Shielding linearised gravity

0.18 MB·

by  R. Beig

Shielding linearised-gravity∗
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1 Robert Beig†, Piotr T. Chru´sciel‡
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University of Vienna
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r
March 14, 2017
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M

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Abstract

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c We present an elementary argument that one can shield linearised
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gravitational fields using linearised gravitational fields. This is done
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r by using third-order potentials for the metric, which avoids the need
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to solve singular equations in shielding or gluing constructions for the
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linearised metric.

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v
6 Contents
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4
0 1 Introduction 2
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1. 2 Shielding linearised gravity 3
0 2.1 The Cauchy problem for linearised gravity . . . . . . . . . . . 4
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2.2 Third order potentials . . . . . . . . . . . . . . . . . . . . . . 6
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: 2.3 Shielding gravitational Cauchy data . . . . . . . . . . . . . . 7
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X 3 Shielding Maxwell fields 9
r
a
4 The Weyl tensor formulation 11
A Integrating 2-forms on R3 12
B Construction of the potential u 13
C A potential for the linearised Riemann tensor 15
∗Preprint UWThPh-2016-31
†[email protected]
‡[email protected],URLhomepage.univie.ac.at/piotr.chrusciel
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1 Introduction
A fundamental property of Newtonian gravity is that the gravitational field
cannot be localised in a bounded region. This is a simple consequence of
the equation
∆φ = 4πGρ,
where φ is the gravitational potential, G is Newton’s constant and ρ is the
matter density: The requirement that ρ≥ 0, and the asymptotic behaviour
−M/r of φ, where M is the total mass, implies that φ vanishes at large
distances along a curve extending to infinity if and only if there is no matter
whatsoever and φ ≡ 0. It is therefore extremely surprising that in general
relativity, gravitational fields can be shielded away by gravitational fields, as
proved recently in a remarkable paper by Carlotto and Schoen [5].
Since Newtonian gravity is part of the weak-field limit of general relativ-
ity(indeed,thisisweak-fieldGRwithsmallvelocities), onewondersifasim-
ilar screeningcan occur forlinearised relativity. As itturnsout, theanalysis
of Carlotto and Schoen can bereadily generalised to linearised gravitational
fields on cone-like sets as considered in [5] (compare [9]). This, however,
requires sophistical mathematical machinery which imposes restrictions to
the sets considered and, as an intermediate step, uses solutions blowing-up
at the relevant boundaries, which leads to difficulties when trying to imple-
ment the method numerically. The object of this note is to point out an
alternative elementary method to perform gluings, or achieve screening of
linearised gravity by linearised gravitational fields near a Minkowski back-
ground. In particular, we give here a very simple proof that at any given
time t, and givenanyopen setΩ ⊂ R3, everylinearised vacuumgravitational
field h on {t}×R3 can be deformed to a new linearised vacuum field h˜
µν µν
so that h˜ coincides with h on Ω and vanishes outside a slightly larger
µν µν
set. In other words, the gravitational field has been screened away outside
of Ω, and this by using gravitational fields only: no matter fields, whether
with positive or negative density, are needed.
We emphasise that the construction of Carlotto-Schoen switches-off the
gravitational field in sets which have a cone-like structure, whether in the
linearised case or in the full treatment. In our approach no restrictions on
the geometry of Ω occur, so that the screening can be done near any set.
Our construction is likely to be useful for the numerical construction of
initial data sets with interesting properties, by providing an efficient way
of making gluings in the far-away zone, where nonlinear corrections become
inessential. Here, as already pointed out, both the Corvino-Schoen and the
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Carlotto-Schoen gluings require solving elliptic equations in spaces of func-
tionswhicharesingularattheboundaryofthegluingregion(see[10,8]fora
review), while our gluings are performed by explicit elementary integrations
((2.31)below),multiplicationbyacut-offfunction,andapplyingderivatives,
once the metric has been put into transverse and traceless (TT)-gauge.
The above leads one naturally to ask similar questions for electric and
magnetic fields. Here we provide a simple proof that Maxwell fields can be
shielded by Maxwell fields. Last but not least, we show how to perform the
screeninginpractice,inthatweprovethatallsolutionsofsourcelessMaxwell
equations in a bounded space-time region can be realised by manipulating
charges and currents in an enclosing bounded region.
