In the study of the Gödel spacetime model, a tetrad was suggested in the literature [1]. Alas, upon entering the tetrad in question, Maple's Tetrad's package complained that that matrix was not a tetrad! What went wrong? After an exchange with Edgardo S. Cheb-Terrab, Edgardo provided us with awfully useful comments regarding the use of the package and suggested that the problem together with its solution be presented in a post, as others may find it of some use for their work as well.

The Gödel spacetime solution to Einsten's equations is as follows.

Working with Cartesian coordinates,

the Gödel line element is

Setting the metric

The problem appeared upon entering the matrix M below supposedly representing the alleged tetrad.

Each of the rows of this matrix is supposed to be one of the null vectors . Before setting this alleged tetrad, Maple was asked to settle the nature of it, and the answer was that M was not a tetrad! With the Physics Updates v.857, a more detailed message was issued:

So there were actually three problems:

The issue, then, became how to correct the matrix M to be a valid tetrad: either change the setup, or change the matrix M. Below the two courses of action are provided.

First the simplest: change the settings. According to the message (7), setting the tetrad to be null, changing the signature to be (+---) and indicating that M represents a tetrad with its spacetime index contravariant would suffice:

The null tetrad metric is now as in the reference used.

Checking now with the spacetime index contravariant

At this point, the command IsTetrad  provided with the equation (15), where the left-hand side has the information that the spacetime index is contravariant

Great! one can now set the tetrad M exactly as entered, without changing anything else. In the next line it will only be necessary to indicate that the spacetime index, , is contravariant.

The tetrad is now the matrix M. In addition to checking this tetrad making use of the IsTetrad command, it is also possible to check the definitions of tetrads and null vectors using TensorArray.

For the null vectors:

From its Weyl scalars, this tetrad is already in the canonical form for a spacetime of Petrov type "D": only

Attempting to transform it into canonicalform returns the tetrad (17) itself

Let's now obtain the correct tetrad without changing the signature as done in (13).

Start by changing the signature back to

So again, M is not a tetrad, even if the spacetime index is specified as contravariant.

By construction, the tetrad M has its rows formed by the null vectors with the ordering . To understand what needs to be changed in M, define those vectors, independent of the null vectors  (with underscore) that come with the Tetrads package.

and set their components using the matrix M taking into account that its spacetime index is contravariant, and equating the rows of M  using the ordering :

Check the covariant components of these vectors towards comparing them with the lines of the Maple's tetrad

This shows the  null vectors (with underscore) that come with Tetrads package

So (29) computed from M is the same as (30) computed from Maple's tetrad.

But, from (30) and the form of Maple's tetrad

for the current signature

we see the ordering of the null vectors is , not  used in [1] with the signature (+ - - -). So the adjustment required in  M, resulting in , consists of reordering M's rows to be

Comparing  with the tetrad computed by Maple ((24) and (31), they are actually the same.

References

[1]. Rainer Burghardt, "Constructing the Godel Universe", the arxiv gr-qc/0106070 2001.

[2]. Frank Grave and Michael Buser, "Visiting the Gödel Universe",  IEEE Trans Vis Comput GRAPH, 14(6):1563-70, 2008.

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.

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

The assumptions on the metric's parameters are

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

Load Tetrads

The Petrov type of this spacetime solution is

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

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  and  different from 0. For the tetrad computed automatically, however, the scalars are

The question is, how to bring the tetrad  (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 of to make , then a transformation of  making , finally use a transformation of  making . 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 , which we see in (8) it is not the case, so we need, in addition, before applying this plan, to perform a transformation of  making

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

The corresponding Weyl scalars free of radicals are

So set this tetrad as the starting point

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:

Note the length of

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 , not the case of (9), so in this step 0 we use a transformation of  making . This transformation introduces a complex parameter E and to get  any value of E suffices. We use :

Indeed, for this tetrad, :

Step 1

Next is a transformation of  to make , that in the case of Petrov type II also implies on .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:

