umbli

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These are questions asked by umbli

Hello,

I want to define an orthonormal tetrad basis of my choice in a spacetime having a metric given in some system of coordinates. My problem is that Maple automatically proposes an orthonormal metric but this is not the one that suits my requirements. So, I would like to specify the tetrad basis manually. As an example, I am trying to reproduce the calculations in sections 6 and 7 of the article https://arxiv.org/abs/gr-qc/0510083 . Here, the metric $g$ is given by the line element $ds^2 = - (c(t,r)^2 - v(t,r)^2) dt^2 + 2 v(t,r) dr dt + dr^2 + r^2 (d\theta^2 + sin(\theta)^2 d\phi^2)$ in $(t, r, \theta, \phi)$ coordinates. My chosen signature is (- + + +). Let, us adopt the convention used by Maple and denote spacetime indices by Greek alphabets and tetrad indices by lowercase Latin letters. Now, I would like to define a tetrad $e_a = (V, S, \Theta, \Phi)$ (as in section 7 of the article referred to above) where:

V^\mu = \frac{1}{c\sqrt{1-\beta(t,r)^2}}[1, - (v + c \beta), 0, 0] \\

S^\mu = \frac{1}{c\sqrt{1-\beta^2}}[-\beta, c + v \beta, 0, 0] \\

\Theta^\mu = [0,0,1,0]

\Phi^\mu = [0,0,0,1].

Here, $|\beta(t,r)| < 1$. I do not know how I may specify this in my worksheet. This may come of use somewhere later. Now, with this choice of the tetrad, we know that $g(e_a, e_b) = \eta_{ab}$ with $\eta$ being the Minkowski metric in spherical coordinates. After defining this tetrad basis, I finally want to calculate Einstein tensor, components of energy-momentum tensr etc. I have problem with constructing this orthonormal tetrad basis myself. It would be great if you could help me with this.

 

An additional curiosity: when we work with multiple tetrad bases, is it possible to denote the the tetrad indices by hatted tetrad labels themselves, as in $\eta_{\hat V, \hat \Theta}$?

 

Thank you.
 

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)

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

 

[IsTetrad, NullTetrad, OrthonormalTetrad, PetrovType, SegreType, TransformTetrad, e_, eta_, gamma_, l_, lambda_, m_, mb_, n_]

(5)

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078438692614)

(6)

``


 

Download dynBH.mw

 


 

 

 


I was starting to set up a curved axisymmetric metric using the Physics package and came across an error message that I could not resolve. I was actually writing the metric in the form given after output line (5) in the code attcahed below. This returned the error message:

Error, (in Physics:-Setup) invalid subscript selector

Then I started fiddling and discovered that somehow braces and order of coefficients are making a difference in the metric. I have written the flat space metric in three different ways after output line (2). The difference is only in the coefficient of the last $d\phi^2$ term. For some reason, $r^2 (sin(\theta))^2$ is shown as $r (sin(\theta))^4$ in output line (3). Removing the brackets around $sin(theta)$ or writing $r^2$ after it is resolving the problem. Is this in someway related to the whole square operation? Can you please help me understand why this is happening?

The original error message I was getting went away after I similarly changed the order of coefficients in the second term of the curved metric to get output (6). Here again, there was a whole square operation!

Thank you!

 

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_ = -dt^2+dr^2+r^2*dtheta^2+r(sin(theta))^4*dphi^2)

[metric = {(1, 1) = -1, (2, 2) = 1, (3, 3) = r^2, (4, 4) = r(sin(theta))^4}]

(3)

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

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

(4)

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

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

(5)

Setup(g_ = -exp(2*nu(r, theta))*dt^2+(exp(2*psi(r, theta)))(dphi-omega(r, theta)*dt)^2+(exp(2*mu(r, theta)))(dtheta)^2+exp(2*lambda(r, theta))*dr^2)

Error, (in Physics:-Setup) invalid subscript selector

 

Setup(g_ = -exp(2*nu(r, theta))*dt^2+(dphi-omega(r, theta)*dt)^2*exp(2*psi(r, theta))+exp(2*mu(r, theta))*dtheta^2+exp(2*lambda(r, theta))*dr^2)

[metric = {(1, 1) = -exp(2*nu(r, theta))+omega(r, theta)^2*exp(2*psi(r, theta)), (1, 4) = -omega(r, theta)*exp(2*psi(r, theta)), (2, 2) = exp(2*lambda(r, theta)), (3, 3) = exp(2*mu(r, theta)), (4, 4) = exp(2*psi(r, theta))}]

(6)

``


 

Download qstn.mw

While using the Physics package in Maple 2018, I am facing a few hurdles. In the file tov.mw ,

1. I have defined a dual-vector with components $u_a$ in step 12, but when I am trying to compute $g(u,u)$ using SumOverRepeatedIndices  in step 15, it is not evaluating the sum.

2. I want to define a tensor $B_{ab}=\nabla_a u_b$. Do I just write Define(B{a,b}=D_[a] u[b]) ?

3. Is there any command to find the symmetric and antisymmetric components and trace of the tensor B or do we evaluate them by writing out the expressions?

Thank you.

I am working with the Physics package in Maple 2018. I have a spacetime with a metric g and the vector field l is the tangent to a null curve. I have two problems:

(1) I want to check in maple that indeed g(l,l)=0. For this I wrote:

SumOverRepeatedIndices(g_[mu,nu]l[~mu]l[~nu])

Firstly, when I try to execute this command, the program keeps on evaluating for a long time. So, I have to `interrupt the current operation', undo this command, save and reopen the file, execute the entire worksheet again and then somehow it works. However, as you can see in step(13) of the attached worksheet, Maple returns an expression for this command but doesn't cancel out terms and show that it is indeed zero. How can I do this?

(2) I wish to find $(\nabla_l l)^\nu$. Is there a way to directly do this? Or should I do something like:

Define(L[mu,~nu]=D_mu l[~nu])

SumOverRepeatedIndices(l[~mu]L[mu,~nu])

Thanks a lot.

Edit: I actually carried out these steps too. First, I defined $L_\mu^\nu$ in step (14). Then, when I ask for the non-zero components of L, I find that the L\mu,\nu components are given. How can I get the non-zero L\mu\nu components? Also, in step (16), I did the `SumOverRepeatedindices' but it returned only a symbolic result without evaluationg.

try2.mw

Hello,

I am trying to work with the Physics package to calculate Christoffel symbols. I have just started using Maple today and am coming up against a few hurdles.

I want to change the signature of the metric from (+---) to (-+++). I am typing the following in math mode:

Setup( signature = '-+++')

However, this is returning the following error:

"Error, invalid sum/difference" with my command rewritten within a red box together with the last '+" within another red box. Can you please help me out with this?

Thank you.

 

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