Generalized regularly discontinuous solutions of the Einstein equations
Generalized regularly discontinuous solutions
of the Einstein equations
Gianluca Gemelli
L.S. B. Pascal, V. P. Nenni 48, Pomezia (Roma), Italy.
e-mail: gianluca.gemelli@poste.it
Abstract
The physical consistency of the match of piecewise-C0 metrics is
discussed. The mathematical theory of gravitational discontinuity hy-
persurfaces is generalized to cover the match of regularly discontinuous
metrics. The mean-value differential geometry framework on a hy-
persurface is introduced, and corresponding compatibility conditions
are deduced. Examples of generalized boundary layers, gravitational
shock waves and thin shells are studied.
Submitted to Int. J. Theor. Phys.
1 Introduction
Is it possible to define weak solutions of the Einstein equations of class
piecewise-C0, i.e. to generalize the compatibility conditions which replace
the field equations on a singular hypersurface to the case when the metric is
regularly discontinuous?
To reach this goal would probably mean to define the most general class
of regularly discontinuous weak solutions of the Einstein equations. It seems
that this problem was never studied before in the literature. But, before we
proceed, we need to discuss whether we are talking of something mathemat-
ically and physically consistent or not.
A fundamental concept of Riemannian geometry is that at any point of
a submanifold there are coordinate choices for which the metric reduces to
http://arxiv.org/abs/0704.0103v1
the Minkowski flat metric. Clearly, if this choice is made on both sides of
the discontinuity surface, any ”jump” in the metric disappears. Thus, the
metric discontinuity appears as a coordinate dependent concept, which is
neither geometrically (nor physically) acceptable in the context of General
Relativity.
But we also have to consider that regularity of the global coordinates plays
an important role in our approach, which is that of [1] and of the literature
cited therein. In particular, since here the spacetime is only C0, we are led to
considering (C0, piecewise C1) coordinate transformations. If the metric is
discontinuous in some globally C0 chart, it is in general impossible to obtain
the vanishing of the metric jump on both sides of a hypersurface with a C0
coordinate transformation (see section 2). Moeover in the following we are
led in a natural way to considering C1 coordinate transformations; the metric
discontinuity is a tensor with respect to such coordinate changes!
In other words the jump of the metric has a precise mathematical mean-
ing, if we consider it in connection with global regular coordinates.
In a well consolidated procedure, the assumption of continuity for the
metric across a gravitational interface is usually taken for granted; however
it follows from the limiting process of the thin sandwich modelization, in
consequence of the hypothesis that the external derivatives of the metric
are bounded [2]. Yet in this paper we are going to see that, even removing
the assumption of continuity, it is still possible to define a generalized inner
geometry of the discontinuity hypersurface; one thus can consistently find
a corresponding generalized set of compatibility conditions, which obviously
reduces to the usual ones when the continuity hypothesis is restored.
Yet, which are the physical motivations to move to such generalization?
Actually gravitational shock waves and thin shells are usually defined by the
presence of singular curvature with a “delta” component concentrated on a
hypersurface, situation which is well cast within the classic C0 piecewise-C1
match of metrics [1, 3].
We were originally led to consider solutions of class piecewise-C0, as pos-
sible generalizations of shock waves and thin shells, by the sake of mathe-
matical completeness, with the idea that phisical interpretation would follow.
We actually found a reacher framework than the usual one, with some in-
teresting new features (and even some rather undesiderable ones), which we
display in this paper.
There are two main theories in the literature for solutions of class C0
piecewise-C1, i.e. that in terms of the second fundamental form (heuristic
theory, see e.g. [4, 5]) and that in terms of the curvature tensor-distribution
(axiomatic theory, see e.g. [6, 1]); such are equivalent through appropiate
extensions (for a self-contained overview see e.g. [1]).
The axiomatic theory appears to be inappropriate to the study of general-
ized solutions, since the theory of distributions is basically linear. Even if we
could in principle replace the discontinuous metric with its associated distri-
bution gD, then it would be impossible to define, for example, replacements
for the Christoffel symbols, since this would involve product of distributions,
which, as it is generally believed, is impossible to define. In fact it was proved
by Schwarz [7] that, under reasonable hypothesis, there can be no definition
of commutative and associative operation on distributions which reduce to
ordinary multiplication on integrable distributions (say on regular functions);
thus in a word it is impossible to define product of distributions.
Or is it? Colombeau [8, 9, 10] developed a theory which apparently con-
tradicts Schwarz’s result. He introduced a very broad space of generalized
functions, which extends the usual space of distributions, a subspace of which
corresponds, in a certain sense (the correspondance is not 1 to 1), to usual
distributions. Colombeau’s formalism permits multiplication of generalized
functions; but the contradiction with the impossibility theorem is only ap-
parent, in fact Schwarz’s hypothesis are violated, since the operation does
not coincide with ordinary multiplication on regular functions nor with mul-
tiplication of a regular function times a distribution (although it does at least
for C∞ functions).
Such theory, however, does not fit in a natural way in general relativity,
since it is impossible to define covariantly invariant geometrical objects; in
fact Colombeau’s space is not invariant for smooth coordinate transforma-
tions, unless they are linear. Such difficulty, however, seems to have been
overcome in subsequent adjustments of the theory, with the introduction of
a richer mathematical framework [11, 12], so that the generalized functions
current apparatus can be used in general relativity, and indeed it has been
applied at least to the calculation of singular curvatures of the spacetimes
of Kerr [13], Reissner-Nordstrom [14], and so-called cosmic-string spacetime
[15]. In such literature Colombeau’s theory is adapted to the handling of
curvature when the metric has a singularity in the sense of functions, i.e. the
ordinary curvature would explode, at a singular event-point or at a singular
worldline. There seems to be no particular reason to forbid Colombeau’s
method also for defining the match of piecewise-C0 regularly discontinuous
metrics at a singular hypersurface; however, as far as the author is aware, no
attempt has been made yet to use it in this framework.
