Shielding linearised-gravity††thanks: Preprint UWThPh-2016-31
We present an elementary argument that one can shield linearised gravitational fields using linearised gravitational fields. This is done by using third-order potentials for the metric, which avoids the need to solve singular equations in shielding or gluing constructions for the linearised metric.
- 1 Introduction
- 2 Shielding linearised gravity
- 3 Shielding Maxwell fields
- 4 The Weyl tensor formulation
- A Integrating 2-forms on
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
where is the gravitational potential, is Newton’s constant and is the matter density: The requirement that , and the asymptotic behaviour of , where is the total mass, implies that vanishes at large distances along a curve extending to infinity if and only if there is no matter whatsoever and . 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 .
Since Newtonian gravity is part of the weak-field limit of general relativity (indeed, this is weak-field GR with small velocities), one wonders if a similar screening can occur for linearised relativity. As it turns out, the analysis of Carlotto and Schoen can be readily generalised to linearised gravitational fields on cone-like sets as considered in  (compare ). 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 implement 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 background. In particular, we give here a very simple proof that at any given time , and given any open set , every linearised vacuum gravitational field on can be deformed to a new linearised vacuum field so that coincides with on and vanishes outside a slightly larger set. In other words, the gravitational field has been screened away outside of , 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 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 Carlotto-Schoen gluings require solving elliptic equations in spaces of functions which are singular at the boundary of the gluing region (see [10, 8] for a review), while our gluings are performed by explicit elementary integrations ((2.31) below), multiplication by a cut-off function, and applying derivatives, 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 screening in practice, in that we prove that all solutions of sourceless Maxwell 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 with a metric which, in the natural coordinates on , takes the form
where denotes the Minkowski metric. Suppose that there exists a small constant such that we have
If we use the metric to raise and lower indices one has
Coordinate transformations , with
preserve (2.2), and lead to the gauge-freedom
Imposing the wave-coordinates condition up to terms,
as well as
2.1 The Cauchy problem for linearised gravity
In what follows we ignore all -terms in the equations above and consider the theory of a tensor field with the gauge-freedom (2.5) and satisfying the equations
Solving the following wave equation
where is the wave-operator of the Minkowski metric, and performing (2.5) leads to a new tensor , still denoted by the same symbol, such that
together with the usual wave equation for :
It follows that (2.10) will hold if and only if
Equivalently, taking (2.11) into account,
The last two equations are of course the linearisations of the usual scalar and vector constraint equations.
There remains the freedom of choosing and . We choose
where in Cartesian coordinates. Indeed, given any and , the first equation can be solved for under suitable natural conditions on the data; the second defines ; the third defines ; finally, the last equation is an elliptic equation for the vector field which can be solved  if one assumes that the field
belongs to a suitable weighted Sobolev or Hölder space, the precise requirements being irrelevant for our purposes. We simply note that if some components of behave as , then will behave like in general, which is likely to introduce terms in the gauge-transformed metric. After performing this gauge-transformation, we end up with a tensor field which satisfies
The further requirement that goes to zero as tends to infinity together with the maximum principle gives
We conclude (compare ) that at any given time every linearised gravitational initial data set can be gauge-transformed to the TT-gauge: writing , we have
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
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 will be non-trivial, with the usual expansion in terms of inverse powers of , starting with -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 , which can be summarised as follows: Let be a symmetric, transverse and traceless tensor on ,
Then there exists a symmetric traceless “third order potential” such that
where (here denotes the Euclidean metric and the associated covariant derivative)
and where can be constructed by the following procedure: Letting
(This is clearly symmetric, and tracelessness is not very difficult to check. Other third-order-potentials are possible, differing by an element of the kernel of .) One way to see how (2.30) arises is to note that is, apart from a numerical factor, the linearisation at the flat metric of the Cotton-York tensor in the direction of the trace-free tensor . For (2.32), the formulae follow by successively integrating thrice the two-forms given in , at each step using the Poincaré formula (3.7) below. We sketch the construction in Appendix A.
The converse is also true: given any symmetric trace-free tensor , the tensor field 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 describing a plane gravitational wave in TT-gauge propagating in direction ,
with possibly complex coefficients , where denotes the real part . Then
As another example, consider the family of fields
We also note that if is compactly supported to start with, then 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 more general version of our previous claim, that at any given time , and given any region , every vacuum initial data set for the gravitational field can be deformed to a new vacuum initial data set which coincides with on and vanishes outside of a slightly larger set.
where is the third-order differential operator of (2.30). Let be any open subset of and let be any open set containing . Let be any smooth function which is identically equal to one on and which vanishes outside of . Then the initial data set
satisfies the vacuum constraint equations everywhere, coincides with in and vanishes outside of .
