ecterrab

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These are answers submitted by ecterrab


 

Download position_of_the_timelike_component.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi
Both the Physics and DifferentialGeometry packages have PetrovType and SegreType commands. Take a look at the corresponding help pages, and there you will find examples. In particular, the SegreType command returns the Plebanski-Petrov and Segre classifications of the Ricci tensor corresponding to the spacetime metric set.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

If you input just Geodesics(), you get the same as when using output = equations, that is, the set of ODEs - the equations themselves - that you need to integrate. If you add initial conditions, you can numerically integrate those geodesic equations - see the help page ?dsolve/numeric.

By the way, it is with(Physics), (not physics, recall that Maple is case-sensitive), and Physics is a modern package developed and maintained by Maplesoft, that comes with your copy of Maple; it is not related in any way to GRTensor, which is an older package, not developed by Maplesoft and that does not come with your copy of Maple.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Given a worksheet that contains functions with dependencies, no, there is not a single command that will make "all the functions in the worksheet" have the their functionality suppressed.

There is however something closer. Suppose you have equation labels for all the output of your worksheet. Then, provided you are willing to type them (or click the equation labels one by one), the following will suppress the functionality of all the functions entering those equation labels:

PDEtools:-declare((1), (2), (3), ... (25), ...)  # list all the equation labels separated by commas.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Call your expression eee; then go with:

PDEtools:-Solve(eee, {a[3],b[2],c[0],c[2]}, independentof = {k[1], k[2], k[3], omega[1], omega[2], omega[3]});

This is what I get: {a[3] = 0, b[2] = 0, c[0] = c[0], c[2] = c[2]}

I suppose you could try also solve/identity

Edgardo S. Cheb-Terrab
Physics, DifferentialEquations and MathematicalFunctions, Maplesoft

 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Download Tricky_example.mw

UPDATE MAY 1: The code in dsolve is now slightly adjusted according with the comment above, so that now dsolve(eqn) automatically produces the simpler general solution, including the singular solutions, as if you had entered dsolve(eqn, way = 4). So this issue is resolved, and the adjustment is included in the Maplesoft Physics Updates for Maple 2023, v.1437 or newer.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Very nice challenge: your tensorial expression being defined as a tensor is of length > 50,000. One way to speed up matters is Setup(tensorsimplifier = normal), because normal is faster than the internal tensor simplifier (not so good as a simplifier though).

But in your example, something more radical is necessary. The underlying issue: when you define a tensor as a tensorial expression before proceeding, a test is performed to assure that the components can be computed. In your example, that goes a long way out of the road in the jungle of computer algebra normalizations, gcds and simplifications. One way to go is to allow for optionally "skipping" that step in Define. I'd need to see whether that has consequences (the verification of components is used in other places, what if it is not done).

I will write again in a few days, hopefully with a solution or workaround.

UPDATE May/1: in the Maplesoft Physics Updates v.1436 or newer, there is a new option for the Define command: computetensordependency = false (default value is true), which skips the most time-consuming verification step. Using this option, the Define(ition) of this tensorial expression of length > 50,000 is performed in ~100 seconds in a two-year-old Mac (M1 chip).

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft.

Hi
You didn't post a worksheet, so assuming "One_Form" is - say - a rank 1 contravariant tensor, see for instance the example show in ?RaiseLowerIndices with equation label (2.3), you have One_Form being "X := evalDG(D_x + 3*D_y)", the operation 
RaiseLowerIndices(g, X, [1])
produces the covariant version, and the operation 
RaiseLowerIndices(g, X, [1, 1])
applies recursively, as if the output was also a contravariant tensor. For example, 
RaiseLowerIndices(g, X, [1, 1, 1, 1, 1, 1])
applies recursively six times resulting in 
a^6 dx + 3 b^6 dy

By the way, what computations do you have in mind that would require this? Generally speaking, computations with tensors are easier using the Physics package. See the help page ?Physics,Tensors.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

As soon as you set a prefix for entering noncommutative objects, eg. Setup(noncommutativeprefix = Z), you can work with symbolic/general/unknown matrices, e.g. Z1 * Z2 - Z2 * Z1 without receiving 0, including the possibility of setting commutator and anticommutator rules for them. See the help pages for Physics:-`*` (the multiplication operator that is at work after you input with(Physics)) and also for Physics:-Setup. Additionally, you have Physics:-Inverse to represent (or compute) the multiplicative inverse, Physics:-Gtaylor for computing series involving these non or anti-commutative objects, and Physics:-diff for differentiatiing expressions involving them and Physics:-Simplify to simplify these algebraic expressions. If at some point you prefer to actually compute products of matrices, use Physics:-`.`. The corresponding help pages have illustrative examples.

Alternatively, you can compute with tensors with two indices representing these matrices (use genericindices to represent matrices of unknown dimensions see ?Physics:-Setup), and if you use this approach, also, Physics:-KroneckerDelta indexed with the indices of generic type you set represents a generic identity matrix.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

No, you cannot have two Riemann tensors "set at the same time".

It is however easy to have as many "Riemann tensors" as you want for different metrics; not the way you suggest but as follows.

 

with(Physics)

Start setting any non-flat metric, only to be able to work with formulas that do not automatically evaluate to zero when there is no curvature. E.g.

g_[sc]

Physics:-g_[mu, nu] = Matrix(%id = 36893488152008277756)

(1)

So now, how is the Riemann tensor computed?

Riemann[definition]

Physics:-Riemann[alpha, beta, mu, nu] = Physics:-g_[alpha, lambda]*(Physics:-d_[mu](Physics:-Christoffel[`~lambda`, beta, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~lambda`, beta, mu], [X])+Physics:-Christoffel[`~lambda`, upsilon, mu]*Physics:-Christoffel[`~upsilon`, beta, nu]-Physics:-Christoffel[`~lambda`, upsilon, nu]*Physics:-Christoffel[`~upsilon`, beta, mu])

(2)

Express everything in terms of the metric

lhs(Physics[Riemann][alpha, beta, mu, nu] = Physics[g_][alpha, lambda]*(Physics[d_][mu](Physics[Christoffel][`~lambda`, beta, nu], [X])-Physics[d_][nu](Physics[Christoffel][`~lambda`, beta, mu], [X])+Physics[Christoffel][`~lambda`, upsilon, mu]*Physics[Christoffel][`~upsilon`, beta, nu]-Physics[Christoffel][`~lambda`, upsilon, nu]*Physics[Christoffel][`~upsilon`, beta, mu])) = convert(rhs(Physics[Riemann][alpha, beta, mu, nu] = Physics[g_][alpha, lambda]*(Physics[d_][mu](Physics[Christoffel][`~lambda`, beta, nu], [X])-Physics[d_][nu](Physics[Christoffel][`~lambda`, beta, mu], [X])+Physics[Christoffel][`~lambda`, upsilon, mu]*Physics[Christoffel][`~upsilon`, beta, nu]-Physics[Christoffel][`~lambda`, upsilon, nu]*Physics[Christoffel][`~upsilon`, beta, mu])), g_)