The direct method we will introduce in the following sections, however,
is so conceptually simple that we prefer not to experiment with Colombeau’s
generalized functions, which would instead mean introducing a far more com-
plicated and unfamiliar mathematical apparatus.
In this paper in fact we propose a new, generalized theory for regularly
discontinuous solutions, covering also the match of piecewise-C0 metrics. Our
theory is heuristic, as it is constructed in a way similar to the heuristic the-
ory of C0, piecewise-C1 solutions originated from the studies of Israel, but
we completely avoid the traditional or projectional Gauss-Codazzi framework
(which either does not include the lightlike case [4, 5], or needs a special adap-
tation for it [16, 1]) and introduce what we called “mean-value differential
geometry” framework, instead (see section 3). This is conceptually very sim-
ple, and permits to construct in a natural way a generalized theory, where
the main role (which used to be that of the jump of the secund fundamental
form) is here played by the jump of the Christoffel symbols. The theory is an
extension of the theory of gravitational discontinuity hypersurfaces we have
studied in [1], to which it reduces when the metric is C0. Even if we should
restrict to C0 solution, by adding the traditional assumption of continuity for
the metric, our theory would undoubtedly have at least the good qualities of
not needing the timelike and the lightlike case to be distinguished (different
from usual heuristic theory), and of just requiring C0 continuity for the co-
ordinates (different from the axiomatic theory). Moreover, it is completely
cast in the framework of general coordinates of the ambient (glued) space-
time, with no use of parametric equations of the hypersurface, nor of inner
coordinates and holonomic 3-basis, which could be considered a good quality
in some applications as well.
Piecewise-C0 weak solutions of the Einstein equations, as far as the au-
thor is aware, have never been considered previously in the literature. They
generalize the corresponding C0 solutions, as examples in this paper show;
however there is more. Apparently in fact the theory allows the propagation
of free gravitational discontinuity at lower speed than the speed of light (sec-
tion 8); or rather, we still have no general proof that the absence of stress
energy concentrated on Σ should, in the timelike case, necessarily imply the
degeneracy of a generalized solution to a boundary layer, although it does
at least for a wide class of spherical matchs (see section 6). Moreover, non-
simmetric stress-energy is allowed on the hypersurface (section 9), like e.g. in
Einstein-Cartan dynamics. This possible link to classical unification theories
is surprising, since in our framework we have nothing similar to Einstein-
Cartan torsion. We therefore see a lot of space for future investigation.
2 Discontinuous metrics
Let us suppose V4 an oriented differentiable manifold of dimension 4, of class
(C0, piecewise C2), provided with a strictly hyperbolic metric of signature
–+++ and class piecewise-C0. Let Ω ⊂ V4 be an open connected subset with
compact closure. Let units be chosen in order to have the speed of light in
empty space c ≡ 1. Greek indices run from 0 to 3.
Let Σ ⊂ Ω be a regular hypersuperface of equation f(x) = 0; let Ω+ and
Ω− denote the subdomains distinguished by the sign of f . We suppose the
metric and its first and second partial derivatives to be regularly discontin-
uous on Σ in all charts of class C0(Ω). Let f ∈ C0(Ω) ∩ C2(Ω\Σ), and let
second and third derivatives of f be regularly discontinuous on Σ. Finally,
let ℓα ≡ ∂αf denote the gradient of f .
Let the metric be a solution of the ordinary Einstein equations on each of
the two domains Ω+ and Ω−. In this situation Σ is the interface between two
general relativistic spacetimes and it is called a (generalized) gravitational
discontinuity hypersurface.
In the following we will develope a theory to justify the introduction of
suitable generalized compatibility conditions to replace the Einstein equa-
tions on Σ (section 5); if such conditions are satisfied the match across the
generalized gravitational hypersurface Σ will be called a generalized regularly
discontinuous solution of the Einstein equations.
Now let us briefly recall some basics notions on regularly discontinuous
fields, which we will use as tools. In any case, for notation and terminology
we refer to [1].
A field ϕ is said to be regularly discontinuous on Σ if its restrictions to
the two subdomains Ω+ and Ω− both have a finite limit for f −→ 0; we
denote such limits by ϕ+ and ϕ−, respectively.
In this case the jump [ϕ] across Σ and its arithmetic mean value ϕ are
well defined on the hypersurface:
[ϕ] = ϕ+ − ϕ−
ϕ = (1/2)(ϕ+ + ϕ−)
If ϕ is continuous across Σ, we obviously have: [ϕ] = 0, ϕ = ϕ. We also have
the converse formulae:
ϕ+ = ϕ+ (1/2)[ϕ]
ϕ− = ϕ− (1/2)[ϕ].
As for the product of two functions ϕ and ψ, we have:
[ϕψ] = [ϕ]ψ + ϕ[ψ]
ϕψ = ϕψ + (1/4)[ϕ][ψ]
If a field ϕ is regularly discontinuous on Σ, its jump [ϕ] is sometimes called
its discontinuity of order 0.
The jump of a regularly discontinuous function has support on Σ, but in
general, the partial derivative of the jump is well defined as the jump of the
derivative of the function (see [17, 18]). In particular, the derivative of the
jump of a continuous field is not null, unless the field is also C1.
Similarly, we define the partial derivative of the mean value as the mean
value of the partial derivative. We can also use regular prolongations to
extend, in a sense, the definition of ϕ and [ϕ] to the whole domain Ω. Thus
they can be regarded as regular and derivable fields in Ω, but their values
(and those of their derivatives) are well defined only on Σ, while in Ω\Σ
they depend on the choice of the prolongation. For details on the method of
regular prolongations see e.g. [17, 18].
We moreover define the covariant derivative of a field with support on Σ
by means of the mean value Γβρ
σ of the Christoffel symbols. For the jump of a
regularly discontinuous vector, for example, with this definition one has that
the jump of the covariant derivative is different than the covariant derivative
of the jump. Thus, by definition, we have:
β] = ∂α[V
β] + Γασ
β[V σ] (4)
and in consequence of (3):
β] = [∇αV
β]− [Γασ
, (5)
and similarly for the jump of any regularly discontinuous tensor.
