MathTensor(tm) Information Sheet (Retail Pricing) Prices Valid January 1, 1993 [Academic discounts are available. Contact MathSolutions.] Hardware Requirements: MathTensor requires approximately 2 megabytes of disk space. It is recommended that a workstation have at least 8 megabytes of RAM memory. The Macintosh version runs on machines with 8 megabytes or more. DOS versions require at least 5 megabytes of memory, but more is highly recommended. The Windows 3.x version works with 8 megabyes of RAM or more. Software Requirements: MathTensor requires Mathematica 1.2, 2.0, or greater. Contact: MathSolutions, Inc. P.O. Box 16175 Chapel Hill, NC 27516 Telephone/Answering Machine/FAX: 919-967-9853 Email: mathtensor@wri.com * MathTensor Product Price List: (Prices include media/manual/shipping via first class mail, Upgrade contracts provide upgrades to MathTensor for one year - a minimum of two upgrades - from date of shipment.) Macintosh/DOS/Windows: No Upgrade $550.00 With Upgrade $700.00 Single Processor Workstations: No Upgrade $800.00 With Upgrade $1000.00 Convex/Multi-Processor Servers No Upgrade $3000.00 With Upgrade $4000.00 Extra Manuals: $25.00 Federal Express Delivery add: $35.00 Upgrades and updates are only sent to users with Upgrade Contracts. Volume discounts available. ** North Carolina residents add 6% sales tax ** Purchase orders are accepted. ** Payment terms - NET 15 ** Prices, availability and details of functionality of any products or services mentioned on this page are subject to change without notice. ** No credit card orders Availability: MathTensor is available in 1/4" tape (QIC-24) tar format, MS-DOS 3.5" floppies, Macintosh 800K floppies, and Sun 3.5" bar format floppies. Other formats will be made available as needed but may require more time for shipment. Copy Protection: MathTensor, like Mathematica, requires a password on UNIX machines. This password must be obtained from MathSolutions, Inc. Authors: Leonard Parker, Ph.D. and Steven M. Christensen, Ph.D., founders of MathSolutions, Inc. -------------------- Order Form Cut Here --------------------------------- MathTensor 2.1.5 Ordering Information (January 1, 1993) Price Number Total o Personal Computers [ ] Macintosh (without upgrade, 800K 3.5" floppies) $550 _____ _______ [ ] Macintosh (with upgrade, 800K 3.5" floppies) $700 _____ _______ [ ] DOS/Windows (without upgrade, 1.44 Meg 3.5" floppies) $550 _____ _______ [ ] DOS/Windows (with upgrade, 1.44 Meg 3.5" floppies) $700 _____ _______ o Single Processor Workstations** [ ] NeXT Workstation (without upgrade, 1.44 Meg 3.5" DOS floppies) $800 _____ _______ [ ] NeXT Workstation (with upgrade, 1.44 Meg 3.5" DOS floppies) $1000 _____ _______ [ ] Sun Sparcstation (without upgrade, 3.5" Sun bar floppies) $800 _____ _______ [ ] Sun Sparcstation (with upgrade, 3.5" Sun bar floppies) $1000 _____ _______ [ ] UNIX Workstation (without upgrade, 1/4" QIC-24 tar tape) $800 _____ _______ [ ] UNIX Workstation (with upgrade, 1/4" QIC-24 tar tape) $1000 _____ _______ [ ] UNIX Workstation (without upgrade, 1/4" QIC-150 tar tape) $800 _____ _______ [ ] UNIX Workstation (with upgrade, 1/4" QIC-150 tar tape) $1000 _____ _______ Specify UNIX Workstation Type _______________________________ [ ] VMS Workstation (without upgrade, 1/4" QIC-24 tar tape) $800 _____ _______ [ ] VMS Workstation (with upgrade, 1/4" QIC-24 tar tape) $1000 _____ _______ Give the Mathematica $MachineID by typing $MachineID while in Mathematica __________________________ If your workstation for MathTensor does not have a $MachineID (usually if you have a network license), type hostid at the UNIX or VMS prompt and enter it here ______________________ o Convex/Multiprocessor Servers** [ ] Convex (without upgrade, 1/4" QIC-24 tar tape) $3000 _____ _______ [ ] Convex (with upgrade, 1/4" QIC-24 tar tape) $4000 _____ _______ [ ] Multiprocessor Server (without upgrade, 1/4" QIC-24 tar tape) $3000 _____ _______ [ ] Multiprocessor Server (with upgrade, 1/4" QIC-24 tar tape) $4000 _____ _______ Specify Machine Type _______________________________ Give the Mathematica $MachineID by typing $MachineID while in Mathematica __________________________ If your workstation for MathTensor does not have a $MachineID (usually if you have a network license), type hostid at the UNIX prompt and enter it here ______________________ [ ] Extra Manuals $25 _____ _______ [ ] Federal Express Delivery (give shipping address and phone number) $35 _______ Subtotal _______ North Carolina residents add 6% sales tax _______ Order Total (Send PO or check in US Dollars) allow 4 weeks for delivery, no credit card orders, terms NET 15) _______ All prices include shipping and one manual. Prices are subject to change without notice. Upgrade gives the purchaser one year of MathTensor updates (minimum of 2). ** If your computer is on the Internet, it is also possible for us to do a remote installation. Contact MSI. o Shipping Information: Name: ________________________________________________ Address: __________________________________________________________ __________________________________________________________ City: ___________________________ State: _________________________ Postal/Zip Code: __________________ Country: _____________________ PO Number __________________________ Telephone: ________________________________ Fax: ________________________________ Email: ________________________________________ --------------------- End of Order Form ---------------------------------- Some of the users of MathTensor can be found at: Wake Forest University The University of Maryland Insituto de Fisica Fundamental - Madrid, Spain City College of New York The University of Winnipeg -Canada Stanford University Caltech The University of California - Santa Barbara Texas A&M University The University of North Carolina - Chapel Hill Universitat Konstanz - Germany Schlumberger KK - Japan Sumitomo Corporation - Japan Oakland University Louisiana Tech University Polaroid Lawrence Livermore Labs Los Alamos Labs The University of New Brunswick -Canada United Technologies Research Center Universitat Hannover - Germany Reed College Yale University The University of Bergen - Norway Cotton, Inc. NASA - Langley NASA - Goddard NASA - JPL National Center for Supercomputing Applications Wolfram Research The University of Wisconsin - Milwaukee The Hebrew University of Jerusalem -Israel Martin Marietta Cornell University Central Connecticut State University Universite de Liege - Belgium Pune University - India Cal State University - Fullerton Queen's University - Canada Utah State University Southwestern University - Texas University of South Florida Deutsches Klimarenzentrum - Hamburg, Germany University of Oklahoma - Norman Montana State University University of Chicago University of Milan - Italy Wellesley College University of San Francisco University of Cologne - Germany University of California - San Diego University of Berne - Switzerland University d. Bundeswehr - Hamburg, Germany University of Thessaloniki - Greece University of California - San Diego Royal Insitute of Technology (KTH) - Sweden Univeritat d. Bundeswehr - Hamburg University of Cologne - German Hamamatsu Photonics - Japan Linkoeping University - Sweden University of Washington - Seattle University of Cincinnati W.R. Grace, Co. Aberdeen Proving Ground Nissin-High Voltage Co, Ltd. - Japan Ibaraki University - Japan University of California - Berkeley CNR-FISBAT - Italy Austin College Mercyhurst College Universidad De Puerto Rico Unviversity of Alberta University of Oregon NeXT Computer Corporation Queen's University - Canada Tokyo Institute of Technology - Japan Univ. Autonoma Metropolitana - Mexico S3IS - France PICA Software - Australia RITME Informatique - France Mount Sinai Hospital - New York IVIC - Venezuela Carnegie Mellon University Choong Puk National University - Korea MathTensor has been mentioned in articles in MacWorld, the Mathematica Journal, Computers In Physics, Science, and PC Magazine. "The idea of packages and special tools has been taken to a kind of modern limit in the form of MathTensor, a tensor- analysis package built on Mathematica." - Richard Crandall, Howard Vollum Professor of Science, Reed College and Chief Scientist, NeXT Computer, Inc., from Computers in Physics, Nov/Dec 1991. "I would like to take this