2 Shielding linearised gravity
Consider R3+1 with a metric which, in the natural coordinates on R3+1,
takes the form
g = η +h , (2.1)
µν µν µν
where η denotes the Minkowski metric. Suppose that there exists a small
constant ǫ such that we have
|h |, |∂ h |, |∂ ∂ h |= O(ǫ). (2.2)
µν σ µν σ ρ µν
If we use the metric η to raise and lower indices one has
1
R = [∂ {∂ hα +∂ hα −∂αh }−∂ ∂ hα ]+O(ǫ2). (2.3)
βδ α β δ δ β βδ δ β α
2
Coordinate transformations xµ 7→ xµ+ζµ, with
|ζ |, |∂ ζ |, |∂ ∂ ζ |, |∂ ∂ ∂ ζ | = O(ǫ), (2.4)
µ σ µ σ ρ µ σ ρ ν µ
preserve (2.2), and lead to the gauge-freedom
h 7→ h +∂ ζ +∂ ζ . (2.5)
µν µν µ ν ν µ
Imposing the wave-coordinates condition up to O(ǫ2) terms,
✷ xα = O(ǫ2), (2.6)
g
leads to
1
∂ hβ = ∂ hβ +O(ǫ2), (2.7)
β α α β
2
as well as
1
R = − ✷ h +O(ǫ2). (2.8)
βδ η βδ
2
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2.1 The Cauchy problem for linearised gravity
InwhatfollowsweignoreallO(ǫ2)-termsintheequationsaboveandconsider
the theory of a tensor field h with the gauge-freedom (2.5) and satisfying
µν
the equations
0= ∂ {∂ hα +∂ hα −∂αh }−∂ ∂ hα . (2.9)
α β δ δ β βδ δ β α
Solving the following wave equation
1
✷ζ = −∂ hβ + ∂ hβ ,
α β α α β
2
where✷ ≡ ✷ isthewave-operator oftheMinkowskimetric,andperforming
η
(2.5) leads to a new tensor h , still denoted by the same symbol, such that
µν
1
∂ hβ = ∂ hβ , (2.10)
β α α β
2
together with the usual wave equation for h:
✷h = 0. (2.11)
βδ
Solutions of this last equation are in one-to-one correspondence with their
Cauchy data at t = 0. However, those data are not arbitrary, which can be
seen as follows: Equations (2.10)-(2.11) imply
1
✷(∂ hβ − ∂ hβ )= 0. (2.12)
β α α β
2
It follows that (2.10) will hold if and only if
1 1
∂ hβ − ∂ hβ = 0= ∂ ∂ hβ − ∂ hβ . (2.13)
β α α β 0 β α α β
(cid:18) 2 (cid:19)(cid:12)t=0 (cid:18) 2 (cid:19)(cid:12)t=0
(cid:12) (cid:12)
(cid:12) (cid:12)
Equivalently, taking (2.11) into account,
∂ (h +hi )| = 2∂ hi | , (2.14)
0 00 i t=0 i 0 t=0
∂ h | = ∂ hj + 1∂ (h −hj ) , (2.15)
0 0i t=0 j i 2 i 00 j t=0
∆h(cid:0)i | = ∂ ∂ hij| , (cid:1)(cid:12) (2.16)
i t=0 i j t=0 (cid:12)
∂ ∂ hj −∂ hk δj = ∆h −∂ ∂ hj . (2.17)
j 0 i 0 k i t=0 0i i j 0 t=0
(cid:0) (cid:1)(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
The last two equations are of course the linearisations of the usual scalar
and vector constraint equations.