Since the spacetime is only C0, we are led to considering (C0, piecewise
C1) coordinate transformations, with regularly discontinuous first deriva-
tives; the metric discontinuity [gαβ] is not a tensor with respect to such
changes of coordinates. In fact we have:
[gαβ] = [gα′β′]
+ qαβ′
+ qα′β
where:
qα′β =
[gα′β′ ]
+ ḡα′β′
We therefore can simulate all (C0, piecewise C1) coordinate changes by com-
bining C1 changes with metric gauge changes:
[gαβ ]←→ [gαβ] + qαβ′
+ qα′β
which generalize usual gravitational gauge changes of the theory of (C0, piece-
wise C1) solutions [1].
Is it always possible to make [gαβ] vanish with an appropriate C
0 trans-
formation? Clearly the answer is negative. In fact it suffices to consider the
case when [gαβ] and ḡαβ are both definite positive in a given chart to see that
the equation obtained from (6) by replacing the first hand side with 0 has no
solution for [∂xα
/∂xα] and ∂xα
/∂xα. Thus the set of effective generalized
gravitational discontinuity hypersurfaces is non empty.
Moreover it will occur in many applications to have ℓα ∈ C
0. Therefore
it will be often desiderable to work in the framework of (C1, piecewise C2)
coordinate transformations, which preserve such condition. The metric dis-
continuity is a tensor with respect to such changes of coordinates, but the
jump of the Christoffel symbols, which appear to play a main role in the
following, is not; we have in fact:
σ] = [Γα′β′
∂xα∂xβ
If the coordinates are C0 and so is the form ℓα we can write:
∂xα∂xβ
= ℓαℓβ∂
where ∂2 denotes the weak discontinuity of order 2 (see e.g. [17, 18]). Thus on
Σ we can generate all (C1, piecewise C2) transformations for [Γ] combining
C2 transformations (with respect to which Γ is a tensor) and Christoffel
gauges transformations, i.e. of the kind:
σ]↔ [Γαβ
σ] + ℓαℓβQ
σ (11)
with some analogy with the case of C0 metrics (where the main role is played
by the first order metric discontinuity ∂g, see [1] section 3).
In any case neither the mean value of the metric g or its jump [g] now
have null covariant derivatives. Consider in fact the identity ∇αgβρ = 0 in
the domain Ω+; from the limit f −→ 0+, on Σ we have:
βρ − (Γαβ
ν)+g+νρ − (Γαρ
ν)+g+βν = 0 (12)
Here, with obvious meaning of the symbols, we denote: g+βρ = (gβρ)
+, gβρ =
gβρ, etc. Consequently on Σ, from (2)1 we have:
∂αgβρ + (1/2)∂α[gβρ]− Γαβ
νgνρ − Γαρ
νgνβ+
−(1/2)([Γαβ
ν ]gνρ + Γαβ
ν [gρν ] + [Γαρ
ν ]gνβ + Γαρ
ν [gβν ])+
−(1/4)([Γαβ
ν ][gνρ] + [Γαρ
ν ][gβν ]) = 0
Similarly, from the limit f −→ 0− and from (2)2 we also have on Σ:
∂αgβρ − (1/2)∂α[gβρ]− Γαβ
νgνρ − Γαρ
νgνβ+
+(1/2)([Γαβ
ν ]gνρ + Γαβ
ν [gρν ] + [Γαρ
ν ]gνβ + Γαρ
ν [gβν ])+
−(1/4)([Γαβ
ν ][gνρ] + [Γαρ
ν ][gβν ]) = 0
From the sum of expressions (13) and (14) we thus have:
∇αgβρ = (1/4)([Γαβ
ν ][gνρ] + [Γαρ
ν ][gβν ]) (15)
and, from difference:
∂α[gβρ] = [Γαβρ] + [Γαρβ ] (16)
From (16), (3), and from the definition of covariant derivative over Σ, we
then have:
∇α[gβρ] = [Γαβ
ν ]gνρ + [Γαρ
ν ]gβν (17)
As for the jump and the mean value of the Christoffel symbols we have, from
ν = (1/2){gνσ(∂αgβσ + ∂βgσα − ∂σgαβ)+
+(1/4)[gνσ](∂α[gβσ] + ∂β[gσα]− ∂σ[gαβ ])}
ν ] = (1/2){gνσ(∂α[gβσ] + ∂β[gσα]− ∂σ[gαβ ])+
+[gνσ](∂αgβσ + ∂βgσα − ∂σgαβ)
or, from (15) and (17):
ν ]gνρ = (1/2)(∇α[gβρ] +∇β[gρα]−∇ρ[gαβ])
ν ][gνρ] = 2(∇αgβρ +∇βgρα −∇ρgαβ)
3 Mean-value geometry on a hypersurface
Let us consider a 4-vector V α, regularly discontinuous on Σ, the jump and the
mean value of which will work as a prototype of vectors with Σ as support.
