Page Index

Classical Problems In Computer Algebra

Bondi Metric

The Bondi metric (Proc. Roy. Soc.A269 21) plays an interesting role in the history of algebraic computing in general relativity (see d'Invero in General Relativity and Gravitation ed. A. Held ISBN 0-306-40265-3 (v. 1)).

Demonstration 1 (bondi): One step calculation of the covariant Riemann, Ricci and Einstein tensors for the Bondi metric. (This calculation, which once took from 10 to 1000 seconds of computation time on mainframe computers, now runs in less than 1 second on a common PC in many, but not all, computer algebra systems.)

bondig1.mpl || bondi.mws || bondi.html || bondi.pdf

Ricci Tensor-Number of terms.

The idea is to calculate the number of terms in the covariant Ricci tensor in n dimensions. The most general form of the metric tensor is used, that is n(n+1)/2 functions of n variables. The number of terms grows rapidly with n (see G.J. Fee, R.G. McLenaghan and R. Pavelle GR12 Contributed Papers p296.)

Diagonal components174169,990298,134
Off-diagonal components1751913,280410,973
In the following examples the same worksheet is used in each dimension. Whereas n=4 can now be handled on most PCs, the case n=5 requires a dedicated resource (several hours computation time at 600 MHz and about 1 GB of RAM).

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General Relativity

Bel-Robinson tensor

The Bel-Robinson tensor is constructed out of the curvature somewhat analogously to the construction of the stress-energy tensor for the electromagnetic field (e.g. Wald General Relativity). It has a number of applications not only in general relativity, but also in supergravity theories (e.g. Deser in Gravitation and Geometry (ed. by Rindler and Trautman)).

Demonstration 1 (br): The Bel-Robinson tensor is defined (with the symmetry Tabcd=T(abcd)) and evaluated in the Kerr-Newman metric. The trace is reduced to zero, and setting the charge to zero, the divergence is also reduced to zero.

newkn.mpl || br.mws || br.html || br.pdf

Gödel Metric

The Gödel (1948) metric is of historical interest in that it provided a stimulus to the study of exact solutions of Einstein's equations. (See, for example, Hawking and Ellis The Large Scale Structure of Space-Time Section5.7)

Demonstration 1 (godel): An elementary study of the Gödel metric.

godel1.mpl || godel.mws || godel.html || godel.pdf

Kerr (and Kerr-Newman) Spacetimes

The Kerr metric (Kerr Phys. Rev. Letters,11,237) is arguably the most important exact solution of the Einstein equations known (e.g. Chandrasekhar The Mathematical Theory of Black Holes). The demonstrations given here provide a brief introduction to the spacetime and its charged counterpart.

Demonstration 0 (horizon): Geometry of the Kerr horizon: The Gauss curvature, area and Euler characteristic of the Kerr horizon are evaluated for r=R where R=m +/- (m2-a2)1/2 in Boyer-Lindquist coordinates at constant t. Also see ergo.

twod.mpl || horizon.mws || horizon.html || horizon.pdf

Demonstration 1 (kerr): An introduction to the Kerr metric: Short comparison of the time taken to show that the solution is vacuum in two coordinate systems, calculation of the Kretschmann scalar and Weyl scalars, coordinates adapted to two Killing vectors, Frobenius theorem, Ricci and Weyl scalars from a null tetrad.

kerr.mpl || newkerr.mpl || npdnkerr.mpl || kerr.mws || kerr.html || kerr.pdf

Demonstration 2 (em): Einstein-Maxwell equations in the Kerr-Newman spacetime: vector potential, electromagnetic field tensor, invariants, 4-current, Maxwell equations, Einstein-Maxwell equations.

newkn.mpl || em.mws || em.html || em.pdf

Demonstration 3 (statlim): An invariant (Rabcd;eRabcd;e) which vanishes on the stationary limit surface of the Kerr spacetime. (See Karlhede, Lindström and Åman Gen. Rel and Grav. 14, 569.)

newkerr.mpl || statlim.mws || statlim.html || statlim.pdf

Demonstration 4 (geww): A generalization of the above considered recently in gr-qc/9808055.

rnds.mpl || sdiaga.mpl || newkerr.mpl || gammas.mpl || bzts.mpl || geww.mws || geww.html || geww.pdf

Mixmaster Cosmology

For an introduction see Misner, Thorne and Wheeler Gravitation. In these demonstrations we show how to enter the spacetime in various forms, and how to perform calculations with it.

Demonstration 1 (mix1in): We enter a "Frame-Field" with constant basis inner product. To see the result look at mix1.mpl. The worksheet, when executed, is ready to work with the mixmaster spacetime.

mix1.mpl || mix1in.mws || mix1in.html || mix1in.pdf

Demonstration 2 (mix1): In the bases created by demonstration 1 we calculate the bases components of the Ricci and Weyl tensors. This is virtually instantaneous. We go on to evaluate the Ricci and Weyl invariants. This takes a very few seconds.

mix1.mpl || mix1.mws || mix1.html || mix1.pdf

Demonstration 3 (mix1c): Suppose you require the coordinate components of (say) Ricci and Weyl, but you don't want to type in the metric. In this demonstration the metric is generated from the bases created in demonstration 1. The coordinate components are then calculated.

mix1.mpl || mix1c.mws || mix1c.html || mix1c.pdf

Nariai Spacetime

This spacetime is a vacuum solution of Einstein's equations with nonzero cosmological constant. It is of interest here because of the form of the constraint.

