denoted by xµ, µ,ν = 0,1,2,3 as usual) in a pseudo-Euclidean space EN of
dimension N. The dimension N, yet unspecified, depends onthe topologyof theRieman-
nian space under consideration, and may be quite high, as acknowledged in [12]. Locally,
any n-dimensional Riemannian manifold can be embedded in a (pseudo)-Euclidean space
of dimension N = n(n + 1)/2. Here we are interested in global embeddings, which
may require a relatively low dimension of the “host” space if the Riemannian space to
be embedded possesses some particular symmetry. For example, the de Sitter space
can be embedded globally in a five-dimensional pseudo-Euclidean space with signature
(+ ), and both exterior and interior Schwarzschild solutions can be embedded
− − −−
globallyinasix-dimensional EN with signatures(++ ) or(+ ). Consider
−−−− −−−−−
a global embedding of a Riemannian space V given by the following set of embedding
4
functions zA:
A,B,... = 1,2,...N,
zA = zA(xµ), with (2)
µ,ν = 0,1,2,3.
2
The metric tensor of V is the induced metric defined as
4
o
g = η ∂ zA∂ zB (3)
µν AB µ ν
o
The inverse metric tensor gµν cannot be obtained directly from the embedding func-
tions, but should be computed from the covariant components as their inverse matrix.
From now on we use the superscript notation in order to make difference between the
o
“basic”inducedmetricg whichwillbeconsidered asabackground, anditsinfinitesimal
µν
deformations expanded in terms of an infinitesimal parameter ε as follows:
o 1 2
g = g +εg +ε2g +... (4)
µν µν µν µν
induced by the following deformation of the initial embedding functions:
zA(xµ) zA(xµ)+εv1A(xµ)+ε2v2A(xµ)+... (5)
→
When seen from the ambient pseudo-Euclidean space, the new embedded manifold
V˜ is the result of an infinitesimal deformation of the initial manifold V induced by a
4 4
vector field in EN . It is quite obvious that on the embedded manifold such a field
(p,q)
can be decomposed into its normal part (in the sense of the pseudo-Euclidean metric)
and a part tangent to V . This last part induces an internal diffeomorphism of V and
4 4
can be always implemented as a local coordinate transformation. Such deformations do
not have any physical meaning, but it is not always necessary to consider exclusively
the deformations orthogonal to the embedded V ; sometimes a deformation having non-
4
vanishing both parallel and orthogonal parts can have less non-zero components in the
ambient space EN than its part orthogonal to the embedded V manifold.
(p,q) 4
2.2 Expressions for connection and curvature
Our first aim is to express all important geometrical quantities e.g. the connection
coefficients andthecurvaturetensor, intermsofembedding functionszA andtheirpartial
derivatives. Let us start with Christoffel connection
o 1 o o o o
Γλ = gλρ ∂ g +∂ g ∂ g . (6)
µν 2 µ νρ ν µρ − ρ µν
(cid:18) (cid:19)
o
¿From the definition of g (3) we have the expression for its partial derivatives:
µν
o
∂ g = η ∂2 zA∂ zB +∂ zA∂2 zB . (7)
λ µν AB λµ ν µ λν
(cid:18) (cid:19)
When substituted into the definition (6) it gives
o o
Γλ = η gλρ∂ zA∂2 zB (8)
µν AB ρ µν
3
Consider now the second covariant derivative of zB:
o
zB = ∂2 zB Γλ ∂ zB. (9)
∇µ∇ν µν − µν λ
Therefore, we have
o
∂2 zB = zB +Γλ ∂ zB, (10)
µν ∇µ∇ν µν λ
and of course, ∂ zB = zB. Substituting (10) into (8) we obtain the following identity:
µ µ
∇
o o o o
Γλ = η gλρ zA zB +η gλρΓσ zA zB. (11)
µν AB ∇ρ ∇µ∇ν AB µν ∇σ ∇ρ
But in the last term we note that
o o o
gλρη zA zB = gλρg = δλ, (12)
AB∇σ ∇ρ σρ σ
reducing (11) to
o o o
Γλ = η gλρ zA zB +Γλ , (13)
µν AB ∇ρ ∇µ∇ν µν
which shows clearly that
η zA zB = 0 (14)
AB ρ µ ν
∇ ∇ ∇
which may be considered as an alternative (although implicit) definition of Christoffel
symbols, and could be also derived as a direct consequence of the fact that g = 0.
