# zbMATH — the first resource for mathematics

Invariants of the Riemann tensor for class $$B$$ warped product space-times. (English) Zbl 1002.83503
Summary: We use the computer algebra system GRTensorII to examine invariants polynomial in the Riemann tensor for class $$B$$ warped product space-times - those which can be decomposed into the coupled product of two 2-dimensional spaces, one Lorentzian and one Riemannian, subject to the separability of the coupling $$ds^2 = ds^2_{\Sigma_1} (u,v) + C(x^\gamma)^2 ds^2_{\Sigma_2} (\theta,\phi)$$, with $$C(x^\gamma)^2=r(u,v)^2 w(\theta,\phi)^2$$ and $$\text{sig}(\Sigma_1)=0$$, $$\text{sig}(\Sigma_2)=2\varepsilon$$ $$(\varepsilon=\pm 1)$$ for class $$B_1$$ space-times and $$\text{sig}(\Sigma_1)=2\varepsilon$$, $$\text{sig}(\Sigma_2)=0$$ for class $$B_2$$. Although very special, these spaces include many of interest, for example, all spherical, plane, and hyperbolic space-times. The first two Ricci invariants along with the Ricci scalar and the real component of the second Weyl invariant J alone are shown to constitute the largest independent set of invariants to degree five for this class. Explicit syzygies are given for other invariants up to this degree. It is argued that this set constitutes the largest functionally independent set to any degree for this class, and some physical consequences of the syzygies are explored.

##### MSC:
 83-08 Computational methods for problems pertaining to relativity and gravitational theory 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 53C80 Applications of global differential geometry to the sciences
##### Software:
NP; SHEEP; GRTensorII
Full Text:
##### References:
 [1] Kramer, D.; Stephani, H.; Herlt, E.; MacCallum, M.; Schmutzer, E., Exact solutions of Einstein’s equations, (1980), CUP Cambridge · Zbl 0449.53018 [2] Narlikar, V.V.; Karmarkar, K.R., (), 91 [3] Koutras, A.; McIntosh, C., Class. quantum grav., 13, L47, (1996) [4] MacCallum, M.A.H.; Skea, J.E.F., SHEEP: A computer algebra system for general relativity, () · Zbl 0829.53057 [5] Tipler, F.J.; Clarke, C.J.S.; Ellis, G.F.R., Singularities and horizons — A review article, () [6] Siklos, S.T.C., Gen. rel. grav., 10, 1003, (1979) [7] Carminati, J.; McLenaghan, R.G., J. math. phys., 32, 3135, (1991) [8] Sneddon, G.E., J. math. phys., 37, 1059, (1996) [9] Zakhary, E.; McIntosh, C.B.G., Gen. rel. grav., 29, 539, (1997) [10] D. Pollney, report, unpublished (1996). [11] Carot, J.; da Costa, J., Class. quantum. grav., 10, 461, (1993) [12] Nakahara, M., Geometry, topology and physics, (1990), IOP Bristol · Zbl 0764.53001 [13] Haddow, B.M.; Carot, J., Class. quant. grav., 13, 289, (1996) [14] Stephani, H., General relativity, (1990), CUP Cambridge · Zbl 0733.53044 [15] Penrose, R.; Rindler, W., () [16] Musgrave, P.; Pollney, D.; Lake, K., Grtensorii, (1998), Queen’s University Kingston, Ontario [17] D. Pollney, report, unpublished (1995). [18] P. Musgrave, report, unpublished (1996). [19] K. Santosuosso, report, unpublished (1997). [20] Gurevich, G.B., Foundations of the theory of algebraic invariants, (1964), Noordhoff Groningen · Zbl 0128.24601 [21] Haddow, B.M., Gen. rel. grav., 28, 481, (1996) [22] Bel, L., LES théories relativistes de la gravitation, (1962), CNRS Paris [23] Misra, R.M.; Singh, R.A., J. math. phys., 8, 1065, (1967) [24] Zakharov, V.D., Gravitational waves in Einstein’s theory, (1973), Halsted Press New York · Zbl 0261.35002 [25] Bonnor, W.B., Class. quantum grav., 12, 499, (1995), and corrigendum [26] McIntosh, C.B.G.; Arianrhod, R.; Wade, S.T.; Hoenselaers, C., Class. quantum grav., 11, 1555, (1994) [27] Maartens, R.; Maharaj, M.S., J. math. phys., 31, 151, (1990) [28] Ellis, G.F.R., Relativistic cosmology, (), 1971 · Zbl 0337.53058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.