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In this section, we will review the ideal MHD. For more detials, one can see Ref. [25-28, 30].
In relativistic hydrodynamics, the electromagnetic strength tensor
$ F^{\mu\nu} $ will be decomposed by the fluid velocity$ u^{\mu} $ ,$$ F^{\mu\nu} = E^{\mu}u^{\nu}-E^{\nu}u^{\mu}+\epsilon^{\mu\nu\alpha\beta}u_{\alpha}B_{\beta}, $$ (1) which gives,
$$ E^{\mu} = F^{\mu\nu}u_{\nu},\;\;\;\;B^{\mu} = \frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}. $$ (2) Generally, we can parametrize the velocity as,
$ u^{\mu} = \gamma(1,{ {v}}), $ with$ \gamma $ being$ 1/\sqrt{1-|{ {v}}|^{2}} $ . We can prove that,$$ u\cdot E = u \cdot B = 0. $$ (3) In the comoving frame of a fluid cell, i.e. local rest frame,
$ u^{\mu} = (1,{\bf{0}}) $ , the$ E^\mu $ and$ B^\mu $ will reduce to$$ E^\mu = (0,{ {E}}),B^\mu = (0,{ {B}}). $$ (4) The ideal MHD includes the energy-momentum and charge conservation equations coupled to the Maxwell's equations.
The energy-momentum conservation is given by,
$$ \partial_{\mu}T^{\mu\nu} = 0, $$ (5) where
$ T^{\mu\nu} $ is the energy-momentum tensor. In an ideal fluid, this tensor can be decomposed as two parts,$$ T^{\mu\nu} = T^{\mu\nu}_{\rm {matter}}+T^{\mu\nu}_{\rm EM}. $$ (6) The first term is the part for the matter
$$ T^{\mu\nu}_{\rm {matter}} = (\epsilon+P)u^{\mu}u^{\nu}-(P+\Pi)g^{\mu\nu}+\pi^{\mu\nu}, $$ (7) where
$ \epsilon,P $ are energy density and pressure.$ \Pi $ and$ \pi^{\mu\nu} $ denote the bulk pressure and shear viscous tensor, respectively. In the first order relativistic hydrodynamics,$ \Pi $ and$ \pi^{\mu\nu} $ can be parameterized as,$$ \Pi = \zeta (\partial \cdot u), $$ (8) $$ \begin{split} \pi^{\mu\nu} = & 2\eta \Big[\frac{1}{2} \Delta^{\mu\alpha}\Delta^{\nu\beta} + \frac{1}{2} \Delta^{\nu \alpha}\Delta^{\mu \beta} - \\ &\frac{1}{3}\Delta^{\mu \nu}\Delta^{\alpha \beta}\Big]\partial_\alpha u_\beta, \end{split} $$ (9) where
$ \zeta $ and$ \eta $ stand for the bulk and shear viscosities, respectively. The projector is defined as,$$ \Delta^{\mu\nu} = g^{\mu\nu}-u^\mu u^\nu. $$ (10) It is easy to prove that
$$ \begin{split} \Delta^{\mu \nu}u_\nu = &\Delta^{\mu \nu} u_\mu = 0, \\ \Delta^{\mu \alpha} \Delta^{\nu}_{\alpha} = &\Delta^{\mu\nu}, \end{split} $$ (11) where we have used that
$ u^\mu u_\nu = 1 $ .Both
$ \Pi $ and$ \pi^{\mu\nu} $ are dissipative effects. In an ideal fluid, all the dissipative terms vanish, i.e.$ \Pi = 0 $ and$ \pi^{\mu\nu} = 0 $ .The second part for the energy-momentum tensor comes from the electromagnetic fields,
$$ \begin{split} T^{\mu\nu}_{\rm EM} = &-F^{\mu\lambda}F^{\nu}_\lambda+\frac{1}{4}g^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma} \\ =&-E^{\mu}E^{\nu}-B^{\mu}B^{\nu}+(E^{2}+B^{2})u^{\mu}u^{\nu} - \\ & \frac{1}{2}g^{\mu\nu}(E^{2}+B^{2})+u^{\mu}\epsilon^{\nu\lambda\rho\sigma}E_{\lambda}u_{\rho}B_{\sigma}+ \\ & u^{\nu}\epsilon^{\mu\lambda\rho\sigma}E_{\lambda}u_{\rho}B_{\sigma}, \end{split} $$ (12) where