µ λρ
∇
Using this result, let us form the following combination of covariant derivatives which
vanishes identically:
η ( zA zB) ( zA zB) = 0
AB µ ρ ν σ ν ρ µ σ
∇ ∇ ∇ ∇ −∇ ∇ ∇ ∇
(cid:20) (cid:21)
Applying the derivation and using the Leibniz rule we get:
η zA zB zA zB) +
AB µ ρ ν σ ν ρ µ σ
∇ ∇ ∇ ∇ −∇ ∇ ∇ ∇
(cid:20) (cid:21)
+ η zA zB zB = 0. (15)
AB ρ µ ν σ ν µ σ
∇ ∇ ∇ ∇ −∇ ∇ ∇
(cid:20) (cid:21)
Recalling that
o
zB = R λ zB, (16)
∇µ∇ν −∇ν ∇µ ∇ρ µν ρ∇λ
(cid:20) (cid:21)
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so that we can write
o
R = η zA zB zA zB (17)
µνλρ AB µ λ ν ρ ν λ µ ρ
− ∇ ∇ ∇ ∇ −∇ ∇ ∇ ∇
(cid:20) (cid:21)
which is the well known Gauss-Codazzi equation.
The definition of the Riemann tensor by means of derivatives of the embedding func-
tions given by formula (17) looks very compact, but is in fact highly non-linear and quite
complicated. This is so because it contains many Christoffel symbols involved in the
o
second covariant derivatives, which contain in turn the contravariant metric tensor gµν.
The components of the contravariant metric tensor are obtained as rational expressions
in third and fourth powers of zA. Nevertheless, the most important point here is that
µ
∇
the Riemann tensor depends only on first and second derivatives of embedding functions,
so that the Einstein equations expressed in terms of the embedding functions will lead
to second-order partial differential equations.
The expressions derived in this section will be very useful in the development of a
power series expansion of infinitesimally deformed embedding.
3 Infinitesimal deformations of embeddings
3.1 General setting
o
Let us consider an isometric embedding of an Einsteinian manifold V in a pseudo-
4
Euclidean space EN with signature (p+,q ), with p+q = N:
p,q −
o
V EN A,B,... = 1,2,...N,
4 → p,q with (18)
zA = zA(xµ) µ,ν = 0,1,2,3.
Consider now an infinitesimal deformation of the embedding defined by a converging
series of terms proportional to the consecutive powers of a small parameter ε. The
deformed embedding defines an Einsteinian space V˜ ;
4
zA(xµ) z˜A(xµ) = zA(xµ)+εvA(xµ)+ε2wA(xµ)+...
→
The induced metric on V˜ can also be developed in series of powers of ε:
4
o 1 2
g˜ = g +εg +ε2g +...
µν µν µν µν
= η ∂ zA∂ zB +ε ∂ zA∂ vB +∂ vA∂ zB +
AB µ ν µ ν µ ν
"
(cid:18) (cid:19)
+ ε2 ∂ vA∂ vB +∂ zA∂ wB +∂ wA∂ zB (19)
µ ν µ ν µ ν
#
(cid:18) (cid:19)
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Among all possible infinitesimal deformations of the embedding functions zA(xµ)+
εvA(xµ) there is a large class of functions vA(xµ) which will not alter the intrinsic
geometry of the embedded manifold. Infinitesimal translations vA = Const. obviously
o
do not change the internal metric g = η ∂ zA∂ zB. Also the generalized Lorentz
µν AB µ ν
transformations of the pseudo-Euclidean space EN keep the internal metric unchanged.
(p,q)
Indeed, if we set
zA z˜A = zA +εΛA zB, (20)
→ B
with ΛA constant matrix. Then the first-order deformed metric is:
B
o 1
g˜ = g +εg +... =
µν µν µν
= η ∂ zA∂ zB +ε ∂ zAΛB ∂ zC +ΛA ∂ zC∂ zB =
AB µ ν µ C ν C ν µ
" #
(cid:18) (cid:19)
o
= g +ε η ΛB +η ΛC ∂ zA∂ zB ,
µν AB C CB A µ ν
(cid:20) (cid:21)
Then the first-order correction vanishes if the matrices ΛA satisfy the identity
B
η ΛB +η ΛC = 0
AB C CB A
whichdefinestheinfinitesimalrigidrotations(Lorentztransformations)ofthepseudo-
Euclidean space EN .
