<Text-field style="Heading 1" layout="Heading 1">Why computer algebra?</Text-field>... and why computer algebra?We can concentrate more on the ideas instead of on the algebraic manipulationsWe can extend results with easeWe can explore the mathematics surrounding a problemWe can share results in a reproducible way<Text-field style="Heading 1" layout="Heading 1">Representation issues that were preventing the use of computer algebra in Physics</Text-field>
Notation and related mathematical methods that were missing:
coordinate free representations for vectors and vectorial differential operators, covariant tensors distinguished from contravariant tensors, functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indicesBras, Kets, projectors and all related to Dirac's notation in Quantum MechanicsInert representations of operations, mathematical functions, and related typesetting were missing:inert versus active representations for mathematical operationsability to move from inert to active representations of computations and viceversa as necessaryhand-like style for entering computations and textbook-like notation for displaying resultsKey elements of the computational domain of theoretical physics were missing: ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functionsability to perform computations taking into account custom-defined algebra rules of different kinds (commutator, anticommutator and bracket rules, etc.)Examples<Text-field style="Heading 1" layout="Heading 1">The Maple computer algebra environment</Text-field>In the presentation that follows we use the Maple worksheet mode, where input lines are identified by a prompt, as in:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We communicate with the computer entering our computation typing in this input line. The output is the result of our computation and automatically gets an equation number than we can later refer toLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNictSSNtaUdGJDYlUSUlaW50RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYxNiVRJGludEYnRjRGNy8lK2V4ZWN1dGFibGVHRkJGPkY+LUYsNiQtRiM2KC1GMTYlUSRjb3NGJy9GNUZCRj4tRiw2JC1GIzYlLUYxNiVRInhGJ0Y0RjdGV0Y+Rj4tRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU1EsMC4zMzMzMzMzZW1GJ0Zfb0ZXRj5GPkZXRj4=To refer to an equation, you enter the equation label by pressing Command + L, then typing the equation number as you see 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<Text-field style="Heading 1" layout="Heading 1">Classical Mechanics</Text-field><Text-field style="Heading 3" layout="Heading 3">Inertia tensor for a triatomic molecule</Text-field>ProblemDetermine the Inertia tensor of a triatomic molecule that has the form of an isosceles triangle with two masses LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUkjbW5HRiQ2JlEiMUYnLyUlc2l6ZUdRIzEwRidGMi9GNlEnbm9ybWFsRidGMkZCLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= in the extremes of the base and mass LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUkjbW5HRiQ2JlEiMkYnLyUlc2l6ZUdRIzEwRidGMi9GNlEnbm9ybWFsRidGMkZCLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= in the third vertex. The distance between the two masses LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUkjbW5HRiQ2JlEiMUYnLyUlc2l6ZUdRIzEwRidGMi9GNlEnbm9ybWFsRidGMkZCLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= is equal to a, and the height of the triangle is equal to h.<Text-field style="Heading 4" bold="true" layout="Heading 4"><Font bold="true">Solution</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=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 general 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There are 3 particlesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUkjbW5HRiQ2JFEiM0YnRjk=Create an indexing 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 inerta tensor 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Choose a system of reference (not at the center of mass). The vectors 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 are related to 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 and 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Choose the origin at the middle of the segment connecting the two atoms of equal 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Two masses are 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The "center of mass"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The positions of the three particles viewed from the center of massLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEpcG9zaXRpb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=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 abstract IT_Matrix at these values of the vectors 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Try changing the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EnYG1fXzNgRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= in and re-execute the lines after that definition and you see the corresponding answer instantly - computer algebra allows for these simple recalculations without having to reformulate anything.<Text-field style="Heading 1" layout="Heading 1">Classical Field Theory</Text-field><Text-field style="Heading 3" layout="Heading 3">*The field equations for the <Equation executable="false" style="Heading 3" input-equation="" display="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">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</Equation> model</Text-field>
The Lagrangian of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSJ+RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSy1JJW1zdXBHRiQ2JS1GLDYmUScmIzkzNDtGJ0YvRjJGNC1GIzYmLUkjbW5HRiQ2JVEiNEYnRjJGNC9GMFEldHJ1ZUYnRjIvRjVRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRjQ= model, the corresponding Action, and the field equations: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoY29udmVydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJWRpZmZGJ0YvRjJGQQ==<Text-field style="Heading 3" layout="Heading 3">*Maxwell equations departing from the 4-dimensional Action for Electrodynamics</Text-field>Maxwell equations result from equating to zero the functional derivative o the Action with respect to the 4-D potential 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4-D electromagnetic potentialLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnRGVmaW5lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiQUYnRi9GMi1GNjYmLUYjNiMtRiw2JVEjbXVGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRidGRi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GNjYkLUYjNiMtRiw2JVEiWEYnRi9GMkZGRkY=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEvQ29tcGFjdERpc3BsYXlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJBRidGL0YyLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlQtRjY2JC1GLDYlUSJYRidGL0YyRkFGQQ==The electromagnetic field tensor 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 functional derivative of the corresponding 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
Simplify the contracted spacetime 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
The system of differential equations behind this formula in standard Maple notation: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1"><Font size="14" italic="true">*The Gross-Pitaevskii field equations for a quantum system of identical particles</Font></Text-field>
Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE). This equation is useful to describe Bose-Einstein condensates (BEC).<Text-field style="Heading 4" layout="Heading 4"><Font bold="true">Solution</Font></Text-field>Two steps:
Construct the Lagrangian for the system, and with it write the action functionalMinimize the action by equating to zero its functional derivative with respect to the boson field.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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