We have, by definition:
[∇β∇αV
σ] = ∇β[∇αV
σ]− [Γβα
ν ]∇νV σ + [∇βν
σ]∇αV ν (21)
where [∇αV
σ] = ∇α[V
σ]+ [Γαν
and where, again by definition, we have:
∇νV σ =
{∂ν(V
+)σ + (Γ+)νλ
σ(V +)λ + ∂ν(V
−)σ + (Γ−)νλ
σ(V −)λ} (22)
Thus, from (2) we have:
∇νV σ = ∇νV σ + (1/4)[Γνλ
σ][V λ], (23)
which, incidentally, is the same result we could get from the formal applica-
tion of (3), wich actually can be applied to covariant derivatives, provided
one interpretes ∇ = ∇. We therefore have:
[∇α∇βV
σ] = ∇α∇β[V
σ] +∇β[Γαν
+ [Γαν
σ]∇βV
−[Γβα
ν ]∇νV
− (1/4)[Γβαν][Γνλ
σ][V λ]+
+[Γβν
σ]∇αV
+ (1/4)[Γβν
σ][Γαλ
ν ][V λ]
and thus, by antisymmetrization:
[∇[β∇α]V
σ] = ∇[β∇α][V
σ] +∇[β[Γα]νσ ]V
[Γν[β
σ][Γα]λ
ν ][V λ] (25)
Now, from the Ricci identity we have: [∇[β∇α]V
σ] = [Rαβρ
σV ρ] and then, by
[∇[β∇α]V
σ] = [Rαβρ
+Rαβρ
σ[V ρ], (26)
and thus from a well known identity which follows from (3) as a consequence
our definition (5) for the covariant derivative on Σ, i.e. (see [1]):
[Rαβρ
σ] = ∇β[Γαρ
σ]−∇α[Γβρ
σ] (27)
we have that the commutator of the covariant derivatives of the jump of a
generic regularly discontinuous vector obeys the following Ricci-like formula:
∇[βα][V
(1/2)Rαβρ
σ − (1/4)[Γν[β
σ][Γα]ρ
[V ρ]. (28)
Not surprisingly, working in a similar way starting from ∇β∇αV σ and anti-
symmetrizing, we find again:
∇[βα]V
(1/2)Rαβρ
σ − (1/4)[Γν[β
σ][Γα]ρ
; (29)
in fact any given field with support on Σ can be considered, by the help of
suitable prolongations, as the jump (or as the mean value of) some regularly
discontinuous field. Thus, for any vector V with support on Σ, we can
introduce the following mean-value geometry Ricci-like formula on Σ:
(∇[β∇α])V
σ = (RΣ)αβρ
σV ρ; (30)
where we have introduced the mean-value geometry curvature (RΣ), defined
by the following mean-value geometry first Gauss-Codazzi identity:
(RΣ)αβρ
σ = Rαβρ
σ − (1/4)([Γβν
σ][Γαρ
ν ]− [Γαν
σ][Γβρ
ν ]) (31)
Notice that, for the sake of simplicity, we have introduced a slight abuse
of notation, since in [1] and [16] the same symbol RΣ instead denotes the
inner curvature defined with the help of projections. Actually anything goes
like in [1] section 4 with the Gauss-Codazzi framework, with the difference
that here we don’t have to make projections, which would involve product
times a discontinuous tangent metric. Moreover here we don’t even have
to distinguish between the cases of Σ timelike or lightlike. In other words
our mean-value differential geometry on a hypersurface is a very simple, in
conceptual terms, analogue of the Gauss-Codazzi apparatus.
Thus, with the heuristic theory of [1] section 6 (see also [4] for the timelike
case) in mind as a prototype, we expect the jump of the Christoffel symbols to
play the main role, in place of the secund fundamental form, in the definition
of compatibility conditions for very weak solutions of the Einstein equations.
Indeed, this happens, as it will be shown in the following.
4 Complex mean-value formalism
The metric being dicontinuous on Σ, we are missing the fundamental tool
to rise and lower indices, and to construct curvature in the traditional way.
This is the reason why sometimes one is tempted to introduce some hybrid
metric object on Σ to replace the metric, even in the (C0, piecewise C1) case
(see e.g. [5]). It is reassuring to find out that the framework of the preceeding
section can be confirmed by such a kind of approach.
It would be desiderable to simply replace g with g on Σ, but it is easy to
check that g has not the necessary algebraic requisites; in particular we have
αρ 6= δβ
ρ. Consider instead:
g̃αβ = gαβ + i(1/2)[gαβ], g̃
αβ = gαβ − i(1/2)[gαβ] (32)
where i is the imaginary unit (i.e. we have i2 = −1). It is easy to check,
with the help of (3), that we have:
g̃αβ g̃
αρ = δα
ρ + i[gαβ ]g
αρ (33)
i.e., in particular: ℜ(g̃αβ g̃
αρ) = δβ
ρ. For the sake of brevity in the following
we will denote A ≈ B the relation ℜ(A) = ℜ(B). Thus the pair g̃αβ and
g̃αβ is a good candidate replacement for the metric on Σ, for the purposes of
rising and lowering indices. Now, similar to (32) let us introduce:
Γ̃αβν = Γαβν + i(1/2)[Γαβν ], Γ̃αβ
σ = Γαβ
σ − i(1/2)[Γαβ
σ] (34)
so that we have: Γ̃αβ
σ ≈ Γ̃αβν g̃
σν and conversely: Γ̃αβν ≈ Γ̃αβ
σg̃νσ. Let us
now introduce the differential operator ∇̃ on Σ, which makes use of Γ̃ in
place of Γ. As we could expect we have:
∇̃ρg̃αβ ≈ 0, ∇̃ρg̃
αβ ≈ 0 (35)
which is the replacement on Σ for the covariant conservation of the metric
tensor.
Now let us construct on Σ the complex curvature tensor R̃ in the familiar
way, but with Γ̃ in place of the ordinary Christoffel symbols (which are
undefined on Σ):
R̃αβρ
σ = ∂βΓ̃αρ
σ − ∂αΓ̃βρ
σ + Γ̃βµ
σΓ̃αρ
µ − Γ̃αµ
σΓ̃βρ
µ (36)
We rather unespectedly find out that
R̃αβρ
σ = (RΣ)αβρ
σ + i(1/2)[Rαβρ
σ] (37)
i.e. in particular we have: R̃αβρ
σ ≈ (RΣ)αβρ
σ, where RΣ is given by (31).
This is just another reason for identifying RΣ as the replacement for the
curvature tensor of Σ, which is the first step of our path to the generalized
compatibility conditions.