$$ \begin{split} E^2 &\equiv - E^\mu E_\mu, \\ B^{2}&\equiv-B^{\mu}B_{\mu}. \end{split} $$ (13) Inserting Eq. (1) into the above equation yields,
$$ \begin{align} T^{\mu\nu} = & (\epsilon\!+\!p\!+\!E^{2}\!+\!B^{2})u^{\mu}u^{\nu}\!-\! (p\!+\!\frac{1}{2}E^{2}\!+\!\frac{1}{2}B^{2})g^{\mu\nu}-\\ &E^{\mu}E^{\nu}-B^{\mu}B^{\nu}+ u^{\mu}\epsilon^{\nu\lambda\rho\sigma}E_{\lambda}u_{\rho}B_{\sigma}+\\ &u^{\nu}\epsilon^{\mu\lambda\rho\sigma}E_{\lambda}u_{\rho}B_{\sigma}. \end{align} $$ (14) The charge conservation equation reads,
$$ \partial_{\mu}j^{\mu} = 0. $$ (15) The charge current
$ j^{\mu} $ can be decomposed as,$$ j^{\mu} = nu^{\mu}+\sigma E^{\mu}, $$ (16) where
$ n $ is the electric charge density and$ \sigma $ is the electric conductivity.For simplicity, we consider a charge neutral fluid, i.e.
$$ n = 0. $$ (17) In order to simplify the calculations, we will use the ideal MHD limit. The ideal MHD limit means that the electric conductivity is infinite, i.e.
$$ \sigma\rightarrow\infty. $$ (18) To avoid the divergence in
$ \sigma E^{\mu} $ term in Eq. (16), the four vector form of$ E^{\mu} $ must vanish,$$ E^{\mu} = 0. $$ (19) In the ideal limit, i.e.
$ E^\mu = 0 $ , the energy-momentum tensor in Eq. (14) reduces to,$$ T^{\mu\nu} = (\epsilon+P+B^{2})u^{\mu}u^{\nu}-\Big(P+\frac{1}{2}B^{2}\Big)g^{\mu\nu}-B^{\mu}B^{\nu}. $$ (20) Now, we will discuss the covariant form of Maxwell's equation,
$$ \begin{split} \partial_{\mu}(\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}) & = 0, \\ \partial_{\mu}F^{\mu\nu} & = j^{\nu}. \end{split} $$ (21) In ideal MHD, the covariant equations for magnetic fields are given by
$$ \partial_{\nu}(B^{\mu}u^{\nu}-B^{\nu}u^{\mu}) = 0. $$ (22) In the local rest frame, i.e.
$ u^{\mu} = (1,{\bf{0}}) $ , the above equation reduces to$$ \frac{\rm {d}}{\rm {d t}}{ {B}}+{ {B}}(\nabla\cdot{ {v}})-({ {B}}\cdot\nabla){{ {v}}} = 0. $$ (23) Contracting Eq. (22) with
$ u_{\mu} $ , yields,$$ \partial_{\mu}B^{\mu} = -B^{\mu}(u\cdot\partial)u_{\mu}, $$ (24) which gives
$$ \nabla\cdot{ {B}} = 0, $$ (25) in the local rest frame. Contracting Eq. (22) with
$ B_{\mu} $ , yields,$$ \frac{1}{2}(u\cdot\partial)B^{2}+B^{2}(\partial\cdot u)+B^{\mu}B^{\nu}\partial_{\nu}u_{\mu} = 0. $$ (26) The thermodynamical relations reads,
$$ \epsilon+p = Ts+\mu n, $$ (27) where
$ T,\; s,\; \mu,\; n $ are temperature, entropy density, chemical potential and charge number density, respectively. In our case, since we have considered the charge neutral fluid$ n = 0 $ , the thermodynamical relation becomes,$$ \begin{align} \epsilon+p = Ts. \end{align} $$ (28) Its differential form is given by
$$ \begin{split} {\rm {d}}\epsilon & = T{\rm {d}}s, \\ {\rm {d}} p & = s{\rm {d}}T. \end{split} $$ (29) -
In the section, we will derive the general expression for the relativistic Kelvin circulation theorem for ideal MHD.
