Question: How do I define an orthonormal tetrad of my choice?


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 . 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.



[`*`, `.`, 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_]


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 = `- + + +`]


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}]


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



`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_]



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









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