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Jun. 24th, 2015 10:21 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
leblonЯ тут время от времени подумываю над таким метафизическим вопросом: почему в теории относительности пространство-время имеет псевдориманову метрику, и является ли эта формальная математическая структура фундаментальной, или это некое приближение, или эффективное описание, чего-то более интуитивного.
Why is it an interesting question to think about? On one hand, one needs metric to write down the equations of motion in field theory (whether classical or quantum), so if not metric, then what? On the other hand, there are various indications that something is fishy here.
(1) Despite many attempts, it appears impossible to interpret the Beckenstein-Hawking entropy of a black hole as an entaglement entropy for fields propagating in a black hole background. The problem is, this entaglement entropy is infinite even for the vacuum state of a free field in Minkowski space. It seems that one needs some sort of minimal length of order of the Planck length, but this is impossible to achieve within the framework of pseudoriemannian geometry.
(2) The metric is not directly measurable. What is measurable, for a single observer, is proper time along his/her worldline. That is, lengths of time-like curves. Lengths of space-like curves are not directly measurable. It would be nice to reformulate field theory using measurable things only. In particular, note that the lenghts of time-like geodesics in Minkowski space satisfy the opposite of the triangle inequality. Why is that?
(3) The causal structure is directly measurable, in the sense that the position of light-cones can be deduced by asking ourselves which observables (anti)-commute.
(4) In view of (2) and (3), it is not clear why the usual topology on Minkowski space is the most physically relevant one. One would expect the weakest topology such that all causal (i.e. time-like or light-like) curves are continuous to play some role too. Geometric data should be natural from the point of view of this topology, not the usual one.
(4') The Minkowski space with the topology from (4) is not paracompact, if I remember correctly. Is this a problem? Not sure. The partition of unit is not all that useful in any case, since one cannot use it to construct pseudoriemannian metrics (and in fact, unlike in the riemannian case, not all smooth manifolds admit -pseudoriemannian metrics).
(5) It would be great to resurrect the Mach principle and somehow regard the geometry of space-time as the way of organizing our experiences. Then both causal and metric structures should be emergent, not fundamental. The causal structure is "easy" (see (3)), but what about the metric? In view of (3), one needs to interpret the proper time along time-like curves in terms of experiences of a system of observers which can communicate between each other. In particular, is there some natural interpretation of the anti-triangle inequality as some sort of consistency condition?
(6) What is the role of quantum mechanics in all this? Causal structure has to do with commutators of observables, which a quantum notion. What about the metric structure? If geometry is really emergent, what is the physical meaning of the Planck length? The ER=EPR paper seems relevant here.
Why is it an interesting question to think about? On one hand, one needs metric to write down the equations of motion in field theory (whether classical or quantum), so if not metric, then what? On the other hand, there are various indications that something is fishy here.
(1) Despite many attempts, it appears impossible to interpret the Beckenstein-Hawking entropy of a black hole as an entaglement entropy for fields propagating in a black hole background. The problem is, this entaglement entropy is infinite even for the vacuum state of a free field in Minkowski space. It seems that one needs some sort of minimal length of order of the Planck length, but this is impossible to achieve within the framework of pseudoriemannian geometry.
(2) The metric is not directly measurable. What is measurable, for a single observer, is proper time along his/her worldline. That is, lengths of time-like curves. Lengths of space-like curves are not directly measurable. It would be nice to reformulate field theory using measurable things only. In particular, note that the lenghts of time-like geodesics in Minkowski space satisfy the opposite of the triangle inequality. Why is that?
(3) The causal structure is directly measurable, in the sense that the position of light-cones can be deduced by asking ourselves which observables (anti)-commute.
(4) In view of (2) and (3), it is not clear why the usual topology on Minkowski space is the most physically relevant one. One would expect the weakest topology such that all causal (i.e. time-like or light-like) curves are continuous to play some role too. Geometric data should be natural from the point of view of this topology, not the usual one.
(4') The Minkowski space with the topology from (4) is not paracompact, if I remember correctly. Is this a problem? Not sure. The partition of unit is not all that useful in any case, since one cannot use it to construct pseudoriemannian metrics (and in fact, unlike in the riemannian case, not all smooth manifolds admit -pseudoriemannian metrics).
(5) It would be great to resurrect the Mach principle and somehow regard the geometry of space-time as the way of organizing our experiences. Then both causal and metric structures should be emergent, not fundamental. The causal structure is "easy" (see (3)), but what about the metric? In view of (3), one needs to interpret the proper time along time-like curves in terms of experiences of a system of observers which can communicate between each other. In particular, is there some natural interpretation of the anti-triangle inequality as some sort of consistency condition?
(6) What is the role of quantum mechanics in all this? Causal structure has to do with commutators of observables, which a quantum notion. What about the metric structure? If geometry is really emergent, what is the physical meaning of the Planck length? The ER=EPR paper seems relevant here.