When is bounded, the new fields can clearly be chosen to vanish outside of a bounded set. For example, consider a plane wave solution as in (2.33). Multiplying the potentials (2.36) by a cut-off function which equals one on and vanishes outside of provides compactly supported gravitational data which coincide with the plane-wave ones in . (Alternatively one can replace in the first line of (2.34), or in (2.2)-(2.36), by .) In the limit , so that is constant and, e.g., , one obtains data which are Minkowskian in , and outside of , and describe a burst of radiation localised in a spherical 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  which are flat inside a ball of radius , and which can be Corvino-Schoen deformed to be Schwarzschildean outside of the ball of radius ; here will be much larger than in general, but can be made as close to as desired by making the free data available in Bartnik’s construction sufficiently small.
For ’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 and the fall-off of the initial field:
For cone-like geometries, as considered in [5, 9], and with , the gravitational field in the screening region will fall-off again as . This is rather surprising, as the gluing approach of  leads to a loss of decay even for the linear problem. One should, however, keep in mind that the transition to the TT-gauge for a metric which falls-off as is likely to introduce terms in the transformed metric, which will then propagate to the gluing region.
As another example, consider the set , which is not covered by the methods of . Our procedure in this case applies but if , and if the cut-off function is taken to depend only upon the first variable of the product , one obtains a gravitational field vanishing outside a slab , with , which might grow as when receding to infinity within the slab.
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 interpolating the respective ’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 version of the Carlotto-Schoen construction which applies to the Maxwell equations; 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 shielding 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 ; these are simply the electric field and the magnetic field at . These fields are not arbitrary, but satisfy the constraints
On , these imply the existence of vector potentials and such that
In fact there is an explicit formula for ,
similarly for . Using (3.2) it is straightforward to show that at any given time , and given any region , every sourceless Maxwell fields can be deformed to new sourceless Maxwell fields which coincide with on and vanish outside a slightly larger set. Indeed, letting and be as in the paragraph following (2.38), the new Maxwell fields at
are divergence-free, coincide with the original fields on , and vanish outside of .
One can solve the Cauchy problem for the Maxwell equations with the new initial data (3.4) to obtain the associated space-time fields, if desired.
The question then arises111We are grateful to Peter Aichelburg for pointing-out the issue to us. if every such configurations can be realised by an experimentalist in the lab. Here “an experimentalist” is defined as someone whose laboratory equipment can produce any desired electric charges and currents subject to the conservation law
These, in turn, will produce Maxwell fields as dictated by the Maxwell equations written in their tensorial special-relativistic form:
More specifically, let us describe the lab as the following “world-volume”:
The region within the lab where the desired Maxwell fields need to be produced will be the set
with and . Let be a source-free solution of the Maxwell equations in , as needed to carry out the desired experiments.
The following prescription tells us what are the charges and currents outside of which will produce a Maxwell field coinciding with in , out of a vacuum configuration at : Let be a smooth function which is identically one on and which vanishes outside of . Let be any four-vector potential associated with , e.g.
Set , , and
Then vanishes outside of the lab world-volume , and coincides with the desired field in the world-volume of the experiment. If the experimenter can produce the four-current (3.8) with her apparatus, she will be able to create the desired Maxwell field in the region where the experiment will take place.
It would be of interest to devise an analogous procedure for the gravitational field, keeping in mind the supplementary difficulty of maintaining positivity of energy density.
4 The Weyl tensor formulation
As is well-known, the vacuum Einstein equations imply a system of equations for the metric and the Weyl tensor ,
which implies a symmetrizable-hyperbolic system of equations in dimension (cf., e.g., ). In the linearised case the equations for the metric and the Weyl tensor decouple, so that one can consider the Weyl tensor equations linearised on Minkowski space-time on their own. We show in Appendix C the equivalence of this approach to the metric one, in the sense that a linearised Weyl tensor is always accompanied by a linearised metric (the reverse property being obvious).
are complemented by the constraint equations
Here is the electric part and the magnetic part of the Weyl tensor:
The symmetry and tracelessness of , as well as tracelessness of are obvious from the symmetries of the Weyl tensor. The symmetry of follows from the less-obvious double-dual symmetry of the Weyl tensor (cf., e.g., [6, Proposition 4.1])
We show in Appendix C how the vanishing of the divergence of relates to the linearised scalar constraint equation, and how the symmetry of relates to the vector constraint equation.
Since both and are transverse and traceless, each of them comes with its own third-order potential as described in Section 2.2, so that shieldings and gluings can be performed on each of them directly, without having to invoke the metric tensor.
