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The Schwarzschild solution could be an good model for a planet and a star, without rotation and electric and magnetic charge. Besides the fact that this solution is spherically symmetric.

The metric of this spacetime is, then, given by the following metric tensor:

$$ \mathrm{g} = -\Big(1- \frac{r_{S}}{r}\Big)c^{2}\mathrm{dt}\otimes \mathrm{dt}+\Big(1- \frac{r_{S}}{r}\Big)^{-1} \mathrm{dr}\otimes \mathrm{dr} + r^{2}\Big[\mathrm{d}\theta\otimes \mathrm{d}\theta + sin^{2}(\theta)\mathrm{d}\phi\otimes \mathrm{d}\phi\Big] $$

or,

$$ds^{2} = -\Big(1- \frac{r_{S}}{r}\Big)c^{2}dt^{2}+\Big(1- \frac{r_{S}}{r}\Big)^{-1} dr^{2}+ r^{2}\Big[d\theta^{2} + sin^{2}(\theta)d\phi^{2}\Big] \tag{1}$$

And, of course that:

$$ \mathrm{Ric(g)} = 0 \iff R_{ab} = 0$$

i.e. the metric is a (vaccum) solution of EFE.

Another solution of EFE, which ensambles electric and magnetic charge, is called Reissner-Nordström solution (R-Ns):

$$ds^{2} = -\Big[1- \frac{r_{S}}{r}+G\Big(\frac{E^{2}}{r^{2}}+\frac{B^{2}}{r^{2}}\Big) \Big]c^{2}dt^{2}+\Big[1- \frac{r_{S}}{r}+G\Big(\frac{E^{2}}{r^{2}}+\frac{B^{2}}{r^{2}}\Big) \Big]^{-1} dr^{2}+ r^{2}\Big[d\theta^{2} + sin^{2}(\theta)d\phi^{2}\Big] \tag{2}$$

Where $E$ and $B$ are, respectively, the total electric charge and total magnetic charge.

This metric is a solution of Einstein-Maxwell Field Equations (E-MFE):

$$\mathrm{G(g)} \equiv \mathrm{Ric(g) - \frac{1}{2}Sg} = \frac{8\pi G}{c^{4}}\mathrm{T}_{EM} \iff R_{ab} - \frac{1}{2}Rg_{ab} = \frac{8\pi G}{c^{4}}\Big[\frac{1}{\mu _{0}} \Big( F_{ac}F^{c}\hspace{1mm}_{b} - \frac{1}{4}g_{ab}F_{ab}F^{ab}\Big) \Big] $$

where $\mathrm{S}$ is the scalar curvature.

$$* * *$$

So, the metric $(2)$ describes some kind of spherical object with charge, such as a black hole. But my question deals with another astrophysical object called Magnetar [1],[2]. Basically, a Magnetar is a type of Pulsar (which is basically a Neutron Star with a high angular momentum) with a huge magnetic field strength [$\approx (10^{8} - 10^{11})$T ] [3].

Now, in order to describe the spacetime on the vicinity of this object, which energy-momentum tensor have I must to consider? My guess:

$$T_{ab} = \Big[ \frac{1}{\mu _{0}} \Big( F_{ac}F^{c}\hspace{1mm}_{b} - \frac{1}{4}g_{ab}F_{ab}F^{ab}\Big) \Big] + \Big[ (\rho + p)u_{a}u_{b}+pg_{ab} \Big]$$

$$ * * * $$

[1] https://arxiv.org/pdf/1503.06313.pdf

[2] https://www.youtube.com/watch?v=jwPZus7fT74

[3] https://en.wikipedia.org/wiki/Magnetar

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    $\begingroup$ Since magnetars are rotating Kerr Newman solution will be suitable to describe the space time. See: en.m.wikipedia.org/wiki/Kerr-Newman_metric $\endgroup$ – Manvendra Somvanshi Apr 27 at 21:54
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    $\begingroup$ Magnetars do not have electric (or magnetic) monopole charges. (Unless they are much weirder than we think!). So Kerr-Newmann is not much help. $\endgroup$ – mmeent Apr 29 at 8:24

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