Register to reply

No. of Independent Components of Riemann Tensor in Schwartzchild Metric

by ObsessiveMathsFreak
Tags: components, independent, metric, riemann, schwartzchild, tensor
Share this thread:
Jan14-12, 12:06 PM
P: 406
In general 4d space time, the Riemann tensor has 20 independent components.

However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have?

(I think it is 4, but I cannot find a source to confirm this)
Phys.Org News Partner Science news on
Scientists discover RNA modifications in some unexpected places
Scientists discover tropical tree microbiome in Panama
'Squid skin' metamaterials project yields vivid color display
Jan14-12, 12:21 PM
Sci Advisor
P: 8,787
Once the metric is specified, there are no independent components of the Riemann tensor, since the metric completely determines the Riemann tensor.

Also, the counting of 20 independent components of the Riemann tensor doesn't take into account the requirement that one should be able to "integrate" the Riemann tensor to get a metric. These additional constraints are listed in Stephani's Relativity: an introduction to special and general relativity, on p143 in the section beginning with "The determination of the metric from a specified curvature tensor amounts ... to the solution of a system of twenty second-order differential equations for the ten metric compoenents ... In general such a system will posess no solutions ..."

Register to reply

Related Discussions
No. of field equations and components or Riemann tensor? Special & General Relativity 5
Riemann tensor: indipendent components Special & General Relativity 5
Riemann Curvature Tensor and working out independent components of a tensor generally Calculus & Beyond Homework 2
Independent Components in Riemann-Christoffel Tensor Special & General Relativity 3
Independent components of tensor Calculus & Beyond Homework 1