爱因斯坦-反麦克斯韦理论的 Killing 向量与幻影度量

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"杀死Spinors的虚拟指标" 在物理学领域，特别是广义相对论和量子场论的研究中，"杀死向量"（Killing Vector）和"杀死旋量"（Killing Spinor）是极其重要的概念。杀死向量指的是在某种流形上保持度量不变的矢量场，它们描述了流形的对称性。杀死旋量则是其在 spinor 空间中的推广，对应于保持洛伦兹变换不变的旋量场。这些对象对于理解和分类具有对称性的时空非常重要。本文"杀死Spinors的虚拟指标"探讨了爱因斯坦-反麦克斯韦理论（Einstein-Anti-Maxwell theory）的度量解，这是一个结合了广义相对论和电磁场理论的框架。这个理论考虑了引力和电磁相互作用，其中杀死旋量的存在揭示了时空的特定对称性。作者 W.A. Sabra 提到了一个与 IWP 度量类似的解，IWP（Imbert-Wagoner-Peterson）度量是一个在特定情况下具有空间向量 Killing 对称性的度量。在这个研究中，他发现了一个允许存在类空间 Killing 向量的度量，并且可以表示为一个满足平面（2+1）维时空波动方程的复函数。波动方程在这里可能指的是线性波动方程，它在平坦时空中描述了波的传播。通过让解决方案仅依赖时间坐标，Sabra 构建了电气和磁性 Kasner 空间。Kasner 空间是一种特殊的同胚于三维欧几里得空间的四维时空解，其特点在于各方向的尺度因子随时间以不同的指数增长或减小，这在宇宙学中有一定的应用，特别是在描述早期宇宙的某些阶段。此外，他还提出了一些欧几里得解。在广义相对论中，欧几里得解通常指的是将时空从洛伦兹度规转换到欧几里得度规的情况，这在某些问题中提供了更直观的理解，例如在黑洞热力学中。文章最后提到了开放访问（Open Access）和SCOAP3（Supporting the Open Access Publishing for Particle Physics）的资助，这意味着这篇研究是公开可获取的，所有感兴趣的学者和研究人员都可以自由阅读和引用。同时，论文遵循 CC BY 许可证，允许他人在遵守署名条件的情况下自由使用、分享和改编内容。这项工作深化了我们对具有杀死旋量的爱因斯坦-反麦克斯韦理论度量解的理解，特别是在（2+1）维时空中的波动方程和对称性方面，对于理论物理的研究有着重要的贡献。

Physics Letters B 750 (2015) 237–241

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Phantom metrics with Killing spinors

W.A. Sabra

Centre for Advanced Mathematical Sciences and Physics Department, American University of Beirut, Lebanon

a r t i c l e i n f o a b s t r a c t

Article history:

Received

25 July 2015

Accepted

10 September 2015

Available

online 16 September 2015

Editor:

M. Cveti

We study metric solutions of Einstein–anti-Maxwell theory admitting Killing spinors. The analogue of the

IWP metric which admits a space-like Killing vector is found and is expressed in terms of a complex

function satisfying the wave equation in ﬂat (2 + 1)-dimensional space–time. As examples, electric and

magnetic Kasner spaces are constructed by allowing the solution to depend only on the time coordinate.

Euclidean solutions are also presented.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The ﬁrst systematic classiﬁcation of metrics admitting super-

covariantly

constant spinors in Einstein–Maxwell theory was per-

formed

many years ago by Tod in [1]. The analysis of Tod was to

some extent motivated by the results of Gibbons and Hull [2]. The

metrics found in [1] are bosonic solutions of minimal N = 2super-

gravity

theory admitting half of the supersymmetry. In the context

of the supergravity theory, the Killing spinor equation represents

the vanishing of the gravitini supersymmetry transformation in a

bosonic background. The metrics with a time-like Killing vector are

the known Israel–Wilson–Perjés (IWP) metrics [3] with the static

limit given by the Majumdar–Papapetrou (MP) metrics [4]. The

second class of metrics with a null-Killing vector is given by plane-

wave

space–times [5]. In recent years, a considerable amount of

research activities has been devoted to the understanding and the

systematic classiﬁcation of supersymmetric solutions in ungauged,

gauged and fake (de Sitter) supergravity theories in various di-

mensions

(see for example [6]). Fake de Sitter supergravity can be

obtained by analytic continuation of anti de Sitter supergravity. We

also note that de Sitter supergravities can also be obtained as gen-

uine

low energy effective theories of the so called

∗

theories of

[7]. For instance, a non-linear Kaluza Klein reduction arising of IIB

∗

string theory and M

∗

theory produce four and ﬁve-dimensional de

Sitter supergravities with vector multiplets. However these theo-

ries

have actions where some of the gauge ﬁelds kinetic terms

have the non-conventional sign [8]. Black hole solutions with anti

or phantom Maxwell ﬁelds

have been studied and analyzed in [9].

E-mail address: waﬁcsabra@gmail.com.

There anti or phantom Maxwell ﬁeld refers to an electromagnetic ﬁeld of the

opposite sign from usual.

Black hole solutions with phantom ﬁelds and their relations to

astrophysics and dark matter were also considered by many au-

thors

(see [10] and references therein). However, to our knowledge

phantom solutions with Killing spinors have not yet been dis-

cussed.

our present work, we shall study metrics admitting Killing

spinors in gravitational theories with anti-Maxwell ﬁelds. We shall

only focus on the simplest theory of four-dimensional Einstein

gravity coupled to a Maxwell ﬁeld as a ﬁrst step for a future

study of supergravity theories with many anti-Maxwell and scalar

ﬁelds in various space–time dimensions. We will consider both the

Lorentzian and the Euclidean theory. The action of the theory is

given by

S =



√

−g



R +κ

μν



(1.1)

where F

μν

is the U (1) gauge ﬁeld strength. We have introduced a

parameter κ which for κ = i, corresponds to the standard Einstein–

Maxwell

theory and for κ = 1 corresponds to the Einstein–anti-

Maxwell

theory, i.e., where the Maxwell ﬁeld kinetic term comes

with the wrong sign. The signature of the metric is taken to be

(−, +, +, +). For κ = 1, this action can be thought of as the

bosonic part of a fake minimal N = 2, D =4supergravity. The Ein-

stein

and gauge ﬁeld equations derived from (1.1) are

μν

=−κ



μρ

−

μν

αβ



d ∗ F =0 . (1.2)

Here F is the two form representing the gauge ﬁeld strength F

μν

The Killing spinor equation is given by



∂

μ,ν



ε = 0. (1.3)

http://dx.doi.org/10.1016/j.physletb.2015.09.025

0370-2693/

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