5 Generalized compatibility conditions
Let us now consider limit f → 0+ of the curvature tensor of the subdomain
Ω+; by (2) we have:
(Rαβρ
σ)+ = Rαβρ
σ + (1/2)[Rαβρ
σ] (38)
and, by (27):
(Rαβρ
σ)+ = Rαβρ
σ +∇[β[Γα]ρ
σ] (39)
We also have, by (31):
(Rαβρ
σ)+ = (RΣ)αβρ
σ +∇[β[Γα]ρ
σ] + [Γν[β
σ][Γα]ρ
ν ] (40)
We see that R and RΣ only differ by terms proportional to [Γ], and not
involving derivatives of it. Thus, in view of neglecting these tems, in the
following we will consider R instead ofRΣ; this simply avoids the introduction
of the symbol “ ∼= ”, with the meaning of equality but for terms not involving
derivatives of [Γ] (which here replaces the second fundamental form K) as in
[1] section 6. Then for the Ricci tensor Rβρ = Rαβρ
α we have:
(Rβρ)
+ = Rβρ + (1/2)∇µ
µ[Γνρ
ν ]− [Γβρ
and for the curvature scalar R = Rα
R+ = R + (1/2)∇µ
µν ]− [Γν
Now, to construct the Einstein tensor G+ we have to remember that, since
the metric is also discontinuous:
(gαβ)
+ = gαβ + (1/2)[gαβ] (43)
so that we have:
(Gβρ)
+ = Gβρ + (1/2)∇µ
µ − (1/8)[gβρ]
µν ]− [Γν
where we have denoted, for the sake of brevity:
µ[Γνρ
ν ]− [Γβρ
µ]− (1/2)gβρ
µν ]− [Γν
Let us fix a coordinate chart and consider a generic (for the moment) regular
prolongation for G, so that its mean value is defined in the whole Ω. Now
consider the Riemann 4-volume integral of G+ over the domain Ω+; from
the Green theorem we obtain (for the general definition of integral on a
hypersurface see [6] p. 6):
Gβρ =
Gβρ + (1/2)
ℓ+µHβρ
µ − (1/8)
ℓ+µ [gβρ]
µν ]− [Γν
The analogous formula for Ω− involves −ℓ− as the outgoing normal vector
and the metric g−αβ = gαβ − (1/2)[gαβ], so we have:
Gβρ =
Gβρ + (1/2)
ℓ−µHβρ
µ + (1/8)
ℓ−µ [gβρ]
µν ]− [Γν
and consequently we have:
Gβρ =
Gβρ +
ℓµHβρ
µ (48)
Thus reasons similar to those of the heuristic theory (see [4] and [1] section
6) lead to the reasonable hypothesis that G remain bounded in the neigh-
bourhood of Σ, for any admissible prolongation, so that from the volume
integral of the Einstein equations, with the presence of an eventual source
term concentrated on Σ:
Gαβ = −χ
Tαβ − χ
T̆αβ (49)
where χ denotes the gravitational constant, we conclude that
ℓµHβρ
µ = −χ
T̆βρ (50)
which is our heuristic reason for considering the following set of general-
ized compatibility conditions to hold on Σ as a replacement for the Einstein
equations:
ℓµHβρ
µ = −χT̆βρ (51)
Here T̆ represents the stress-energy content of the hypersurface.
In the simpler case ℓα ∈ C
0, it is very easy to check that the object ℓµHβρ
is gauge-invariant in the sense of (11), as we could hope.
Turning now to the comparison with the C0 case, we see from eq.s (71)
and (85) of [1] that our generalized conditions (51) are formally identical to
ordinary compatibility conditions [eq. (110) of the same paper], if expressed
in terms of [Γ] (which in the general case is a function of the jump of the
metric [g] as well as of its weak discontinuity ∂g). Therefore it is clear
that generalized compatibility conditions reduce to ordinary ones in case the
metric is continuous, i.e. in case [gαβ] = 0.
In particular, let us suppose g ∈(C0, piecewise C1) and f ∈ C0(Ω); let
us moreover suppose (ℓ · ℓ) > 0, i.e. Σ timelike. By definition of Christoffel
symbols, and from (11) of [1], we have:
σ] = (ℓ · ℓ)−1/2(NβGρ
σ +NρGβ
σ −NσGβρ) + (ℓ · ℓ)
1/2NβNρQ
σ (52)
Q is a vector which can be set to zero with a suitable gauge choice; it plays
no role in (51), as one would expect, in fact we have:
ℓµ[Γβρ
µ] = −Gβρ + (ℓ · ℓ)Nβρ(Q ·N)
ℓβ[Γνρ
ν ] = NβNρGν
ν + (ℓ · ℓ)NβNρ(Q ·N)
ℓµ[Γν
µν ] = Gν
ν + (ℓ · ℓ)(Q ·N)
ℓµ[Γν
νµ] = −Gν
ν + (ℓ · ℓ)(Q ·N)
and, since g = g = h(N) +N ⊗N , we have from (45):
ℓµHβρ
µ = Gβρ − h(N)βρGν
ν (54)
i.e., according to (88) of [1]:
ℓµHβρ
µ = Hβρ (55)
as expected. Now let us instead suppose (ℓ · ℓ) = 0, i.e. Σ lightlike. Let
u ∈ C0(Ω) be a given auxiliary reference frame. From eq.s (21) and (16) of
[1] we have:
σ] = (u ·ℓ)−1(−LβF(u)ρ
σ−LρF(u)β
σ+LσF(u)βρ)+(u ·ℓ)
2LβLρQ̂
σ (56)
and consequently, from (18) and (19) of the same paper:
ℓµ[Γβρ
µ] = LβB(u, n)ρ + LρB(u, n)β − (u · ℓ)
3LβLρ(Q̂ · L)
ℓβ[Γνρ
ν ] = LβLρG(u, n)ν
ν − (u · ℓ)3LβLρ(Q̂ · L)
ℓµ[Γν
µν ] = ℓµ[Γν
νµ] = 0
We therefore have:
ℓµHβρ
µ = G(u, n)ν
νLβLρ − LβB(u, n)ρ − LρB(u, n)β (58)
i.e. again, according to (83) of [1], we have: ℓµHβρ
µ = Hβρ, as expected.