Appendix A Integrating 2-forms on
In this Appendix we address the question of asymptotic behaviour of potentials for closed two-forms. The analysis below has obvious generalizations to -forms on () with .
Let be a closed 2-form on with , . Then there exists a 1-form with satisfying if , otherwise.
Consider first the case . Then
and when . To see this, use spherical coordinates in the argument of and substitute for . When , consider
which converges and has the right decay at infinity, but blows up at the origin. The previous expression is still defined and, in the shell , differs from by a closed 1-form. Since this shell is simply connected, this difference satisfies for some function . Now extend smoothly to a function on all of . Then the 1-form given by in the interior and by in the exterior satisfies our requirements.
An essentially identical argument shows that if has compact support, then can also be chosen with compact support (which also follows from standard results in algebraic topology [4, Corollary 4.7.1]).
Appendix B Construction of the potential
For the convenience of the reader we review the construction in , and take this opportunity to correct a minor mistake in the presentation there, namely the second sentence after (3.12) there. Let us define
Since , there exists a tensor field such that
Symmetry of implies that all traces of vanish, which implies in turn that
Hence there exists a tensor field , which can be chosen to be antisymmetric in , so that
From tracelessness of one finds , which shows that there exists a vector field such that
which implies existence of a potential :
a lengthy calculation shows that
Thus neither , nor , nor contribute to and we finally obtain (2.30). For (2.31), we have to successively write down expressions for (i) , (ii) , and (iii) , at each step using formula (A.1), and take the symmetric, tracefree part of at the end. In going from (i) to (ii) and (ii) to (iii) one uses the identities
respectively. The rest is index gymnastics.
If for large , from what has been said here and in Section A, or by analysing (2.31) for , we find that can be chosen to be of when , and of otherwise. Furthermore, if is compactly supported, then can also be chosen to be compactly supported.
We end this Appendix with an analysis of the kernel of on a simply connected region. For this we follow through the steps starting from (B.2) with , which implies existence of a potential such that
Next, from (B.4), there exists an antisymmetric tensor field such that
Equation (B.5) implies the existence of a function such that
Inserting into (B.7) we find that the terms involving cancel so that
Setting , we conclude that any tensor field satisfying on a simply connected region can be written as
Appendix C A potential for the linearised Riemann tensor
In this appendix we show that every linearised Riemann tensor on a star-shaped subset of arises from a linearised metric , in arbitrary dimension , where is defined uniquely up to the usual gauge transformations. We leave it as an exercise to the reader to obtain an explicit formula for by following the steps of our calculation below.
Suppose, thus, that is a field on Minkowski space-time having the algebraic symmetries of the Riemann tensor and satisfying the Bianchi identity . Then
with . But, since ,
Inserting the identity
The right-hand side of (C.4) multiplied by is, up to terms, the Riemann tensor of the metric . Equivalently, is the linearised Riemann tensor associated with .
The addition of a pure-trace tensor to does not change the trace-free part of . So, for a tensor with Weyl-symmetries satisfying , there exists a second-order potential as in (C.4), which is trace-free.
As already pointed out, this implies existence of a symmetric tensor field such that
But the right-hand side of (C.6) is the linearised Riemann tensor associated with the linearised metric perturbation . Since the left-hand side of (C.6) has vanishing traces, we conclude that the linearised Ricci tensor associated with vanishes. Equivalently, satisfies the linearised Einstein equations.
To understand the nature of the divergence constraint , let us denote by the linearised Riemann tensor of the three-dimensional metric , with the associated linearised Ricci tensor . We have just seen that for solutions of , which gives for such solutions
Here we have used the fact that the three-dimensional Riemann tensor differs from the four-dimensional one by quadratic terms in the extrinsic curvature, hence both tensors coincide when linearised at Minkowski space-time. The vanishing of the divergence of the Einstein tensor implies
which together with (C.7) shows that the constraint equation is, for asymptotically flat solutions, equivalent to the linearised scalar constraint .
Let us show that symmetry of is equivalent to the vector constraint equation. For this let
denote the linearised extrinsic curvature tensor of the slices . By a direct calculation, or by linearising the relevant embedding equations, we find
Again for solutions of it holds that
Let us finally consider the kernel of the map sending into . Namely, when , from (C.4) we infer
where . But, since ,
Now defining , there results
so that , whence . Finally, using the symmetry of , it follows that
Acknowledgements: Supported in part by the Austrian Science Fund (FWF) project P29517-N16. Useful discussions with Peter Aichelburg and Jérémie Joudioux are gratefully acknowledged.
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