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arXiv 1809.00293gr-qcmath-phmath.MP
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嵌入电磁弦流体的时空的对称性

Symmetries of spacetimes embedded with an Electromagnetic String Fluid

Michael Tsamparlis, Antonios Mitsopoulos, Andronikos Paliathanasis

备注：
27 pages, no figures, version accepted for publication in GRG
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摘要：

电磁弦流体（EMSF）是一种各向异性的带电弦流体，与强磁场相互作用。在这种流体中，我们考虑由流体的4-速度$u^a$和沿磁场的单位矢量$n^a$定义的双同余。使用由矢量$u^a$定义的标准1+3分解和由双同余${u^a,n^a}$定义的1+1+2分解，我们确定EM弦流体在两种分解中的运动学和动力学量。为了解决由此产生的场方程，我们考虑简化假设的形式collineations。我们将类属量$L_{X}g_{ab}$分解为迹$ψ$和无迹部分$H_{ab}$。因为所有的直射都可以用量$L_{X}g_{ab}$表示，所以可以计算由度规定义的所有张量的李导数，即里奇张量，外尔张量等。这是按照如下方式完成的。利用微分几何中的有关恒等式，我们用两个不可约部分$ψ,H_{ab}$表示量$L_{X}R_{ab}$，其中$R_{ab}$是Ricci张量.随后使用引力场方程，我们计算相同的量$L_{X}R_{ab}$的动力学变量的李导数。我们将这两个结果等同起来，并找到$L_{X}{\text{Dynamic variable}}=F(ψ,H_{ab},{\text{ dynamic variables}})$形式的场方程。这个结果是普遍的，对所有的引力系统，特别是EMSF都成立。随后，我们在两个层面上专门研究。我们考虑的情况下，共形杀伤矢量（CKV）平行于$u^a$和CKV平行于$n^a$。

英文摘要：

The electromagnetic string fluid (EMSF) is an anisotropic charged string fluid interacting with a strong magnetic field. In this fluid we consider the double congruence defined by the 4-velocity of the fluid $u^a$ and the unit vector $n^a$ along the magnetic field. Using the standard 1+3 decomposition defined by the vector $u^a$ and the 1+1+2 decomposition defined by the double congruence ${u^a,n^a}$ we determine the kinematic and the dynamic quantities of an EM string fluid in both decompositions. In order to solve the resulting field equations we consider simplifying assumptions in the form of collineations. We decompose the generic quantity $L_{X}g_{ab}$ in a trace $ψ$ and and a traceless part $H_{ab}$. Because all collineations are expressible in terms of the quantity $L_{X}g_{ab}$ it is possible to compute the Lie derivative of all tensors defined by the metric i.e. the Ricci tensor, the Weyl tensor etc. This makes possible the effects of any assumed collineation on the gravitational field equations. This is done as follows. Using relevant identities of Differential Geometry we express the quantity $L_{X}R_{ab}$ where $R_{ab}$ is the Ricci tensor in terms of the two irreducible parts $ψ,H_{ab}$. Subsequently using the gravitational field equations we compute the same quantity $L_{X}R_{ab}$ in terms of the Lie derivative of the dynamic variables. We equate the two results and find the field equations in the form $L_{X}{\text{Dynamic variable}}=F(ψ,H_{ab},{\text{ dynamic variables}})$. This result is general and holds for all gravitational systems and in particular for the EMSF. Subsequently we specialize our study at two levels. We consider the case of a Conformal Killing Vector (CKV) parallel to $u^a$ and a CKV parallel to $n^a$.

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