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There remains the freedom of choosing ζ | and ∂ ζ | . We choose
α t=0 t α t=0
∂ hk −2∂ hk −2∆ζ = 0,
0 k k 0 0 t=0
(cid:0) (cid:1)(cid:12)
h00+2∂0ζ0 t=0 =(cid:12)0,
(cid:0) (cid:1)(cid:12)
h0i +∂iζ0+∂0ζ(cid:12)i t=0 = 0,
Di hij − 13hk(cid:0)kδji +Diζj +Djζ(cid:1)i(cid:12)(cid:12)− 32Dkζkδji t=0 = 0, (2.18)
(cid:0) (cid:1)(cid:12)
where D ≡ Di ≡ ∂ in Cartesian coordinates. Indeed(cid:12), given any h and
i i µν
∂ h | , the first equation can be solved for ζ | under suitable natural
0 µν t=0 0 t=0
conditionsonthedata; theseconddefines∂ ζ | ;thethirddefines∂ ζ | ;
0 0 t=0 0 i t=0
finally, the last equation is an elliptic equation for the vector field ζ |
i t=0
which can be solved [7] if one assumes that the field
1
∂ hi − hk δi (2.19)
i j 3 k j t=0
(cid:0) (cid:1)(cid:12)
(cid:12)
belongs to a suitable weighted Sobolev or H¨older space, the precise require-
ments being irrelevant for our purposes. We simply note that if some com-
ponents of h behave as 1/r, then ζ will behave like lnr in general, which is
ij
likely to introduce lnr/r terms in the gauge-transformed metric. After per-
forming this gauge-transformation, we end up with a tensor field h which
µν
satisfies
1
∂ hk = h = h = ∂ hi − hk δi = 0. (2.20)
0 k t=0 00 t=0 0i t=0 i j 3 k j t=0
(cid:12) (cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Inserting this into (2.14)-(2.17) we find
∂ h | = 0, (2.21)
0 00 t=0
∂ h | = −1∂ hj | , (2.22)
0 0i t=0 6 i j t=0
∆hi | = 0, (2.23)
i t=0
∂ ∂ hj −∂ hk δj = 0. (2.24)
j 0 i 0 k i t=0
(cid:0) (cid:1)(cid:12)
The further requirement that hi goes to zer(cid:12)o as r tends to infinity together
i
with the maximum principle gives
hi | = 0. (2.25)
i t=0
We conclude (compare [1]) that at any given time t = t every linearised
0
gravitational initial data set (h ,∂ h )| can be gauge-transformed to
µν t µν t=t0
the TT-gauge: writing k = ∂ h , we have
ij 0 ij
hk | = ∂ hi | = kk = ∂ ki = 0. (2.26)
k t=t0 i j t=t0 k i j
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From what has been said and from uniqueness of solutions of the wave
equation we also see that in this gauge we will have for all t
h = h = hk = ∂ hi = 0, (2.27)
00 0i k i j
which further implies that (2.26) is preserved by evolution.
It should be pointed out that when the construction is carried out on
the complement of a ball, e.g. because sources are present, or because we
perform the construction at large distances only where the non-linearities
become negligible, then (2.25) will not hold in general, and the trace of h
ij
will be non-trivial, with the usual expansion in terms of inverse powers of r,
starting with 1/r-terms associated with the total mass of the configuration.
In such cases our construction below still applies to the transverse-traceless
part of the metric.
2.2 Third order potentials
We will need the following result from [3], which can be summarised as
follows: Let h be a symmetric, transverse and traceless tensor on R3,
ij
∂ hi = 0 = hi . (2.28)
i j i
Thenthereexists asymmetrictraceless “thirdorderpotential” u suchthat
ij
h = P(u) , (2.29)
mℓ mℓ
where(hereg denotestheEuclideanmetricandDi theassociatedcovariant
ij
derivative)
1 1
P(u) := ǫ ij∂ ∆u −2∂ Dnu + g DnDku , (2.30)
mℓ m i jℓ (ℓ j)n jℓ nk
2 2
(cid:0) (cid:1)
and where u = u dxidxj can be constructed by the following procedure:
ij
Letting
1
σ (~x) := ǫ ℓh (λ~x)λ(1−λ)2dλ, (2.31)
ijk ij ℓk
Z
0
we set
u = 2xmxnx σ +r2xmσ . (2.32)
jℓ (j ℓ)mn m(jℓ)
(This is clearly symmetric, and tracelessness is not very difficult to check.
Other third-order-potentials u are possible, differing by an element of the
kernel of P.) One way to see how (2.30) arises is to note that P(u) is,
apart from a numerical factor, the linearisation at the flat metric of the
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Cotton-York tensor in the direction of the trace-free tensor u. For (2.32),
the formulae follow by successively integrating thrice the two-forms given
in [3], at each step using the Poincar´e formula (3.7) below. We sketch the
construction in Appendix A.
The converse is also true: given any symmetric trace-free tensor u , the
ij
tensor field P(u) defined by (2.30) is symmetric, transverse and traceless (of
which only the last property and the vanishing of the divergence on the first
index are obvious).