Physics:-Riemann[alpha, beta, mu, nu] = Physics:-g_[alpha, lambda]*((1/2)*Physics:-d_[mu](Physics:-g_[`~lambda`, `~sigma`], [X])*(Physics:-d_[nu](Physics:-g_[beta, sigma], [X])+Physics:-d_[beta](Physics:-g_[nu, sigma], [X])-Physics:-d_[sigma](Physics:-g_[beta, nu], [X]))+(1/2)*Physics:-g_[`~lambda`, `~sigma`]*(Physics:-d_[mu](Physics:-d_[nu](Physics:-g_[beta, sigma], [X]), [X])+Physics:-d_[beta](Physics:-d_[mu](Physics:-g_[nu, sigma], [X]), [X])-Physics:-d_[mu](Physics:-d_[sigma](Physics:-g_[beta, nu], [X]), [X]))-(1/2)*Physics:-d_[nu](Physics:-g_[`~kappa`, `~lambda`], [X])*(Physics:-d_[mu](Physics:-g_[beta, kappa], [X])+Physics:-d_[beta](Physics:-g_[kappa, mu], [X])-Physics:-d_[kappa](Physics:-g_[beta, mu], [X]))-(1/2)*Physics:-g_[`~kappa`, `~lambda`]*(Physics:-d_[mu](Physics:-d_[nu](Physics:-g_[beta, kappa], [X]), [X])+Physics:-d_[beta](Physics:-d_[nu](Physics:-g_[kappa, mu], [X]), [X])-Physics:-d_[kappa](Physics:-d_[nu](Physics:-g_[beta, mu], [X]), [X]))+(1/4)*Physics:-g_[`~lambda`, `~tau`]*(Physics:-d_[upsilon](Physics:-g_[mu, tau], [X])+Physics:-d_[mu](Physics:-g_[tau, upsilon], [X])-Physics:-d_[tau](Physics:-g_[mu, upsilon], [X]))*Physics:-g_[`~omega`, `~upsilon`]*(Physics:-d_[nu](Physics:-g_[beta, omega], [X])+Physics:-d_[beta](Physics:-g_[nu, omega], [X])-Physics:-d_[omega](Physics:-g_[beta, nu], [X]))-(1/4)*Physics:-g_[`~chi`, `~lambda`]*(Physics:-d_[upsilon](Physics:-g_[chi, nu], [X])+Physics:-d_[nu](Physics:-g_[chi, upsilon], [X])-Physics:-d_[chi](Physics:-g_[nu, upsilon], [X]))*Physics:-g_[`~psi`, `~upsilon`]*(Physics:-d_[mu](Physics:-g_[beta, psi], [X])+Physics:-d_[beta](Physics:-g_[mu, psi], [X])-Physics:-d_[psi](Physics:-g_[beta, mu], [X])))

(3)

You see you have a sort of template here. Define a new tensor R and call g_ as G

Define(R[alpha, beta, mu, nu])

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

(4)

subs(Riemann = R, g_ = G, Physics[Riemann][alpha, beta, mu, nu] = Physics[g_][alpha, lambda]*((1/2)*Physics[d_][mu](Physics[g_][`~lambda`, `~sigma`], [X])*(Physics[d_][nu](Physics[g_][beta, sigma], [X])+Physics[d_][beta](Physics[g_][nu, sigma], [X])-Physics[d_][sigma](Physics[g_][beta, nu], [X]))+(1/2)*Physics[g_][`~lambda`, `~sigma`]*(Physics[d_][mu](Physics[d_][nu](Physics[g_][beta, sigma], [X]), [X])+Physics[d_][beta](Physics[d_][mu](Physics[g_][nu, sigma], [X]), [X])-Physics[d_][mu](Physics[d_][sigma](Physics[g_][beta, nu], [X]), [X]))-(1/2)*Physics[d_][nu](Physics[g_][`~kappa`, `~lambda`], [X])*(Physics[d_][mu](Physics[g_][beta, kappa], [X])+Physics[d_][beta](Physics[g_][kappa, mu], [X])-Physics[d_][kappa](Physics[g_][beta, mu], [X]))-(1/2)*Physics[g_][`~kappa`, `~lambda`]*(Physics[d_][mu](Physics[d_][nu](Physics[g_][beta, kappa], [X]), [X])+Physics[d_][beta](Physics[d_][nu](Physics[g_][kappa, mu], [X]), [X])-Physics[d_][kappa](Physics[d_][nu](Physics[g_][beta, mu], [X]), [X]))+(1/4)*Physics[g_][`~lambda`, `~tau`]*(Physics[d_][upsilon](Physics[g_][mu, tau], [X])+Physics[d_][mu](Physics[g_][tau, upsilon], [X])-Physics[d_][tau](Physics[g_][mu, upsilon], [X]))*Physics[g_][`~omega`, `~upsilon`]*(Physics[d_][nu](Physics[g_][beta, omega], [X])+Physics[d_][beta](Physics[g_][nu, omega], [X])-Physics[d_][omega](Physics[g_][beta, nu], [X]))-(1/4)*Physics[g_][`~chi`, `~lambda`]*(Physics[d_][upsilon](Physics[g_][chi, nu], [X])+Physics[d_][nu](Physics[g_][chi, upsilon], [X])-Physics[d_][chi](Physics[g_][nu, upsilon], [X]))*Physics[g_][`~psi`, `~upsilon`]*(Physics[d_][mu](Physics[g_][beta, psi], [X])+Physics[d_][beta](Physics[g_][mu, psi], [X])-Physics[d_][psi](Physics[g_][beta, mu], [X]))))

R[alpha, beta, mu, nu] = G[alpha, lambda]*((1/2)*Physics:-d_[mu](G[`~lambda`, `~sigma`], [X])*(Physics:-d_[nu](G[beta, sigma], [X])+Physics:-d_[beta](G[nu, sigma], [X])-Physics:-d_[sigma](G[beta, nu], [X]))+(1/2)*G[`~lambda`, `~sigma`]*(Physics:-d_[mu](Physics:-d_[nu](G[beta, sigma], [X]), [X])+Physics:-d_[beta](Physics:-d_[mu](G[nu, sigma], [X]), [X])-Physics:-d_[mu](Physics:-d_[sigma](G[beta, nu], [X]), [X]))-(1/2)*Physics:-d_[nu](G[`~kappa`, `~lambda`], [X])*(Physics:-d_[mu](G[beta, kappa], [X])+Physics:-d_[beta](G[kappa, mu], [X])-Physics:-d_[kappa](G[beta, mu], [X]))-(1/2)*G[`~kappa`, `~lambda`]*(Physics:-d_[mu](Physics:-d_[nu](G[beta, kappa], [X]), [X])+Physics:-d_[beta](Physics:-d_[nu](G[kappa, mu], [X]), [X])-Physics:-d_[kappa](Physics:-d_[nu](G[beta, mu], [X]), [X]))+(1/4)*G[`~lambda`, `~tau`]*(Physics:-d_[upsilon](G[mu, tau], [X])+Physics:-d_[mu](G[tau, upsilon], [X])-Physics:-d_[tau](G[mu, upsilon], [X]))*G[`~omega`, `~upsilon`]*(Physics:-d_[nu](G[beta, omega], [X])+Physics:-d_[beta](G[nu, omega], [X])-Physics:-d_[omega](G[beta, nu], [X]))-(1/4)*G[`~chi`, `~lambda`]*(Physics:-d_[upsilon](G[chi, nu], [X])+Physics:-d_[nu](G[chi, upsilon], [X])-Physics:-d_[chi](G[nu, upsilon], [X]))*G[`~psi`, `~upsilon`]*(Physics:-d_[mu](G[beta, psi], [X])+Physics:-d_[beta](G[mu, psi], [X])-Physics:-d_[psi](G[beta, mu], [X])))

