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Featured Post


In a recent question in Mapleprimes, a spacetime (metric) solution to Einstein's equations, from chapter 27 of the book of Exact Solutions to Einstein's equations [1] was discussed. One of the issues was about computing a tetrad for that solution [27, 37, 1] such that the corresponding Weyl scalars are in canonical form. This post illustrates how to do that, with precisely that spacetime metric solution, in two different ways: 1) automatically, all in one go, and 2) step-by-step. The step-by-step computation is useful to verify results and also to compute different forms of the tetrads or Weyl scalars. The computation below is performed using the latest version of the Maplesoft Physics Updates.

 

with(Physics)

Physics:-Version()

`The "Physics Updates" version in the MapleCloud is 851 and is the same as the version installed in this computer, created 2020, October 19, 13:47 hours Pacific Time.`

(1)

The starting point is this image of page 421 of the book of Exact Solutions to Einstein's equations, formulas (27.37)

 

Load the solution [27, 37, 1] from Maple's database of solutions to Einstein's equations

g_[[27, 37, 1]]

_______________________________________________________

 

`Systems of spacetime coordinates are:`*{X = (z, zb, r, u)}

 

`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (z, zb, r, u)}

 

`The `*`Robinson and Trautman (1962)`*` metric in coordinates `*[z, zb, r, u]

 

`Parameters: `*[P(z, zb, u), H(X)]

 

"`Comments: ` admits geodesic, shearfree, twistfree null congruence, rho=-1/r=rho_b"

 

`Resetting the signature of spacetime from `*`- - - +`*` to `*`+ + + -`*` in order to match the signature in the database of metrics`

 

_______________________________________________________

 

`Setting `*lowercaselatin_is*` letters to represent `*space*` indices`

 

Physics:-g_[mu, nu] = Matrix(%id = 18446744078276690638)

(2)

"CompactDisplay(?)"

H(X)*`will now be displayed as`*H

 

P(z, zb, u)*`will now be displayed as`*P

(3)

The assumptions on the metric's parameters are

Assume(P(z, zb, u) > 0, (H(X))::real, r >= 0)

 

The line element is as shown in the second line of the image above

g_[lineelement]

2*r^2*Physics:-d_(z)*Physics:-d_(zb)/P(z, zb, u)^2-2*Physics:-d_(r)*Physics:-d_(u)-2*H(X)*Physics:-d_(u)^2

(4)

Load Tetrads

with(Tetrads)

_______________________________________________________

 

`Setting `*lowercaselatin_ah*` letters to represent `*tetrad*` indices`

 

((`Defined as tetrad tensors `*`see <a href='http://www.maplesoft.com/support/help/search.aspx?term=Physics,tetrads`*`,' target='_new'>?Physics,tetrads`*`,</a> `*`&efr;`[a, mu]*`, `)*eta[a, b]*`, `*gamma[a, b, c]*`, `)*lambda[a, b, c]

 

((`Defined as spacetime tensors representing the NP null vectors of the tetrad formalism `*`see <a href='http://www.maplesoft.com/support/help/search.aspx?term=Physics,tetrads`*`,' target='_new'>?Physics,tetrads`*`,</a> `*l[mu]*`, `)*n[mu]*`, `*m[mu]*`, `)*conjugate(m[mu])

 

_______________________________________________________

(5)

The Petrov type of this spacetime solution is

PetrovType()

"II"

(6)

The null tetrad computed by the Maple system using a general algorithms is

Setup(tetrad = null)

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078178770326)

(7)

 

According to the help page TransformTetrad , the canonical form of the Weyl scalars for each different Petrov type is

 

So for type II, when the tetrad is in canonical form, we expect only `&Psi;__2` and `&Psi;__3` different from 0. For the tetrad computed automatically, however, the scalars are

Weyl[scalars]

psi__0 = -P(z, zb, u)*(2*(diff(P(z, zb, u), z))*(diff(H(X), z))+P(z, zb, u)*(diff(diff(H(X), z), z)))/(r^2*(H(X)^2+1)^(1/2)), psi__1 = ((1/2)*I)*(-(diff(diff(H(X), r), z))*P(z, zb, u)^2*r+2*P(z, zb, u)^2*(diff(H(X), z))-(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r+(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u))/(P(z, zb, u)*r^2*(H(X)^2+1)^(1/4)), psi__2 = (1/6)*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X))/r^2, psi__3 = 0, psi__4 = 0

(8)