As an example, consider h describing a plane gravitational wave in
ij
TT-gauge propagating in direction ~k,
h (~k) = ℜ H ei~k·~x , ∂ H = 0 = Hi = H kj, (2.33)
ij ij ℓ ij i ij
(cid:0) (cid:1)
withpossiblycomplex coefficients H , whereℜdenotes therealpart. Then
ij
1
σ = ℜ ǫ Hℓ eiλ~k·~x(λ−2λ2+λ3)dλ
ijk ijℓ k
Z
(cid:0) 0 (cid:1)
= ℜ Wǫ Hℓ , where (2.34)
ijℓ k
2i(cid:0)ei~k·~x(~k·~x+(cid:1) 3i)−~k·~x(~k·~x−4i)+6
W(~x) = (2.35)
(~k·~x)4
(which tends to 1/12 when ~k·~x tends to zero),
u = ℜ W 2xmxix ǫ Hk −r2xiǫ Hk . (2.36)
jℓ (j ℓ)mk i ik(j ℓ)
(cid:16) (cid:17)
(cid:0) (cid:1)
As another example, consider the family of fields
2
u = ln(1+r2) D λ +D λ − Dkλ g . (2.37)
ij i j j i k ij
3
(cid:0) (cid:1)
Tensors of the form (2.37) with the ln(1+r2) term removed form the kernel
of P for any λ (cf. Appendix A), which easily implies that if λ ∼ O(rσ) for
i
large r then h ∼ O(rσ−4), for all σ ∈ R.
ij
We also note that if h is compactly supported to start with, then u
ij ij
can also be chosen to be compactly supported; compare Appendix A.
2.3 Shielding gravitational Cauchy data
We are ready to prove now a somewhat moregeneral version of our previous
claim, that at any given time t, and given any region Ω ⊂ R3, every vacuum
initial data set for the gravitational field (h ,k ) can be deformed to a new
ij ij
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vacuum initial data set (h˜ ,k˜ ) which coincides with (h ,k ) on Ω and
ij ij ij ij
vanishes outside of a slightly larger set.
Indeed, consider any linearised gravitational field in the gauge (2.27).
Denote by (h ,k ) theassociated Cauchy data at t, andlet (u ,v )denote
ij ij ij ij
the corresponding potentials discussed in Section 2.2, thus
(h ,k ) = (P(u) ,P(v) ), (2.38)
ij ij ij ij
where P is the third-order differential operator of (2.30). Let Ω beany open
subset of R3 and let Ω be any open set containing Ω. Let χ be any smooth
Ω
function which is identically equal to one on Ω and which vanishes outside
e
of Ω. Then the initial data set
e (h˜ ,k˜ ) = (P(χ u) ,P(χ v) ) (2.39)
ij ij Ω ij Ω ij
satisfiesthevacuumconstraintequationseverywhere,coincideswith(h ,k )
ij ij
in Ω and vanishes outside of Ω.
When Ω is bounded, the new fields (h˜ ,k˜ ) can clearly be chosen to
ij ij
e
vanish outside of a bounded set. For example, consider a plane wave so-
lution as in (2.33). Multiplying the potentials (2.36) by a cut-off function
χ which equals one on B(R ) and vanishes outside of B(R ) provides
B(R1) 1 2
compactly supported gravitational data which coincide with the plane-wave
ones in B(R ). (Alternatively one can replace~k·~x in the first line of (2.34),
1
or in (2.35)-(2.36), by ~k·~x χ .) In the limit ~k = 0, so that h is con-
B(R1) ij
stant and, e.g., k = 0, one obtains data which are Minkowskian in B(R ),
ij 1
and outside of B(R ), and describe a burst of radiation localised in a spher-
2
ical shell. Note that the Minkowskian coordinates for the interior region
are distinct from the ones for the outside region. The closest full-theory
configuration to this would be Bartnik’s time symmetric initial data set [2]
which are flat inside a ball of radius R , and which can be Corvino-Schoen
1
deformed to be Schwarzschildean outside of the ball of radius R ; here R
2 2
willbemuchlargerthanR ingeneral, butcanbemadeasclosetoR asde-
1 1
sired by making the free data available in Bartnik’s construction sufficiently
small.
For Ω’s which are not bounded it is interesting to enquire about fall-off
properties of the shielded field. This will depend upon the geometry of Ω
and the fall-off of the initial field:
For cone-like geometries, as considered in [5, 9], and with h = O(1/r),
µν
the gravitational field in the screening region will fall-off again as O(1/r).