The energy density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for a quantum system of identical boson particles is (see [3])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 is a complex field, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNistRiw2JlEieEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEieUYnRi9GMkY1RkAtRiw2JlEiekYnRi9GMkY1RkAtRiw2JlEidEYnRi9GMkY1RjJGREYyRkRGMkZE an external potential, the term 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 is the atom-atom interaction.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmU2V0dXBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2LS1GLDYlUSxyZWFsb2JqZWN0c0YnRi9GMi1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1GPjYtUSI9RidGQUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRmVuRj0tRjY2Ji1GIzYuLUYsNiVRInRGJ0YvRjItRj42LVEiLEYnRkFGQy9GR0YxRkhGSkZMRk5GUEZSL0ZWUSwwLjMzMzMzMzNlbUYnRj0tRiw2JVEibUYnRi9GMkZeby1GLDYlUSkmaHNsYXNoO0YnRi9GMi1GPjYuRmBvRi9GMkZDRmFvRkhGSkZMRk5GUEZSRmJvLUYsNiVRIkdGJ0YvRjItSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGVC8lJmRlcHRoR0ZkcC8lKmxpbmVicmVha0dRJWF1dG9GJ0Zeby1GLDYlUSJWRidGL0YyLUY2NiQtRiM2KS1GLDYlUSJ4RidGL0YyRl5vLUYsNiVRInlGJ0YvRjJGXm8tRiw2JVEiekYnRi9GMkZeb0Zbb0ZBRkEvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRl5vLUYsNiVROGF1dG9tYXRpY3NpbXBsaWZpY2F0aW9uRidGL0YyRj1GV0Y9LUYsNiVGMUYvRjJGQS1GPjYtUSI6RidGQUZDRkZGSEZKRkxGTkZQRlpGZm4=The Lagrangian density LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in terms of the Energy 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Construct the action and equate to zero the functional 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Make the Laplacian 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The standard form of the Gross\342\200\223Pitaevskii equation: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For a continuation of this computation deriving a continuity equation for a system of identical particles, see the Mapleprimes post "Quantum Mechanics using Computer Algebra" .For the Bogoliubov spectrum and dispersion relations of this problem above see the Mapleprimes post "Quantum Mechanics II".For a derivation of the Landau criterion for superfluidity in a system of identical particles see the Mapleprimes post "Superfluidity in Quantum Mechanics"LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1">Quantum mechanics</Text-field><Text-field style="Heading 2" italic="true" size="14" layout="Heading 2"><Font size="14" italic="true">*The quantum operator components of </Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbW92ZXJHRiQ2JS1JI21pR0YkNihRIkxGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMTI4LDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYwUScmcmFycjtGJ0YyRjVGO0Y+LyUmZmVuY2VHRj0vJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVUZRRjJGNUY7Rj4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbW92ZXJHRiQ2JS1JI21pR0YkNihRIkxGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMTI4LDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYwUScmcmFycjtGJ0YyRjVGO0Y+LyUmZmVuY2VHRj0vJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVUZRRjJGNUY7Rj4=</Equation><Font size="14" italic="true"> satisfy </Font><Equation executable="false" style="2D Math" input-equation="" 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r and p as tensors of the 3-D Euclidean space embedded in 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Now set the related Commutator rules for the algebra in tensor 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 how these algebra rules work: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 definition of 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rule to be 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 now the operator 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2KFEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMTI4LDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvSSttc2VtYW50aWNzR0YkUS5QaHlzaWNzVGVuc29yRictRiw2JlEibUYnRjVGO0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRKlswLDAsMjU1XUYnRjsvJSlyZWFkb25seUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnL0Y/USdub3JtYWxGJw== in the commutator 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Simplify all in one go:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSlTaW1wbGlmeUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlRistSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FmX28vLUksJUNvbW11dGF0b3JHNiI2JComJkkrTGV2aUNpdml0YUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlSSJqRzYiSSJhRzYiSSJtRzYiIiIiLUkiKkc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkRzYkJkkickc2IjYjSSJhRzYiJkkicEc2IjYjSSJtRzYiIiIiKiYmSStMZXZpQ2l2aXRhRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii9JMWhpZGVtb2R1bGVwcmVmaXhHNiJJJXRydWVHJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiVJImtHNiJJImJHNiJJImNHNiIiIiItSSIqRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii9JMWhpZGVtb2R1bGVwcmVmaXhHNiJJJXRydWVHJSpwcm90ZWN0ZWRHNiQmSSJyRzYiNiNJImJHNiImSSJwRzYiNiNJImNHNiIiIiIqKl4jIiIiIiIiJkkrTGV2aUNpdml0YUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlSSJtRzYiSSJhRzYiSSJiRzYiIiIiLUkiKkc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkRzYkJkkickc2IjYjSSJhRzYiJkkicEc2IjYjSSJiRzYiIiIiJkkrTGV2aUNpdml0YUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlSSJqRzYiSSJrRzYiSSJtRzYiIiIiRicvJSthY3Rpb250eXBlR1E8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMjAxODApRicvJSV2aWV3R1EmbGFiZWxGJ0YrL0Y4USdub3JtYWxGJ0YrThe same one step at a time,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JSFH<Text-field style="Heading 2" layout="Heading 2"><Font size="14" italic="true">Quantization of the energy of a particle in a magnetic field</Font></Text-field>Show that the energy of a particle in a constant magnetic field oriented along the z axis can be written as 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where 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiw2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMTI4LDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGOC9GPFEnbm9ybWFsRic= are creation and anihilation operators.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 4" layout="Heading 4">Solution</Text-field>The classical Hamiltonian is given 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 underlying quantum mechanics algebra rules 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The 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In Coulomb's gauge, the following vector potential gives the magnetic field of the problem, 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Indeed we 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Derive now the commutation rule for 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Time to bring in annihilation and creation 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 the normalization of this 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To express the Hamiltonian in terms of 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This is the Hamiltonian of an harmonic oscillator with frequency 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. The possible values for the energy are known: 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is a positive integer.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" italic="true" size="14" layout="Heading 2"><Font size="14" italic="true">Unitary Operators in Quantum Mechanics</Font></Text-field><Text-field style="Heading 1" layout="Heading 1"><Font size="12">*Eigenvalues of an unitary operator and exponential of Hermitian operators</Font></Text-field>Show that the eigenvalues of an unitary operator are all on the unit circle, their modulus is 1.Show that an operator 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 is unitary provided LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is Hermitian (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) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMi8lMGZvbnRfc3R5bGVfbmFtZUdRLEJ1bGxldH5JdGVtRidGNA== is any real parameter.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If 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 is a normalized eigenvector of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTkYyRjw=with eigenvalue 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Multiplying sides by 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 that 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 unitaryLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoTGlicmFyeUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzotRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVE0SXNIZXJtaXRpYW5PcGVyYXRvckYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRIkhGJ0YvRjJGOQ==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYsNiVRJ0RhZ2dlckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlRistSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1Fmby9JIlZHNiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyooXiMiIiIiIiJJJ2xhbWJkYUc2IiIiIkkiSEc2IiIiIkYnLyUrYWN0aW9udHlwZUdRPG1hcGxlc29mdC5jb206bGFiZWwoTDE4NzkzKUYnLyUldmlld0dRJmxhYmVsRidGKy9GOFEnbm9ybWFsRidGKw==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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is unitaryJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1"><Font size="12">Properties of unitary operators</Font></Text-field>Consider two set of kets 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 and 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, each of them constituting a complete orthonormal basis of the same space.