The question is, how to bring the tetrad `&efr;`[a, mu] (equation (7)) into canonical form. The plan for that is outlined in Chapter 7, by Chandrasekhar, page 388, of the book "General Relativity, an Einstein centenary survey", edited by S.W. Hawking and W.Israel. In brief, for Petrov type II, use a transformation ofClass[2] to make Psi[0] = `&Psi;__1` and `&Psi;__1` = 0, then a transformation of Class[1] making Psi[4] = 0, finally use a transformation of Class[3] making Psi[3] = 1. For an explanation of these transformations see the help page for TransformTetrad . This plan, however, is applicable if and only if the starting tetrad results in `&psi;__4` <> 0, which we see in (8) it is not the case, so we need, in addition, before applying this plan, to perform a transformation of Class[1] making `&psi;__4` <> 0.

 

In what follows, the transformations mentioned are first performed automatically, in one go, letting the computer deduce each intermediate transformation, by passing to TransformTetrad the optional argument canonicalform. Then, the same result is obtained by transforming the starting tetrad  one step at at time, arriving at the same Weyl scalars. That illustrates well both how to get the result exploiting advanced functionality but also how to verify the result performing each step, and also how to get any desired different form of the Weyl scalars.

 

Although it is possible to perform both computations, automatically and step-by-step, departing from the tetrad (7), that tetrad and the corresponding Weyl scalars (8) have radicals, making the readability of the formulas at each step less clear. Both computations, can be presented in more readable form without radicals departing from the tetrad shown in the book, that is

e_[a, mu] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (1, 4) = -1, (2, 1) = 0, (2, 2) = r/P(z, zb, u), (2, 3) = 0, (2, 4) = 0, (3, 1) = r/P(z, zb, u), (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = -H(X)}))

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078621688766)

(9)

"IsTetrad(?)"

`Type of tetrad: `*null

 

true

(10)

The corresponding Weyl scalars free of radicals are

"WeylScalars(?)"

psi__0 = P(z, zb, u)*(2*(diff(P(z, zb, u), z))*(diff(H(X), z))+P(z, zb, u)*(diff(diff(H(X), z), z)))/r^2, psi__1 = -(1/2)*(-(diff(diff(H(X), r), z))*P(z, zb, u)^2*r+2*P(z, zb, u)^2*(diff(H(X), z))-(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r+(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u))/(r^2*P(z, zb, u)), psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*H(X))/r^2, psi__3 = 0, psi__4 = 0

(11)

So set this tetrad as the starting point

"Setup(?)"

[tetrad = {(1, 4) = -1, (2, 2) = r/P(z, zb, u), (3, 1) = r/P(z, zb, u), (4, 3) = -1, (4, 4) = -H(X)}]

(12)


All the transformations performed automatically, in one go

 

To arrive in one go, automatically, to a tetrad whose Weyl scalars are in canonical form as in (31), use the optional argument canonicalform:

T__5 := TransformTetrad(canonicalform)

WeylScalars(T__5)

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

(13)

Note the length of T__5

length(T__5)

58242

(14)

That length corresponds to several pages long. That happens frequently, you get Weyl scalars with a minimum of residual invariance, at the cost of a more complicated tetrad.

 

The transformations step-by-step leading to the same canonical form of the Weyl scalars

 

Step 0

 

As mentioned above, to apply the plan outlined by Chandrasekhar, the starting point needs to be a tetrad with `&Psi;__4` <> 0, not the case of (9), so in this step 0 we use a transformation of Class[1] making `&psi;__4` <> 0. This transformation introduces a complex parameter E and to get `&psi;__4` <> 0 any value of E suffices. We use E = 1:

TransformTetrad(nullrotationwithfixedl_)

Matrix(%id = 18446744078634914990)

(15)

"`T__0` := eval(?,E=1)"

Matrix(%id = 18446744078634940646)

(16)

Indeed, for this tetrad, `&Psi;__4` <> 0:

WeylScalars(T__0)[-1]

psi__4 = ((diff(diff(H(X), r), r))*r^2*P(z, zb, u)+P(z, zb, u)^3*(diff(diff(H(X), z), z))+2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^2+2*(diff(diff(H(X), r), z))*P(z, zb, u)^2*r+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)+2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r-4*P(z, zb, u)^2*(diff(H(X), z))-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)^2-2*(diff(H(X), r))*P(z, zb, u)*r-2*(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u)+2*H(X)*P(z, zb, u))/(r^2*P(z, zb, u))