(5)

 

And that is basically all. This approach is of use not only for the Riemann tensor but for everything that has a definition - pre-existing one or that you construct. To compute the components of this R tensor, indicate (define) G[mu, nu] to represent whatever metric (4 x 4 symmetric matrix) you want, and the formula above represents the corresponding Riemann. For example,

G[mu, nu] = (Matrix(4, 4, {(1, 1) = r^2/(e^2-2*m*r+r^2), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = (-e^2+2*m*r-r^2)/r^2}))

G[mu, nu] = Matrix(%id = 36893488152081444124)

(6)

"Define(?)  "

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

(7)

These are the components of R

TensorArray(R[alpha, beta, mu, nu] = G[alpha, lambda]*((1/2)*Physics[d_][mu](G[`~lambda`, `~sigma`], [X])*(Physics[d_][nu](G[beta, sigma], [X])+Physics[d_][beta](G[nu, sigma], [X])-Physics[d_][sigma](G[beta, nu], [X]))+(1/2)*G[`~lambda`, `~sigma`]*(Physics[d_][mu](Physics[d_][nu](G[beta, sigma], [X]), [X])+Physics[d_][beta](Physics[d_][mu](G[nu, sigma], [X]), [X])-Physics[d_][mu](Physics[d_][sigma](G[beta, nu], [X]), [X]))-(1/2)*Physics[d_][nu](G[`~kappa`, `~lambda`], [X])*(Physics[d_][mu](G[beta, kappa], [X])+Physics[d_][beta](G[kappa, mu], [X])-Physics[d_][kappa](G[beta, mu], [X]))-(1/2)*G[`~kappa`, `~lambda`]*(Physics[d_][mu](Physics[d_][nu](G[beta, kappa], [X]), [X])+Physics[d_][beta](Physics[d_][nu](G[kappa, mu], [X]), [X])-Physics[d_][kappa](Physics[d_][nu](G[beta, mu], [X]), [X]))+(1/4)*G[`~lambda`, `~tau`]*(Physics[d_][upsilon](G[mu, tau], [X])+Physics[d_][mu](G[tau, upsilon], [X])-Physics[d_][tau](G[mu, upsilon], [X]))*G[`~omega`, `~upsilon`]*(Physics[d_][nu](G[beta, omega], [X])+Physics[d_][beta](G[nu, omega], [X])-Physics[d_][omega](G[beta, nu], [X]))-(1/4)*G[`~chi`, `~lambda`]*(Physics[d_][upsilon](G[chi, nu], [X])+Physics[d_][nu](G[chi, upsilon], [X])-Physics[d_][chi](G[nu, upsilon], [X]))*G[`~psi`, `~upsilon`]*(Physics[d_][mu](G[beta, psi], [X])+Physics[d_][beta](G[mu, psi], [X])-Physics[d_][psi](G[beta, mu], [X]))), simplifier = simplify, output = setofequations)