opportunity to tell you how pleased I have been with MathTensor. I have been using it to study higher-dimensional cosmological solutions of Einstein's Equations. It has been an indispensable tool for me. You have done a great job." - Dwight Vincent, University of Winnipeg - Canada, Feb 1992. "Regarding my comments on MathTensor, you can be assured that they were coming out of my heart, and that they are TRUE. Yes, it is a very nice and useful tool. Many people, like myself, who are getting older and loosing the patience (and in some cases the ability to do long tedious computations) to deal with the "nitty gritty" of things can find here REAL HELP. Also, for those of us who are daring and curious it opens up a door to tackle problems that we otherwise wouldn't." Juan Perez Mercader, Insituto de Fisica Fundamental - Madrid, Spain, Feb 1992. "MathTensor is an outstanding package for doing tensor calculus. Best of all is the support. Questions receive an immediate reply by email from both Leonard and Steve, including advice which frequently goes far beyond the original point. Part of the MathTensor package is the continuing insight of two researchers who use this mathematics as their tool." - George Ruppeiner, Associate Professor of Physics, University of South Florida, Sept 1992. -----------------------General Information-------------------------- MathTensor Tensor analysis is extensively used in applications in physics, mathematics, engineering, and many other areas of scientific research. Problems involving tensors often are extraordinarily large and can be some of the most difficult computations in all of science. Equations with thousands of terms are common and can only be manipulated by computer mathematics systems like Mathematica. MathTensor is the largest Mathematica package yet developed outside of Wolfram Research. It adds over 250 new functions and objects to Mathematica to give the user both elementary and advanced tensor analysis functionality. MathTensor is a general tool for handling both indicial and concrete tensor indices. Standard objects like Riemann tensor, Ricci tensor, metric and others are built into the system along with common functions like the covariant derivative, index commutation, raising and lowering of indices, and various differential forms operations. MathTensor has been under development by Leonard Parker and Steven M. Christensen since the first alpha test release of Mathematica. It contains over 25,000 lines of Mathematica code contained in nearly 100 files totalling approximately 2.0 Megabytes of disk space. MathTensor will run on any machine that runs Mathematica and has sufficient RAM memory (generally 8 Megabytes or more) and disk space for file storage and swap. MathTensor runs under versions 1.2 and 2.X of Mathematica. MathSolutions, Inc. was formed by Leonard Parker and Steven M. Christensen. MathSolutions has its office in Chapel Hill, North Carolina. Both Parker and Christensen are theoretical physicists, specializing in research in Einstein's Special and General Theory of Relativity, quantum field theory, black hole theory, and cosmology. Christensen is a Contributing Editor to the Mathematica Journal and is the founder and Moderator of the MathGroup, Mathematica mailing list. Parker obtained his Ph.D. in Physics from Harvard University and Christensen his Ph.D. in Physics from the University of Texas at Austin. MathTensor is a trademark of MathSolutions, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. ----------------------MathTensor Examples ---------------------------- MathTensor provides commands for simplifying and manipulating tensor expressions, as well as a knowledge base of transformation rules and definitions required for dealing with some of the more important tensors. MathTensor is designed to work along with the functions of Mathematica to provide users with the functions and objects they need to devise their own custom tensor analysis programs. MathTensor provides most of the basic structures needed for doing tensor computations and for programming new functions. As MathTensor is used and special applications are developed, they will be added to future