(17)

Step 1

Next is a transformation of Class__2 to make `&Psi;__0` = 0, that in the case of Petrov type II also implies on `&Psi;__1` = 0.According to the the help page TransformTetrad , this transformation introduces a parameter B that, according to the plan outlined by Chandrasekhar in Chapter 7 page 388, is one of the two identical roots (out of the four roots) of the principalpolynomial. To see the principal polynomial, or, directly, its roots you can use the PetrovType  command:

PetrovType(principalroots = 'R')

"II"

(18)

The first two are the same and equal to -1

R[1 .. 2]

[-1, -1]

(19)

So the transformed tetrad T__1 is

T__1 := eval(TransformTetrad(T__0, nullrotationwithfixedn_), B = -1)

Matrix(%id = 18446744078641721462)

(20)

Check this result and the corresponding Weyl scalars to verify that we now have `&Psi;__0` = 0 and `&Psi;__1` = 0

IsTetrad(T__1)

`Type of tetrad: `*null

 

true

(21)

WeylScalars(T__1)[1 .. 2]

psi__0 = 0, psi__1 = 0

(22)

Step 2

Next is a transformation of Class__1 that makes `&Psi;__4` = 0. This transformation introduces a parameter E, that according to Chandrasekhar's plan can be taken equal to one of the roots of Weyl scalar `&Psi;__4`that corresponds to the transformed tetrad. So we need to proceed in three steps:

a. 

transform the tetrad introducing a parameter E in the tetrad's components

b. 

compute the Weyl scalars for that transformed tetrad

c. 

take `&Psi;__4` = 0 and solve for E

d. 

apply the resulting value of E to the transformed tetrad obtained in step a.

 

a.Transform the tetrad and for simplicity take E real

T__2 := eval(TransformTetrad(T__1, nullrotationwithfixedl_), conjugate(E) = E)

Matrix(%id = 18446744078624751238)

(23)

"IsTetrad(?)"

`Type of tetrad: `*null

 

true

(24)

b. Compute `&Psi;__4` for this tetrad

simplify(WeylScalars(T__2)[-1])

psi__4 = (r^2*P(z, zb, u)*(E-1)^2*(diff(diff(H(X), r), r))-2*r*P(z, zb, u)^2*(E-1)*(diff(diff(H(X), r), z))+P(z, zb, u)^3*(diff(diff(H(X), z), z))-2*P(z, zb, u)^2*(E-1)^2*(diff(diff(P(z, zb, u), z), zb))+2*r*P(z, zb, u)*(E-1)*(diff(diff(P(z, zb, u), u), z))-2*r*P(z, zb, u)*(E-1)^2*(diff(H(X), r))+4*P(z, zb, u)^2*(E+(1/2)*(diff(P(z, zb, u), z))-1)*(diff(H(X), z))+2*((P(z, zb, u)*(E-1)*(diff(P(z, zb, u), zb))-(diff(P(z, zb, u), u))*r)*(diff(P(z, zb, u), z))+H(X)*P(z, zb, u)*(E-1))*(E-1))/(r^2*P(z, zb, u))

(25)

c. Solve `&Psi;__4` = 0 discarding the case E = 0 which implies on no transformation

simplify(solve({rhs(psi__4 = (r^2*P(z, zb, u)*(E-1)^2*(diff(diff(H(X), r), r))-2*r*P(z, zb, u)^2*(E-1)*(diff(diff(H(X), r), z))+P(z, zb, u)^3*(diff(diff(H(X), z), z))-2*P(z, zb, u)^2*(E-1)^2*(diff(diff(P(z, zb, u), z), zb))+2*r*P(z, zb, u)*(E-1)*(diff(diff(P(z, zb, u), u), z))-2*r*P(z, zb, u)*(E-1)^2*(diff(H(X), r))+4*P(z, zb, u)^2*(E+(1/2)*(diff(P(z, zb, u), z))-1)*(diff(H(X), z))+2*((P(z, zb, u)*(E-1)*(diff(P(z, zb, u), zb))-(diff(P(z, zb, u), u))*r)*(diff(P(z, zb, u), z))+H(X)*P(z, zb, u)*(E-1))*(E-1))/(r^2*P(z, zb, u))) = 0, E <> 0}, {E}, explicit)[1])