{R[1, 1, 1, 1] = 0, R[1, 1, 1, 2] = 0, R[1, 1, 1, 3] = 0, R[1, 1, 1, 4] = 0, R[1, 1, 2, 1] = 0, R[1, 1, 2, 2] = 0, R[1, 1, 2, 3] = 0, R[1, 1, 2, 4] = 0, R[1, 1, 3, 1] = 0, R[1, 1, 3, 2] = 0, R[1, 1, 3, 3] = 0, R[1, 1, 3, 4] = 0, R[1, 1, 4, 1] = 0, R[1, 1, 4, 2] = 0, R[1, 1, 4, 3] = 0, R[1, 1, 4, 4] = 0, R[1, 2, 1, 1] = 0, R[1, 2, 1, 2] = (2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 2, 1, 3] = 0, R[1, 2, 1, 4] = 0, R[1, 2, 2, 1] = -(2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 2, 2, 2] = 0, R[1, 2, 2, 3] = 0, R[1, 2, 2, 4] = 0, R[1, 2, 3, 1] = 0, R[1, 2, 3, 2] = 0, R[1, 2, 3, 3] = 0, R[1, 2, 3, 4] = 0, R[1, 2, 4, 1] = 0, R[1, 2, 4, 2] = 0, R[1, 2, 4, 3] = 0, R[1, 2, 4, 4] = 0, R[1, 3, 1, 1] = 0, R[1, 3, 1, 2] = 0, R[1, 3, 1, 3] = (2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 1, 4] = 0, R[1, 3, 2, 1] = 0, R[1, 3, 2, 2] = 0, R[1, 3, 2, 3] = 0, R[1, 3, 2, 4] = 0, R[1, 3, 3, 1] = -(2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 3, 2] = 0, R[1, 3, 3, 3] = 0, R[1, 3, 3, 4] = 0, R[1, 3, 4, 1] = 0, R[1, 3, 4, 2] = 0, R[1, 3, 4, 3] = 0, R[1, 3, 4, 4] = 0, R[1, 4, 1, 1] = 0, R[1, 4, 1, 2] = 0, R[1, 4, 1, 3] = 0, R[1, 4, 1, 4] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-32*e^2*m-80*m^3)*r^9+(5*e^4+140*e^2*m^2+160*m^4)*r^8+(-42*e^4*m-312*e^2*m^3-160*m^5)*r^7+(e^6+136*e^4*m^2+352*e^2*m^4+64*m^6)*r^6+(-4*e^6*m-200*e^4*m^3-160*e^2*m^5)*r^5+(-5*e^8+4*e^6*m^2+112*e^4*m^4)*r^4+24*e^8*m*r^3+(-4*e^10-29*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[1, 4, 2, 1] = 0, R[1, 4, 2, 2] = 0, R[1, 4, 2, 3] = 0, R[1, 4, 2, 4] = 0, R[1, 4, 3, 1] = 0, R[1, 4, 3, 2] = 0, R[1, 4, 3, 3] = 0, R[1, 4, 3, 4] = 0, R[1, 4, 4, 1] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(32*e^2*m+80*m^3)*r^9+(-5*e^4-140*e^2*m^2-160*m^4)*r^8+(42*e^4*m+312*e^2*m^3+160*m^5)*r^7+(-e^6-136*e^4*m^2-352*e^2*m^4-64*m^6)*r^6+(4*e^6*m+200*e^4*m^3+160*e^2*m^5)*r^5+(5*e^8-4*e^6*m^2-112*e^4*m^4)*r^4-24*e^8*m*r^3+(4*e^10+29*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[1, 4, 4, 2] = 0, R[1, 4, 4, 3] = 0, R[1, 4, 4, 4] = 0, R[2, 1, 1, 1] = 0, R[2, 1, 1, 2] = -4*(m-(1/2)*r)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[2, 1, 1, 3] = 0, R[2, 1, 1, 4] = 0, R[2, 1, 2, 1] = r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, R[2, 1, 2, 2] = 0, R[2, 1, 2, 3] = 0, R[2, 1, 2, 4] = 0, R[2, 1, 3, 1] = 0, R[2, 1, 3, 2] = 0, R[2, 1, 3, 3] = 0, R[2, 1, 3, 4] = 0, R[2, 1, 4, 1] = 0, R[2, 1, 4, 2] = 0, R[2, 1, 4, 3] = 0, R[2, 1, 4, 4] = 0, R[2, 2, 1, 1] = 0, R[2, 2, 1, 2] = 0, R[2, 2, 1, 3] = 0, R[2, 2, 1, 4] = 0, R[2, 2, 2, 1] = 0, R[2, 2, 2, 2] = 0, R[2, 2, 2, 3] = 0, R[2, 2, 2, 4] = 0, R[2, 2, 3, 1] = 0, R[2, 2, 3, 2] = 0, R[2, 2, 3, 3] = 0, R[2, 2, 3, 4] = 0, R[2, 2, 4, 1] = 0, R[2, 2, 4, 2] = 0, R[2, 2, 4, 3] = 0, R[2, 2, 4, 4] = 0, R[2, 3, 1, 1] = 0, R[2, 3, 1, 2] = 0, R[2, 3, 1, 3] = 0, R[2, 3, 1, 4] = 0, R[2, 3, 2, 1] = 0, R[2, 3, 2, 2] = 0, R[2, 3, 2, 3] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 3, 2, 4] = 0, R[2, 3, 3, 1] = 0, R[2, 3, 3, 2] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 3, 3, 3] = 0, R[2, 3, 3, 4] = 0, R[2, 3, 4, 1] = 0, R[2, 3, 4, 2] = 0, R[2, 3, 4, 3] = 0, R[2, 3, 4, 4] = 0, R[2, 4, 1, 1] = 0, R[2, 4, 1, 2] = 0, R[2, 4, 1, 3] = 0, R[2, 4, 1, 4] = 0, R[2, 4, 2, 1] = 0, R[2, 4, 2, 2] = 0, R[2, 4, 2, 3] = 0, R[2, 4, 2, 4] = -4*(m-(1/2)*r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[2, 4, 3, 1] = 0, R[2, 4, 3, 2] = 0, R[2, 4, 3, 3] = 0, R[2, 4, 3, 4] = 0, R[2, 4, 4, 1] = 0, R[2, 4, 4, 2] = (2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[2, 4, 4, 3] = 0, R[2, 4, 4, 4] = 0, R[3, 1, 1, 1] = 0, R[3, 1, 1, 2] = 0, R[3, 1, 1, 3] = -4*(m-(1/2)*r)^2*sin(theta)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 1, 1, 4] = 0, R[3, 1, 2, 1] = 0, R[3, 1, 2, 2] = 0, R[3, 1, 2, 3] = 0, R[3, 1, 2, 4] = 0, R[3, 1, 3, 1] = sin(theta)^2*r^2*(2*m-r)^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 1, 3, 2] = 0, R[3, 1, 3, 3] = 0, R[3, 1, 3, 4] = 0, R[3, 1, 4, 1] = 0, R[3, 1, 4, 2] = 0, R[3, 1, 4, 3] = 0, R[3, 1, 4, 4] = 0, R[3, 2, 1, 1] = 0, R[3, 2, 1, 2] = 0, R[3, 2, 1, 3] = 0, R[3, 2, 1, 4] = 0, R[3, 2, 2, 1] = 0, R[3, 2, 2, 2] = 0, R[3, 2, 2, 3] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 2, 2, 4] = 0, R[3, 2, 3, 1] = 0, R[3, 2, 3, 2] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 2, 3, 3] = 0, R[3, 2, 3, 4] = 0, R[3, 2, 4, 1] = 0, R[3, 2, 4, 2] = 0, R[3, 2, 4, 3] = 0, R[3, 2, 4, 4] = 0, R[3, 3, 1, 1] = 0, R[3, 3, 1, 2] = 0, R[3, 3, 1, 3] = 0, R[3, 3, 1, 4] = 0, R[3, 3, 2, 1] = 0, R[3, 3, 2, 2] = 0, R[3, 3, 2, 3] = 0, R[3, 3, 2, 4] = 0, R[3, 3, 3, 1] = 0, R[3, 3, 3, 2] = 0, R[3, 3, 3, 3] = 0, R[3, 3, 3, 4] = 0, R[3, 3, 4, 1] = 0, R[3, 3, 4, 2] = 0, R[3, 3, 4, 3] = 0, R[3, 3, 4, 4] = 0, R[3, 4, 1, 1] = 0, R[3, 4, 1, 2] = 0, R[3, 4, 1, 3] = 0, R[3, 4, 1, 4] = 0, R[3, 4, 2, 1] = 0, R[3, 4, 2, 2] = 0, R[3, 4, 2, 3] = 0, R[3, 4, 2, 4] = 0, R[3, 4, 3, 1] = 0, R[3, 4, 3, 2] = 0, R[3, 4, 3, 3] = 0, R[3, 4, 3, 4] = -4*(m-(1/2)*r)^2*sin(theta)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 4, 4, 1] = 0, R[3, 4, 4, 2] = 0, R[3, 4, 4, 3] = sin(theta)^2*(2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 4, 4, 4] = 0, R[4, 1, 1, 1] = 0, R[4, 1, 1, 2] = 0, R[4, 1, 1, 3] = 0, R[4, 1, 1, 4] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(40*e^2*m+80*m^3)*r^9+(-13*e^4-196*e^2*m^2-160*m^4)*r^8+(114*e^4*m+456*e^2*m^3+160*m^5)*r^7+(-17*e^6-368*e^4*m^2-512*e^2*m^4-64*m^6)*r^6+(108*e^6*m+520*e^4*m^3+224*e^2*m^5)*r^5+(-11*e^8-228*e^6*m^2-272*e^4*m^4)*r^4+(48*e^8*m+160*e^6*m^3)*r^3+(-4*e^10-53*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 1, 2, 1] = 0, R[4, 1, 2, 2] = 0, R[4, 1, 2, 3] = 0, R[4, 1, 2, 4] = 0, R[4, 1, 3, 1] = 0, R[4, 1, 3, 2] = 0, R[4, 1, 3, 3] = 0, R[4, 1, 3, 4] = 0, R[4, 1, 4, 1] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-40*e^2*m-80*m^3)*r^9+(13*e^4+196*e^2*m^2+160*m^4)*r^8+(-114*e^4*m-456*e^2*m^3-160*m^5)*r^7+(17*e^6+368*e^4*m^2+512*e^2*m^4+64*m^6)*r^6+(-108*e^6*m-520*e^4*m^3-224*e^2*m^5)*r^5+(11*e^8+228*e^6*m^2+272*e^4*m^4)*r^4+(-48*e^8*m-160*e^6*m^3)*r^3+(4*e^10+53*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 1, 4, 2] = 0, R[4, 1, 4, 3] = 0, R[4, 1, 4, 4] = 0, R[4, 2, 1, 1] = 0, R[4, 2, 1, 2] = 0, R[4, 2, 1, 3] = 0, R[4, 2, 1, 4] = 0, R[4, 2, 2, 1] = 0, R[4, 2, 2, 2] = 0, R[4, 2, 2, 3] = 0, R[4, 2, 2, 4] = (e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 2, 3, 1] = 0, R[4, 2, 3, 2] = 0, R[4, 2, 3, 3] = 0, R[4, 2, 3, 4] = 0, R[4, 2, 4, 1] = 0, R[4, 2, 4, 2] = -(e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 2, 4, 3] = 0, R[4, 2, 4, 4] = 0, R[4, 3, 1, 1] = 0, R[4, 3, 1, 2] = 0, R[4, 3, 1, 3] = 0, R[4, 3, 1, 4] = 0, R[4, 3, 2, 1] = 0, R[4, 3, 2, 2] = 0, R[4, 3, 2, 3] = 0, R[4, 3, 2, 4] = 0, R[4, 3, 3, 1] = 0, R[4, 3, 3, 2] = 0, R[4, 3, 3, 3] = 0, R[4, 3, 3, 4] = (e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, R[4, 3, 4, 1] = 0, R[4, 3, 4, 2] = 0, R[4, 3, 4, 3] = -(e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, R[4, 3, 4, 4] = 0, R[4, 4, 1, 1] = 0, R[4, 4, 1, 2] = 0, R[4, 4, 1, 3] = 0, R[4, 4, 1, 4] = 0, R[4, 4, 2, 1] = 0, R[4, 4, 2, 2] = 0, R[4, 4, 2, 3] = 0, R[4, 4, 2, 4] = 0, R[4, 4, 3, 1] = 0, R[4, 4, 3, 2] = 0, R[4, 4, 3, 3] = 0, R[4, 4, 3, 4] = 0, R[4, 4, 4, 1] = 0, R[4, 4, 4, 2] = 0, R[4, 4, 4, 3] = 0, R[4, 4, 4, 4] = 0}

