# Weyl Curvature, Mach's Principle, and Heisenberg Uncertainty?

1. Jul 26, 2004

### Russell E. Rierson

I have been reading that the quantity called "Weyl curvature" can exist independently of any matter, or energy, in the universe?

This seems to contradict Heisenberg uncertainty which says there can be no 100% vacuum, because uncertainty in position and uncertainty in momentum must be greater than zero:

DxDp >= [Planck's constant]/[2*pi]

Mach's principle seems to say that the distribution of matter-energy determines the geometry of space-time, and if there is no matter-energy then there is no geometry.

The Weyl tensor vanishes for a constant curvature if there are no
tidal forces. So it appears that a Weyl curvature, which is described
as 1/2 of the Riemann curvature tensor[where it is split into two
parts, the Ricci tensor and the Weyl tensor] is dependent on
matter-energy -"existing" in the universe also?

Thanks for the help.

Last edited: Jul 26, 2004
2. Jul 26, 2004

### keebs

3. Jul 27, 2004

### Russell E. Rierson

Thanks yes, classical GR and QM are incompatable, if my interpretation is correct.

The various GR books, seem to explain that the Weyl tensor Cabcd measures tidal distortion.

They start with the 256 = 4^4 component tensor Rabcd, and then impose the symmetries required by curvature Rabcd = R[ab][cd] = R[cd][ab] and
R[abcd] = 0 and Ra[bcd] = 0

Then the Riemann curvature tensor Rabcd has 20 independent components.

Decompose Rabcd into the 10-component symmetric Ricci tensor Rab and the 10-component conformal traceless Weyl tensor Cabcd. Then the Einstein tensor Gab is given by Gab = R_ab - (1/2)R, where R is the scalar curvature?

The Riemann curvature tensor Rabcd obeys the Bianchi identities, and the Einstein tensor Gab is the only contraction that obeys contracted Bianchi identities, which mean from a geometric perspective, that Eli Cartan's boundary of a boundary is zero?

It seems to me that a universe devoid of matter would have energy distortions allowing for uncertainty, giving a tidal distortion[gravity waves? quantum foams?] and hence, a non-zero Weyl tensor?

Last edited: Jul 27, 2004