{E = ((diff(diff(H(X), r), r))*r^2*P(z, zb, u)+(diff(diff(H(X), r), z))*P(z, zb, u)^2*r-2*(diff(H(X), r))*P(z, zb, u)*r-(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u)+(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r-2*P(z, zb, u)^2*(diff(H(X), z))-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)+2*H(X)*P(z, zb, u)+(-P(z, zb, u)^4*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X))*(diff(diff(H(X), z), z))+P(z, zb, u)^4*(diff(diff(H(X), r), z))^2*r^2+(-2*r^2*(diff(diff(P(z, zb, u), u), z))*P(z, zb, u)^3+2*r^2*P(z, zb, u)^2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))-4*r*(diff(H(X), z))*P(z, zb, u)^4)*(diff(diff(H(X), r), z))-2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(H(X), r), r))*r^2+P(z, zb, u)^2*(diff(diff(P(z, zb, u), u), z))^2*r^2+(-2*r^2*P(z, zb, u)*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))+4*r*(diff(H(X), z))*P(z, zb, u)^3)*(diff(diff(P(z, zb, u), u), z))+4*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(P(z, zb, u), z), zb))+4*(diff(H(X), z))^2*P(z, zb, u)^4+4*P(z, zb, u)^2*(diff(P(z, zb, u), z))*((diff(H(X), r))*P(z, zb, u)*r-(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)-(diff(P(z, zb, u), u))*r-H(X)*P(z, zb, u))*(diff(H(X), z))+(diff(P(z, zb, u), u))^2*(diff(P(z, zb, u), z))^2*r^2)^(1/2))/(P(z, zb, u)*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X)))}

(26)

d. Apply this result to the tetrad (23). In doing so, do not display the result, just measure its length (corresponds to two+ pages)

T__3 := simplify(eval(T__2, {E = ((diff(diff(H(X), r), r))*r^2*P(z, zb, u)+(diff(diff(H(X), r), z))*P(z, zb, u)^2*r-2*(diff(H(X), r))*P(z, zb, u)*r-(diff(diff(P(z, zb, u), u), z))*r*P(z, zb, u)+(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*r-2*P(z, zb, u)^2*(diff(H(X), z))-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)+2*H(X)*P(z, zb, u)+(-P(z, zb, u)^4*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X))*(diff(diff(H(X), z), z))+P(z, zb, u)^4*(diff(diff(H(X), r), z))^2*r^2+(-2*r^2*(diff(diff(P(z, zb, u), u), z))*P(z, zb, u)^3+2*r^2*P(z, zb, u)^2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))-4*r*(diff(H(X), z))*P(z, zb, u)^4)*(diff(diff(H(X), r), z))-2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(H(X), r), r))*r^2+P(z, zb, u)^2*(diff(diff(P(z, zb, u), u), z))^2*r^2+(-2*r^2*P(z, zb, u)*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))+4*r*(diff(H(X), z))*P(z, zb, u)^3)*(diff(diff(P(z, zb, u), u), z))+4*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(P(z, zb, u), z), zb))+4*(diff(H(X), z))^2*P(z, zb, u)^4+4*P(z, zb, u)^2*(diff(P(z, zb, u), z))*((diff(H(X), r))*P(z, zb, u)*r-(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))*P(z, zb, u)-(diff(P(z, zb, u), u))*r-H(X)*P(z, zb, u))*(diff(H(X), z))+(diff(P(z, zb, u), u))^2*(diff(P(z, zb, u), z))^2*r^2)^(1/2))/(P(z, zb, u)*((diff(diff(H(X), r), r))*r^2+2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*(diff(H(X), r))*r-2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)+2*H(X)))}[1]))

length(T__3)

12589

(27)

Check the scalars, we expect `&Psi;__0` = `&Psi;__1` and `&Psi;__1` = `&Psi;__4` and `&Psi;__4` = 0

WeylScalars(T__3); %[1 .. 2], %[-1]

psi__0 = 0, psi__1 = 0, psi__4 = 0

(28)

Step 3

Use a transformation of Class[3] making Psi[3] = 1. Such a transformation changes Psi[3]^` '` = A*exp(-I*Omega)*Psi[3], where we need to take A*exp(-I*Omega) = 1/`&Psi;__3`, and without loss of generality we can take Omega = 0.