Demonstration 1 (nariai): The coordinate Rab and basis R(a)(b) components of the Ricci tensor are calculated and simplified.

nariai.mpl || nariaib.mpl || nariai.mws || nariai.html || nariai.pdf

Reissner-Nordström-de Sitter Spacetime

This spacetime has an interesting global structure (e.g. Brill (grqc 9501023) to appear in Springer Lec. Notes in Phys.).

Demonstration 1 (rnds): Calculation and simplification of a few invariants (R, RabRab,RabcdRabcd,R,a,a,Rab;cRab;c).

rnds.mpl || rnds.mws || rnds.html || rnds.pdf

Demonstration 2 (rnds1): Automatic generation of an NPtetrad from the metric, calculation of the Petrov type and the Ricci scalars.

rnds.mpl || rnds1.mws || rnds1.html || rnds1.pdf

Schwarzschild Spacetime

The Schwarzschild spacetime remains at the gateway to our understanding of Einstein's theory of the gravitational field. Whereas the original spacetime (in "curvature" coordinates) is computationally trivial, an understanding of the spacetime's complete structure came almost fifty years after its original derivation by way of the simultaneous discoveries of Kruskal and Szekeres.

Demonstration 0 (schwsoln): Solve Einstein's vacuum field equations for a spherical static spacetime.

statica.mpl || schwsoln.mws || schwsoln.html || schwsoln.pdf

Demonstration 1 (kruskalo): Transformation from the original Schwarzschild coordinates to the original Kruskal coordinates ( Phys. Rev. 119 1743) with grtansform. Calculation and simplification of the Ricci tensor and Kretschmann scalar in the Kruskal coordinates. This calculation is of interest for computer algebra systems since components of the metric tensor are only transcendental functions of the coordinates. This requires the use of constraints on the metric by way of grconstraint.

kruskalo.mpl || kruskalo.mws || kruskalo.html || kruskalo.pdf

Demonstration 2 (israel): Calculation and simplification of the Ricci tensor and Kretschmann scalar in the Israel coordinates ( Phys. Rev. 143 1016).

israel.mpl || israel.mws || israel.html || israel.pdf

Demonstration 3 (schwbasis): Calculation and simplification of the pre-rotation coefficients, rotation coefficients and structure constants in a general basis. A call is made to the Weyl tensor in the form C (a) (b)(c) (d) as a demonstration.

schwb.mpl || schwbasis.mws || schwbasis.html || schwbasis.pdf

Demonstration 4 (boost): Ricci tensor for a boosted black hole in particularly awkward coordinates.

schcartesian.mpl || boost.mws || boost.html || boost.pdf

Self-Similar Spacetimes

These are spacetimes which contain a homothetic Killing vector. Typically, a self-similar spacetime will have a metric tensor which contains functions with arguments of the form r/t where r and t are coordinates. The purpose of these demonstrations is to show how GRTensorII handles such functions directly. Self-similar spacetimes are widely discussed in the literature. (See, for example Ref. [2] in K. Lake Phys. Rev. Lett. 68,3129.)

Demonstration 1 (ss1): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in curvature coordinates.

ss1.mpl || ss1.mws || ss1.html || ss1.pdf

Demonstration 2 (ss2): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in comoving coordinates.

ss2.mpl || ss2.mws || ss2.html || ss2.pdf

Demonstration 3 (ss3): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in Bondi coordinates. This is a particularly convenient form.

ss3.mpl || ss3.mws || ss3.html || ss3.pdf

Static fluid spheres

In a recent preprint Raham and Visser (gr-qc/0103065) have exhibited a simple and explicit formula for those spacetime metrics characterizing the entire class of static spherically symmetric perfect fluid spacetimes.

Demonstration 1 (visser): We use constraints to obtain their equation (2.3) (both in a basis and with coordinates). The typical computation time is < 1/5 second.

visser.mpl || visserb.mpl || visser.mws || visser.html || visser.pdf

Stephani metric

The Stephani metric (Commun. math. Phys. 4, 137 (1967)) is the most general conformally flat non static perfect fluid solution of Einstein's equations.

Demonstration 1 (steph): Constraints on the derivatives of V(x,y,z,t) are used to show that the space is conformally flat, and to reduce the Einstein tensor to a perfect fluid form. The explicit form of V(x,y,z,t) is then used to reduce the energy density and pressure to the standard forms.

stephani.mpl || stephani.mws || stephani.html || stephani.pdf


There are a number of ways in which GRTensorII can be used to find symmetries of a spacetime.

