EMT_Toshmatov_et_al_Kerr_Newman.ipynb

Kernel: SageMath (stable)

Exact solution to nonlinear Maxwell's equations and Einstein's equations in Toshmatov et al.

This worksheet is devoted to the study of solutions to the Einstein and nonlinear Maxwell equations from Toshmatov et al. (https://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.084037)

%display latex

Spacetime: general M(r)

M = Manifold(4, 'M')
X.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
X

The mass and angular momentum parameters:

M0, a = var('M0 a', domain='real')
assume(M0>0, a>=0)
var('gam', latex_name="\gamma");

The metric:

var('Qm', latex_name="Q_m");
g = M.lorentzian_metric('g')
MM=function('M')(r)
rho2 = r^2 + (a*cos(th))^2
Delta = r^2 -(2*M(r)*r) + a^2
g[0,0] = -(1-(2*M(r)*r)/rho2)
g[0,3] = -a*(2*M(r)*r)*sin(th)^2/rho2
g[1,1], g[2,2] = rho2/Delta, rho2
g[3,3] = (r^2+a^2+(2*M(r)*r)*(a*sin(th))^2/rho2)*sin(th)^2
g.display()

gm1=g.inverse()

Electromagnetic field

We need the 1-forms $\mathrm{d}t$ and $\mathrm{d}\phi$ to form the electromagnetic field; we get them as follows:

X.coframe()

dt = X.coframe()[0]
dr = X.coframe()[1]
dtheta = X.coframe()[2]
dphi = X.coframe()[3]

The potential defined according to Eq. (25) of Toshmatov et al. (2017):

var('Q')
A = -Qm*a*cos(th)/rho2 * dt +  Qm*cos(th)*(r^2+a^2)/rho2 * dphi
A.set_name('A')
A.display()

F = A.exterior_derivative()
F.set_name('F')
F.display()

The electromagnetic field invariant $\mathcal{F} := F_{ab} F^{ab}$:

Fuu = F.up(g)
F2 = F['_ab']*Fuu['^ab']
print(F2)

Scalar field on the 4-dimensional differentiable manifold M

F2.display()

We introduce the Lagrangian density $L(\mathcal{F})$ for the electromagnetic field

L=gam*F2

Similarly, we introduce $\frac{\mathrm{d}L}{\mathrm{d}\mathcal{F}}$ as a lambda function, formed from the derivative of $L(x)$:

Nonlinear Maxwell equation

We introduce the derivative of the Lagrangian density $\mathcal{L_F(F)}$ for the electromagnetic field

LF=gam

The nonlinear Maxwell equations read $$\mathrm{d} H = 0$$ with $H := \star \left(\frac{\mathrm{d}L}{\mathrm{d}\mathcal{F}} F\right)$

H = (LF * F).hodge_dual(g)
H.set_name('H')
H.display()

Nonlinear Maxwell's equations are satisfied independantly of $M(r)$

dH=H.exterior_derivative()
dH==0

Einstein's equations

Energy-momentum tensor of the electromagnetic field

Fud = F.up(g,0)
Tud=-4*LF*F['_ab']*Fuu['^cb']+L*g['_ab']*gm1['^bc']
print(Tud)

Tensor field of type (1,1) on the 4-dimensional differentiable manifold M

Tud.display()

Tud[0,0]+Tud[1,1]+Tud[2,2]+Tud[3,3]

latex(Tud[0,0].factor())

latex(Tud[1,1].factor())

latex(Tud[2,2].factor())

latex(Tud[3,0].factor())

latex(Tud[3,3].factor())

Einstein's tensor

G = g.ricci() - 1/2*g.ricci_scalar()*g
G.set_name('G')
print(G)

Field of symmetric bilinear forms G on the 4-dimensional differentiable manifold M

Gud=G['_ab']*gm1['^bc']
Gud.display()

latex(Gud[0,0].simplify())

latex(Gud[1,1].simplify())

latex(Gud[2,2].simplify())

latex(Gud[3,0].simplify())

latex(Gud[3,3].simplify())

latex(Gud[0,0]+Gud[1,1]+Gud[2,2]+Gud[3,3])

Check that Einstein's equations are satisfied

Gud[0,0].expr().substitute_function(MM,M0+8*gam*pi*Q^2/r)

Gud.substitute_function(MM,M0+8*gam*pi*Q^2/r)==8*pi*Tud

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-33-5c5bdec69eff> in <module>()
----> 1 Gud.substitute_function(MM,M0+Integer(8)*gam*pi*Q**Integer(2)/r)==Integer(8)*pi*Tud

/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4518)()
    491             AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah'
    492         """
--> 493         return self.getattr_from_category(name)
    494 
    495     cdef getattr_from_category(self, name):
/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4627)()
    504         else:
    505             cls = P._abstract_element_class
--> 506         return getattr_from_other_class(self, cls, name)
    507 
    508     def __dir__(self):
/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/cpython/getattr.pyx in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2468)()
    387         dummy_error_message.cls = type(self)
    388         dummy_error_message.name = name
--> 389         raise AttributeError(dummy_error_message)
    390     cdef PyObject* attr = instance_getattr(cls, name)
    391     if attr is NULL:
AttributeError: 'TensorFieldFreeModule_with_category.element_class' object has no attribute 'substitute_function'