Formulating and solving the equivalence problem for Schwarzschild metric in a simple caseIn connection with the digitizing in Maple 2016 of the database of solutions to Einstein's equations of the book Exact Solutions to Einstein Field Equations. I was recently asked about a statement found in the "What is new in Physics in Maple 2016" page:
In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying [Einstein\342\200\231s] partial differential equations. [By combining that functionality with the one in the Physics and Physics:-Tetrads package] you can also formulate and, depending on the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation.
This question posed is a reasonable one: "could you please provide one example?" This post provides that example.First of all the existing science behind: in my opinion, the main reference regarding the equivalence problem is at the paper "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980, by A. Karlhede (University of Stockholm). This approach got refined later by others and, generally speaking, it is currently know as the Cartan-Karlhede method, summarized in chapter 9.2 of the book Exact Solutions to Einstein Field Equations. whose solutions were all digitized within the Physics and DifferentialGeometry packages for Maple 2016. This method of Chapter 9.2, however, is not the only approach to the problem, and sometimes simpler methods can handle the problem faster, or just in simpler forms.The example worked out below is actually the example from Karlhede's paper just mentioned, on pages 704 - 706: "Show that the Schwarzschild metric and its form written in terms of isotropic spherical coordinates are equivalent, and derive the transformation that relates them". Because this problem happens to be simple for nowadays computer algebra, below I also tackle it modified, slightly more difficult variants of it. The approach shown works for more complicated cases as well.Below we tackle Karlhede's paper-problem using: one PDEtools command, the Physics:-TransformCoordinates, the Physics:-Weyl command to compute the Weyl scalars and the Physics:-Tetrads:-PetrovType to see the Petrov type of the metrics involved. The transformation resolving the equivalence is explicitly derived.Start loading the Physics and Tetrads package. To reproduce the computations below, as usual, update your Physics library with the one available for download at the Maplesoft R&D Physics 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To formulate the problem, set first some symbols to represent the changed metric, changed mass and changed coordinates - no mathematics at this 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Set now a new coordinates system, call it Y, involving the new coordinates (in the paper they are represented with a tilde on top of the letters)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According to eq.(7.6) of the paper, the line element of Schwarzschild solution in isotropic spherical coordinates is given 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Set this to be the 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Check it outLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2JS1GLDYlUSFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y9RkA=In connection with the transformation used further below, compute now the Petrov type and the Weyl scalars for this metric, just to have an idea of what is behind this metric.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVErUGV0cm92VHlwZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZARj1GQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElV2V5bEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRKHNjYWxhcnNGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y9RkA=We see that the Weyl scalars are already in canonical form (see post in Mapleprimes about canonical forms): only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 and the important thing: it depends on only one coordinate, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJiM1ODY3MTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Yy .Now: we want to see if this metric is equivalent to Schwarzschild metric in standard spherical coordinatesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2JS1GLDYlUSNzY0YnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRj1GQA==The equivalence we want to resolve is regarding an arbitrary relationship LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EpJiM1ODY2NjtGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEibUYnRjJGNUY4RjUvRjlRJ25vcm1hbEYnRjVGQy1JI21vR0YkNi5RIn5GJ0Y1RkMvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZZRjVGQw==between the masses used in and and a generic change of variables from X to 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Using a differential equation mindset, the formulation of the equivalence between and under the transformation is actually simple: change variables in , using and the Physics:-TransformCoordinates command (this is the command that changes variables in tensorial expressions), then equate the result to , then try to solve the problem for the unknowns LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEmJm1mcjtGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIm1GJ0YyRjUvRjZRJ25vcm1hbEYnRkBGQA==, 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5MzI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIllGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjFGMkYyRkFGMg==. We note at this point, however, that the Weyl scalars for Schwarzschild metric in this standard form are also in canonical form of Petrov type D and also depend on only one variable, r LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVErUGV0cm92VHlwZUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYkRisvRjNRJ25vcm1hbEYnRj9GP0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElV2V5bEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRKHNjYWxhcnNGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y9RkA=The fact that the Weyl scalars in both cases ( and ) are in canonical form (only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 ) and in both cases this scalar depends on only one coordinate is already an indicator that the transformation involved changes only one variable in terms of the other one. So one could just search for a transformation of the form 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 and resolve the problem instantly. Still, to make the problem slightly more general, consider instead a generic transformation for r in terms of all of 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Transform the coordinates in the metric (because of having used PDEtools:-declare, derivatives of the unknowns R are displayed indexed, for compact notation)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Proceed equating to to obtain a set of equations that entirely formulates the 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This problem, shown in Karlhede's paper as the example of the approach he summarized, is solvable using the differential equation commands of PDEtools (in this case casesplit) in one go and no time, obtaining the same solution shown in the paper with equation number (7.10), the problem actually admits two 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By all means this does not mean this differential equation approach is better than the general approach mentioned in the paper (also in section 9.2 of the Exact Solutions book). This presentation above only makes the point of the paragraph mentioned at the beginning of this worksheet "... [in Maple 2016] you can also formulate and, depending on the the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation." In any case this problem above is rather easy for the computer. Consider a slightly more difficult problem, where 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. For example: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Tackle now the same 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The solutions to the equivalence between and are then given 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, despite that the Weyl scalars suggest that a transformation of only one variable is sufficient to solve the problem, one could also consider a more general transformation, of more variables. Provided we exclude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= (because there is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEkY29zRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGPC8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItSShtZmVuY2VkR0YkNiQtRiw2JVEoJnRoZXRhO0YnRjpGPEY8RjxGKy8lK2V4ZWN1dGFibGVHRjtGPA== around and that would take us to solve differential equations for 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, that involve things like 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), and also to speed up matters let's remove the change in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmJnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdGMUYy, consider an arbitrary change in r and 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So our transformation now involve two arbitrary variables, each one depending on all the four coordinates, and a more complicated function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEpJiM1ODY2NjtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYlLUYsNiZRIm1GJ0YvRjJGNUYyL0Y2USdub3JtYWxGJ0YyRkAtRiw2JVEhRidGL0Y1RjJGQA==. Change variables (because of having used PDEtools:-declare, derivatives of the unknowns R and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmJlRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdGMUYy are displayed indexed, for compact notation)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Construct the set of Partial Differential Equations to be 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Solve the problem running a differential elimination (actually without solving any differential equations): there are more than two 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Consider for instance the first oneLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkc29sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPkY+LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkBGPg==Compute the actual solution behind this case :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 fact that the time t appears defined in terms of the transformed time 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 involving an arbitrary constant is expected: the time does not enter the metric, it only enters through derivatives of 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 entering the Jacobian of the transformation used to change variables in tensorial expressions (the metric) in .Summary: the approach shown above, based on formulating the problem for the transformation functions of the equivalence and solving for them the differential equations using the commands in PDEtools, after restricting the generality of the transformation functions by looking at the form of the Weyl scalars, works well for other cases too, specially now that, in Maple 2016, the Weyl scalars can be expressed also in canonical form in one go (see previous Mapleprimes post on "Tetrads and Weyl scalars in canonical form"). Also important: in Maple 2016 it is present the functionality necessary to implement the approach of section 9.2 of the Exact solutions book as well.