(8)

If you want to ignore those with right-hand side equal to zero, go with

remove(proc (u) options operator, arrow; rhs(u) = 0 end proc, TensorArray(R[alpha, beta, mu, nu] = G[alpha, lambda]*((1/2)*Physics[d_][mu](G[`~lambda`, `~sigma`], [X])*(Physics[d_][nu](G[beta, sigma], [X])+Physics[d_][beta](G[nu, sigma], [X])-Physics[d_][sigma](G[beta, nu], [X]))+(1/2)*G[`~lambda`, `~sigma`]*(Physics[d_][mu](Physics[d_][nu](G[beta, sigma], [X]), [X])+Physics[d_][beta](Physics[d_][mu](G[nu, sigma], [X]), [X])-Physics[d_][mu](Physics[d_][sigma](G[beta, nu], [X]), [X]))-(1/2)*Physics[d_][nu](G[`~kappa`, `~lambda`], [X])*(Physics[d_][mu](G[beta, kappa], [X])+Physics[d_][beta](G[kappa, mu], [X])-Physics[d_][kappa](G[beta, mu], [X]))-(1/2)*G[`~kappa`, `~lambda`]*(Physics[d_][mu](Physics[d_][nu](G[beta, kappa], [X]), [X])+Physics[d_][beta](Physics[d_][nu](G[kappa, mu], [X]), [X])-Physics[d_][kappa](Physics[d_][nu](G[beta, mu], [X]), [X]))+(1/4)*G[`~lambda`, `~tau`]*(Physics[d_][upsilon](G[mu, tau], [X])+Physics[d_][mu](G[tau, upsilon], [X])-Physics[d_][tau](G[mu, upsilon], [X]))*G[`~omega`, `~upsilon`]*(Physics[d_][nu](G[beta, omega], [X])+Physics[d_][beta](G[nu, omega], [X])-Physics[d_][omega](G[beta, nu], [X]))-(1/4)*G[`~chi`, `~lambda`]*(Physics[d_][upsilon](G[chi, nu], [X])+Physics[d_][nu](G[chi, upsilon], [X])-Physics[d_][chi](G[nu, upsilon], [X]))*G[`~psi`, `~upsilon`]*(Physics[d_][mu](G[beta, psi], [X])+Physics[d_][beta](G[mu, psi], [X])-Physics[d_][psi](G[beta, mu], [X]))), simplifier = simplify, output = setofequations))

{R[1, 2, 1, 2] = (2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 2, 2, 1] = -(2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 1, 3] = (2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 3, 1] = -(2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 4, 1, 4] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-32*e^2*m-80*m^3)*r^9+(5*e^4+140*e^2*m^2+160*m^4)*r^8+(-42*e^4*m-312*e^2*m^3-160*m^5)*r^7+(e^6+136*e^4*m^2+352*e^2*m^4+64*m^6)*r^6+(-4*e^6*m-200*e^4*m^3-160*e^2*m^5)*r^5+(-5*e^8+4*e^6*m^2+112*e^4*m^4)*r^4+24*e^8*m*r^3+(-4*e^10-29*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[1, 4, 4, 1] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(32*e^2*m+80*m^3)*r^9+(-5*e^4-140*e^2*m^2-160*m^4)*r^8+(42*e^4*m+312*e^2*m^3+160*m^5)*r^7+(-e^6-136*e^4*m^2-352*e^2*m^4-64*m^6)*r^6+(4*e^6*m+200*e^4*m^3+160*e^2*m^5)*r^5+(5*e^8-4*e^6*m^2-112*e^4*m^4)*r^4-24*e^8*m*r^3+(4*e^10+29*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[2, 1, 1, 2] = -4*(m-(1/2)*r)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[2, 1, 2, 1] = r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, R[2, 3, 2, 3] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 3, 3, 2] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 4, 2, 4] = -4*(m-(1/2)*r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[2, 4, 4, 2] = (2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 1, 1, 3] = -4*(m-(1/2)*r)^2*sin(theta)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 1, 3, 1] = sin(theta)^2*r^2*(2*m-r)^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 2, 2, 3] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 2, 3, 2] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 4, 3, 4] = -4*(m-(1/2)*r)^2*sin(theta)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 4, 4, 3] = sin(theta)^2*(2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[4, 1, 1, 4] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(40*e^2*m+80*m^3)*r^9+(-13*e^4-196*e^2*m^2-160*m^4)*r^8+(114*e^4*m+456*e^2*m^3+160*m^5)*r^7+(-17*e^6-368*e^4*m^2-512*e^2*m^4-64*m^6)*r^6+(108*e^6*m+520*e^4*m^3+224*e^2*m^5)*r^5+(-11*e^8-228*e^6*m^2-272*e^4*m^4)*r^4+(48*e^8*m+160*e^6*m^3)*r^3+(-4*e^10-53*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 1, 4, 1] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-40*e^2*m-80*m^3)*r^9+(13*e^4+196*e^2*m^2+160*m^4)*r^8+(-114*e^4*m-456*e^2*m^3-160*m^5)*r^7+(17*e^6+368*e^4*m^2+512*e^2*m^4+64*m^6)*r^6+(-108*e^6*m-520*e^4*m^3-224*e^2*m^5)*r^5+(11*e^8+228*e^6*m^2+272*e^4*m^4)*r^4+(-48*e^8*m-160*e^6*m^3)*r^3+(4*e^10+53*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 2, 2, 4] = (e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 2, 4, 2] = -(e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 3, 3, 4] = (e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, R[4, 3, 4, 3] = -(e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4}