Therefore the set (51) of compatibility conditions, together with ordinary
Einstein Equations to hold on each side of the discontinuity hypersurface,
defines the class of generalized regularly discontinuous solutions of the Ein-
stein equations. And in case [g] = 0, i.e. for continuous metric, from such
conditions we recover the ordinary compatibility conditions for regularly dis-
continuous weak solutions.
However, in the generic case we have some differences, as we are going to
show in the following.
6 A class of spherical boundary layers
Let us consider the match of two piecewise-C0 regularly discontinuous spher-
ical solutions of the vacuum Einstein equations, of the form
ds2 = −a(r, t)dt2 + b(r, t)dr2 + r2dΩ2 (59)
with dΩ2 = dθ2 + sin2 θdϕ2, across a spherical admissible gravitational dis-
continuity hypersurface Σ of equation r = ρ(t), with ρ(t) ∈ C1. Therefore
the form ℓα = δα
r − ρ̇δα
t is continuous (while ℓβ = gβαℓα in general is not).
We suppose globally C0 coordinates, the same form of the metric in both
domains Ω+ and Ω−, and the identification t+ = t−, r+ = r−, θ+ = θ−,
ϕ+ = ϕ− on Σ. Leta, b > 0 and let a, b ∈ piecewise-C0 be regularly dis-
continuous on Σ and with regularly discontinuous first derivatives. Let us
denote by a dot the partial derivative with respect to t, and by a prime that
with respect to r. Let moreover condition a− bρ̇ > 0, i.e. (ℓ · ℓ) > 0, hold on
both sides on Σ.
We have:
[gαβ ] = −[a]δα
t + [b]δα
r (60)
Now let us define the match as a generalized regularly discontinuous solution
by (51), with T̆ = 0, i.e. in the absence of stress-energy concentrated on Σ.
In this case our compatibility conditions reduce to:
ℓβ[Γµρ
µ]− ℓµ[Γβρ
µ] = 0 (61)
which, for a match of metrics of the kind (59), are equivalent to the following
system:
ρ̇[ḃb−1] + [a′b−1] = 0
ρ̇[ḃa−1] + [a′a−1] = 0
ρ̇[a′a−1] + [ȧa−1] = 0
ρ̇[b′b−1] + [ḃb−1] = 0
[b−1] = 0
i.e. we have [b] = 0 and consequently:
ρ̇[ḃ] + [a′] = 0
ρ̇[ḃa−1] + [a′a−1] = 0
ρ̇[a′a−1] + [ȧa−1] = 0
ρ̇[b′] + [ḃ] = 0
and from (3):
ρ̇[ḃ] + [a′] = 0
(ρ̇ḃ+ a′)[a−1] = 0
(ρ̇ a′ + ȧ)[a−1] + (ρ̇[a′] + [ȧ])a−1 = 0
ρ̇[b′] + [ḃ] = 0
Now if we had both ρ̇[ḃ] + [a′] = 0 and ρ̇ḃ + a′ = 0, by (2) we would have
ρ̇ḃ + a′ = 0 on both sides of the hypersurface. We discard for the moment
this singular situation, and from (64)2 we conclude that [a
−1] = 0.
Thus in this case our generalized compatibility conditions imply [a] =
[b] = 0, i.e. they force the match to be C0, piecewise-C1.
In [1] we have already studied some examples of C0, piecewise-C1 matchs
of metrics of the kind (59) at a hypersurface of constant radius r = rb, with
ℓα = δα
r. Namely, we have considered: external Schwarzschild - internal
Schwarzschild; external Schwarzschild - Tolman VI; external Schwarzschild -
Tolman V. Such matchs obviously have ℓα ∈ C
0; moreover condition ρ̇ ḃ+a′ 6=
0 reduce in this case to a′ 6= 0, which is obviously satisfied. In each case we
have verified that condition [a] = [b] = 0 imply ∂a = 0 (where ∂ denotes first
order discontinuity), which then define the match as a boundary layer [1] (it
actually also imply ∂b = 0, as one can verify). Such is a general result, since
for a metric of the kind (59) the completely temporal and radial components
of the Einstein tensor are independent from the second derivatives of the
metric:
Gtt = −a(b
′r + b2 − b)/r2b2
Grr = −(a
′r − ab+ a)/ar2
so that the corresponding vacuum Einstein equations reduce to:
b′r + b2 − b = 0,
a′r − ab+ a = 0.
Now, since in the match of (59) vacuum solutions equations (66) are satisfied
on each side of the interface Σ, their jump is in particular null, and from (3)
we have:
′] + (2b− 1)[b] = 0
′]− a[b]− (b+ 1)[a] = 0
from which it clearly follows that conditions [a] = [b] = 0 imply [a′] = [b′] = 0,
i.e. ∂a = ∂b = 0.
Summarizing, for the match of two piecewise-C0 regularly discontinuous
spherical solutions, in the above hypothesis, generalized compatibility con-
ditions (51) imply [a] = [b] = 0 i.e. they force the match to be C0. On
the other hand conditions [a] = [b] = 0 imply that Σ is a boundary layer.
Therefore for such spherical matchs generalized compatibility conditions (51)
are necessary and sufficient for the match to be a boundary layer.
7 Generalized gravitational shock waves
Let us consider the match of two plane wave metrics of the form
ds2 = −2dξdη + F (ξ)2dx2 +G(ξ)2dy2 (68)
across a hypersurface Σ of equation ξ = 0. Here ξ and η are the two null
coordinates. We suppose continuously matching coordinates and F,G reg-
ularly discontinuous, together with their first and second derivatives. The
gradient vector of Σ is the continuous characteristic (on each side of Σ) vector
ℓα = δα
Generalized compatibility conditions (51) in the case T̆ = 0 (i.e. no stress-
energy concentrated on the hypersurface) reduce to the following single scalar
equation:
[F−1F ′ +G−1G′] = 0 (69)
which characterize the generalized gravitational shock wave. Let us now
study compatibility of (69) with the Einstein Equations. Einstein vacuum
equations also reduce to a single scalar equation:
F−1F ′′ +G−1G′′ = 0 (70)
which we suppose to hold on each side of the hypersurface Σ; thus replacing
F+ and G+ by their expressions in terms of F , [F ], G and [G] according to
(2) gives rise to the following couple scalar conditions:
(2F ′′ + [F ′′])(2G+ [G]) + (2G′′ + [G′′])(2F + [F ]) = 0 (71)
(2F ′′ − [F ′′])(2G− [G]) + (2G′′ − [G′′])(2F − [F ]) = 0 (72)
Equations (71)-(72) are compatible with (69), i.e. the three equations set
can be solved algebraically with respect to F , [G] and to any member of the
pair (F , G), and the solution is not necessarily trivial.