Demonstration 1 (ckvec): The spacetime is in a form adapted to a conformal Killing vector.

ckvec.mpl || ckvec.mws || ckvec.html || ckvec.pdf

Demonstration 2 (cktest): Conformal Killing vectors in conformally flat spacetime (Cartesian Coordinates).

confflat.mpl || cktest.mws || cktest.html || cktest.pdf

Demonstration 3 (kilt): Killing tensor in the Kerr-Newman metric (e.g. Wald General Relativity) .

npdnkn3.mpl || kilt.mws || kilt.html || kilt.pdf

Tolman (dust) Spacetime

The spherically symmetric dust solution to Einstein's equations was discovered by Tolman (Proc. Nat. Acad. Sci. 20,169) and has been the subject of much investigation. In these demonstrations we use the ability of GRTensorII to apply constraints in order to reduce the Kretschmann scalar to the form given by Bondi (Mon. Not. Roy. Astr. Soc. 107, 410). A complete set of the invariants here is {Ricciscalar,R1,R2,W2R} (see gr-qc/9809012)

Demonstration 1 (dust1): Direct application of constraints.

dust1.mpl || dust1.mws || dust1.html || dust1.pdf

Demonstration 2 (dust2): Modified constraints (no radicals) suggested by C.W.Hellaby.

dust2.mpl || dust2.mws || dust2.html || dust2.pdf

Demonstration 3 (dustlambda): Inclusion of the cosmological constant.

lambdadust.mpl || lambdadust.mws || lambdadust.html|| lambdadust.pdf

Tomimatsu-Sato Spacetimes

The Tomimatsu-Sato spacetimes exhibit intermediate expression growth and as a consequence they represent good candidates for testing computer algebra systems.

Demonstration 1 (tosa): The Ricci tensor for the delta=2 Tomimatsu-Sato spacetime (Phys. Rev. Lett. 29, 1344) is reduced to zero using a classical coordinate calculation. (Note: This calculation now takes about 30 seconds on a typical PC.)

tosa.mpl || tosa.mws || tosa.html || tosa.pdf

Triaxial Vacuum (Euclidean Bianchi IX)

In an interesting comment on a paper by Nielsen and Pedersen (SIGSAM Bulletin 22,7) MacCallum (SIGSAM Bulletin 23,22) compared various approaches and platforms to the study of the Einstein equations for a specific Euclidean 4-dimensional space. In this demonstration we show how the solution of Belinskii, Gibbons, Page and Pope (Phys. Letters 76B,433) is entered into GRTensorII, subjected to a coordinated transformation with grtransform, and reduced to vacuum.

nipe.mws || nipe.html || nipe.pdf

Vector Kinematics

Vector operators are now loaded with GRTensorII (see GRTensor Operators ). These objects can also be defined interactively with grdef ( or grdefine ).

Demonstration 1(vector): Acceleration, expansion, shear and rotation for a comoving velocity field in a spherically symmetric background.

sdiag.mpl || vector.mws || vector.html || vector.pdf

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Geometry in Three Dimensions

Riemann Tensor

It is well known that the Riemann tensor can be expressed in terms of the Ricci tensor in three dimensions (e.g. Weinberg Gravitation ).

Demonstration 1 (kawai): Here we take a look at "the black hole that went away" ( Cornish gr-qc/9609016). Note that we avoid use of ro ( r[o] ) to designate a constant in the input since r is used as a coordinate.

kawai.mpl || kawai.mws || kawai.html || kawai.pdf

Cotton-York tensor

The Cotton-York tensor is a two-index symmetric, divergence and trace-free tensor which in three dimensions is zero if and only if the space is conformally flat. (See, e.g. Kramer, Stephani, Herlt, MacCallum and Schmutzer Exact Solutions of Einstein's Field Equations).

Demonstration 1 (hedge): The Cotton-York tensor, divergence and trace for the combed hedgehog metric (see Williams Phys. Rev. D 49 117).

hedge.mpl || hedge.mws || hedge.html || hedge.pdf

Weyl-Schouten tensor

The Weyl-Schouten tensor is a three index tensor which in three dimensions is zero if and only if the space is conformally flat. (See, e.g. Nakahara Geometry, Topology and Physics ).

Demonstration 1 (ws): The Weyl-Schouten tensor and trace are evaluated for the Riemannian three-metric in orthogonal coordinates. ( A smooth Riemannian three-metric always admits such coordinates ( DeTurck and Yang Duke Math. J. 51, 243).)

diag3.mpl || ws.mws || ws.html || ws.pdf

Demonstration 2 (wscy): Relation between the Cotton-York and Weyl-Schouten tensors (CYab=eacdgbeWSecd).

diag3.mpl || wscy.mws || wscy.html || wscy.pdf

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Geometry in Five Dimensions

Demonstration 1 (acm): The Ricci tensor, Ricciscalar and Kretschmann scalar for the five dimensional metric considered by Abolghasem, Coley and McManus ( Gen. Rel. Grav. 30 1569)

fiveds.mpl || acm.mws || acm.html || acm.pdf

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Other Demonstrations

Last update: Feb 2001