One can always build an unitary operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn that maps one basis to the other, i.e.: 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<Text-field style="Heading 2" size="12" layout="Heading 2"><Font size="12">*Verify that </Font><Equation executable="false" style="2D Math" input-equation="" 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size="12"> implies on </Font><Equation executable="false" style="2D Math" input-equation="" 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this operatorial equation to 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style="Heading 2" size="12" layout="Heading 2"><Font size="12">*Show that </Font><Equation executable="false" style="2D Math" input-equation="" 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have 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since 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 and 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 form two complete basis, their Kets satisfy the closure relation, i.e. the right-hand side is equal to 1 and U is unitary.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" size="12" layout="Heading 2"><Font size="12">*Show that the matrix elements of </Font><Equation executable="false" style="Heading 2" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiVUYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0YyL0Y5USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiVUYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0YyL0Y5USdub3JtYWxGJw==</Equation><Font size="12"> in the </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font size="12"> and </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font size="12"> basis are equal</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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<Text-field style="Heading 2" layout="Heading 2"><Font size="12">Show that </Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiw2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzE2RicvJSVib2xkR0Y0RjUvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjJGJy9GOVElYm9sZEYnLyUrZm9udHdlaWdodEdGRA==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiw2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUlc2l6ZUdRIzE2RicvJSVib2xkR0Y0RjUvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjJGJy9GOVElYm9sZEYnLyUrZm9udHdlaWdodEdGRA==</Equation><Font style="Text"> <Font bold="true">and</Font></Font><Font size="12"> </Font><Equation executable="false" style="Heading 2" input-equation="" display="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">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</Equation><Font size="12">have the same spectrum</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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If 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, then the eigenkets of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJkFzY3I7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== are 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 left-hand side can be rewritten performing the 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 conclusion, after an unitary transform, an eigenvector of the initial operator is an eigenvector of the new operator with the same eigenvalue.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1"><Font size="12" encoding="UTF-8">Schr\303\266dinger equation and unitary transform</Font></Text-field>Consider a ket 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 that solves the time-dependant Schr\303\266dinger equation:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbW5HRiQ2JVEiaUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUSgmIzg0NjM7RicvJSdpdGFsaWNHRjFGL0YyRjUtSSZtZnJhY0dGJDYoLUY2Ni5RKyZQYXJ0aWFsRDtGJ0YvRjJGOUY7Rj1GP0ZBRkNGRUZHRkotRiM2J0ZVLUZNNiZRInRGJy9GUVEldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnRi8vJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNvLyUpYmV2ZWxsZWRHRjEtRjY2L0Y4RmduRi9GaW5GOUY7Rj1GP0ZBRkNGRUZHRkotSShtZmVuY2VkR0YkNigtRiM2KS1GNjYvRjhGL0Zbb0YyRjlGO0Y9Rj9GQUZDRkVGR0ZKLUklbXN1YkdGJDYlLUZNNidRJiZwc2k7RidGUEYvRltvRjItRltwNigtRiM2JS1GTTYnRmZuRmduRi9GW29GaW5GL0YyRi9GW29GMi8lJW9wZW5HUSFGJy8lJmNsb3NlR0ZfcS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUZNNiNGX3EtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGXHIvJSpsaW5lYnJlYWtHUSVhdXRvRidGZXFGL0YyRi9GW29GMi9GXnFRKSZ2ZXJiYXI7RicvRmFxUScmcmFuZztGJy1GNjYuUSI9RidGL0YyRjlGO0Y9Rj9GQUZDRkUvRkhRLDAuMjc3Nzc3OGVtRicvRktGXXMtRk02JlEiSEYnRmduRi9GaW4tRltwNiUtRiM2JkZaRi9GW29GMkYvRjJGNUZqb0YvRltvRjI=and consider 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= is a unitary operator. Does 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 evolves according a Schr\303\266dinger 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and if yes, which is the expression of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEvJkhpbGJlcnRTcGFjZTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYlLUYsNiZRInRGJ0YvRjJGNUYyL0Y2USdub3JtYWxGJ0YyRkBGMkZA?<Text-field style="Heading 4" layout="Heading 4"><Font bold="true">Solution</Font></Text-field>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Compute now the evolution of 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 this equation taking into account Schr\303\266dinger's equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI6RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGS0YyRjQ=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Now, from 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this result in 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the coefficient of 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 on the right-hand 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So 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 satisfies a Schrodinger equation and as one can expect, LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEvJkhpbGJlcnRTcGFjZTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is 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Recalling that 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 the time independent case, i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVEYr, LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEvJkhpbGJlcnRTcGFjZTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= reduced to: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiJUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1"><Font size="12">Translation operators using Dirac notation</Font></Text-field>In this section, we focus on the operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYvNiZRImFGJ0YyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LlEpJmVxdWFscztGJ0Y1L0Y5USdub3JtYWxGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctSSVtc3VwR0YkNiUtRkI2LlEvJkV4cG9uZW50aWFsRTtGJ0Y1RkVGR0ZJRktGTUZPRlFGUy9GVlEmMC4wZW1GJy9GWVEsMC4xMTExMTExZW1GJy1JJm1mcmFjR0YkNigtRiM2Ky1JI21uR0YkNiVRKyZ1bWludXMwO2lGJ0Y1RkUtRkI2LVExJkludmlzaWJsZVRpbWVzO0YnRkVGR0ZJRktGTUZPRlFGU0Zqbi9GWUZbb0Y7RmdvLUYvNiZRIlBGJ0YyRjVGOEY1LyUpcmVhZG9ubHlHRjQvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0ZFLUYvNiZRJyZoYmFyO0YnRjJGNUY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZbcS8lKWJldmVsbGVkR0Y3LyUxc3VwZXJzY3JpcHRzaGlmdEdGQC1GLzYjUSFGJ0Y1Rl5wRmBwRkU=<Text-field style="Heading 2" layout="Heading 2"><Font size="12">Settings</Font></Text-field>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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSdBc3N1bWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSkmaHNsYXNoO0YnRjRGNy1JI21vR0YkNi5RIj5GJ0Y0RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1JI21uR0YkNiVRIjBGJ0Y0RjcvRjhRJ25vcm1hbEYnRis=Useful closure relationsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSpQcm9qZWN0b3JGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GSzYlUSRLZXRGJ0ZORlEtRlQ2JC1GIzYlLUZLNiVRIlhGJ0ZORlEtRjM2LVEiLEYnRi9GNi9GOkZQRjtGPUY/RkFGQy9GRlEmMC4wZW1GJy9GSVEsMC4zMzMzMzMzZW1GJy1GSzYlUSJ4RidGTkZRRi9GLw==To have equivalent projectors with different integration variables, we use \356\225\200 as the identity operator: 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style="Heading 2" size="12" layout="Heading 2"><Font size="12">The Action (translation) of the operator </Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRImFGJ0YvRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRImFGJ0YvRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn</Equation><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font size="12"> on a ket</Font></Text-field>Considering a general ket LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXJvd0dGJDYnLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSC1JI21pR0YkNiZRJyYjOTY4O0YnLyUnaXRhbGljR0Y0RjJGNUYuRjJGNUYyRjUvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR1EnJnJhbmc7Ric=, introduce a closure 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 gives after a variable change 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's now evaluate the action of 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 on LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXJvd0dGJDYnLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSC1JI21pR0YkNiZRJyYjOTY4O0YnLyUnaXRhbGljR0Y0RjJGNUYuRjJGNUYyRjUvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR1EnJnJhbmc7Ric= in the 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Comparing the above with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiJUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GU0Y1LUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRXGFsLy1JJHBzaUc2IjYjLCZJInhHNiIiIiJJImFHNiIhIiItSSRJbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwkKioiIiMjIiIiIiIjKiZJI1BpRyUqcHJvdGVjdGVkRyEiIkknJmhiYXI7RzYiISIiIyIiIiIiIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKipeIyIiIiIiIkknJmhiYXI7RzYiISIiLCZJInhHNiIiIiJJImFHNiIhIiIiIiJJInBHNiIiIiIiIiItSVUjbW92ZXIobWkoIiZwc2k7Iixmb250c3R5bGV8aXI9fGlyIm5vcm1hbCIpLG1vKCImc2ltOyIpKUc2IjYjSSJwRzYiIiIiIyIiIiIiIy9JInBHNiI7LCRJKWluZmluaXR5RyUqcHJvdGVjdGVkRyEiIkkpaW5maW5pdHlHJSpwcm90ZWN0ZWRHRicvJSthY3Rpb250eXBlR1E8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMTk4ODkpRicvJSV2aWV3R1EmbGFiZWxGJy1GLDYjUSFGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYjUSFGJy1JKG1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUWJkby8sJi1JKEJyYWNrZXRHNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiL0kxaGlkZW1vZHVsZXByZWZpeEc2IkkldHJ1ZUclKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JS1JJEJyYUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJYRzYiSSJ4RzYiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqKl4jISIiIiIiSSJhRzYiIiIiSSJQRzYiIiIiSScmaGJhcjtHNiIhIiItSSRLZXRHNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiL0kxaGlkZW1vZHVsZXByZWZpeEc2IkkldHJ1ZUclKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kkcHNpRzYiIiIiLUkkcHNpRzYiNiMsJkkieEc2IiIiIkkiYUc2IiEiIiEiIiwmLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJCoqIiIjIyIiIiIiIyomSSNQaUclKnByb3RlY3RlZEchIiJJJyZoYmFyO0c2IiEiIiMiIiIiIiMtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyoqXiMhIiIiIiJJInBHNiIiIiIsJkkiYUc2IiIiIkkieEc2IiEiIiIiIkknJmhiYXI7RzYiISIiIiIiLUlVI21vdmVyKG1pKCImcHNpOyIsZm9udHN0eWxlfGlyPXxpciJub3JtYWwiKSxtbygiJnNpbTsiKSlHNiI2I0kicEc2IiIiIiMiIiIiIiMvSSJwRzYiOywkSSlpbmZpbml0eUclKnByb3RlY3RlZEchIiJJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiIi1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCQqKiIiIyMiIiIiIiMqJkkjUGlHJSpwcm90ZWN0ZWRHISIiSScmaGJhcjtHNiIhIiIjIiIiIiIjLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqKl4jIiIiIiIiSScmaGJhcjtHNiIhIiIsJkkieEc2IiIiIkkiYUc2IiEiIiIiIkkicEc2IiIiIiIiIi1JVSNtb3ZlcihtaSgiJnBzaTsiLGZvbnRzdHlsZXxpcj18aXIibm9ybWFsIiksbW8oIiZzaW07IikpRzYiNiNJInBHNiIiIiIjIiIiIiIjL0kicEc2IjssJEkpaW5maW5pdHlHJSpwcm90ZWN0ZWRHISIiSSlpbmZpbml0eUclKnByb3RlY3RlZEchIiJGJy8lK2FjdGlvbnR5cGVHUTxtYXBsZXNvZnQuY29tOmxhYmVsKEwxOTkxMSlGJy8lJXZpZXdHUSZsYWJlbEYnRjovRjNRJ25vcm1hbEYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoaXNvbGF0ZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIiVGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJiZwc2k7RicvRjBGRUZBLUY2NiQtRiM2JS1GLDYlUSJ4RidGL0YyLUY+Ni1RKCZtaW51cztGJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjIyMjIyMmVtRicvRlZGYm8tRiw2JVEiYUYnRi9GMkZBRkEtRiw2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2"><Font size="12">Action of </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRImFGJ0YvRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRImFGJ0YvRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn</Equation><Font size="12"> on an operator</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>Let's consider an operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiYtRiw2JlEiWEYnRi9GMkY1RjIvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0Y2USdub3JtYWxGJ0YyRkNGMkZARkM=, that can be written as a formal power series: 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.
Its matrix elements are: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Using the closure 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRIlhGJ0YvRjIvRjNRJ25vcm1hbEYn can also be represented in the 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 basis 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Let's now introduce two closure relations to evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEiWEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= in the momentum basis LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXJvd0dGJDYnLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSC1JI21pR0YkNiZRIlBGJy8lJ2l0YWxpY0dRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJ0YuRjJGNUYyRjUvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR1EnJnJhbmc7Ric=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIyUlRidGL0YyRkE=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 now the translation operator LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMTI4LDBdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRImFGJ0YvRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==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Making a variable change 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 the matrix element of this result and compute the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmdmFsdWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1">General Relativity</Text-field><Text-field style="Heading 2" layout="Heading 2">*Exact Solutions to Einstein's Equations <Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation></Text-field>Main reference: - Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. Cambridge University Press, 2003.The authors reviewed more than 4,000 papers containing solutions to Einstein\342\200\231s equations in the literature and organized the material into chapters according to the physical properties of these solutions. These solutions are now digitized within Maple 2016, so that it is now possible to actually compute with them.
The solutions are turned active by a simple call to the g_ spacetime metric.Everything else gets automatically derived on the fly (Christoffel symbols , Ricci and Riemann tensors orthonormal and null tetrads, etc.)Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics and DifferentialGeometry packages.All the mathematics within the Maple library are readily available to work with these solutions.<Text-field style="Heading 2" layout="Heading 2">Examples</Text-field>Load Physics, set the metric to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEuU2Nod2FyenNjaGlsZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiJ0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JlEic0YnRi9GMkY1Ri9GMkY1 solution (and everything else automatically) in one 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSNzY0YnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=And that is all we do. The tensor components of the general relativity tensors related to this solution get derived automatically. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoUmllbWFubkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRK2RlZmluaXRpb25GJ0YvRjIvRjNRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnThe invariants using the formulas by Carminati and McLenaghanLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoUmllbWFubkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRK2ludmFyaWFudHNGJ0YvRjIvRjNRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnThe related Weyl scalars in the context of the Newman-Penrose formalismLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSVXZXlsRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiMtRiw2JVEyc2NhbGFyc2RlZmluaXRpb25GJ0Y0RjcvRjhRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRis=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElV2V5bEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRKHNjYWxhcnNGJ0YvRjIvRjNRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnThe four Killing vectors 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEvS2lsbGluZ1ZlY3RvcnNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJLRidGL0YyL0YzUSdub3JtYWxGJw==These are the 2x2 matrix components of the Christoffel symbols of the second 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This is the tetrad related to Schwarzschild's solutionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRKFRldHJhZHNGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZV9GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYjUSFGJy9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=One can check these things directly; for instance this is the definition of the tetrad, where the right-hand side is the tetrad metricLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZV9GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUStkZWZpbml0aW9uRidGL0YyL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==This shows that, for the components given by , the definition 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These are Einstein's 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where the right-hand side is the energy momentum tensor.Departing from this expression, verify that all the components of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EvYFRfXyZtdTssJm51O2BGJw== are equal to 0 , i.e. that the Schwarzschild metric is a Schwarzschild vacuum solution.