1. Introduction

One of the biggest surprises that General Relativity (GR) has given us is that under certain circumstances the theory predicts its own limitations. There are two physical situations where we expect the theory to break down. The first one is the gravitational collapse of certain massive stars when their nuclear fuel is spent. The second one is the far past of the universe when the density and temperature were extreme. In both cases, we expect that the geometry of spacetime will show some pathological behaviour.

The nature of a gravitational singularity is a delicate issue. It might be tempting to define a gravitational singularity following other physical theories (such as electromagnetism) as the location where the relevant physical quantities are undefined. However, in the gravitational case this prescription does not work due to the identification of the spacetime background with the gravitational field. As a result, the concepts of ‘spatial location’ and ‘temporal duration’ have meaning only in the domain where the gravitational field is defined. This represents a problem because the size, place and shape of singularities can not be straightforwardly characterised by any physical measurement.

The first mathematical description of a gravitational singularity comes from Penrose and Hawking seminal theorems. They characterised singularities as obstructions to geodesic completeness and managed to show that this happens under certain conditions^{1}. Broadly speaking, the theorems establish that a spacetime (

a condition on the curvature,

an appropriate initial or boundary condition,

and a global causal condition,

must be geodesically incomplete^{2}.

One would like to attach a boundary to the incomplete spacetime to understand the singularity better. The procedure to attach a boundary to a Lorentzian manifold can be done in several nonequivalent ways. In this work we will focus on the *b*-boundary method^{3}. This method allows a classification of singularities in terms of parallel propagated frames, it distinguishes between points at infinity and points at a finite distance, and it generalizes the idea of affine length to all curves regardless of them being geodesic or not. Other common techniques to attach boundaries to Lorentzian manifolds are conformal boundaries^{1}^{,}^{4} and the causal boundaries^{1}^{,}^{5} which we describe below. In addition, we would like to mention other constructions such as the *a*-boundary^{6}^{,}^{7} and boundaries constructed from light rays^{8}.

The conformal boundary allows us to study the structure of the metric at “infinity”. The idea of conformal compactification is to bring points at “infinity” on a non-compact pseudo-Riemannian manifold

there exists a smooth scalar field

every null geodesic in

This technique has the evident drawback that it can only be applied to this kind of spacetimes^{4}. Moreover, notice that in Minkowski spacetime the conformal boundary is given by ^{1} formal definitions of ^{4} and to the AdS-CFT correspondence^{9}^{,}^{10}.

On the other hand, the causal boundary of a spacetime consists on attaching a boundary that depends only on the causal structure. However, this implies on this particular construction that one is not able to distinguish between boundary points and points at infinity. Moreover one has to assume that *U* is an IP if it satisfies *V* and *W*, satisfying ^{11}. However, this topology presents some problems that have led to several redefinitions. A full revision of the causal boundary and its relationship with the conformal boundary can be found in^{5}. Also its relation with boundaries in Riemannian and Fislerian manifolds can be found in^{11}.

There is also the *b-boundary* to any incomplete spacetime

The structure of this paper is as follows. In Sec. 2 we give a general overview of the mathematical preliminaries needed. In Sec. 3 we describe how the Schmidt metric and the b-boundary are constructed following the procedure by Schmidt ^{3}. In Sec. 4 we discuss the geometry of the orthonormal-bundles for

2. Preliminaries

As a first step, let us present some of the required concepts of differential geometry. We present the basic concept of fibre bundles, G-principal bundles, solder forms and connections. The manifolds we consider in this paper are paracompact,

*2.1 Fibre Bundles and G-Principal Bundles*

A Fibre bundle with fibre

A G-principal bundle P over a manifold

and satisfies that ^{12}.

Let

Every tangent space

2.2 Solder form and Connections

The solder form of a frame bundle

where Q is an element of

A connection

A connection form

with the following properties:

if

for all g in G and all

for all

Let us remind the reader that connections and connection forms uniquely determine one another.

In coordinates, the connection form

where

The solder form

and ^{12}.

3. The Schmidt metric

If one thinks of a singularity in classical Newtonian gravity, the statement that the gravitational field is singular at a certain location is unambiguous. As an example, take the gravitational potential of a spherical mass M in Cartesian coordinates

where G is the gravitational constant, and the potential exhibits a singularity at the point

However, in the case of GR the prescription given above can not work. This is due to the identification of the background spacetime with the gravitational field. Hence, only in the regions where the gravitational field is defined it is meaningful to talk about locations. Consider the spacetime with the line element

defined on the manifold

on

Another idea is trying to define a singularity in terms of invariant quantities, such as invariant scalars. The reason for this is that if these quantities diverge then it matches our physical idea that objects must suffer stronger and stronger deformations as we encounter the singularity. These scalars are usually constructed from contractions of the Riemann tensor and its derivatives. Unfortunately, these scalars are not well-suited to define the complete geometry. Consider the metric

given in the coordinates ^{13}.

