There's an uncertainty relationship involving position. Is there one for the spacetime metric? If so, what is it?

I don't think there's exactly an uncertainty of measuring distance, but I suppose if there is a particle that comprises the fabric of space time, the uncertainty principle might effect it.

Edited February 24, 201114 yr by steevey

If you have two particles A and B, both A and B have a certain uncertainty in position. Would there then not be uncertainty for the distance between A and B?

If you have two particles A and B, both A and B have a certain uncertainty in position. Would there then not be uncertainty for the distance between A and B?

Wouldn't it be that the more precisely the distance between A and B is known, the less precisely the change in distance between them is known, and vice versa?

 

Is it valid to determine this using the coordinate system of A, so that the position of B is expressed relative to A while A is fixed (relative to itself at least), and the distance between the two is equivalent to the position of B?

The Heisenberg uncertainty principle will generalise to particles living on curved backgrounds. Note, the uncertainty principle really is talking about the phase space, or really a quantum version of this. The coordinates still commute amongst themselves, as do the momenta.

 

Now, this is different to having an uncertainty principle for the underlying space itself, i.e. coordinates that no longer commute. This leads us to noncommutative geometry.

 

I must say that noncommutative geometry is not really a well developed mathematical theory, rather it is lots of bits and pieces from various approaches.

 

As you want a metric, the best way to do this to date is Connes' approach. Here a noncommutative Riemannian manifold is a spectral triple (A,H,D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D on H, (a Dirac operator) with some technical requirements.

 

I don't know how to answer the question of some analogue of the uncertainty principle for the metric, or really the Dirac operator. You should see what you can find in the literature on spectral triples. Please let us know what you find out.