Asymptotic behavior of Maxwell fields in higher dimensions

Abstract

We study the fall-off behaviour of test electromagnetic fields in higher dimensions as one approaches infinity along a congruence of "expanding" null geodesics. The considered backgrounds are Einstein spacetimes including, in particular, (asymptotically) flat and (anti-)de Sitter spacetimes. Various possible boundary conditions result in different characteristic fall-offs, in which the leading component can be of any algebraic type (N, II or G). In particular, the peelingoff of radiative fields F = N r 1−n/2 + Gr −n/2 + . . . differs from the standard four-dimensional one (instead it qualitatively resembles the recently determined behaviour of the Weyl tensor in higher dimensions). General p-form fields are also briefly discussed. In even n dimensions, the special case p = n/2 displays unique properties and peels off in the "standard way" as F = N r 1−n/2 + IIr −n/2 + . . .. A few explicit examples are mentioned.

Asymptotic Charges at Null Infinity in Any Dimension

“…Due to this simplification, we restrict the present analysis to D > 4 and defer the discussion of the special cases D = 3 and D = 4 to a dedicated section. In order to stress the relevant piece of physical information following from our choices of the falloffs, we first consider the leading-order terms in the equations of motion G µ = 0 and analyse the 2 In the three-dimensional case (n = 1), to be discussed in Section 2.1.3, we shall also consider a logarithmic dependence in r. For a discussion of the asymptotic behaviour of Maxwell fields in Einstein spacetimes see [36]. 3 The D-dimensional wave equation −∂ 2 t f + r −n ∂ r (r n ∂ r f ) + ∆ f = 0, where t = u + r, admits spherically symmetric solutions whose large-r behaviour is r −n/2 exp(iku).…”

Asymptotic properties of gravitational and electromagnetic fields in higher dimensions

“…It has been observed [3] that the Weyl tensor peels off differently in n > 4 dimensions. Here, we summarize our recent results [4,5] on the leading-order behavior of gravitational and electromagnetic fields in higher dimensions. Ref.…”

“…Under the above conditions, one is able to determine how the Maxwell and Weyl tensors fall off as r → ∞, as we summarize in sections 2 and 3. However, as an intermediate step, one also needs the r-dependence of the Ricci rotation coefficients and of the derivative operators [6], which is given in [4] (it follows from the Ricci identities [8], also using the commutators [9] and the Bianchi identities [10]). For example, ρ ij = δ ij r + .…”

Black Holes in Higher Dimensions (Black Strings and Black Rings)

“…Turning to Robinson-Trautmann spacetimes in higher dimensions, Marcello Ortaggio recalled that the general metric is known 98 , and that the electrovac solutions essentially reduce to static black holes with an Einstein horizon 98,99 . He remarked that the shearfree condition may be too strong in higher dimensions [100][101][102][103] , and pointed out that in the case of more general p-forms electromagnetic radiation may have different properties than in the standard case 104,105 . Focussing on the Einstein-Maxwell equations with p-forms, Marcello Ortaggio recalled a number of known results for static p-forms like the no dipole hair theorem 106 .…”