- Introduction to
*GRTensorII*

(mws, html, pdf)

(restricted to the Schwarzschild spacetime) - Classical Problems In Computer Algebra
- General Relativity
- Geometry in Three Dimensions
- Geometry in Five Dimensions

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

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.)

Dimension | 2 | 3 | 4 | 5 |

Diagonal components | 17 | 416 | 9,990 | 298,134 |

Off-diagonal components | 17 | 519 | 13,280 | 410,973 |

- twodg1.mpl || cost2.mws || cost2.html || cost2.pdf
- threed.mpl || cost3.mws || cost3.html || cost3.pdf
- fourd.mpl || cost4.mws || cost4.html || cost4.pdf
- fived.mpl || cost5.mws || cost5.html

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
T_{abcd}=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

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

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 +/- (m^{2}-a^{2})^{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 (R_{abcd;e}R^{abcd;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

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

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 R_{a}^{b} and basis R_{(a)(b)} components of the Ricci tensor are calculated and simplified.

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

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, R_{ab}R^{ab},R_{abcd}R^{abcd},R_{,a}^{,a},R_{ab;c}R^{ab;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

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

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

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

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

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

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

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 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

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 r_{o} ( r[o] ) to designate a constant in the input since r is used as a coordinate.

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

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

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
(CY^{ab}=*e*^{acd}g^{be}WS_{ecd}).

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

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