A more troublesome feature of using scalars for defining singularities is that they are ‘too local’ in the sense that they are evaluated at given points. Therefore, if the point is removed, the scalar cannot be computed directly and we need an approximation procedure.

A precise mathematical way to approximate the “missing points” is to use convergent sequences of points on the manifold. In this case the formal statement is: “The sequence

In Riemannian geometry, the notion of distance allows us to define Cauchy sequences ^{14}. This allows us to use Cauchy sequences or sequences of points along geodesics as our sequences of points.

The Riemannian case is an useful example, but as soon as we move to Lorentzian geometry, which we take as the correct geometrical setting for GR, the previous discussion cannot be used as stated. The reason is that Lorentzian metrics do not have a distance function defined and, therefore, Cauchy sequences cannot be defined. Thus, one is restricted to the notion of geodesically complete manifolds in the Lorentzian case.

Moreover, the existence of three kinds of vectors available in any Lorentzian metric defines three nonequivalent notions of geodesic completeness -depending on the character of the tangent vector of the curve- spacelike completeness, null completeness and timelike completeness, which are, unfortunately, not equivalent. It is possible to construct spacetimes with the following characteristics^{15}^{,}^{16}^{,}^{19}:

timelike complete, spacelike and null incomplete,

spacelike complete, timelike and null incomplete,

null complete, timelike and spacelike incomplete,

timelike and null complete, spacelike incomplete,

spacelike and null complete, timelike incomplete, or

timelike and spacelike complete, null incomplete.

Furthermore, there are examples of a geodesically null, timelike and spacelike complete spacetimes with an inextendible timelike curve of finite length^{16}^{,}^{19}. A particle following this trajectory will experience bounded acceleration and in a finite amount of proper time its spacetime location would stop being represented as a point in the manifold.

In order to overcome this, Schmidt provided an elegant way to generalise the idea of affine length to all curves, regardless of such curves being geodesic or not. This construction in the case of incomplete curves allows to attach to the spacetime

where ^{3}. However, as mentioned above, we will use the Levi-Civita connection because we will always assume a metric on the manifold.

Let

which is called the generalised affine-length of

Notice that if there is a curve ^{16} shows a b-incomplete spacetime that is geodesically complete. Therefore, b-incompleteness is a generalisation of geodesic incompleteness.

Now given an incomplete spacetime

We define the quotient space

We repeat the same construction for subgroups of

4. The Schmidt metric of 1+1 spacetimes

In this section, we locally construct the Schmidt metric for general 1+1 spacetimes. Moreover, we find a relationship between the scalar curvature of the Schmidt metric on (

Notation: We use overlines to denote the Riemannian geometric quantities that belongs to

4.1 The Schmidt metric for 1+1 conformal spacetimes

Let ^{20}:

An orthonormal basis is then given by the vector fields

The orthonormal basis prescribed above is not unique. Any other orthonormal basis is of the form

for some

Let us notice that the coefficients of such a basis with respect to

These matrices are important in the sense that they are useful to define local coordinates on

As stated in Sec. 3, the Schmidt metric

where

Now let us consider a curve

and

where we have used [1] and [2]. Then, the line element for the Schmidt metric using a general inner product can be written as:

where

It can be shown that using two different inner products produce two unifomly equivalent metrics^{18}.

In application it is commonly used the Euclidean innner product which give the line element for the Schmidt:

We avoid quoting long tensorial expressions for the curvature tensors and give only the result for the Ricci scalar of (15) in terms of

Taking into account that

This means that Eq. (16) becomes

Notice that in (18) is the relationship between both scalar curvatures. As direct consequence, we can establish the negativity of the Ricci scalar for any Schmidt metric in the Lorentzian signature. In Sec. 5 we give counterexamples that such a condition does not hold in the Riemannian case. Also, Eq. (18) has been obtained using the Levi-Civita connection. Therefore, using another connection, even in the Lorentzian case, may not hold.

Now we calculate such scalar curvatures for some physical spacetimes.

4.2 The Schmidt metric of Minkowski spacetime

We can write the Schmidt metric in the form

Now let us consider the change of coordinates:

or in an equivalent manner

We explicitly calculate

and all other components are zero. The Ricci scalar is then

Hence, the geometry in the bundle is not flat even if Minkowski spacetime is flat.