(9)

To get the components of Riemann for a different metric, naturally, you need to indicate the different metric. You get the idea. Define G and get the component of R using TensorArray.

 

You may prefer to explore these components, as in

TensorArray(rhs(R[alpha, beta, mu, nu] = G[alpha, lambda]*((1/2)*Physics[d_][mu](G[`~lambda`, `~sigma`], [X])*(Physics[d_][nu](G[beta, sigma], [X])+Physics[d_][beta](G[nu, sigma], [X])-Physics[d_][sigma](G[beta, nu], [X]))+(1/2)*G[`~lambda`, `~sigma`]*(Physics[d_][mu](Physics[d_][nu](G[beta, sigma], [X]), [X])+Physics[d_][beta](Physics[d_][mu](G[nu, sigma], [X]), [X])-Physics[d_][mu](Physics[d_][sigma](G[beta, nu], [X]), [X]))-(1/2)*Physics[d_][nu](G[`~kappa`, `~lambda`], [X])*(Physics[d_][mu](G[beta, kappa], [X])+Physics[d_][beta](G[kappa, mu], [X])-Physics[d_][kappa](G[beta, mu], [X]))-(1/2)*G[`~kappa`, `~lambda`]*(Physics[d_][mu](Physics[d_][nu](G[beta, kappa], [X]), [X])+Physics[d_][beta](Physics[d_][nu](G[kappa, mu], [X]), [X])-Physics[d_][kappa](Physics[d_][nu](G[beta, mu], [X]), [X]))+(1/4)*G[`~lambda`, `~tau`]*(Physics[d_][upsilon](G[mu, tau], [X])+Physics[d_][mu](G[tau, upsilon], [X])-Physics[d_][tau](G[mu, upsilon], [X]))*G[`~omega`, `~upsilon`]*(Physics[d_][nu](G[beta, omega], [X])+Physics[d_][beta](G[nu, omega], [X])-Physics[d_][omega](G[beta, nu], [X]))-(1/4)*G[`~chi`, `~lambda`]*(Physics[d_][upsilon](G[chi, nu], [X])+Physics[d_][nu](G[chi, upsilon], [X])-Physics[d_][chi](G[nu, upsilon], [X]))*G[`~psi`, `~upsilon`]*(Physics[d_][mu](G[beta, psi], [X])+Physics[d_][beta](G[mu, psi], [X])-Physics[d_][psi](G[beta, mu], [X])))), simplifier = simplify, explore)

" G[alpha,lambda] ((`∂`[mu](G[]^(lambda,sigma)) (`∂`[nu](G[beta,sigma])+`∂`[beta](G[nu,sigma])-`∂`[sigma](G[beta,nu])))/2+(G[]^(lambda,sigma) (`∂`[mu](`∂`[nu](G[beta,sigma]))+`∂`[beta](`∂`[mu](G[nu,sigma]))-`∂`[mu](`∂`[sigma](G[beta,nu]))))/2-(`∂`[nu](G[]^(kappa,lambda)) (`∂`[mu](G[beta,kappa])+`∂`[beta](G[kappa,mu])-`∂`[kappa](G[beta,mu])))/2-(G[]^(kappa,lambda) (`∂`[mu](`∂`[nu](G[beta,kappa]))+`∂`[beta](`∂`[nu](G[kappa,mu]))-`∂`[kappa](`∂`[nu](G[beta,mu]))))/2+(G[]^(lambda,tau) (`∂`[upsilon](G[mu,tau])+`∂`[mu](G[tau,upsilon])-`∂`[tau](G[mu,upsilon])) G[]^(omega,upsilon) (`∂`[nu](G[beta,omega])+`∂`[beta](G[nu,omega])-`∂`[omega](G[beta,nu])))/4-(G[]^(chi,lambda) (`∂`[upsilon](G[chi,nu])+`∂`[nu](G[chi,upsilon])-`∂`[chi](G[nu,upsilon])) G[]^(psi,upsilon) (`∂`[mu](G[beta,psi])+`∂`[beta](G[mu,psi])-`∂`[psi](G[beta,mu])))/4)`      `(`ordering of free indices`=[alpha,beta,mu,nu])"

(10)

Or get all the non-zero ones in a sort of more compact form

ArrayElems(TensorArray(rhs(R[alpha, beta, mu, nu] = G[alpha, lambda]*((1/2)*Physics[d_][mu](G[`~lambda`, `~sigma`], [X])*(Physics[d_][nu](G[beta, sigma], [X])+Physics[d_][beta](G[nu, sigma], [X])-Physics[d_][sigma](G[beta, nu], [X]))+(1/2)*G[`~lambda`, `~sigma`]*(Physics[d_][mu](Physics[d_][nu](G[beta, sigma], [X]), [X])+Physics[d_][beta](Physics[d_][mu](G[nu, sigma], [X]), [X])-Physics[d_][mu](Physics[d_][sigma](G[beta, nu], [X]), [X]))-(1/2)*Physics[d_][nu](G[`~kappa`, `~lambda`], [X])*(Physics[d_][mu](G[beta, kappa], [X])+Physics[d_][beta](G[kappa, mu], [X])-Physics[d_][kappa](G[beta, mu], [X]))-(1/2)*G[`~kappa`, `~lambda`]*(Physics[d_][mu](Physics[d_][nu](G[beta, kappa], [X]), [X])+Physics[d_][beta](Physics[d_][nu](G[kappa, mu], [X]), [X])-Physics[d_][kappa](Physics[d_][nu](G[beta, mu], [X]), [X]))+(1/4)*G[`~lambda`, `~tau`]*(Physics[d_][upsilon](G[mu, tau], [X])+Physics[d_][mu](G[tau, upsilon], [X])-Physics[d_][tau](G[mu, upsilon], [X]))*G[`~omega`, `~upsilon`]*(Physics[d_][nu](G[beta, omega], [X])+Physics[d_][beta](G[nu, omega], [X])-Physics[d_][omega](G[beta, nu], [X]))-(1/4)*G[`~chi`, `~lambda`]*(Physics[d_][upsilon](G[chi, nu], [X])+Physics[d_][nu](G[chi, upsilon], [X])-Physics[d_][chi](G[nu, upsilon], [X]))*G[`~psi`, `~upsilon`]*(Physics[d_][mu](G[beta, psi], [X])+Physics[d_][beta](G[mu, psi], [X])-Physics[d_][psi](G[beta, mu], [X]))))))