Check first the value of `&Psi;__3` in the last tetrad computed

WeylScalars(T__3)[4]

psi__3 = (1/2)*(-2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(H(X), r), r))*r^2-P(z, zb, u)^4*(diff(diff(H(X), z), z))*(diff(diff(H(X), r), r))*r^2+P(z, zb, u)^4*(diff(diff(H(X), r), z))^2*r^2+4*(diff(P(z, zb, u), z))*(diff(H(X), r))*(diff(H(X), z))*P(z, zb, u)^3*r+2*(diff(H(X), r))*P(z, zb, u)^4*(diff(diff(H(X), z), z))*r-4*(diff(P(z, zb, u), zb))*(diff(P(z, zb, u), z))^2*(diff(H(X), z))*P(z, zb, u)^3+4*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(P(z, zb, u), z), zb))-4*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(H(X), r), z))*r+2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*P(z, zb, u)^2*(diff(diff(H(X), r), z))*r^2-2*(diff(P(z, zb, u), zb))*(diff(P(z, zb, u), z))*P(z, zb, u)^4*(diff(diff(H(X), z), z))+2*P(z, zb, u)^5*(diff(diff(H(X), z), z))*(diff(diff(P(z, zb, u), z), zb))-2*P(z, zb, u)^3*(diff(diff(P(z, zb, u), u), z))*(diff(diff(H(X), r), z))*r^2+4*(diff(H(X), z))^2*P(z, zb, u)^4-4*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^2*r-4*H(X)*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3+4*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(P(z, zb, u), u), z))*r+(diff(P(z, zb, u), u))^2*(diff(P(z, zb, u), z))^2*r^2-2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*P(z, zb, u)*(diff(diff(P(z, zb, u), u), z))*r^2-2*H(X)*P(z, zb, u)^4*(diff(diff(H(X), z), z))+P(z, zb, u)^2*(diff(diff(P(z, zb, u), u), z))^2*r^2)^(1/2)/(r^2*P(z, zb, u))

(29)

So, the transformed tetrad T__4 to which corresponds Weyl scalars in canonical form, with `&Psi;__0` = `&Psi;__1` and `&Psi;__1` = `&Psi;__4` and `&Psi;__4` = 0 and `&Psi;__3` = 1, is

T__4 := simplify(eval(TransformTetrad(T__3, boostsn_l_plane), A = 1/rhs(psi__3 = (1/2)*(-2*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(H(X), r), r))*r^2-P(z, zb, u)^4*(diff(diff(H(X), z), z))*(diff(diff(H(X), r), r))*r^2+P(z, zb, u)^4*(diff(diff(H(X), r), z))^2*r^2+4*(diff(P(z, zb, u), z))*(diff(H(X), r))*(diff(H(X), z))*P(z, zb, u)^3*r+2*(diff(H(X), r))*P(z, zb, u)^4*(diff(diff(H(X), z), z))*r-4*(diff(P(z, zb, u), zb))*(diff(P(z, zb, u), z))^2*(diff(H(X), z))*P(z, zb, u)^3+4*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(P(z, zb, u), z), zb))-4*(diff(H(X), z))*P(z, zb, u)^4*(diff(diff(H(X), r), z))*r+2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*P(z, zb, u)^2*(diff(diff(H(X), r), z))*r^2-2*(diff(P(z, zb, u), zb))*(diff(P(z, zb, u), z))*P(z, zb, u)^4*(diff(diff(H(X), z), z))+2*P(z, zb, u)^5*(diff(diff(H(X), z), z))*(diff(diff(P(z, zb, u), z), zb))-2*P(z, zb, u)^3*(diff(diff(P(z, zb, u), u), z))*(diff(diff(H(X), r), z))*r^2+4*(diff(H(X), z))^2*P(z, zb, u)^4-4*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^2*r-4*H(X)*(diff(P(z, zb, u), z))*(diff(H(X), z))*P(z, zb, u)^3+4*(diff(H(X), z))*P(z, zb, u)^3*(diff(diff(P(z, zb, u), u), z))*r+(diff(P(z, zb, u), u))^2*(diff(P(z, zb, u), z))^2*r^2-2*(diff(P(z, zb, u), u))*(diff(P(z, zb, u), z))*P(z, zb, u)*(diff(diff(P(z, zb, u), u), z))*r^2-2*H(X)*P(z, zb, u)^4*(diff(diff(H(X), z), z))+P(z, zb, u)^2*(diff(diff(P(z, zb, u), u), z))^2*r^2)^(1/2)/(r^2*P(z, zb, u)))))

IsTetrad(T__4)

`Type of tetrad: `*null

 

true

(30)

WeylScalars(T__4)

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

(31)

These are the same scalars computed in one go in (13)