(p,q)
The geometric character of our approach enables us to eliminate unphysical degrees of
freedom using simple geometrical arguments. Remember thatin the traditional approach
leading to linearized equations for gravitational fields the starting point is the following
development of the metric tensor:
o
g = g +εh , (21)
µν µν µν
thus introducing ten components of h as dynamical fields. We know however that most
µν
of them do not represent real dynamical degrees of freedom due to the gauge invariance.
The metric tensor itself does not correspond to any directly measurable quantity. In
fact, its components may be changed by a gauge transformation without changing the
components oftheRiemanntensor whichisthesourceofmeasurable gravitationaleffects.
In particular, the gauge transformation
g g˜ = g + ξ + ξ (22)
µν µν µν µ ν ν µ
→ ∇ ∇
doesnotaltertheRiemanntensorsothatbothg˜ andg describethesamegravitational
µν µν
field.
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The arbitrary vector field ξµ generating gauge transformation (22) represents four
degrees of freedom which are redundant in g ; this is why in the linearized Einstein
µν
equations one may impose four gauge conditions e.g.
hµν = 0. (23)
µ
∇
The unphysical degrees of freedom can be easily eliminated from the embedding de-
formation functions vA(xµ) if we note that any vector field in the embedding space EN
that is tangent to the embedded Riemannian space V describes nothing else but a dif-
4
feomorphism of V , in other words a coordinate change, which has no influence on any
4
physical or geometrical quantities.
Vector fields tangent to the four-dimensional embedded manifold V can be decom-
4
posed along four arbitrarily chosen independent smooth vector fields in EN tangent to
V . On the other hand, vector fields transversal to the embedded hypersurface V must
4 4
satisfy the following obvious orthogonality conditions:
η ∂ zAvB = η vA zB = 0. (24)
AB µ AB µ
∇
For any value of A the four partial derivatives (let us remind that zA = ∂ zA) span a
µ µ
∇
basis of four vector fields in EN tangent to the submanifold V ; therefore any vector vB
4
satisfying the orthogonality condition (24) is transversal to V (as seen in EN).
4
The orthogonality condition (24) imposes four independent equations, which reduce
the number of independent deformation functions vA to N 4. This means that gen-
−
eral non-redundant deformations can be decomposed along N 4 independent fields
−
XA , k = 1,2,...,N 4:
(k) −
N−4
vA(xµ) = vk(xν))XA (xλ). (25)
(k)
k=1
X
The basic fields XA (xλ) can be chosen at will provided they induce a non-singular global
(k)
vector field on V , while the relevant degrees of freedom are contained in N 4 functions
4
−
vk(xλ). To take an example, the de Sitter space can be globally embedded in a five-
dimensional pseudo-Euclidean space E5 with signature (+ ); therefore its global
1,4 −−−−
deformations can be described by a single function v(xλ) (see [10]).
One may ask the following question: if the deformation destroys the initial symme-
try of the embedded manifold so that the deformed manifold cannot be embedded in
the initially sufficient N-dimensional pseudo-Euclidean space but needs a flat embedding
space of higher dimension ? It is known that global embeddings of the Kerr metric
need more than six flat dimensions sufficient for the embedding of the exterior (or in-
terior) Schwarzschild solution (see [19], [22], [23], [24]), although Schwarzschild’s metric
can be obtained from Kerr’s metric as a limit when the Kerr parameter a (the angular
momentum) tends to zero.
The answer is that as long as we investigate only the first-order corrections to geom-
etry, we should not worry about this issue for the following two reasons: first, when a
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global embedding is given, its infinitesimal deformations cannot lead to a global mod-
ification of the embedding; second, if the bigger embedding space was introduced, say
EN+m, it would contain the initial embedding space EN as its linear subspace, so that
EN+m = EN Em,
⊕
and its pseudo-Euclidean metric could be represented as a blockwise reducible matrix
η 0
η = AB , (26)
αβ 0 η
ij
(cid:18) (cid:19)
with A,B,... = 1,2,...N, i,j,... = 1,2,..m, α,β,... = 1,2,....,N +m. Accordingly, any
deformation of the initial embedding can be decomposed in two parts, one contained in
the initial embedding space EN and another one in the complementary subspace Em:
vα = [vA, vm]. (27)
But the initial embedding functions had their components entirely in the first subspace
EN, zα = [zA, 0], therefore the deformed embedding functions can be written as
z˜α = [zA +εvA, εvk], (28)
so that the induced metric of the deformed embedding will be
o 1 2
g = g +εg +ε2g +...
