bikramphy

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These are questions asked by bikramphy

Given a diagonal metric of signature (++- -)  for example the diagonal elements are [1,1,-1,-1], then how to find the Dirac gamma matrix in the standard representation or any representation.

How can I define an arbitrary signature of a space-time metric?

I want to define the signature of the form `++--`.

I am trying  to solve a first order differential equation.  But the maple is not able to solve. Please help me in finding the solutions...

Exact solution is not necessary. Solution in the series expansion is also be very helpful.

Here is the attached maple code..Please do check. Please scroll down to download the given maple file.


 

``

``

restart

with(Student[NumericalAnalysis])

with(PDEtools)

with(DEtools)

with(MmaTranslator)

[FromMma, FromMmaNotebook, Mma, MmaToMaple]

(1)

``

``

``

C1 := 2*M/(4*M*R^2*a-3*R^3*a)

``

D1 := -M*(8*M-9*R)*(3*ln(1+2*M/(4*M-3*R))+2*ln(1-2*M/R))/(3*R^2*(-16*M^2+26*M*R-9*R^2+(16*M^2-30*M*R+11*R^2)*ln(1+2*M/(4*M-3*R))+R^2*ln(-3*R/(4*M-3*R))))

``

A1 := R^2*(3*ln(1+2*M/(4*M-3*R))+2*ln(1-2*M/R))/(-32*M^2+52*M*R-18*R^2+(2*(16*M^2-30*M*R+11*R^2))*ln(1+2*M/(4*M-3*R))+2*R^2*ln(-3*R/(4*M-3*R)))

``

explambda := simplify((-2*C1*a*r^2+1)/(C1*a*r^2+1))

``

density := simplify(a*C1*(2*C1*a*r^2-3)/(-2*C1*a*r^2+1)^2)

``

pr := simplify(-D1+3*a*A1*C1*(2*C1*a*r^2-3)/(-2*C1*a*r^2+1)^2)

``

pt := simplify((D1*(-D1*r^2+4)-4*a^4*C1^4*r^6*(9*(1+A1)^2-(4*(1+3*A1))*D1*r^2+4*D1^2*r^4)+2*a*C1*(18*A1-(16+9*A1)*D1*r^2+4*D1^2*r^4)-3*a^2*C1^2*r^2*(9+16*A1+27*A1^2-(28*(1+A1))*D1*r^2+8*D1^2*r^4)+4*a^3*C1^3*r^4*(18+3*A1*(17+9*A1)-(10*(2+3*A1))*D1*r^2+8*D1^2*r^4))/((4*(C1*a*r^2+1))*(2*C1*a*r^2-1)^3))

``

drhodp := simplify(1/A1)

``

dptdp := (diff(pt, r))/(diff(pr, r))

``

A := simplify(-(-1-explambda)/r+4*explambda*Pi*r*(pr-density))

``

B := -(explambda^2+4*explambda+1)/r^2-64*explambda^2*Pi^2*r^2*pr^2-16*explambda*Pi*((-2+explambda)*pr-pt-density)-(-4*explambda*Pi*(pr+density)-4*explambda*Pi*drhodp*(pr+density))/dptdp

``

sys := {r*(diff(y(r), r))+y(r)^2+(A*r-1)*y(r)+r^2*B = 0}

 

[moderator edited to remove pages of output]
 

Download CALCULATION_OF_Y_v1.mw
 

I am trying to solve a differential equation numerically. But I am facing some difficulties. It is saying that ODE system has a removable singularity at r = 0. How can I remove the singularity?  Please check the attached figure and try to help.

Given a lagrangian  in general relativity , how can I calculate equation of motion using euler lagrangian equation in maple software ?

the lagrangian is 

L = 1/2 g[mu,nu] diff(x[~mu], lambda) diff(x[~nu],lambda)

here lambda is the affine parameter.  and the metric is 

 

 

metric coordinate = (t,u,x,y)

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