versions of MathTensor. The following pages give a few examples of how MathTensor works. (* First, after starting Mathematica 2.X, we load a file which in turn loads many other files containing the MathTensor function and object definitions. *) In[1]:= <ub b Out[5]= R ab; (* The DefineTensor function permits you to define your own tensors. The input name of the object defined below is "tensor", and its print name, which appears in output lines, is "t". The last argument indicates that it will have two indices, which upon interchange result in multiplication of the object by a weight factor of 1 -- that is, it is a symmetric tensor. *) In[6]:= DefineTensor[tensor,"t",{{2,1},1}] PermWeight::sym: Symmetries of t assigned PermWeight::def: Object t defined In[7]:= tensor[la,lb] Out[7]= t ab (* MathTensor now automatically reorders symmetric indices into lexical order. *) In[8]:= tensor[lb,la] Out[8]= t ab (* The standard symmetries of the Riemann tensor are built into its definition. *) In[9]:= RiemannR[lb,la,lc,ld] Out[9]= -R abcd In[10]:= RiemannR[lc,ld,la,lb] Out[10]= R abcd (* MathTensor knows that the appropriate sum of indices on the Riemann tensor gives the Ricci tensor. *) In[11]:= RiemannR[la,lb,ua,lc] Out[11]= R bc (* The sum of the indices on the Ricci tensor gives the Riemann Scalar. *) In[12]:= RicciR[la,ua] Out[12]= R (* Now we define a new tensor with four indices and no symmetries. *) In[13]:= DefineTensor[T,"T",{{1,2,3,4},1}] PermWeight::sym: Symmetries of T assigned PermWeight::def: Object T defined (* We produce a complicated product of seven of these tensor with multiple summations of indices and then add it to another similar object. *) In[14]:= SevenTensorTest := T[la,lb,uc,ud] T[lc,ld,ue,uf] T[le,lf,ug,uh] T[lg,li,ui,uj] * T[lh,lj,uk,ul] T[lk,ll,um,un] T[lm,ln,ua,ub] - T[la,lb,uc,ud] T[lc,le,ue,uf] * T[ld,lf,ug,uh] T[lg,lh,ui,uj] T[li,lj,uk,ul] T[lk,ll,um,un] T[lm,ln,ua,ub] (* Trying to find some simplification of SevenTensorTest is not easy by hand. *) In[15]:= SevenTensorTest cd ef gh ij kl mn ab Out[15]= T T T T T T T - ab cd ef gi hj kl mn cd ef gh ij kl mn ab > T T T T T T T ab ce df gh ij kl mn (* But the MathTensor command Tsimplify rapidly finds that the two terms are equal. *) In[16]:= Tsimplify[%] Out[16]= 0 (* Some terms differ only by the renaming of summation indices. *) In[17]:= T[la,lb,lc,ld] RiemannR[ua,ub,uc,ud] + T[le,lf,lg,lh] RiemannR[ue,uf,ug,uh] abcd efgh Out[17]= R T + R T abcd efgh (* MathTensor's canonicalization functions can rename indices and combine terms. *) In[18]:= Canonicalize[%] pqrs Out[18]= 2 R T pqrs (* MathTensor can symmetrize or antisymmetrize pairs of indices. *) In[19]:= Symmetrize[T[la,lb,lc,ld], {la,lb}] T + T abcd bacd Out[19]= ------------- 2 In[20]:= Expand[%] T T abcd bacd Out[20]= ----- + ----- 2 2 In[21]:= Antisymmetrize[%,{lc,ld}] T T T T abcd abdc bacd badc ----- - ----- + ----- - ----- 2 2 2 2 Out[21]= ----------------------------- 2 In[22]:= Expand[%] T T T T abcd abdc bacd badc Out[22]= ----- - ----- + ----- - ----- 4 4 4 4 (* MathTensor understands how to convert covariant to ordinary partial derivatives with affine connection terms added. *) In[23]:= CD[RicciR[la,lb],lc] Out[23]= R ab;c In[24]:= CDtoOD[%] p p Out[24]= R - G R - G R ab,c bc pa ac pb (* Using positive and negative index values, MathTensor can deal with concrete contravariant or covariant indices. *) In[25]:= RiemannR[1,2,3,4] 1234 Out[25]= R In[26]:= RiemannR[-1,2,-3,4] 4 2 Out[26]= R 3 1 (* We can set the dimension of the spacetime to some value, like 4. *) In[27]:= Dimension = 4 Out[27]= 4 (* Then using the MakeSum function, we can explicitly write out sums in terms of concrete indices. *) In[28]:= MakeSum[RicciR[la,lb] RicciR[lc,ub]] 1 2 3 4 Out[28]= R R + R R + R R + R R 1a c 2a c 3a c 4a c (* Now we define tensor T with two indices that are symmetric. *) In[29]:= DefineTensor[T,"T",{{2,1},1}] PermWeight::sym: Symmetries of T assigned PermWeight::def: Object T defined (* Then the lower components of T can be defined in terms of the components of other tensors like the Ricci tensor. *) In[30]:= SetComponents[T[la,lb],RicciR[la,lc] RicciR[lb,uc]] Components assigned to