{(1, 2, 1, 2) = (4*e^6*m-2*e^6*r-8*e^4*m^3-8*e^4*m^2*r+14*e^4*m*r^2-4*e^4*r^3+16*e^2*m^4*r-24*e^2*m^2*r^3+16*e^2*m*r^4-3*e^2*r^5-16*m^5*r^2+32*m^4*r^3-24*m^3*r^4+8*m^2*r^5-m*r^6)*r^3/(e^2-2*m*r+r^2)^5, (1, 2, 2, 1) = -(4*e^6*m-2*e^6*r-8*e^4*m^3-8*e^4*m^2*r+14*e^4*m*r^2-4*e^4*r^3+16*e^2*m^4*r-24*e^2*m^2*r^3+16*e^2*m*r^4-3*e^2*r^5-16*m^5*r^2+32*m^4*r^3-24*m^3*r^4+8*m^2*r^5-m*r^6)*r^3/(e^2-2*m*r+r^2)^5, (1, 3, 1, 3) = r^2*(4*sin(theta)^2*csc(theta)^2*e^6*m^2+sin(theta)^2*csc(theta)^2*e^6*r^2+3*sin(theta)^2*csc(theta)^2*e^4*r^4+3*sin(theta)^2*csc(theta)^2*e^2*r^6-32*sin(theta)^2*csc(theta)^2*m^5*r^3+80*sin(theta)^2*csc(theta)^2*m^4*r^4-80*sin(theta)^2*csc(theta)^2*m^3*r^5+40*sin(theta)^2*csc(theta)^2*m^2*r^6-10*sin(theta)^2*csc(theta)^2*m*r^7-4*e^6*m^2-3*e^6*r^2-7*e^4*r^4-6*e^2*r^6+16*m^5*r^3-48*m^4*r^4+56*m^3*r^5-32*m^2*r^6+9*m*r^7-r^8-4*sin(theta)^2*csc(theta)^2*e^6*m*r-24*sin(theta)^2*csc(theta)^2*e^4*m^3*r+36*sin(theta)^2*csc(theta)^2*e^4*m^2*r^2-18*sin(theta)^2*csc(theta)^2*e^4*m*r^3+48*sin(theta)^2*csc(theta)^2*e^2*m^4*r^2-96*sin(theta)^2*csc(theta)^2*e^2*m^3*r^3+72*sin(theta)^2*csc(theta)^2*e^2*m^2*r^4-24*sin(theta)^2*csc(theta)^2*e^2*m*r^5+8*e^6*m*r+16*e^4*m^3*r-44*e^4*m^2*r^2+32*e^4*m*r^3-32*e^2*m^4*r^2+96*e^2*m^3*r^3-96*e^2*m^2*r^4+40*e^2*m*r^5+sin(theta)^2*csc(theta)^2*r^8)*sin(theta)^2/(e^2-2*m*r+r^2)^5, (1, 3, 2, 3) = r^3*(sin(theta)^2*csc(theta)^2-1)*cos(theta)*sin(theta)*(2*m-r)^2/(e^2-2*m*r+r^2)^2, (1, 3, 3, 1) = -r^2*(4*sin(theta)^2*csc(theta)^2*e^6*m^2+sin(theta)^2*csc(theta)^2*e^6*r^2+3*sin(theta)^2*csc(theta)^2*e^4*r^4+3*sin(theta)^2*csc(theta)^2*e^2*r^6-32*sin(theta)^2*csc(theta)^2*m^5*r^3+80*sin(theta)^2*csc(theta)^2*m^4*r^4-80*sin(theta)^2*csc(theta)^2*m^3*r^5+40*sin(theta)^2*csc(theta)^2*m^2*r^6-10*sin(theta)^2*csc(theta)^2*m*r^7-4*e^6*m^2-3*e^6*r^2-7*e^4*r^4-6*e^2*r^6+16*m^5*r^3-48*m^4*r^4+56*m^3*r^5-32*m^2*r^6+9*m*r^7-r^8-4*sin(theta)^2*csc(theta)^2*e^6*m*r-24*sin(theta)^2*csc(theta)^2*e^4*m^3*r+36*sin(theta)^2*csc(theta)^2*e^4*m^2*r^2-18*sin(theta)^2*csc(theta)^2*e^4*m*r^3+48*sin(theta)^2*csc(theta)^2*e^2*m^4*r^2-96*sin(theta)^2*csc(theta)^2*e^2*m^3*r^3+72*sin(theta)^2*csc(theta)^2*e^2*m^2*r^4-24*sin(theta)^2*csc(theta)^2*e^2*m*r^5+8*e^6*m*r+16*e^4*m^3*r-44*e^4*m^2*r^2+32*e^4*m*r^3-32*e^2*m^4*r^2+96*e^2*m^3*r^3-96*e^2*m^2*r^4+40*e^2*m*r^5+sin(theta)^2*csc(theta)^2*r^8)*sin(theta)^2/(e^2-2*m*r+r^2)^5, (1, 3, 3, 2) = -r^3*(sin(theta)^2*csc(theta)^2-1)*cos(theta)*sin(theta)*(2*m-r)^2/(e^2-2*m*r+r^2)^2, (1, 4, 1, 4) = -(e^12-10*e^10*m*r+4*e^10*r^2+29*e^8*m^2*r^2-24*e^8*m*r^3+5*e^8*r^4-4*e^6*m^2*r^4+4*e^6*m*r^5-e^6*r^6-112*e^4*m^4*r^4+200*e^4*m^3*r^5-136*e^4*m^2*r^6+42*e^4*m*r^7-5*e^4*r^8+160*e^2*m^5*r^5-352*e^2*m^4*r^6+312*e^2*m^3*r^7-140*e^2*m^2*r^8+32*e^2*m*r^9-3*e^2*r^10-64*m^6*r^6+160*m^5*r^7-160*m^4*r^8+80*m^3*r^9-20*m^2*r^10+2*m*r^11)/(r^4*(e^2-2*m*r+r^2)^5), (1, 4, 4, 1) = (e^12-10*e^10*m*r+4*e^10*r^2+29*e^8*m^2*r^2-24*e^8*m*r^3+5*e^8*r^4-4*e^6*m^2*r^4+4*e^6*m*r^5-e^6*r^6-112*e^4*m^4*r^4+200*e^4*m^3*r^5-136*e^4*m^2*r^6+42*e^4*m*r^7-5*e^4*r^8+160*e^2*m^5*r^5-352*e^2*m^4*r^6+312*e^2*m^3*r^7-140*e^2*m^2*r^8+32*e^2*m*r^9-3*e^2*r^10-64*m^6*r^6+160*m^5*r^7-160*m^4*r^8+80*m^3*r^9-20*m^2*r^10+2*m*r^11)/(r^4*(e^2-2*m*r+r^2)^5), (2, 1, 1, 2) = -r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, (2, 1, 2, 1) = r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, (2, 3, 1, 3) = r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)*cos(theta), (2, 3, 2, 3) = (-r^2*(2*m-r)^2*sin(theta)^2+(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^2+(e^2-2*m*r+r^2)*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (2, 3, 3, 1) = -r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)*cos(theta), (2, 3, 3, 2) = (r^2*(2*m-r)^2*sin(theta)^2-(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^2-(e^2-2*m*r+r^2)*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (2, 4, 2, 4) = -(2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), (2, 4, 4, 2) = (2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), (3, 1, 1, 3) = sin(theta)^2*(-(e^2-2*m*r+r^2)^3*csc(theta)^2*sin(theta)^2+(e^2-2*m*r+r^2)^3*sin(theta)^4*csc(theta)^4-(2*m-r)^2*r^2*(e^2-m*r)*csc(theta)^2*sin(theta)^2)/(e^2-2*m*r+r^2)^3, (3, 1, 2, 3) = r*(sin(theta)^2*cos(theta)*csc(theta)^2-2*sin(theta)*cot(theta)+cos(theta))*csc(theta)^2*sin(theta)^3, (3, 1, 3, 1) = sin(theta)^2*((e^2-2*m*r+r^2)^3*csc(theta)^2*sin(theta)^2-(e^2-2*m*r+r^2)^3*sin(theta)^4*csc(theta)^4+(2*m-r)^2*r^2*(e^2-m*r)*csc(theta)^2*sin(theta)^2)/(e^2-2*m*r+r^2)^3, (3, 1, 3, 2) = -r*(sin(theta)^2*cos(theta)*csc(theta)^2-2*sin(theta)*cot(theta)+cos(theta))*csc(theta)^2*sin(theta)^3, (3, 2, 1, 3) = r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)^3*cos(theta)*csc(theta)^2, (3, 2, 2, 3) = sin(theta)^2*(r^2*(2*m-r)^2*csc(theta)^2*sin(theta)^2+(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^4-2*(e^2-2*m*r+r^2)*r^2*sin(theta)*cos(theta)*csc(theta)^2*cot(theta)-(e^2-2*m*r+r^2)*csc(theta)^2*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (3, 2, 3, 1) = -r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)^3*cos(theta)*csc(theta)^2, (3, 2, 3, 2) = sin(theta)^2*(-r^2*(2*m-r)^2*csc(theta)^2*sin(theta)^2-(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^4+2*(e^2-2*m*r+r^2)*r^2*sin(theta)*cos(theta)*csc(theta)^2*cot(theta)+(e^2-2*m*r+r^2)*csc(theta)^2*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (3, 3, 1, 2) = 2*r*(sin(theta)*cot(theta)-cos(theta))*csc(theta)^2*sin(theta)^3, (3, 3, 2, 1) = -2*r*(sin(theta)*cot(theta)-cos(theta))*csc(theta)^2*sin(theta)^3, (3, 4, 3, 4) = -(2*m-r)^2*sin(theta)^4*(e^2-m*r)*csc(theta)^2/(r^2*(e^2-2*m*r+r^2)), (3, 4, 4, 3) = (2*m-r)^2*sin(theta)^4*(e^2-m*r)*csc(theta)^2/(r^2*(e^2-2*m*r+r^2)), (4, 1, 1, 4) = -(e^12-10*e^10*m*r+4*e^10*r^2+53*e^8*m^2*r^2-48*e^8*m*r^3+11*e^8*r^4-160*e^6*m^3*r^3+228*e^6*m^2*r^4-108*e^6*m*r^5+17*e^6*r^6+272*e^4*m^4*r^4-520*e^4*m^3*r^5+368*e^4*m^2*r^6-114*e^4*m*r^7+13*e^4*r^8-224*e^2*m^5*r^5+512*e^2*m^4*r^6-456*e^2*m^3*r^7+196*e^2*m^2*r^8-40*e^2*m*r^9+3*e^2*r^10+64*m^6*r^6-160*m^5*r^7+160*m^4*r^8-80*m^3*r^9+20*m^2*r^10-2*m*r^11)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), (4, 1, 4, 1) = (e^12-10*e^10*m*r+4*e^10*r^2+53*e^8*m^2*r^2-48*e^8*m*r^3+11*e^8*r^4-160*e^6*m^3*r^3+228*e^6*m^2*r^4-108*e^6*m*r^5+17*e^6*r^6+272*e^4*m^4*r^4-520*e^4*m^3*r^5+368*e^4*m^2*r^6-114*e^4*m*r^7+13*e^4*r^8-224*e^2*m^5*r^5+512*e^2*m^4*r^6-456*e^2*m^3*r^7+196*e^2*m^2*r^8-40*e^2*m*r^9+3*e^2*r^10+64*m^6*r^6-160*m^5*r^7+160*m^4*r^8-80*m^3*r^9+20*m^2*r^10-2*m*r^11)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), (4, 2, 2, 4) = (e^2-2*m*r+r^2)*(e^2-m*r)/r^4, (4, 2, 4, 2) = -(e^2-2*m*r+r^2)*(e^2-m*r)/r^4, (4, 3, 3, 4) = (e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, (4, 3, 4, 3) = -(e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4}