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpRWluc3RlaW5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUStkZWZpbml0aW9uRidGL0YyL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==Substituting in 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Define this expression as a tensor, and compute its 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiNRIUYnL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==One can query the database, directly from the spacetime metrics command (g_). For example, these are the solutions (metrics) to Einstein's equations that appear in the book and related to Levi-Civita, the Italian mathematicianLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSVjaXZpRidGL0YyL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==These solutions can be set in one go from the metrics command, just by indicating the number with which it appears in the "Exact Solutions to Einstein's Equations" book.The ability to query rapidly and set things in one go change the game. One can also query visually and by properties. For example, search for solutions in the database that are of Pure Radiation type, with Petrov Type "D", Plebanski-Petrov Type "O" and that have Isometry Dimension equal to 1: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 one of these solutions and everything related in one 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The amount of solutions/cases found in the book and digitized in Maple 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">*"Physical Review D" 87, 044053 (2013)</Text-field>Given the spacetime metric, 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a) Compute the Ricci and Weyl scalarsb) Compute the trace 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RKCZlcXVpdjtGJ0YyRjQvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZLRistSShtZmVuY2VkR0YkNiUtRiw2JlEickYnL0YwUSV0cnVlRidGMi9GNVEnaXRhbGljRidGMkY0LUYsNiNRIUYnRjJGNA== is some function of the radial coordinate, 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 is the Ricci tensor, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJkRzY3I7RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRi82JlEoJmFscGhhO0YnL0YzRjdGNS9GOVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y1Rj8= is the covariant derivative operator and 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 is the stress-energy 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c) Compute the components of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JlEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictRi82JlEoJmFscGhhO0YnL0YzRjdGNS9GOVEnbm9ybWFsRictSSltcGhhbnRvbUdGJDYkLUYvNiZRJyZiZXRhO0YnRkBGNUZBLyUsY29uc3RyYWludHNHUSVib3RoRictRi82I1EhRidGNUZBLUYjNidGTC1GRDYkRj1GSUZGRjVGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGVS9JK21zZW1hbnRpY3NHRiRRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRidGNUZBLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1GLDYuUSgmZXF1aXY7RidGL0YyRjVGN0Y5RjtGPUY/RkEvRkRRLDAuMjc3Nzc3OGVtRicvRkdGTEYvRjI=the traceless part of LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRKCZhbHBoYTtGJy9GMEY0RjIvRjZRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYmUScmYmV0YTtGJ0Y+RjJGPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYsNiNRIUYnRjJGPy1GOTYnRkotRkI2JEY7RkdGREYyRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlMvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn of item b)d) Compute an exact solution to the nonlinear system of differential equations conformed by the components of LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRKCZhbHBoYTtGJy9GMEY0RjIvRjZRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYmUScmYmV0YTtGJ0Y+RjJGPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYsNiNRIUYnRjJGPy1GOTYnRkotRkI2JEY7RkdGREYyRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlMvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn obtained in c)Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.<Text-field style="Heading 3" layout="Heading 3">a) The Ricci and Weyl scalars</Text-field>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Set the 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
The square of the line element and the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZV9GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxudWxsdmVjdG9yc0YnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElV2V5bEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRMnNjYWxhcnNkZWZpbml0aW9uRidGL0YyL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSVXZXlsRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiMtRiw2JVEoc2NhbGFyc0YnRjRGNy9GOFEnbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGKw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmU2V0dXBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSpzaWduYXR1cmVGJ0YvRjIvRjNRJ25vcm1hbEYnJSFH<Text-field style="Heading 3" layout="Heading 3">b) The trace of <Equation executable="false" style="2D Math" input-equation="" 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indicated stress-energy 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item b) in one go, that is the trace of Z, defining the tensorial equation 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answer to a), that is the trace of 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style="Heading 3" layout="Heading 3">b) The components of <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1GLDYuUSgmZXF1aXY7RidGL0YyRjVGN0Y5RjtGPUY/RkEvRkRRLDAuMjc3Nzc3OGVtRicvRkdGTEYvRjI=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1GLDYuUSgmZXF1aXY7RidGL0YyRjVGN0Y5RjtGPUY/RkEvRkRRLDAuMjc3Nzc3OGVtRicvRkdGTEYvRjI=</Equation>the traceless part of <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1JKG1zdWJzdXBHRiQ2KC1JI21pR0YkNiZRIlpGJy8lJ2l0YWxpY0dRJXRydWVGJ0YvL0YzUSdpdGFsaWNGJy1GIzYnLUZMNiZRKCZhbHBoYTtGJy9GUEYxRi9GMi1JKW1waGFudG9tR0YkNiQtRkw2JlEnJmJldGE7RidGWUYvRjIvJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GTDYjUSFGJ0YvRjItRiM2J0Zdby1GZW42JEZWRmpuRmduRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGZm8vSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0YnRi9GMg==</Equation></Text-field>Define a tensor 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 with the traceless part of 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Verify that W is tracelessLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRI211RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y6Rj8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=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The nonzero components for the traceless 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 are thenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYoLUYsNiVRI211RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GQjYtUSJ8aXJGJ0Y/RkUvRkhGPkZJRktGTUZPRlEvRlRRLDAuMTExMTExMWVtRicvRldGaG4tRiw2JVEjbnVGJ0Y9Rj9GQS1GLDYlUShub256ZXJvRidGL0YyRj8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=JSFH<Text-field style="Heading 3" layout="Heading 3">c) An exact solution for the nonlinear system of differential equations conformed by the components of <Equation executable="false" style="2D Math" input-equation="" display="LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRKCZhbHBoYTtGJy9GMEY0RjIvRjZRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYmUScmYmV0YTtGJ0Y+RjJGPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYsNiNRIUYnRjJGPy1GOTYnRkotRkI2JEY7RkdGREYyRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlMvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn">LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRKCZhbHBoYTtGJy9GMEY0RjIvRjZRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYmUScmYmV0YTtGJ0Y+RjJGPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYsNiNRIUYnRjJGPy1GOTYnRkotRkI2JEY7RkdGREYyRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlMvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn</Equation></Text-field>Create an ODE system with the nonzero components of 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a differential elimination process towards identifying singular cases, frequently simpler to solve: there are three 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 third one, a singular case, is of reasonably small size ...LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkbWFwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEnbGVuZ3RoRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GPjYtUSJ+RidGQUZDL0ZHRkVGSEZKRkxGTkZQRlIvRlZGVC1GLDYlUSZjYXNlc0YnRi9GMkZBAn exact solution for 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Specialize one of these constants using the 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 exact 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Verifying this 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JSFH<Text-field style="Heading 2" layout="Heading 2">*The Equivalence problem between two metrics</Text-field>From 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.
