Using worksheets to solve the Einstein equation

N.

 

Christensen

and

T.

 

Moore

, “,”  (), – ().

J. B.

 

Hartle

, (, , ).

E. F.

 

Taylor

and

J. A.

 

Wheeler

, (, , ).

T.

 

Cheng

, (, , ).

B. F.

 

Schutz

, , 2nd ed. (, , ).

R. J. A.

 

Lambourne

, (, , ).

T. A.

 

Moore

, (, , ).

See the links under “Curvature and the Einstein Equation” on the website <http://web.physics.ucsb.edu/~gravitybook/mathematica.html>.

See, for example, the computer tools “GRTensor II” (<http://grtensor.phy.queensu.ca>), “Maxima” (<http://maxima.sourceforge.net>), or “Ricci” (<http://www.math.washington.edu/~lee/Ricci/>). See also the list at <https://en.wikipedia.org/wiki/Tensor_software>, though this may not be up to date.

One can find a variety of general relativity simulation programs (as well as other resources) at <http://www.compadre.org/relativity/search/search.cfm?gs=1374&SS=1384&b=1>.

Programs for simulating orbits around black holes and the effective potential energy for objects falling in Kerr spacetime are available at <http://pages.pomona.edu/~tmoore/grw/resources.html>.

W.

 

Rindler

, , 2nd ed. (, , ), pp. –. Note that although Rindler outlines a method for tabulating Ricci tensor components, he does not explicitly provide the tabulation.

See supplementary material at http://dx.doi.org/10.1119/1.4939908, which includes PDF versions of the Diagonal Metric Worksheet and the Off-Diagonal Metric Worksheet, the Xojo source code, and a PDF listing of the source code. To run the computer code, download the Xojo IDE (free) from <http://xojo.com>. This program extensively uses a Xojo feature called “operator overloading,” which allows for the definition of what the +, −, and $*$ operators do when they appear between variables pointing to instances of the Expression class. Please note that the program's output still required quite a bit of human post-processing to yield the Off-Diagonal Metric Worksheet. Also note that the program uses “G” where this paper and the worksheet use “H” (which we substituted in the post-processing stage because “G” could easily be confused with the universal gravitational constant).

For a somewhat more detailed argument for this metric, see

R. P.

 

Gruber

 et al, “,”  , – ().

For example, Ref. 12, p. 230;

M. P.

Hobson

,

G.

Efstathiou

, and

A. N.

Lasenby

, (, , ), p. ;

H. C.

 

Ohanian

and

R.

 

Ruffini

, , 2nd ed. (, , ), p. ; Ref. 7, pp. 268–269; Ref. 4, p. 90.

Ref. 7, pp. 267–268.

Ref. 7, problem P23.1 (page 276).

Ref. 7, problem P23.6 (page 278).

Ref. 7, pp. 292–302, 306.

Ref. 7, pp. 376–378.

Ref. 7, p. 271ff.

Ref. 3, pp. B-4, B-12 through B-15.

Ref. 7, pp. 180–182.

Kruskal–Szekeres coordinates are discussed in Ref. 7, pp. 182–183, 186–187; Ref. 2, pp. 269–275; and many other general relativity textbooks.

For example, Ref. 2, p. 198; Ref. 15 (Hobson et al.), p. 210; Ref. 7, p. 174 and p. 184.

Ref. 15 (Hobson et al.), p. 256.