T (* We can ask for a specific covariant component of T. *) In[31]:= T[-1,-1] 1 2 3 4 Out[31]= R R + R R + R R + R R 11 1 21 1 31 1 41 1 (* One example application built into MathTensor does variations with respect to the metric tensor of structures that are functions of the metric tensor, Metricg. The variation of the square root of the determinant of the metric times the Riemann scalar gives terms in the variation of the metric, called h. *) In[32]:= Sqrt[Detg] ScalarR Out[32]= Sqrt[g] R In[33]:= Variation[%,Metricg] MetricgFlag::off: MetricgFlag is turned off by this operation pq p q Out33]= Sqrt[g] h - Sqrt[g] h + pq; p ;q pq Sqrt[g] R g h pq pq ----------------- - Sqrt[g] R h 2 pq (* A more complicated variation is just as easily found. *) In[34]:= Sqrt[Detg] RicciR[la,lb] RicciR[ua,ub] ab Out[34]= Sqrt[g] R R ab In[35]:= Variation[%,Metricg] pq r -(Sqrt[g] h R ) ;r pq p rq Out[35]= --------------------- + Sqrt[g] h R - 2 q; pr p qr r pq Sqrt[g] h R Sqrt[g] h R p ; qr pq;r ------------------ - ------------------ + 2 2 p qr Sqrt[g] h R q pr p ;qr Sqrt[g] h R - ------------------ + pq;r 2 pq rs Sqrt[g] g R R h rs pq qr p ----------------------- - Sqrt[g] R R h - 2 pq r pr q Sqrt[g] R R h pq r (* MathTensor's ApplyRules function permits the user to define large sets of rules that can be applied to expressions to simplify them. One set of rules that is provided as an example are the RiemannRules. Familiar rules are applied in the next two examples. MathTensor includes several functions DefUnique and RuleUnique that help the user devise their own rules and save them for later use. *) In[36]:= CD[RicciR[la,lb],ub] b Out[36]= R ab; In[37]:= ApplyRules[%,RiemannRules] R ;a Out[37]= --- 2 In[38]:= RiemannR[la,lb,lc,ld] RiemannR[ua,uc,ub,ud] acbd Out[38]= R R abcd In[39]:= ApplyRules[%,RiemannRules] pqrs R R pqrs Out[39]= ----------- 2 (* Suppose we define a tensor with four indices. *) In[40]:= DefineTensor[tensor,"t",{{1,2,3,4},1}] PermWeight::sym: Symmetries of t assigned PermWeight::def: Object t defined (* MathTensor's multiple index facility permits the user to add indices with one, two or three primes to be used as extra non-spacetime indices. *) In[41]:= AddIndexTypes In[42]:= tensor[ala,blb,clc,ld] Out[42]= t a'b''c'''d (* Tools for building rules involving these extra indices are provided. Future releases of MathTensor will extend this functionality. *) (* Shipped with MathTensor is the file Components.m which may be run separately from MathTensor. Components.m takes a file like SchwarzschildIn.m, listed next, and computes the components of the affine connection, the Riemann tensor, Ricci tensor, Riemann scalar, Weyl tensor, Einstein tensor and several other objects. SchwarzschildIn.m is the input file containing information about the famous Schwarzschild metric. This metric represents the curved space of a gravitating spherically symmetric object with mass M in otherwise empty space. *) ---------------------- SchwarzschildIn.m file listing -------------------- (* Copyright (c) 1992 MathSolutions, Inc.*) (* SchwarzschildIn.m *) Dimension = 4 x/: x[1] = r x/: x[2] = theta x/: x[3] = phi x/: x[4] = t Metricg/: Metricg[-1, -1] = (1 - (2*G*M)/r)^(-1) Metricg/: Metricg[-2, -1] = 0 Metricg/: Metricg[-3, -1] = 0 Metricg/: Metricg[-4, -1] = 0 Metricg/: Metricg[-2, -2] = r^2 Metricg/: Metricg[-3, -2] = 0 Metricg/: Metricg[-4, -2] = 0 Metricg/: Metricg[-3, -3] = r^2*Sin[theta]^2 Metricg/: Metricg[-4, -3] = 0 Metricg/: Metricg[-4, -4] = -(1 - (2*G*M)/r) Rmsign = 1 Rcsign = 1 CalcEinstein = 1 CalcRiemann = 1 CalcWeyl = 1 SetOptions[Expand, Trig->True] SetOptions[Together, Trig->True] CompSimp[a_] := Expand[Together[a/.CompSimpRules[1] ] ] CompSimpRules[1] = {} (* End of file SchwarzschildIn.m *) ------------ (* If we load Components.m into Mathematica we can run the following command to produce from SchwarzschildIn.m two new files, SchwarzschildOut.m, which contains results that can be used immediately in MathTensor, and SchwarzschildOut.out which can be printed. The computation below takes from just a few seconds to less than one minute on typical workstations. More complex metrics can take a bit longer. *) In[1]:= <