The projection of Eq. (5) along the velocity
$ u^{\nu} $ reads,$$ u_{\nu}\partial_{\mu}T^{\mu\nu} = 0. $$ (30) Inserting Eq. (20) into above equation yields,
$$ \begin{split} (u\cdot\partial)\epsilon+&(\epsilon+p)(\partial\cdot u)+ (u\cdot\partial)\frac{1}{2}B^{2}+B^{2}(\partial\cdot u)-\\ &u_{\mu}(B\cdot\partial)B^{\mu} = 0, \end{split} $$ (31) where we have used,
$ u\cdot B = 0 $ . Applying Eq. (26), we eventually obtain,$$ \begin{align} (u\cdot\partial)\epsilon+(\epsilon+p)(\partial\cdot u) = & 0. \end{align} $$ (32) Using the thermodynamical relations (28, 29), we get,
$$ \partial_{\mu}(su^{\mu}) = 0. $$ (33) We can take a volume integral over Eq. (33) and obtain that
$$ \frac{{\rm {d}}S}{{\rm {d}}t} = \int\nolimits_V {\rm {d}}^3 x \partial_{\mu}(su^{\mu}) = 0, $$ (34) where
$ S $ is the total entropy for the whole system. The above equation mean the total entropy is conserved.Another equation for the energy-momentum conservation is
$$ \Delta_{\alpha \nu} \partial_\mu T^{\mu\nu} = 0, $$ (35) which gives us,
$$ \begin{split} (u\cdot\partial)u_{\alpha} = & \frac{1}{(\epsilon+p+B^{2})}\Big[\Delta_{\alpha}^{\nu}\partial_{\nu}\Big(P+\frac{1}{2}B^{2}\Big)+\\ &\Delta_{\mu\alpha}(B\cdot\partial)B^{\mu}+B_{\alpha}(\partial\cdot B)\Big]. \end{split} $$ (36) It tells us that the magnetic field may accelerate the fluid velocity. However, in our previous work[25-28, 30], we have found a kind of force-free type configuration for the magnetic fields. In such cases, the fluid velocity will not be acceptilated.
We can also rewrite the
$ T^{\mu\nu} $ as,$$ T^{\mu\nu} = (Ts+B^{2})u^{\mu}u^{\nu}-\Big(P+\frac{1}{2}B^{2}\Big)g^{\mu\nu}-B^{\mu}B^{\nu}. $$ (37) Eq. (5) reads, with the help of Eq. (28),
$$ \begin{split} su^{\mu}&\partial_{\mu}(Tu^{\nu})-\partial^{\nu}\Big(P+\frac{1}{2}B^{2}\Big) + \\ &\partial_{\mu}(B^{2}u^{\mu}u^{\nu})-\partial_{\mu}(B^{\mu}B^{\nu}) = 0. \end{split} $$ (38) Using the thermodynamic relation (28, 29) again, we get,
$$ \begin{split} & u^{\mu}\partial_{\mu}(Tu^{\nu})-\partial^{\nu}T \\ & = \frac{1}{s}\left[\partial^{\nu}\Big(\frac{1}{2}B^{2}\Big)-\partial_{\mu}\big(B^{2}u^{\mu}u^{\nu}\big)+ \partial_{\mu}\big(B^{\mu}B^{\nu}\big)\right]. \end{split} $$ (39) For convenience, we can introduce an antisymmetric
$ T $ -vorticity tensor$ \Xi^{\mu\nu} $ by[31-33],$$ \Xi^{\mu\nu} = \partial^{\nu}(Tu^{\mu})-\partial^{\mu}(Tu^{\nu}), $$ (40) which is like
$ F^{\mu\nu} $ in an electromagnetic field. In this case, we can use a compact form to rewrite Eq. (39),$$ \begin{align} \Xi^{\mu\nu}u_{\nu} = \frac{1}{s}\left[\partial^{\mu}\Big(\frac{1}{2}B^{2}\Big)-\partial_{\nu}(B^{2}u^{\mu}u^{\nu})+ \partial_{\nu}(B^{2}b^{\mu}b^{\nu})\right]. \end{align} $$ (41) Now, we will discuss the integral form of the