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

psi__0 = 0, psi__1 = 0, psi__2 = -(1/6)*(-(diff(diff(H(X), r), r))*r^2+2*(diff(H(X), r))*r-2*(diff(P(z, zb, u), z))*(diff(P(z, zb, u), zb))+2*(diff(diff(P(z, zb, u), z), zb))*P(z, zb, u)-2*H(X))/r^2, psi__3 = 1, psi__4 = 0

(32)

``


 

Download The_metric_[27_37_1]_in_canonical_form.mw

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

Featured Post

A few weeks ago a television station in Toronto asked me if I’d share some tips on how parents could help their kids stay engaged with remote learning. My initial reaction was to run for the hills – appearing on live TV is not my cup of tea. However my colleagues persuaded me to accept. You can see a clip of that segment here - I’ve included it in this post because otherwise someone on the marketing team would have ;-)

My tips are based on a wide variety of experiences. My role at Maplesoft requires me to speak with educators at all levels, and remote learning has been a hot topic of conversation lately, as you can imagine. As well, in my past life (i.e. life before kids) I was a high school math tutor, and now as a parent I’m in the thick of it helping my son navigate Kindergarten remotely.

So here are my 5 tips on how parents of elementary and high-school aged children can help their kids stay engaged with remote learning. If you have other tips, including suggestions for university students, feel free to leave them in the comments sections. And if these tips help you, please let me know. It will have made the stress of my appearance on TV worthwhile!

 

Tip 1: Look for the positives

These are unprecedented times for kids, parents and teachers. Over the course of the last 6-7 months, learning as we’ve grown to know it has changed radically. And while the change has been incredibility difficult for everyone, it’s helpful to look for the positives that remote learning can bring to our children:

  • Remote learning can help some kids focus on their work by minimizing the social pressures or distractions they may face at school.
  • Older kids are appreciating the flexibility that remote learning can offer with respect to when and how they complete their work.  
  • Younger kids are loving the experience of learning in the presence of mom and dad. My 4 year old thinks it’s awesome that I now know all the lyrics to the songs that he learns in school.
  • As many remote learning classrooms include students from across the school board, this can provide kids with the opportunity to connect with their peers from different socio-economic backgrounds living across the city.

 

Tip 2: Don’t shy away from your kid’s teacher

While some kids are thriving learning from home, we know that others are struggling.

If your high school student is struggling at school, do whatever it takes to convince them to connect with their teacher. If your child is younger, make the connection yourself.

In my role, I’ve had the opportunity to work with many teachers, and rest assured, many of them would welcome this engagement.  They want our kids to succeed, but without the face-to-face classroom interaction it’s becoming increasingly more difficult for them to rely on visual cues to see how your child is doing and if they are struggling with a concept.

So I encourage you to reach out to your kid’s teacher especially if you notice your child is having difficulty.

 

Tip 3: Get creative with learning

Another benefit of remote learning is that it presents us with a unique opportunity to get creative with learning.

Kids, especially those in middle school and high school, now have the time and opportunity to engage with a variety of different online learning resources. And when I say online learning resources, I mean more than just videos. Think interactive tools (such as Maple Learn), that help students visualize concepts from math and science, games that allow students to practice language skills, repositories of homework problems and practice questions that allow kids to practice concepts, the list goes on.

Best of all, many content providers and organizations, are offerings these resources and tools available for free or at a substantially reduced cost to help kids and parents during this time.

So if your child is having difficulty with a particular subject or if they are in need of a challenge, make sure to explore what is available online.

 

Tip 4: Embrace the tech

To be successful, remote learning requires children to learn a host of new digital skills, such as how to mute/unmute themselves, raise their hands electronically, turn on and off their webcam, toggle between applications to access class content and upload homework, keep track of their schedule via an electronic calendar, etc. This can be daunting for kids who are learning remotely for the first time.

As a parent you can help your child become more comfortable with remote learning by setting aside some time either before or after class to help them master these new tools. And since this is likely new to you, there are some great videos online that will show you how to use the system your school has mandated be it Microsoft Teams, Google Classroom or something else.  

 

Tip 5: It’s a skill

Remember that remote learning is a skill like any other skill, and it takes time and practice to become proficient.

So remember to be patient with yourself, your kids, and their teachers, as we embark on this new journey of learning. Everyone is trying their best and I truly believe a new rhythm will emerge as we progress through the school year.

We will find our way.



nonlinear PDE third order

Maple asked by Mo_Jalal 15 October 23