µν µν µν µν
= η ∂ zA∂ zB +εη ∂ zA∂ vB +∂ vA∂ zB (29)
AB µ ν AB µ ν µ ν
(cid:18) (cid:19) (cid:18) (cid:19)
+ ε2 η ∂ vA∂ vB +∂ zA∂ wB +∂ wA∂ zB +η ∂ vi∂ vj ...
AB µ ν µ ν µ ν ij µ ν
" #
(cid:18) (cid:19) (cid:18) (cid:19)
¿From this one can see that the deformations towards the extra dimensions do not con-
tribute to the first-order corrections of any geometrical quantities obtained from the de-
formed embedding functions. This is why we shall not consider such deformations while
investigating at first only the terms linear in the infinitesimal parameter ε. Our principal
aim now is to establish the explicit form of connection and curvature components in-
˜
duced on the infinitesimally deformed embedding V . To this end we must calculate the
4
approximate expression of the contravariant metric tensor gµν. If the covariant metric is
decomposed as
o 1 2
g = g +εg +ε2g +... (30)
µν µν µν µν
then we have the following formulae defining the corresponding decomposition of gµν:
o 1 2
gµν = gµν +εgµν +ε2gµν +...
o o 1 o o 2 o o o 1 1
= gµν εgµρgµσg ε2 gµρgµσg +ε2 gµρgµσgλκ g g . (31)
ρσ ρσ ρλ σκ
− −
" # " #
8
3.2 The first order corrections to Einstein equations
In what follows we shall keep only the first order terms linear in ε.
Let us start by computing the first (linear) correction to the components of the
Christoffel connection, which develops in Taylor series as
o 1 2
Γλ = Γλ +εΓλ +Γλ +..., (32)
µν µν µν µν
then by definition we have:
1 1 o 1 1 1 1 1 o o o
Γλ = gλρ ∂ g +∂ g ∂ g + gλρ ∂ g +∂ g ∂ g (33)
µν 2 µ νρ ν µρ − ρ µν 2 µ νρ ν µρ − ρ µν
(cid:18) (cid:19) (cid:18) (cid:19)
1 o 1 1 1 1 o o 1 o o o
= gλρ ∂ g +∂ g ∂ g gλσgρκg ∂ g +∂ g ∂ g .
µ νρ ν µρ ρ µν σκ µ νρ ν µρ ρ µν
2 − 2 −
(cid:18) (cid:19) (cid:18) (cid:19)
One easily checks that
1 o 1 1 1 o
gλρ ∂ g +∂ g ∂ g = η gλρ ∂ zA∂2 vB +∂ vA∂2 zB , (34)
2 µ νρ ν µρ − ρ µν AB ρ µν ρ µν
(cid:18) (cid:19) (cid:20) (cid:21)
while the first term after some algebra gives
1 1 o o o o o o
gλρ ∂ g +∂ g ∂ g = η gλρ ∂ zAΓσ ∂ vB +∂ vAΓσ ∂ zB . (35)
2 µ νρ ν µρ − ρ µν − AB ρ µν σ ρ µν σ
(cid:18) (cid:19) (cid:20) (cid:21)
Combining together (34), (34) and (35) we find the final expression
1 o
Γλ = η gλρ zA vB + vA zB (36)
µν AB ∇ρ ∇µ∇ν ∇ρ ∇µ∇ν
(cid:20) (cid:21)
This expression has a tensorial character as it should be, because by definition both
quantities
Γλ and Γ˜λ
µν µν
transformasconnection coefficients, thereforetheir difference must transformas atensor,
and this is true for any term of the development into series of powers of ε.
1
The coefficients Γλ will be useful for the derivation of geodesic equations in the
µν
deformed space-time, but they are not necessary for the computation of the first-order
deformation of the Riemann tensor, which can be determined as follows.
Let us develop second covariant derivatives of the deformed embedding functions z˜A
yields:
˜ ˜ z˜A = ˜ ˜ zA +ε ˜ ˜ vA +O(ε2)
µ ν µ ν µ ν
∇ ∇ ∇ ∇ ∇ ∇
1
= zA +ε vA Γλ zA +O(ε2). (37)
∇µ∇ν ∇µ∇ν − µν ∇λ
#
(cid:20)
9