This is rather surprising, as the gluing approach of [5] leads to a loss of
decay even for the linear problem. One should, however, keep in mind that
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the transition to the TT-gauge for a metric which falls-off as 1/r is likely to
introduce lnr/r terms in the transformed metric, which will then propagate
to the gluing region.
As another example, consider the set Ω = (a,b) × R2, which is not
covered by the methods of [5]. Our procedure in this case applies but if
h = O(1/r), and if the cut-off function is taken to depend only upon the
µν
firstvariable of the productΩ = (a,b)×R2, one obtains a gravitational field
h˜ vanishingoutside a slab Ω = (c,d)×R2, with [a,b] ⊂ (c,d), which might
µν
grow as r2lnr when receding to infinity within the slab.
e
So far we have been concentrating on “shielding”. But of course the
above can be used to glue linearised field across a gluing region, by inter-
polating the respective u’s to each other in the gluing zone. Equivalently,
screen each of the fields which are glued to zero across the gluing region,
and add the resulting new fields.
3 Shielding Maxwell fields
Maxwell equations in Minkowski space-time share at least two features with
linearised gravity: existence of constraint equations, and existence of gauge
transformations. It might therefore be unsurprising that there exists a ver-
sion oftheCarlotto-Schoen construction which appliesto theMaxwell equa-
tions; compare [11, 10] for a discussion of the Maxwell equivalent of the
Corvino-Schoen construction, which generalises without further ado to the
Carlotto-Schoen setting. We wish to show here how to carry-out the shield-
ing of Maxwell fields with Maxwell fields in an elementary way.
Recall that solutions of the source-free Maxwell equations are in one-to-
one correspondence with their initial data at time t; these are simply the
electric field E~ and the magnetic field B~ at t. These fields are not arbitrary,
but satisfy the constraints
divE~ = divB~ = 0. (3.1)
On R3, these imply the existence of vector potentials ~ω and A~ such that
E~ = curl~ω, B~ = curlA~. (3.2)
In fact there is an explicit formula for ~ω,
1
ω = ǫ xj Ek(λx)λdλ, (3.3)
i jik
Z
0
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similarly for A~. Using (3.2) it is straightforward to show that at any given
time t, and given any region Ω ⊂ R3, every sourceless Maxwell fields (E~,B~)
can be deformed to new sourceless Maxwell fields which coincide with (E~,B~)
on Ω and vanish outside a slightly larger set. Indeed, letting Ω and χ be
Ω
as in the paragraph following (2.38), the new Maxwell fields at t
e
E~ = curl(χ ~ω) and B~ = curl(χ A~) (3.4)
Ω Ω
aredivergence-free, coincidewiththeoriginalfieldsonΩ,andvanishoutside
of Ω.
One can solve the Cauchy problem for the Maxwell equations with the
e
new initial data (3.4) to obtain the associated space-time fields, if desired.
The question then arises1 if every such configurations can be realised by
an experimentalist in the lab. Here “an experimentalist” is defined as some-
one whose laboratory equipment can produce any desired electric charges ρ
and currents~j subject to the conservation law
∂ρ
+div~j = 0. (3.5)
∂t
These, inturn,willproduceMaxwell fieldsas dictated by theMaxwell equa-
tions written in their tensorial special-relativistic form:
∂ Fµν = 4πjµ, where (jµ) = (ρ,~j). (3.6)
ν
More specifically, let us describe the lab as the following “world-volume”:
U := [t ,t ]×Ω ⊂ R4.
0 3
The region within the labfwhere the desireed Maxwell fields need to be pro-
duced will be the set
U := [t ,t ]×Ω⊂ U ,
1 2
with t0 < t1 ≤ t2 < t3 and Ω ⊂ Ω. Let Fµν befa source-free solution of the
Maxwell equations in U, as needed to carry out the desired experiments.
e
The following prescription tells us what are the charges and currents
outside of U which will produce a Maxwell field F˜ coinciding with F in
µν µν
U, out of a vacuum configuration at t ≤ t0: Let χU be a smooth function
which is identically one on U and which vanishes outside of U. Let A be
µ
any four-vector potential associated with F , e.g.
µν f
1
A (xα)= xν F (λxα)λdλ. (3.7)
µ νµ
Z
0
1Weare grateful to Peter Aichelburgfor pointing-out theissue to us.
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