The first two are the same and equal to -1

So the transformed tetrad  is

Check this result and the corresponding Weyl scalars to verify that we now have  and

Step 2

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

a.Transform the tetrad and for simplicity take E real

b. Compute  for this tetrad

c. Solve  discarding the case  which implies on no transformation

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

Check the scalars, we expect

Step 3

Use a transformation of  making . Such a transformation changes , where we need to take , and without loss of generality we can take

Check first the value of  in the last tetrad computed

So, the transformed tetrad  to which corresponds Weyl scalars in canonical form, with  and , is

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

 The domain is tranformed from Z = X+IY to w. The points tranformed are

 a is not known apriori. The value of a should be found to make sure [1,1] in the Z coordinate is transformed to [1+a] in the w coordinate.

 Value of K1 is found using the transformation of [1,0] to 1 in the w coordinate

 The height Y in the Z coordinate is found by integrating from 1 to 1+a in the w coordinate.

 The choice of a = sqrt(2)-1 gives the height of 1 for Z coordinate.

 integration from 0 to 1+a in the w coordinate gives 1,1 in the Z coordinate.

 Integrating w from 0 to wmid =1/sqrt(2) gives the point 0.5,0 in the Z coordinate

 Next w domain is transformed to Z2 domain Z2 = X2+IY2

 K2 is found based on the transformation of wmid to [1,0] in Z2 coordinate.

 The corner 0,1 in the Z coordinate is mapped by integrating eq2 from 0 to 1 in the w coordinate

 This is the point 1,0 in the original coordinate.

 The height in the Z2 coorinate is found by integrating eq2 from wmid to 1+a.

 The magnitude in the Y direction is given by the coefficient of I, the imaginary number

 The analytial solution in the Z2 corordinate is a line in Y2 to satisfy simple zero flux conditions at X2 = 0 and X2 = 1.

 The value of phi is zero at Y2 = Ytot (originally the cathode domain in the Z domain)

 The analytical solution is given by

 The potential at the corner is given by substituting the imaginary value of corner for Y2 in phinanal)

local flux/current density calculation, written in terms of w is

average flux/current density calculation for the anode

 The average current density at Y =0, local current density at X = 0,Y=0 and potential at X=1,Y=0 (Corner) can be used to study convergence of FEM and other numerical methods

This FEM code is for solving Laplace's equation with primary current distribution considered in Model 1.
This code is based on FEM weak-form. Biquadratic Lagrange shape functions (9nodes in an element) are used.

Procedure to perform numerical integration on shape functions to obtain local matrices (can be replaced with analytical integration for this particular problem)
  -Shape functions are also used as weight functions in applying weak formulation. Numerical integration is done using Simpson's rule.
  -Local cartesian coordinates x,y are converted to natural coordinates zeta and eta. This transformation is not required for this simple geomerty but useful in general. zeta and eta are obtained by scaling x and y with hx/2 and hy/2, respectively, in this code.

DataFrames: An example from the 2020 U.S. Presidential election

(Or why DataFrames are more powerful and readable than spreadsheets.)

In this example of working with DataFrames, the goal is to use a spreadsheet from a website, which contains polling data, to estimate the probability each of the two candidates from the major parties will win the US Presidential election in November.  I first tried doing the calculations with a spreadsheet, but I discovered DataFrames was far more powerful. Warning: This worksheet uses live data. Hence the outcome at the end of the worksheet is likely to change daily. A more extensive example with even more common DataFrame operations should be available soon.

How the US Presidential election works - highly simplified version: In the US there are only two parties for which their candidate could win the election:  the Democratic party and Republican party. The Republican party is often referred to as the "Grand Old Party", or GOP. Each state executes its own election. The candidate who receives the most votes wins the states "electoral votes" (EV). The number of the electoral votes for each state is essentially proportional to the population of the state. A candidate who receives a total of 270 or more EVs out of 538, is declared the president of the US for the next term, which starts January 20 of 2021.

