A Metric with Harmonic Coefficient and General Relativity

[Alexander Pigazzini]

Goodmorning everyone,
is there any implies to use in general relativity a metric whose coefficients are harmonic functions?
For example in (1+1)-dimensions, is there any implies for using a metric ds²=E(du²+dv²) with E a harmonic function?
In (1+1)-dimensions is well-know that the Einstein Tensor is null, and the field equation becomes Λg_ij=8π GT_ijwhere Λ is the cosmological constant and G is the gravitational constant.
Here there is a direct correspondence (without considering the constants) of the metric tensor (g_ij) and stress-energy tensor (T_ij).
In this case, if the coefficients of metric tensor are harmonic function, then also the coefficients of the stress-energy tensor are harmonic too.
What it means / implies that the metric coefficients and the stress-energy tensor coefficients are harmonic functions?

 

[Alexander Pigazzini]

1. Is there any possible implies or interest to use in general relativity a metric whose coefficients are harmonic functions?
2. What is the meaning (physical) if the stress-energy tensor (T_ij ) has the components that are harmonic functions?