(11)

 

There are other ways of accomplishing the same, basically exploring that you can Define tensors represneting any tensorial equation you want. See ?Physics,Tensors, section 2.c amd 2.j.

NULL

NULL


 

Download Riemann_for_different_metrics.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

That command redefines the tetrad (either the one set, or any other one that you pass) according to the signature you want - check ?Physics,Redefine and the Examples section.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft.

This is fixed, and the fix is distributed to everybody using Maple 2023 within the Maplesoft Physics Updates v.1430 or newer. 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft.

Entering 

subs(-e - r = s*(e + r), s = -1, %)

 

You get - (e + r) * cos(alpha/2-90), which is one thing. Is it possible to address this systematically? Yes, and I would vote for it. How? It would require a change in the so-called kernel, maybe in a new version of Maple; or maybe I find a way to resolve this at the so-called library level and distribute this within the Maplesoft Physics Updates.

The second issue is, as mentioned by @acer : there is a normalization for all mathematical functions; in the case of cot, you have cot(90 - z) -> -cot(z - 90). This can also be changed to be the reverse of that, but why would one do that? Your expression is very particular; for a different expression, the current normalization may look more convenient; besides that, too many things were coded over the years that may or not work as in the past if you change the normalization of a function. I wouldn't vote for such a change in the normalization of cot.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Add to your initialization file:
 

`simplify/trig/from_sincos/do/22` := eval(`simplify/trig/from_sincos/do/2`):
`simplify/trig/from_sincos/do/2` := () -> eval(`simplify/trig/from_sincos/do/22`(args), [csc = 1/sin, sec = 1/cos])

 

That suffices to achieve what you are asking. Now, changes like this would require testing to be sure that there are no other things in Maple 2023 that rely on the new csc and sec output by simplify; I suppose through using this or equivalent approaches you will discover.
 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions.

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