4.3 The Schmidt metric of Friedmann-Robertson-Walker (FRW) spacetime

For simplicity, let us consider the case of the 1+1 FRW cosmological model which can be obtained from the 4-dimensional one by collapsing two spatial coordinates. This is equivalent to considering the injection map

where

for any value of

From Eq. the Schmidt metric in

Using computer algebra we calculated the Ricci tensor for the line element . In components it is given by

And the Ricci scalar is

which can equivalently be obtained from Eq.(18).

4.4 The Schmidt metric of De Sitter and Anti-De Sitter spacetimes

Let us now consider the De Sitter and Anti-De Sitter models and study the behaviour of the corresponding curvature scalars. First, consider the De Sitter case. The two dimensional De Sitter spacetime for closed spatial sections is defined with the line element

To obtain the conformal form we make the change

In these coordinates ^{1}. The Ricci scalar for is then

Using Eq.(15) we get

The Ricci tensor is computed by taking

Thus the Ricci scalar is

Notice that in the limit as

Now let us look at the Anti-De Sitter spacetime. The two dimensional Anti-De Sitter metric has the line element

with

We identify

Where the non-vanishing components of the Ricci tensor are

and its trace is given by

Notice that for spacetimes that behave asymptotically as Anti-De Sitter spacetime, the curvature would behave similarly as in the Anti-De Sitter case as one approaches the asymptotic region. Moreover, in many applications such as in the AdS/CFT correspondance one uses a conformal compactification. In those cases it is neccesary to compute the curvature again because the curvature is not a conformal invariant.

5. Discussion

In our exposition, we obtained Eq.(15) which is the line element of the Schmidt metric for all 1+1 Lorentzian manifolds ^{24}. This is, for example, the case when ^{23}. The degeneracy of the fibre also affects the topology of ^{23}. The b-boundary has also given some results that link the geometry^{27} of principal bundles with that of the base manifold and with non-commutative geometry^{26}. Moreover, it has been shown that in four dimensions the Friedmann-Robertson-Walker and Schwarzschild b-completion ^{21}^{,}^{23}^{,}^{28}.

The notion of b-incomplete spaces allows us to describe incomplete curves in manifolds with connections. Our initial motivation to study this, was to develop the language to describe pathologies in the geometry as we approach points that in some sense are “boundary points" of the manifold. One can describe how the main manifestation of gravity in GR, the curvature of the manifold, can behave along 𝑏-incomplete curves. This is the scheme proposed by Ellis and Schmidt to classify singularities^{29}^{,}^{30}. In particular, they defined that if

Theorem 1. Let

The proof follows directly from Eq. [18].

Notice that the hypothesis of this theorem together with the hypothesis of any of the Hawking and Penrose theorems gives a singularity theorem in which it is guaranteed that curvature blow up exist. This is in contrast with the usual singularity theorem in which only geodesic incompleteness can be shown.

The theorem above and the singularity theorems implicitly assume a characterisation of singularities in terms of incomplete curves. This notion of singularity captures the idea that there are ‘obstructions’ within the history of point-like observers. In the future one would like extending those theorems to relate these obstructions to curvature blow-up and ill-possessedness of initial value problems of field equations. This approach constitutes most of the research program on the Strong Cosmic Censorship conjecture^{31}^{,}^{32}, the idea behind generalised hyperbolicity^{33}^{,}^{34}^{,}^{35}^{,}^{36} and field regularity^{37}^{,}^{38}^{,}^{39}^{,}^{40}^{,}^{41}^{,}^{42}.

Appendix

A. The Riemannian case

As it was mentioned in Sec. 5, in the Riemannian case there are three conformally distinctly connected Riemann surfaces (the disc, the plane and the sphere). Moreover, in this case the fibres in

In the Riemannian case it is a well know fact that if

An orthonormal basis is given by the vector fields

Any other orthonormal basis is constructed as a linear combination of as

for some

Notice the main difference with the Lorentzian case in the definition of the matrix

The Schmidt metric

for

and

giving the line element for the Schmidt metric:

The plane

The euclidean metric on the plane is given by the line element

which is characterised by

Then using Eq.(A.10) we have that the line element for the corresponding Schmidt metric is

which is just the flat metric in

The sphere

The round metric on the sphere is given by the line element

Eq. (A.13) can be expressed in terms of isothermal coordinates

This metric is characterised by

In a similar manner as we did for the plane metric we use Eq.(A.10) to get the line element for the Schmidt metric:

with curvature scalar