Conclusions

The goal of this thesis was to investigate the inheritance of symmetry property for the Einstein’s equations when matter/energy models are included. Indeed, we showed the validity of this property in the context of asymptotically flat electrovac- uum spacetimes: An asymptotic time-like symmetry to all orders at infinity is indeed a (local) symmetry of both gravity and electromagnetism, Proposition 3.3.1. We also provided weaker conditions for the first condition to hold, namely, an asymptotic time-like symmetry to first order in a non-radiating spacetime must be an asymp- totic time-like symmetry to all orders, Proposition 2.1.4. We also sketched the proof for the same results when a massless Klein-Gordon field is also present, but we stressed that the conclusion no longer holds for the positive-mass case.

The assumed regularity assumptions are still artificial and probably restrict too much the class of spacetimes satisfying them. Nevertheless, we expect to extend the techniques employed here to asymptotic expansions including time-independent logarithmic terms. However, the precise class of regularity conditions compatible with physical systems are still not well understood and a generalisation in this di- rection seems to need a different approach. Also, it is important to remark that the regularity assumptions used in this thesis can be deduced from regular initial data if the resulting development is time-periodic, provided the preservation of regularity property at spatial infinity holds, see Section 4.1.3.

also propose a unique continuation from infinity result for the Einstein’s equation themselves in Conjecture 4.3.2. The regularity assumptions for such a result will also have to be assumed at this point.

Appendix A

Exact solutions

A.1

Minkowski spacetime

ConsiderR4 with the standard Lorentzian metric,

gM =−dt2+dx2+dy2+dz2.

We are interested in the causal structure of this spacetime and its behaviour at infinity. It is possible to gain a deep insight of the global structure by conformally embedding Minkowski spacetime into a bounded region of some other Lorentzian manifold. We start by writing the metric in spherical coordinates,

gM =−dt2+dr2+r2˘γ,

where ˘γ = dθ2 + sin2(θ)dφ2 is the standard round metric on S2. Then define the

null coordinates

u=t−r, v=t+r, q= arctan(u), p= arctan(v),

the latter being just a re-parametrization to bring infinity to a bounded region. Note that working with null coordinates guarantees the conformality of the coordinate transformation.

Finally we go back to time-radial coordinates,

t0= 1

2(p+q), r

0

= 1

It can be checked that the metric in these new coordinates is

gM = sec2(t0+r0) sec2(t0−r0)(−dt02+dr02+ sin2(r0)˘γ

| {z }

¯

g

).

So, except for the sec2 factors, we recognise it as the canonical metric for the space

R×S3.

Summarising, we have constructed an embedding Φ : (R4, η) → (R×S3,¯g)

and a function Ω :R4 →Rsuch that:

¯

g= Ω2Φ∗(gM).

This means that all the causal structure is preserved by the embedding, that is, a vector is time-like (null or space-like) with respect tog if and only if it is time-like (null or space-like) with respect to ¯g.

Now, we observe that the image of R4 under the embedding is the region

where −π₂ < t0+r0 < π₂ and −π₂ < t0 −r0 < π₂. The boundary of this region is precisely where the conformal factor, Ω, vanishes and it consists of three points and two null surfaces, see Figure A.1. Note that i−, i+ can be regarded as past and future infinities, since any future directed time-like geodesic must start and end at these points; in the same manner i0 is called spatial infinity. Similarly, the null hypersurfacesJ− and J+ are called past and future null infinities, respectively.

Remark. It is worth noticing that the metric ¯g extends smoothly to infinity. This will not be the case for the following constructions involving mass (see remark fol- lowing the description of the Schwarzschild spacetime). To the author’s knowledge it remains an open question whether this condition characterises completely flat spacetime.

It is illustrative to compute the connection coefficients of the Minkowski spacetime in outgoing null coordinates. These values correspond to the leading or- ders of the connection coefficients of any asymptotically flat spacetime in this gauge.

Consider the Minkowski metric in outgoing null coordinates,

t0 t0 = π₂ t0 = 0 t0 =−π₂ J+ J− i+ i− i0 r0 = 0 r0 =π i+ i− i0 r 0 = 0 J+ J−

Figure A.1: Left: Embedding of Minkowski space into R×S3. Coordinates (θ, φ)

have been suppressed and each point represents one half of a 2-sphere of area 4πsin2r0. Note that points on the dashed lines are truly points, due to the sin- gularity of polar coordinates. Right: The diagram in the (t0, r0)-plane is known as

a Penrose-Carter diagram. Each point (with the exception of the line r0 = 0 and

i0) represents a sphere. Dashed lines correspond to {r =constant} and dotted to {t=constant}. Radial null curves are at 45◦.

and choose the null frame, e0=L=∂r, e1 =L= 2∂u−∂r, e2= 1 r∂θ, e3 = 1 rsinθ∂φ.

That is, the orthonormalisation matrix is given by

hµν =       1 −1 0 0 0 2 0 0 0 0 1_r 0 0 0 0 _rsin1 _θ       .

The null second fundamental forms are then,

χij =−χ_ij = 1 r 1 0 0 1 ! .

The other non-vanishing connection coefficients are ω223 and ω332, that is, those

corresponding to the induced connection on the sphere of radius r.