<Text-field style="Heading 3" layout="Heading 3">Example from: A. Karlhede, "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980</Text-field>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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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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYjUSFGJy9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVErUGV0cm92VHlwZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiNRIUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElV2V5bEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRKHNjYWxhcnNGJ0YvRjIvRjNRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnWe see that the Weyl scalars are already in canonical form, only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 and the important thing: it depends on only one coordinate, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJiM1ODY3MTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Yy .We want to see if this metric is equivalent to Schwarzschild metric in standard spherical coordinatesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSNzY0YnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIllGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA=. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUStQZXRyb3ZUeXBlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiNGKy9GOFEnbm9ybWFsRidGKw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElV2V5bEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRKHNjYWxhcnNGJ0YvRjIvRjNRJ25vcm1hbEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnThe 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. 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To make the problem slightly more general, consider instead a generic transformation for r in terms of all of 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, and also allow the time to change, so we search for two transformation functions resolving the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEvQ29tcGFjdERpc3BsYXlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYjUSFGJy1JKG1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUU88JC9JInJHNiItSSJSRzYiNiNJIllHNiIvSSJ0RzYiLUkiVEc2IjYjSSJZRzYiRicvJSthY3Rpb250eXBlR1E8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMjAzMTkpRicvJSV2aWV3R1EmbGFiZWxGJ0Y6L0YzUSdub3JtYWxGJw==Transform the coordinates in the metric 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Change also the relationship between the masses so that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmJm1mcjtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJtRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlRGOg==, for instance:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSVzdWJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiotRiw2JVEjbXRGJ0Y0RjctSSNtb0dGJDYtUSJ+RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRkovJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlktRkM2LVEiPUYnRkZGSEZLRk1GT0ZRRlNGVS9GWFEsMC4yNzc3Nzc4ZW1GJy9GZW5Gam5GQi1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGRi1GIzYjLUklbXN1cEdGJDYlRj8tRiM2Iy1GYG82JFEiMkYnRkYvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUubGluZXRoaWNrbmVzc0dGYm8vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZHAvJSliZXZlbGxlZEdGSi1GQzYtUSIsRidGRkZIL0ZMRjZGTUZPRlFGU0ZVRlcvRmVuUSwwLjMzMzMzMzNlbUYnLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRZXcvJkkjZ19HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiL0kxaGlkZW1vZHVsZXByZWZpeEc2IkkldHJ1ZUclKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkjbXVHNiJJI251RzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHNiIiNV06IVFEelN1WSU9RicvJSthY3Rpb250eXBlR1E8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMjAzMzMpRicvJSV2aWV3R1EmbGFiZWxGJ0YrRkZGKw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoY29udmVydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRJHJoc0YnRi9GMi1GNjYkLUYjNiYtRiw2I1EhRidGQS1JKG1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUWV3LyZJI2dfRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ii9JMWhpZGVtb2R1bGVwcmVmaXhHNiJJJXRydWVHJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJI211RzYiSSNudUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjL0kkJWlkRzYiIjU1akNYJXpTdVklPUYnLyUrYWN0aW9udHlwZUdRPG1hcGxlc29mdC5jb206bGFiZWwoTDIwMjg2KUYnLyUldmlld0dRJmxhYmVsRidGQS9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSI9RidGUS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGWS8lKXN0cmV0Y2h5R0ZZLyUqc3ltbWV0cmljR0ZZLyUobGFyZ2VvcEdGWS8lLm1vdmFibGVsaW1pdHNHRlkvJSdhY2NlbnRHRlkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Ziby1GRTYlLUZINiNRZ24tSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEc2IiI1VXAhW1d6U3VZJT1GJy9GTFE8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMjAzNzQpRidGTi1GVDYtUSJ+RidGUUZXRlpGZm5GaG5Gam5GXG9GXm8vRmFvUSYwLjBlbUYnL0Zkb0ZgcC1GVDYtUSIsRidGUUZXL0ZlbkYxRmZuRmhuRmpuRlxvRl5vRl9wL0Zkb1EsMC4zMzMzMzMzZW1GJy1GLDYlUS9zZXRvZmVxdWF0aW9uc0YnRi9GMkZRThis problem, shown in Karlhede's paper as the example of the approach he summarized, is solvable using only differential elimination, in no time, obaining the same solution shown in the paper with equation number (7.10)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The fact that the time t appears defined in terms of the transformed time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2Ki1GIzYqLUYsNiVRJiZUYXU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjkvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQLUkobWZlbmNlZEdGJDYkLUYsNiZRIllGJy9GN1EldHJ1ZUYnL0Y6USdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRLlBoeXNpY3NUZW5zb3JGJ0Y5LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0ZaLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGOS1GPTYtUSkmZXF1YWxzO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EsMC4yNzc3Nzc4ZW1GJy9GUkZoby1GIzYrLUY9Ni1RKiZ1bWludXMwO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZgcC1GLDYlUSYmdGZyO0YnRllGZW4tRj02LVEnJnBsdXM7RidGOUZARkJGREZGRkhGSkZMRl9wRmFwLUYsNiVRJF9DMUYnRllGZW5Gam5GXW9GX29GYW9GOUZqbkZdb0Zfb0Zhb0Y5RitGam5GXW9GX29GYW9GOQ== 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.JSFH<Text-field style="Heading 3" layout="Heading 3">*Equivalence for Schwarzschild metric (spherical and Krustal coordinates)</Text-field>This problem is interesting because:a) It is well known in the literatureb) It involves departing from a metric expressed in "mixed coordinates"c) When writting the metric entirely in Krustal coordinates, the dependence involves special functions (LambertW)<Text-field style="Heading 4" layout="Heading 4"><Font size="14" bold="true">Formulation of the problem (remove mixed coordinates)</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictRiw2I1EhRic=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 departure point, Schwarzschild metric in spherical coordinatesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSNzY0YnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=Introduce now Krustal coordinates following the literature (see wikipedia) and the corresponding line element involving "mixed" 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 mixing of variables is visible: in the line element above is in Krustal coordinates but you also see r , which belongs to the X (not K) coordinates. For the purpose of formulating problem free of this mixing of coordinates, set the metric now to be 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To remove the mix of coordinates, introduce a transformation with unknown transformation functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYnLUYjNictSSNtaUdGJDYmUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJoRidGNEY3RjpGN0ZBRjdGQS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGN0ZB, change variables, and resolve for the transformation functions 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 (this in itself is resolving a form of equivalence problem).LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EoYHRyX18wYEYnLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYwNi1RKiZjb2xvbmVxO0YnRjNGNkY5RjtGPUY/RkFGQy9GRlEsMC4yNzc3Nzc4ZW1GJy9GSUZORi8tSShtZmVuY2VkR0YkNiYtRiM2LC1GLDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvRjRRJ2l0YWxpY0YnLUYwNi1RIj1GJ0YzRjZGOUY7Rj1GP0ZBRkNGTUZPLUYsNiVRImZGJ0ZYRmVuLUZRNiQtRiM2JS1GLDYlUSJyRidGWEZlbi1GMDYtUSIsRidGM0Y2L0Y6RlpGO0Y9Rj9GQUZDRkUvRklRLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRlhGZW5GM0Zkb0YvLUYsNiVRInZGJ0ZYRmVuRmduLUYsNiVRImhGJ0ZYRmVuRl1vRjMvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEvQ29tcGFjdERpc3BsYXlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYjUShgdHJfXzBgRicvRjNRJ25vcm1hbEYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVE1VHJhbnNmb3JtQ29vcmRpbmF0ZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYjUShgdHJfXzBgRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI2dfRidGL0YyLUY2NiYtRiM2JS1GLDYlUSNtdUYnL0YwRkVGQUY9LUYsNiVRI251RidGXG9GQUZBLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkE=Equate to and 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Without loss of generality, set 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 it out:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYjUSFGJy9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=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 is where things become computationally challenguing: compute the inverse of the transformation 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 itr involves the LambertW function. Set now the metric to be the standard Schwarzschild's metric in spherical coordinates and compute use itr to get he form of the metric entirely in Krustal coordinates - no more mixingsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSNzY0YnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=So this is Schwarzschild's solution all in Krustal 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This metric involves the LambertW function in a non-simplifiable form (to avoid that is the reason for people to use the mixed coordinates version ). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 4" layout="Heading 4"><Font size="14" bold="true">Solving the Equivalence</Font></Text-field>We now have the two forms: in spherical and in Krustal coordinates, so we can formulate the equivalence problem from one coordinate system to the other one.The transformation to be resolved does not need to involve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 because neither LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 nor LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn enter either of the two metrics. The transformation does not need to involve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5Nzc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRjEvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjI= because they enter the metrics in exactly the same position and with the same dependence. In addition the Weyl scalars of both metrics are in canonical form and the only scalar different from zero, that is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn does not depend on any of 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) So we look for a generic transformation from spherical to Krustal of the 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The metric set in this moment is in spherical coordinates, , so change using and equate to in Krustal 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Again, this is a nonlinear, non-rational PDE system in two unknowns depending on two independent variables (see ). You can now either call pdsolve on , solving the problem in one step, or first split into cases without solving any differential equation, just doing differential elimination, to see the 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So by only using differential elimination we removed all nonlinearities. This problem is actually easy for the differential equation 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the transformation of coordinates resolving the equivalence between and 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it transforming fully written in spherical coordinates into fully written in Krustal 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style="Heading 2" layout="Heading 2">Tetrads and Weyl scalars in canonical form</Text-field>Generally speaking a canonical form is obtained using transformations that leave invariant the tetrad metric in a tetrad system of references, so that theWeyl scalars are fixed as much as possible (conventionally, either equal to 0 or to 1).Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics.