Creating DataFrame from web based data:

First I download the data from the website. It is a CSV spreadsheet.

Each row contains information about a poll conducted in one of the states.  The first poll starts on row 2, hence the number of polls are:

Now I want to create a new DataFrame containing only the most useful information. In web_data, many are the columns are not important. However I do want to keep the column label names from those columns I wish to retain.

Because  the first poll in web_data is labeled 2, I would like to relabel all the polls starting from 1

Creating a DataFrame from a Matrix or another DataFrame:  (with row labels and column labels)

Now I can build the DataFrame that I will be working with:

What each column means

* "Day" - day of the year in 2020 when the poll within the state was halfway completed. The larger the value, the more recent the poll.

* "State" - the state in the US where the poll was conducted. The candidate that receives the most votes "wins the state".

* "EV" - the number of electoral votes given to the candidate who receives the most votes within the state.

* "Dem" - the percentage of people who said they are going to vote for the candidate from the Democratic party.

* "GOP" - the percentage of people who said they are going to vote for the candidate from the Republican party.

Sorting:

By using the sort function, using the `>` operator, I can see which polls are the more recent. (If you run the worksheet yourself, the outcome will change as more polls are added to the website spreadsheet.)

Selecting Unique entries - by column values:

For the my simple analysis, I will use only the most recent poll, one from each state. Hence, using AreUnique, I can pull the first row that matches a state name. This new DataFrame called states.

(Note, one of the "states" is the District of Columbia, D.C., which is why there are 51 rows.)

Removing a column: (and relabeling rows)

This next example isn't necessary, but shows some of the cool features of DataFrames.

Since there is only 1 entry per state, I'm going to remove the "State" column and relabel all the rows with the state names

Indexing by row labels:

This allow me to to display information by individual states. What is the data for California, Maine and Alaska?

Mathematics with multiple-columns:

My preference is to work with fractions, rather than percentages. Hence I want all the values in the "Dem" and "GOP" to be divided by 100 (or multiplied by 1/100).  Treating each column like a vector, the multiplication is performed individually on each cell. This is what the tilda, "~", symbol performs.

Mathematics: using a function to calculate a column

For the next action, I want to use the power of the Statistics package to create a "probability of winning the state" function.

For simplicity, I will assume the outcome of the voting in a state is purely random, but is conditional to popularity of each candidate as measured by the polls. I'll assume the likelihood of an outcome follows a normal (Gaussian) distribution with the peak being at point where the difference of the polling of the two candidates is zero. (Note, other than 2016, where there was an unusually larger percentage of undecided voters on election day, this simple model is reasonable accurate. For example, in 2012, of the states which appeared to be the "closest", the winner over-performed his polling in half of them, and under-performed in the other half with a mean difference of nearly zero.)  From previous elections, the standard deviation of differences between polling values and the actual outcome is at most 0.05, however, it does increase with the fraction of undecided voters.

To mathematically model this situation, I have chosen to use the "Cumulative Density Function" CDF in the Statistics package. It will calculate the probability that a candidate polling with fraction f1 wins the election if the other candidate is polling with fraction f2.  The variable u is the fraction of undecided voters. It is included in the calculation to increase the spread of the possible outcomes.

Converting this expression into a function using the worst named function in Maple, unapply:

Now I can calculate a DataFrames column of the "win probability", in this case, for the candidate from the Democratic platy. By apply the function, individually, using the columns "Dem" and "GOP", I produce:

Appending a column:

I can add this column to the end of the states with the label "DemWinProb":

Mathematics of adding the entries of a column:

How many electoral votes are available? add them up.

While the number of EV a candidate wins is discrete, I can use the "win probability" from each state to estimate the total number of EV each of the candidates might win. This means adding up number of EV in each state times, individually, the probability of winning that state:

Currently, the candidate from the Democratic party is likely to win more then 300 electoral vtes.

What about for the candidate from the Republican / "GOP" party?

Summing the two EV values, we obtain the total number of electoral votes.