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# Mathematica for General Relativity and Gravity research

Posted 11 years ago
 Dear community members,I'm currently try to use Wolfram Mathematica to some gravity research. But i can't find built-in methods do differential geometry calculations in Mathematica.For example, is there any way to compute Einstein or Ricci tensor by metric? Or something more complicate, like create manifold, some medium with fixed state equation and write Einstein equations for this system?Thanks,Boris Latosh
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Posted 11 years ago
 I did not need this type of functionality in the past, so unfortunately I cannot tell you which packages are the most useful.But you'll find a list of tensor-related packages here on Mathematica.SE.The list can be edited by the public wiki-style, so if you put some time into trying one of these packages, you can add a paragraph to the wiki post describing your experiences.
Posted 11 years ago
 Here's my rewrite of one of the notebooks at http://web.physics.ucsb.edu/~gravitybook/mathematica.html. rewritten from http://web.physics.ucsb.edu/~gravitybook/math/curvature.pdf   In[5]:= n = 4;  In[6]:= coord = {r, \[Theta], \[Phi], t};  In[8]:= metric = DiagonalMatrix[{(1 - 2 m/r)^-1, r^2, r^2 Sin[\[Theta]]^2,      2 m/r - 1}]; In[10]:= inversemetric = Simplify @ Inverse[metric];In[12]:= affine = Simplify @    Table[1/2 Sum[      inversemetric[[i,         s]]*(D[metric[[s, j]], coord[[k]]] + D[metric[[s, k]], coord[[j]]] -          D[metric[[j, k]], coord[[s]]]), {s, n}], {i, n}, {j, n}, {k, n}];In[14]:= riemann = Simplify @    Table[D[affine[[i, j, l]], coord[[k]]] - D[affine[[i, j, k]], coord[[l]]] +      Sum[affine[[s, j, l]] affine[[i, k, s]] -        affine[[s, j, k]] affine[[i, l, s]], {s, n}], {i, n}, {j, n}, {k,      n}, {l, n}];In[15]:= ricci = Simplify @ Table[Sum[riemann[[i, j, i, l]], {i, n}], {j, n}, {l, n}]Out[15]= {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}In[16]:= scalar = Simplify[Sum[inversemetric[[i, j]] ricci[[i, j]], {i, n}, {j, n}]]Out[16]= 0In[17]:= einstein = ricci - 1/2 scalar*metricOut[17]= {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}It shows that the Schwarzschild metric satisifes the Einstein equation.  I didn't show the components of the 3rd and 4th rank tensors here.
Posted 11 years ago
 You may want to look through this thread on Mathematica StackExchange:http://mathematica.stackexchange.com/questions/8895/how-to-calculate-scalar-curvature-ricci-tensor-and-christoffel-symbols-in-mathem/which has material that may help you get going (as well as having some additional possibly useful links)
Posted 11 years ago
 I looked at the TensorProduct page and didn't see a component-based definition of TensorProduct.
Posted 11 years ago
 Please take a look at the new Symbolic Tensors functionality introduced in version 9. Specifically:Symbolic Tensors - guideTensors - guideSymbolic Tensors - tutorialTensor Symmetries - tutorialSymmetrized Arrays - tutorialTo get started with basic examples read through this page: TensorProduct
Posted 11 years ago
 Great! Thank to everybody! This will do for beginning!
Posted 11 years ago
 You might want to have a look at this:http://www.wolfram.com/books/profile.cgi?id=7182M.
Posted 11 years ago
 If you want a full tensor analysis system that will also do what you mention, seehttp://smc.vnet.net/mathtensor.htmlhttp://www.amazon.com/MathTensor-System-Tensor-Analysis-Computer/dp/0201569906/ref=sr_1_1?ie=UTF8&qid=1400179230&sr=8-1&keywords=mathtensorSteve C.
Posted 11 years ago
 There are some old Mathematica (v 4) notebooks for doing GR at this link:http://web.physics.ucsb.edu/~gravitybook/mathematica.htmlI've updated and run one of them myself and it worked fine,working from the pdf version.  It may be possible to run the notebook itselfbut I didn't try that.
Posted 11 years ago