The implementation is as in "General Relativity, an Einstein century survey", edited by S.W. Hawking (Cambridge) and W. Israel (U. Alberta, Canada), specifically Chapter 7 written by S. Chandrasekhar, page 388:
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 invariancePetrov type 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 type II00LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRS1JI21uR0YkNiVRIjBGJ0YvRjJGL0Yy10nonePetrov type III00010nonePetrov type 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 remains invariant under rotations of Class IIIPetrov type N00001LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1ErYCZQc2k7X180YEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn remains invariant under rotations of Class II
The transformations (rotations of the tetrad system of references) used are of Class I, II and III as defined in Chandrasekar's chapter - equations (7.79) in page 384, (7.83) and (7.84) in page 385. Transformations of Class I can be performed with the command Physics:-Tetrads:-TransformTetrad using the optional argument nullrotationwithfixedl_, of Class II using nullrotationwithfixedn_ and of Class III by calling TransformTetrad(spatialrotationsm_mb_plan, boostsn_l_plane), so with the two optional arguments simultaneously. The determination of appropriate transformation parameters to be used in these rotations, as well as the sequence of transformations happens all automatically by using the optional argument, canonicalform of TransformTetrad .LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGTC8lJmRlcHRoR0ZaLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZWNiZGWEZlbkZnbi9Gam5RJWF1dG9GJy1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEoUGh5c2ljc0YnRi9GMkY5LUY2Ni1RIjpGJ0Y5RjtGU0ZARkJGREZGRkgvRktGT0ZNRlVGXG9GYG8tRmRvNiQtRiM2Iy1GLDYlUShUZXRyYWRzRidGL0YyRjlGNQ==<Text-field style="Heading 3" layout="Heading 3">Petrov type I</Text-field>The numbers below used to enter the metric always refer to the equation number in the "Exact solutions to Einstein's field equations" textbookJkkjZ19HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRis2IzclIiM3IiNAIiIiThe default tetrad computed by the Physics package routinesJkkjZV9HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShUZXRyYWRzRzYlRiUvRidJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGL0YuRig=The corresponding Weyl scalarsJkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0Yo... there is abs around. Let's assume everything is positive to simplify the presentation of formulasLUknQXNzdW1lRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJTYlLUkiPkdGJTYkSSJ4R0YoIiIhLUYxNiRJInlHRihGNC1GMTYkSSJhR0YoRjQ=The scalars are now simpler, although still not in "canonical form" because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X180YEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18zYEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjFGJ0YzRjZGM0Y2.JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoThe Petrov typeLUkrUGV0cm92VHlwZUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFRldHJhZHNHNiVGJS9GJ0koUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJUYvRi5GKA==In this case the Weyl scalars are in canonical form when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2I1ErYCZQc2k7X18wYEYnLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjYtRjA2LlEiLEYnRjNGNkY5L0Y8USV0cnVlRidGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLUSwwLjMzMzMzMzNlbUYnLUYwNi5RIn5GJ0YzRjZGOUY7Rj1GP0ZBRkNGRUZVL0ZLRlYtRiw2I1ErYCZQc2k7X180YEYnRi9GTEYzRjY= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18zYEYnLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjFGJ0YzRjZGM0Y2.LUkwVHJhbnNmb3JtVGV0cmFkRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoVGV0cmFkc0c2JUYlL0YnSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRi9GLjYjSS5jYW5vbmljYWxmb3JtR0YoDespite the fact that the result is a much more complicated tetrad, this is an amazing result in that the resulting Weyl scalars are all fixed (see below). Let's first verify that this is indeed a tetrad, and that now the Weyl scalars are in canonical formLUkpSXNUZXRyYWRHNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShUZXRyYWRzRzYlRiUvRidJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGL0YuNiMtSSdNYXRyaXhHRi02Iy9JJCVpZEdGKCI1QSVcUVZ6U3VZJT0=Set to be the tetrad in use and recompute the Weyl scalarsQyQtSSZTZXR1cEc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJkkoX3N5c2xpYkdGKS9JMWhpZGVtb2R1bGVwcmVmaXhHRilJJXRydWVHRiY2Iy9JJ3RldHJhZEdGKS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjVBJVxRVnpTdVklPSEiIg==Inded we now have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1ErYCZQc2k7X18wYEYnLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjYtRjA2LlEiLEYnRjNGNkY5L0Y8USV0cnVlRidGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLUSwwLjMzMzMzMzNlbUYnLUYsNiNRK2AmUHNpO19fNGBGJ0YvRkxGM0Y2 and 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 Weyl scalars computed after setting the canonical tetrad to be the tetrad in use are in canonical form. Great! NOTE: computing the canonicalWeyl scalars is not really the difficult part, and within the code, these scalars are computed before arriving at the tetrad . What is really difficult (from the point of view of computational complexity and simplifications) is to compute the actual canonical form of the tetrad . LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 3" layout="Heading 3">Petrov type II</Text-field>Consider this other solution to Einstein's equation (again, the numbers in g_[[24,37,7]] always refer to the equation number in the "Exact solutions to Einstein's field equations" textbook)JkkjZ19HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRis2IzclIiNDIiNQIiIoCheck the Petrov typeLUkrUGV0cm92VHlwZUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFRldHJhZHNHNiVGJS9GJ0koUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJUYvRi5GKA==The starting tetradJkkjZV9HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShUZXRyYWRzRzYlRiUvRidJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGL0YuRig=results in Weyl scalars not in canonical form:JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoFor Petrov type "II", the canonical form is as for type "I" but in addition LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18xYEYnLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIzAuRidGM0Y2RjNGNg== Again let's assume positive, not necessary, but to get simpler formulas aroundLUknQXNzdW1lRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJTYmLUkiPkdGJTYkLUkiZkdGKDYjSSJ4R0YoIiIhLUYxNiRGNkY3LUYxNiRJInlHRihGNy1GMTYkSSJhR0YoRjc=Compute now a canonical form for the tetrad, to be used instead of LUkwVHJhbnNmb3JtVGV0cmFkRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoVGV0cmFkc0c2JUYlL0YnSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRi9GLjYjSS5jYW5vbmljYWxmb3JtR0YoSet this tetrad and check the Weyl scalars againQyQtSSZTZXR1cEc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJkkoX3N5c2xpYkdGKS9JMWhpZGVtb2R1bGVwcmVmaXhHRilJJXRydWVHRiY2Iy9JJ3RldHJhZEdGKS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjVFJD5fZ3pTdVklPSEiIg==JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoThis result is fantastic. Compare these Weyl scalars with the ones before transforming the tetrad.