Q1: Can we think of the Minkowski metric as inducing a topology (the corresponding "metric topology")?
A1: It seems that one cannot. A metric topology requires a positive-semidefinite metric, with the additional property that two points can only be separated by zero distance in the metric if they are coincident. The Minkowski metric clearly does not satisfy these requirements: it can give negative distances (for timelike separation) and two null-separated points need not be coincident. So one defines the topology of a Lorentzian manifold independently of a metric (or perhaps using a Euclidean-signature metric), and then looks for Lorentzian metrics to put on it.
Q2: If a point on a manifold has a well-defined meaning, why are there no local observables in GR?
A2: The following two papers discuss observables in classical GR:
(i) "Construction of a complete set of observables in the general theory of relativity", A. Komar, Phys. Rev. 111, 1182 (1958).
(ii) "Observables in general relativity", P. Bergmann, Rev. Mod. Phys. 33, 510 (1961).
In these papers the authors actually construct a complete set of coordinate-independent observables in pure GR. See Eq 5 of the second paper for a choice of a basis. I will admit there are lots of words (specially in the second paper) and it is not easy for me to make out precisely what the author wants to conclude. But so as far as I can see, the real problem of defining observables seems related to quantum gravity, where we sum over all metrics.
In any case both classical and quantum gravity have much more structure than just differential geometry of manifolds (they are theories of fields on manifolds). So, just the fact that a point on a manifold is a meaningful concept does not tell us much about what we can or cannot do in the field theory.
I may revise the answer to this question if some further thoughts/inputs come up.
Sir when can one ensure that a manifold is smooth and differentiable? In Einstein Theory of General relativity one assumes spacetime is smooth and differentiable everywhere but given any spacetime with a causal structure we cannot ensure that points which are spacelike seperated can be smoothly connected ...
ReplyDeleteWhen we require spacetime to satisfy certain conditions, this means that we reject spacetimes that do not satisfy those conditions. I'm not sure what is meant by "when can one ensure". We ensure it by rejecting the unphysical cases. I also didn't understand the last sentence. In a connected differentiable manifold, any two points can be smoothly connected. It has nothing to do with whether they are space-like or time-like separated. Maybe you have some additional assumptions about the path, such as whether it can be traversed by a physical particle -- but this question has not been discussed so far in the lectures at all.
DeleteOk thank you Sir for your kind reply 🙂🙏🏻🙏🏻🌼🌼...
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