Shear-Free Proofs in General Relativity download page
This page contains the links to download the MAPLE-based original proofs of the shear-free conjecture, as presented in covariant formalism.
The original equations are published in;
Senovilla, J.M.M., Sopuerta, C.F., Szekeres, P. Theorems on shear-free perfect fluids with their Newtonian analogues, Gen.Rel.Grav, 30, 389-411 (1998).
The independent proofs attached here are the original work of Peter Huf and John Carminati.( see Huf, P.A.a* & Carminati, J.a (2014). Elucidation of covariant proofs in general relativity: example of the use of algebraic software in the shear-free conjecture in MAPLE (in preparation).
There are several modes of being able to view the proofs:
1. View the proofs as pdf files:
:
| FILE | link to pdf proofs | last updated | comments | ||
| GENERAL SHEAR-FREE EQUATIONS (1-31) | |||||
| Equations 1-6 | Equations1-6-pdf | April 7,2017 | |||
| Equations 7-10 | Equations 7-10 | April 7,2017 | |||
| Equations 11-12 | Equations 11-12 | April 7,2017 | |||
| Equations 13-14 | Equations 13-14 | April 7,2017 | |||
| Equations 15-19 | Equations 15-19 | April 7,2017 | |||
| Equations 20-25 | Equations 20-25 | April 7,2017 | |||
| Equations 26-28 | Equations 26-28 | April 7,2017 | |||
| Equation 29 |
HC51 HC52 Equation 29 |
April 7,2017 | |||
| Equations 30-31 | Equations 30-31 | April 7,2017 | |||
| SHEAR-FREE EQUATIONS FOR DUST (a=du=p=0) (32-50) | |||||
| Equations 32-37 |
|
April 14,2017 | |||
| Equation 38 | Equation 38 | April 14,2017 | |||
| Equation 39 | Equation 39 | April 14,2017 | |||
| Equation 40 | Equation 40 | April 14,2017 | |||
| Equation 41 | Equation 41 | April 14,2017 | |||
| Equation 41bc | Equations 41bc | April 14,2017 | |||
| Equation 42 | Equation 42 | April 14,2017 | |||
| Equation 43 | Equation 43 | April 14,2017 | |||
| Equations 44-48 | Equations 44-48 | April 14,2017 | |||
| Equations 49-50 | Equations 49-50 | April 14,2017 | |||
| SHEAR-FREE EQUATIONS FOR ACCLERATION PARALLEL TO VORTICITY (a//w) (52-80) | |||||
| Equation 51 | not included | ||||
| Equations 52-53 | Equations 52-53 | April 21,2017 | |||
| Equations 52-54, including 56-57 | Equations 52-54 | April 21,2017 | |||
| Equation 55 | Equation 55 | April 21,2017 | |||
| Equations 56-57 see Equations 52-54 above | April 21,2017 | ||||
| Equations 58 | Equation 58 | April 21,2017 | |||
| Equation 59 | Equation 59 | April 21,2017 | |||
| Equation 60 | Equation 60 | April 21,2017 | |||
| Equation 61 | Equation 61 | April 21,2017 | |||
| Equations 62 | Equation 62 | April 21,2017 | |||
| Equation 63 | Equation 63 | April 21,2017 | |||
| Equations 64 | Equation 64 | April 21,2017 | |||
| Equation 65 | Equation 65 | April 21,2017 | |||
| Equation 66 | Equation 66 | April 21,2017 | |||
| Equations 67 | Equation 67 | April 21,2017 | |||
| Equations 68 | Equation 68 | April 21,2017 | |||
| Equation 69 | Equation 69 | April 21,2017 | |||
| Equation 70 | Equation 70 | April 21,2017 | |||
| Equations 71 | Equation 71 | April 22,2017 | |||
| Equation 72 | Equation 72 | April 22,2017 | |||
| Equation 72a | Equation 72a | April 22,2017 |
not explicity stated in Senovilla et al. 1998 under review |
||
| Equation 72b | Equation 72b | April 22,2017 | not explicity stated in Senovilla et al. 1998 | ||
| Equation 72c | Equation 72c | April 22,2017 | not explicity stated in Senovilla et al. 1998 | ||
| Equations 73 | Equation 73 | April 22,2017 | |||
| Equation 74a | Equation 74a | April 22,2017 | under review | ||
| Equation 74b | Equation 74b | April 22,2017 | |||
| Equation 75 | Equation 75 | April 22,2017 | |||
| Equation 76 | Equation 76 | April 22,2017 | |||
| Equation 77 | Equation 77 | April 22,2017 | |||
| Equation 78 | Equation 78 | April 22,2017 | |||
| Equation 79a | Equation 79a | April 22,2017 | under review | ||
| Equation 79b | Equation 79b | April 22,2017 | |||
| Equation 80 | Equation 80 | April 22,2017 | under review | ||
| Equation 81 | Equation 81 | April 22,2017 |
not explicity stated in Senovilla et al. 1997 under review |
||
| Equation 82 | Equation 82 | April 22,2017 | not explicity stated in Senovilla et al. 1997 |
2. Direct download of worksheets:
The proofs are presented using the TensorPack software package, which runs in the MAPLE envoronment.
Download the proofs as a zip file from the link below:
| LINK | UPDATED | FEATURES |
| Shear-free-general-conditions-1-31 | April 13, 2017 | contains independent proofs general equations of shear-free conjecture as in Senovilla et al. 1998 |
| Shearfree theorem for dust download | April 21, 2017 | contains independent proofs condtition of shear-free conjecture as in Senovilla et al. 1998 |
| Shearfree theorem for a//w download | April 22, 2017 | contains independent proofs acceleration parallel to vorticity of shear-free conjecture as in Senovilla et al. 1998 |
Related links:
Summary of GR tensor software packages packages
OTHER REFERENCES:
Ellis, G.F.R., Maartens R., MacCallum, M.A.H., Relativistic Cosmology, Cambridge University Press, Cambridge, UK (2012)
Ehlers, J., Contributions to the Relativistic Mechanics of Continuous Media, Gen. Relativ. Gravitation, 25 (12) , 1225-66 (1993)
Misner, C.W., Thorne, K.S., Wheeler, J.A. Gravitation, Freeman, San Francisco (1973)
Peter
Huf
phone +61429380524
email peterhuf@deakin.edu.au
email peter@bach2roq.com
April 11, 2017