Finally let us notice that, if the additional condition [F ] = [G] = 0 holds,
i.e. if the solution is C0, condition (69) reduces to F−1[F ′]+G−1[G′] = 0 i.e.:
F−1∂F +G−1∂G = 0 (73)
which is the analogous condition for the ordinary shock wave, according to
[1] section 10.5.
8 Slow generalized gravitational waves
Let us start trying to match two vacuum solutions of the kind (68) across
the timelike (on both sides) hypersurface Σ of equation ξ = ζ . Again we
suppose continuously matching coordinates, F,G regularly discontinuous to-
gether with their first and second derivatives, and T̆ = 0. This times gener-
alized compatibility conditions include (69) and the following two additional
scalar conditions:
[FF ′] = 0, [GG′] = 0 (74)
i.e., in terms of F , [F ], G and [G], according to (3):
F [F ′] + [F ]F ′ = 0 (75)
G[G′] + [G]G′ = 0 (76)
It is easy to check that the system (75)-(76) is not compatible with (71)-(72),
in the sense that the whole system does not admit non-trivial solutions for
F , [F ], G and [G].
On the other hand we have proved in section 6 that a wide class of gen-
eralized spherical matchs at a hypersurface of constant radius necessarily
degenerate to a C0 match.
Other examples of degeneracy have not been included in the paper for
the sake of brevity, but at least it seems to be a hard task to construct a non-
trivial generalized match across a timelike (on each side) hypersurface, with
no stress-energy content. Such difficulty is certainly not a proof that this is
an impossible task, but it makes us wonder whether such a solution should
necessarily degenerate to a boundary layer, just like it happens for ordinary
C0 solutions (see e.g. [1]). This would forbid the existence of generalized
solutions which propagate at a speed slower than light. Such would be a
desiderable prohibition under certain respect, since one could expect that
gravitational interactions in vacuo must necessarily propagate at the speed
of light also in a generalized theory.
In general terms, since for generalized solutions the metric is discontinu-
ous, a hypersurface can in principle have different signatures on the different
sides; for this reason we cannot simply distinguish between the timelike and
the lightlike case, as for usual C0 solutions. We should rather distinguish
between three cases: timelike-timelike, timelike-lightlike (or conversely) and
lightlike-lightlike.
In any case it is legitimate to expect that, at least in the timelike-timelike
case, similar to the timelike case of (C0, piecewise C1) solutions, absence of
stress-energy concentrated on Σ should imply the solution to degenerate to
a boundary layer [1].
Unfortunately for generalized solutions we still have no proof that absence
of stress energy concentrated on Σ does necessarily imply the degeneracy of
the solution to a boundary layer.
Therefore, although the examples considered in this paper seem to sug-
gest that such property could hold true also in the generalized case, for the
moment such result is still a conjecture; we thus have to admit that the the-
ory in principle allows propagation of generalized gravitational shock waves
at lower speed than the speed of light. We would call such waves “slow gen-
eralized gravitational shock waves”. It would be reasonable to forbid this
situation as unphysical, but for now this can only be done ad hoc, by means
of a corresponding additional hypothesis.
9 Non-symmetric stress-energy
Notice that ℓµHβρ
µ is not necessarily symmetric; from identity:
ν = (1/2)g−1∂αg (77)
where g denotes the determinant of the contravariant metric, we have:
ℓµH[βρ]
µ = (1/4)(ℓβ[g
−1∂ρg]− ℓρ[g
−1∂βg]) (78)
Thus the generalized scheme allows in principle the presence of non sym-
metric stress-energy on the discontinuity hypersurface. We will display non-
trivial examples of non-symmetry in the following section. Notice that the
right hand side of (78) is identically null in case g ∈ C0 and ℓα ∈ C
0, since
in this case we have [g−1∂βg] = ℓβg
−1∂g.
A non-symmetric Einstein tensor is a feature of Einstein-Cartan theory of
gravitation (see [19], see also [3] section 7.2), where it is due to the presence
of torsion in the non-symmetric connection used to construct generalized
curvature. Thus the generalized theory can be interpreted, at least to some
extent, as introducing a torsion equivalent tool on the shell only, even if there
are no geometrical objects in our theory which can be directly interpretated
as torsion. However, Einstein-Cartan theory also has a spin - angular mo-
mentum field equation in addition to the Einstein equations, which here is
missing.
In the literature, compatibility conditions for C0 solutions of boundary
layers [20], and recently of shock waves and thin shells [21], have been studied
also in the framework of Einstein-Cartan theory; actually this can lead to
non-symmetric stress-energy on the shell. But in that theory this feature
is inherited from the ambient spacetime, which is not here: non-symmetric
stress-energy arises on the shell only, in consequence of the theory. This
interesting feauture probably is worth investigating.
10 Generalized thin shells
Now let us consider a more general form of the spherical metric:
ds2 = −a(r, t)dt2 + b(r, t)dr2 + c(r, t)dΩ2 (79)
Let us consider a match of two spherical solutions of the Einstein equations
across a timelike (on each side) hypersurface of equation r = ρ(t). Again we
suppose ρ(t) ∈ C1 and therefore ℓα = δα
r − ρ̇δα
t ∈ C0. Let the coordinates
be C0 globally, and let the metric have the same form (79) in both domains
Ω+ and Ω−, with the identification t+ = t−, r+ = r−, θ+ = θ−, ϕ+ = ϕ− on
Let moreover a, b, c > 0 and let a, b, c ∈ piecewise-C0 be regularly dis-
continuous on Σ and with regularly discontinuous first derivatives. Again we
denote by a dot the partial derivative with respect to t, and by a prime that
with respect to r.
