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The matrix E and defining vectors...

umbli 60
October 09 2020
0 0

@ecterrab 

Thank you. I am trying to incorporate your suggestions but am stuck trying to do a simple thing. Can you please tell me how I may define the matrix E[a,~mu] (i.e. $E_a^\mu$) whose rows are given by the vectors $V,S,Theta,Phi$ mentioned in the opening question or in the worksheet attached below?

I have a  confusions about the steps in my worksheet given below. It is highlighted in yellow.
 

> 

restart
> 

with(Physics)

[`*`, `.`, Annihilation, AntiCommutator, Antisymmetrize, Assume, Bra, Bracket, Cactus, Check, Christoffel, Coefficients, Commutator, CompactDisplay, Coordinates, Creation, D_, Dagger, Decompose, Define, Dgamma, Einstein, EnergyMomentum, Expand, ExteriorDerivative, Factor, FeynmanDiagrams, Fundiff, Geodesics, GrassmannParity, Gtaylor, Intc, Inverse, Ket, KillingVectors, KroneckerDelta, LeviCivita, Library, LieBracket, LieDerivative, Normal, Parameters, PerformOnAnticommutativeSystem, Projector, Psigma, Redefine, Ricci, Riemann, Setup, Simplify, SpaceTimeVector, StandardModel, SubstituteTensor, SubstituteTensorIndices, SumOverRepeatedIndices, Symmetrize, TensorArray, Tetrads, ThreePlusOne, ToFieldComponents, ToSuperfields, Trace, TransformCoordinates, Vectors, Weyl, `^`, dAlembertian, d_, diff, g_, gamma_]

 (1)
> 

Setup(signature = `-+++`, coordinates = (X = [t, r, theta, phi]))

`* Partial match of 'coordinates' against keyword 'coordinatesystems'`

  

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (t, r, theta, phi)}

  

`Systems of spacetime Coordinates are: `*{X = (t, r, theta, phi)}

  

[coordinatesystems = {X}, signature = `- + + +`]

 (2)
> 

Setup(g_=-(c(t,r)^2 - v(t,r)^2)*dt^2 + 2*v(t,r)*dt*dr + dr^2 + r^2*dtheta^2 + r^2*sin(theta)^2*dphi^2)

[metric = {(1, 1) = -c(t, r)^2+v(t, r)^2, (1, 2) = v(t, r), (2, 2) = 1, (3, 3) = r^2, (4, 4) = r^2*sin(theta)^2}]

 (3)
> 

PDETools:-declare(c(t, r), v(t, r))

` c`(t, r)*`will now be displayed as`*c

  

` v`(t, r)*`will now be displayed as`*v

 (4)
> 

Define(beta(t,r));

`Defined objects with tensor properties`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (5)
> 

PDETools:-declare(beta(t,r))

` beta`(t, r)*`will now be displayed as`*beta

 (6)
> 

clear(V[~mu])

clear(V[`~mu`])

 (7)

First, I know this was a wrong command.Sorry!

> 

Define()

`Defined as tensors`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], S[`~mu`], V[`~mu`], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (8)
> 

Define(redo,S[~mu]=[-beta/(c*sqrt(1-beta^2)),(c+v*beta)/(c*sqrt(1-beta^2)),0,0]);

`Defined objects with tensor properties`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], S[`~mu`], V[`~mu`], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (9)
> 

S[`~mu`, matrix]

S[`~mu`] = Array(%id = 18446744078148776654)

 (10)
> 

Define(redo,V[~mu]=[1/(c*sqrt(1-beta^2)),-(v+c*beta)/(c*sqrt(1-beta^2)),0,0]);

`Defined objects with tensor properties`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], S[`~mu`], V[`~mu`], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (11)
> 

V[`~mu`, matrix]

V[`~mu`] = Array(%id = 18446744078253469198)

 (12)

 

In both equations (11) and (12), it can be seen that there is a common factor of $\frac{1}{c \sqrt{1-\beta^2}}$ multiplying each entry of the vector fields $V^\mu$  and $S^\mu$. How can I define a vector field by taking the common factor outside the matrix, as in say $A^\mu = a(t,r) (1, 0,1,1)$? This is just easier to input.

 

> 

Define(redo,Theta[~mu]=[0,0,1,0]);

`Defined objects with tensor properties`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], S[`~mu`], Theta[`~mu`], V[`~mu`], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (13)
> 

Define(redo,Phi[~mu]=[0,0,0,1]);

`Defined objects with tensor properties`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Phi[`~mu`], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], S[`~mu`], Theta[`~mu`], V[`~mu`], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (14)
> 

with(LinearAlgebra):
> 

Define()

`Defined as tensors`

  

{beta, Physics:-D_[mu], Physics:-Dgamma[mu], Phi[`~mu`], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], S[`~mu`], Theta[`~mu`], V[`~mu`], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (15)
> 

Define(E, query)

`Not defined as a tensor within the framework of the Physics package`

 (16)
> 

with(Tetrads)

`Setting lowercaselatin_ah letters to represent tetrad indices `

  

0, "%1 is not a command in the %2 package", Tetrads, Physics

  

0, "%1 is not a command in the %2 package", Tetrads, Physics

 (17)
> 

Define(E[a,~mu])

Error, bad index into Matrix

  
> 

``


I now want to construct the tensor $E_a^\mu$ whose rows are the vectors $V,S,\Theta,\Phi$ respectively i.e. $E_1^\mu = V^\mu$ and so on. I am not able to do this.

Finally, if I had defined the vector field V using the command

V := <1/(c*sqrt(1-beta^2)),-(v+c*beta)/(c*sqrt(1-beta^2)),0,0>

in the way matrices are defined within LinearAlgenra, would there be a difference in the way it is interpreted by the Physics package?

 

I am sorry if these are very naive questions. Any help would be greatly appreciated. Thank you.

Download dynBH.mw

@tomleslie So, it is a good idea to put ...

umbli 60
March 06 2020
0 0

@tomleslie So, it is a good idea to put a `.' between functions always!

@ecterrab This is great. Thanks a lot...

umbli 60
February 11 2020
0 0

@ecterrab This is great. Thanks a lot

So, functional dependence should always ...

umbli 60
June 04 2019
0 0

@ecterrab 

Thank you. It is a beautiful experience to work with the Physics package with your patient help. Thanks a lot. I have understood where I was going wrong. It is also nice to see how we can simply define the directional derivative in the direction of some vector field.

However, I have two further queries:

1. How do i type the nabla symbol for covariant derivative that you have used?

2. This is a curiosity: I notice that the tetrads are given by the notation l_,n_ etc. The metric is also given by g_. Do predefined tensors always have an underscore in the Physics package?

Thanks again.

Thank you...

umbli 60
June 03 2019
0 0

@ecterrab 

Thanks a lot for the explanation. I did not know that on a computer, lines and columns can be referenced by positive integers only.

changing values taken by spacetime indic...

umbli 60
June 02 2019
0 0

 

I discovered that Christoffel[0,0,0] and Christoffel[1,1,1] are the same. Both of them stand for \Gamma_{ttt}. Is there a way to change the values the spacetime indices take to (0,1,2,3) instead of (1,2,3,4)?

Using 0 as an index...

umbli 60
June 02 2019
0 0

Thanks a lot for the elaborate example.

However, I did not understand your warning, "Just be careful with using 0 as an index, since it always points to the timelike position,which now is 1". Suppose, I want to find $g_{tt}$. In Maple, should I write g_[00] or g_[tt] for that.?

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