$ T $ -vorticity. By introducing the proper time$ \tau $ with the relation$ {\rm d}/{\rm d}\tau = u^{\mu}\partial_{\mu} $ , the above equation can be expressed as a circulation integral along a covariant loop$ L(\tau) $ ,$$ \begin{split} & \frac{\rm {d}}{{\rm {d}}\tau}\oint_{L(\tau)}Tu^{\mu}{\rm {d}}x_{\mu} = \oint_{L(\tau)}\frac{{\rm {d}}}{{\rm {d}}\tau}(Tu^{\mu}){\rm {d}}x_{\mu} \\ = & \oint_{L(\tau)}u^{\nu}\partial_{\nu}(Tu^{\mu}){\rm {d}}x_{\mu} = \oint_{L(\tau)}\Xi^{\mu\nu}u_{\nu}{\rm {d}}x_{\mu} \\ = & \oint_{L(\tau)}\frac{1}{s}\bigg[\partial^{\nu}\Big(\frac{1}{2}B^{2}\Big)-\partial_{\mu}(B^{2}u^{\mu}u^{\nu}) +\\ & \partial_{\mu}(B^{2}b^{\mu}b^{\nu})\bigg]{\rm {d}}x_{\nu} \end{split} $$ (42) In the first line, the differential acts on the
$ {\rm {d}}x^{\mu} $ is nonzero, i.e.$ \frac{{\rm {d}}}{{\rm {d}}\tau}{\rm {d}}x^{\mu} = {\rm {d}}\left(\frac{{\rm {d}}x^{\mu}}{{\rm {d}}\tau}\right) = {\rm {d}}u^{\mu} = {\rm {d}}x^{\nu}\partial_{\nu}u^{\mu} $ , but it vanishes since$ u^{\mu}\partial_{\nu}u_{\mu}{\rm {d}}x^{\nu} = 0 $ . In the second line we have used the Stokes theorem,$$ \oint_{L(\tau)}\partial^{\mu}T{\rm {d}}x_{\mu} = \oint_{S}\big[\partial^{\mu}\partial^{\nu}T- \partial^{\nu}\partial^{\mu}T\big]{\rm {d}}\sigma_{\mu\nu} = 0. $$ (43) Eq. (42) is the relativistic Kelvin circulation theorem for ideal MHD. It means that the magnetic fields may modify the
$ T $ -vorticity consideration. -
We will apply the main result in Eq. (42) to the one of the analytic solution found in Refs. [25-26].
In our work[25-26], we consider the ideal MHD with transverse magnetic fields in a Bjorken flow. The Bjorken flow means that the fluid velocity is given by,
$$ u^{\mu} = \left(\frac{t}{\tau},0,0,\frac{z}{\tau}\right), $$ (44) where
$ \tau $ is the proper time defined as,$$ \tau = \sqrt{t^{2}-z^{2}}. $$ (45) All the thermodynamic quantities, such as energy density, pressure, temperature, entropy density etc., will only depend on the proper time. The magnetic fields decays as,
$$ B^{\mu} = \left(0,0,B_{0}\frac{\tau_{0}}{\tau},0\right), $$ (46) where
$ B_{0} $ and$ \tau_{0} $ are the initial magnetic field and initial proper time, respectively.Inserting this solution into Eq. (42), we find that the
$$ \frac{{\rm {d}}}{{\rm {d}}\tau}\oint_{L(\tau)}Tu^{\mu}{\rm {d}}x_{\mu} = 0. $$ (47) for this special solutions. In this case, the
$ T $ -vorticity is still conserved.
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摘要: 探讨了理想磁流体力学中的相对论性Kelvin圈积分定理。相对论性Kelvin圈积分定理是指温度涡旋在圈积分下是一个守恒量。首先简略的回顾了相对论重离子碰撞中的理想磁流体力学相关结果。本文的核心是推导出理想磁流体力学中的相对论性Kelvin圈积分定理的表达形式。同时,也将已知的理想磁流体力学的解析解运用到该定理上。这一主要结果也可以运用到相对论重离子碰撞中磁流体力学的研究中。Abstract: We have studied the relativistic Kelvin circulation theorem for ideal Magnetohydrodynamics. The relativistic Kelvin circulation theorem is a conservation equation for the called T-vorticity, We have briefly reviewed the ideal magnetohydrodynamics in relativistic heavy ion collisions. The highlight of this work is that we have obtained the general expression of relativistic Kelvin circulation theorem for ideal Magnetohydrodynamics. We have also applied the analytic solutions of ideal magnetohydrodynamics in Bjorken flow to check our results. Our main results can also be implemented to relativistic magnetohydrodynamics in relativistic heavy ion collisions.
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