In this case for the left hand side of the generalized compatibility condi-
tions ℓµHβρ
µ we obtain:
ℓµHβρ
µ = −
[a′b−1/2] + ρ̇[ḃb−1/2 + ċc−1]
+([a′a−1/2 + c′c−1] + ρ̇[ḃa−1/2])δβ
+([ȧa−1/2 + ċc−1] + ρ̇[a′a−1/2])δβ
−([ḃb−1/2] + ρ̇[b′b−1/2 + c′c−1])δβ
+([c′b−1/2] + ρ̇[ċa−1/2])(δβ
θ + sin2 θδβ
[b−1(a′a−1/2 + c′c−1)] + ρ̇[a−1(ḃb−1/2 + ċc−1)]
where, obviously:
gβρ = −aδβ
t + bδβ
r + c(δβ
θ + sin2 θδβ
ϕ) (81)
We now mean to interpret (80) as the matter-energy of a thin shell. Let
us first get back to the particular case ρ̇ = 0 (static shell) and ȧ = ḃ =
ċ = 0, to make the interpretation simpler by eliminating the non-symmetric
component; rearranging some terms we in fact obtain:
ℓµHβρ
µ = (−[a′b−1/2] + ac−1[c′b−1/2])δβ
+([a′a−1/2 + c′c−1]− bc−1[c′b−1/2])δβ
c−1[c′b−1/2]− [b−1(a′a−1/2 + c′c−1)]
This can be interpretated as a perfect isotropic magneto-fluid thin shell with
infinite conductivity, i.e. we can solve the compatibility conditions by con-
sidering the following stress-energy as the right hand side:
T̆αβ = (ρ0 + p+ µh
2)UαUβ + (p+ (1/2)µh
2)gαβ − µhαhβ (83)
where ρ0 is the proper density, h the magnetic field and µ the magnetic
permeability [22, 23, 24, 25, 6]; here we define h2 = hαhβg
αβ . In fact it
suffices to define the following 4-velocity vector:
a− (1/4)[a2]a−1δα
t = (a−1)1/2δα
t (84)
which by construction is unitary on Σ, in the following sense: UαUβg
αβ = −1,
and the following magneto-hydrodynamical variables:
ρ0 = χ
−1a−1[a′b−1]/2 + χ−1c−1(b b−1/2− aa−1 + 1)[c′b−1]/2+
−χ−1[b−1](a′a−1/2 + c′c−1)− (3/2)χ−1b−1[a′a−1/2 + c′c−1]
p = χ−1c−1(bb−1/2− 1)[c′b−1/2] + (1/2)χ−1b−1[a′a−1/2 + c′c−1]+
+χ−1[b−1](a′a−1/2 + c′c−1)
hα = ±
b−1χ−1([a′a−1/2 + c′c−1]− bc−1[c′b−1/2])δα
to match (82) and (83) via ℓµHβρ
µ = −χT̆βρ. If [a] = [b] = [c] = 0 then the
generalized shell (85) degenerates to the C0 magnetohydrodynamical shell
considered in [1] section 10.1, in the particular case ρ̇ = 0.
The slightly more general case of ȧ = ḃ = ċ = 0, but ρ̇ 6= 0, displays
non-symmetric terms in (80); however it is not difficult to see that the per-
fect magnetofluid interpretation still holds, provided such additional non-
symmetric terms are interpreted, or neglected. In fact in this case we have:
ℓµHβρ
µ = (−[a′b−1/2] + a c−1[c′b−1/2])δβ
+([a′a−1/2 + c′c−1]− bc−1[c′b−1/2])δβ
+(1/2)ρ̇[a′a−1/2− b′b−1/2− c′c−1](δβ
t + δβ
+(1/2)ρ̇[a′a−1/2 + b′b−1/2 + c′c−1](δβ
t − δβ
c−1[c′b−1/2]− [b−1(a′a−1/2 + c′c−1)]
Now let us consider, for the sake of brevity, the following quantities:
ρ̇2[ a
]2b−1 − (a
]− [ a
])2a−1
]− [ a
]− [ a
and let us suppose that inequality α < 0 holds, which is necessary for the
physical interpretation. In fact in this case the following vector:
]− [ a
t + 1
ρ̇[ a
]− [ a
is a unit timelike vector on Σ, in the sense that UαUβg
αβ = −1. Rearranging
terms, (86) now reads:
ℓµHβρ
µ = αUβUρ + βδβ
r + 1
ρ̇[ a
t − δβ
c−1[ c
]− [b−1( a
which can be matched via ℓµHβρ
µ = −χT̆βρ with a stress-energy tensor of
the following kind:
T̆αβ = (ρ0 + p+ µh
2)UαUβ + (p+ (1/2)µh
2)gαβ − µhαhβ + Aαβ (91)
where A denotes the anti-symmetric term. We have:
ρ0 = −χ
−1α + χ−1 1
]− χ−1[b−1( a
)]− 1
p = −χ−1 1
] + χ−1[b−1( a
)]− 1
µh2 = χ−1βb−1
while the anti-symmetric term A reads:
Aαβ = −χ
t − δβ
r) (93)
The interpretation of such term is still missing; alternatively it could be
neglegted by adding the further hypothesys:
= 0 (94)
which is equivalent to [g−1g′] = 0, as we could expect from (78).
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	Introduction
	Discontinuous metrics
	Mean-value geometry on a hypersurface
	Complex mean-value formalism
	Generalized compatibility conditions
	A class of spherical boundary layers
	Generalized gravitational shock waves
	Slow generalized gravitational waves
	Non-symmetric stress-energy
	Generalized thin shells