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 3" layout="Heading 3">Petrov type III</Text-field>JkkjZ19HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRis2IzclIiM3IiNOIiIiLUknQXNzdW1lRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJTYkLUkiPkdGJTYkSSJ6R0YoIiIhLUYxNiRJJ0xhbWJkYUdGKEY0The Petrov type and the original tetradLUkrUGV0cm92VHlwZUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFRldHJhZHNHNiVGJS9GJ0koUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJUYvRi5GKA==JkkjZV9HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShUZXRyYWRzRzYlRiUvRidJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGL0YuRig=This tetrad results in the following scalarsJkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0Yothat are not in canonical form, which for Petrov type III is as in Petrov type II but in addition we should have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2.Compute now a canonical form for the tetradLUkwVHJhbnNmb3JtVGV0cmFkRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoVGV0cmFkc0c2JUYlL0YnSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRi9GLjYjSS5jYW5vbmljYWxmb3JtR0YoSet this one to be the tetrad in use and recompute the Weyl scalarsQyQtSSZTZXR1cEc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJkkoX3N5c2xpYkdGKS9JMWhpZGVtb2R1bGVwcmVmaXhHRilJJXRydWVHRiY2Iy9JJ3RldHJhZEdGKS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjUlSFsiSCV6U3VZJT0hIiI=JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 3" layout="Heading 3">Petrov type N</Text-field>JkkjZ19HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRis2IzclIiM3IiInIiIiLUknQXNzdW1lRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJTYkLUkiPkdGJTYkSSJ5R0YoIiIhLUYxNiRJJ0xhbWJkYUdGKEY0LUkrUGV0cm92VHlwZUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFRldHJhZHNHNiVGJS9GJ0koUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJUYvRi5GKA==The original tetrad and related Weyl scalars are not in canonical form:JkkjZV9HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShUZXRyYWRzRzYlRiUvRidJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGL0YuRig=JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoFor Petrov type "N", the canonical form has LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X180YEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 and all the other LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X19uYEYnLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2.Compute a canonical form, set it to be the tetrad in use and recompute the Weyl scalarsLUkwVHJhbnNmb3JtVGV0cmFkRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoVGV0cmFkc0c2JUYlL0YnSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRi9GLjYjSS5jYW5vbmljYWxmb3JtR0YoQyQtSSZTZXR1cEc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJkkoX3N5c2xpYkdGKS9JMWhpZGVtb2R1bGVwcmVmaXhHRilJJXRydWVHRiY2Iy9JJ3RldHJhZEdGKS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjVVdCZIZnpTdVklPSEiIg==JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 3" layout="Heading 3">Petrov type D</Text-field>JkkjZ19HNiYlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRis2IzclIiM3IiIpIiIlLUknQXNzdW1lRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJTYmLUkiPkdGJTYkSSJBR0YoIiIhLUYxNiRJIkJHRihGNC1GMTYkSSJ4R0YoRjQzMUY0SSJ6R0YoMUY9LUkiKkdGJDYkSSNQaUdGJS1JIl5HRiQ2JCIiJSEiIg==LUkrUGV0cm92VHlwZUc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFRldHJhZHNHNiVGJS9GJ0koUGh5c2ljc0c2JEYlSShfc3lzbGliR0YoL0kxaGlkZW1vZHVsZXByZWZpeEdGKEkldHJ1ZUdGJUYvRi5GKA==The default tetrad and related Weyl scalars are not in canonical form, which for Petrov type "D" is with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 and all the other 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JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoTransform the tetrad, set it and recompute the Weyl scalarsLUkwVHJhbnNmb3JtVGV0cmFkRzYmJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoVGV0cmFkc0c2JUYlL0YnSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlRi9GLjYjSS5jYW5vbmljYWxmb3JtR0YoQyQtSSZTZXR1cEc2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJkkoX3N5c2xpYkdGKS9JMWhpZGVtb2R1bGVwcmVmaXhHRilJJXRydWVHRiY2Iy9JJ3RldHJhZEdGKS1JJ01hdHJpeEdGKzYjL0kkJWlkR0YpIjUlKkcnKioqKXpTdVklPSEiIg==JkklV2V5bEc2JiUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiRGJUkoX3N5c2xpYkdGKC9JMWhpZGVtb2R1bGVwcmVmaXhHRihJJXRydWVHRiVGKzYjSShzY2FsYXJzR0YoAgain the expected canonical form of the Weyl scalars, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1ErYCZQc2k7X18yYEYnLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjUvJSpzZXBhcmF0b3JHRjUvJSlzdHJldGNoeUdGNS8lKnN5bW1ldHJpY0dGNS8lKGxhcmdlb3BHRjUvJS5tb3ZhYmxlbGltaXRzR0Y1LyUnYWNjZW50R0Y1LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiVRIjBGJ0YzRjZGM0Y2 remains invariant under transformations of Class III.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc5NDUyNDUwNDM4WCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjKyIlIiUhIiIiIiEqJiUiYUciIiMlInhHISIjRilGKUYqLCQlInlHRixGKUYpeic2JEYtRjBbW1tbW19bW1tbcyIiIl1bW1tbW1tbW1tzISIlRic=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc5NTAyOTcwMjYyWCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjKyIlIiUsJComJSJ4RyIiIiwmLSUiZkc2I0YqRisqJiUieUdGKyUiYUdGK0YrRishIiMsJEYqISIiIiIhRjZGNiokRiojRjUiIiNGNkY2RjZGN0YnTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc5OTIzNDI3MDc4WCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjKyIlIiUiIiEtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUiekchIiMtRio2IywkRjAiIiVGKCwkLUYqNiNGMCIiIywkRilGOUYoRihGKCwkKiQtJSRhYnNHNiMlJ0xhbWJkYUchIiIiIiRGJw==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc5OTIzMDg2NTEwWCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjKyIlIiUiIiEsJComJSJ5RyEiIyUnTGFtYmRhRyEiIiMhIiQiIiMsJComRi1GLkYrISIlIiIkRihGKCwkRipGNUYoRihGKEY2Ric=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc5OTU4MjE5MTAyWCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjKyIlIiUsJComJSJCRyIiIy1JJHNpbkc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUiekdGKyEiIiIiISokJSJBR0YrRjRGNComRjZGKyUieEdGK0Y0RjRGNCokRipGK0Yn