(no subject)
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.
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(no subject)
Date: 2015-06-24 03:00 pm (UTC)(no subject)
Date: 2015-06-24 03:13 pm (UTC)Well, it's 21st century now; maybe it's time to move on from locally euclidian manifolds over R (which R, by the way)...
(no subject)
Date: 2015-06-24 04:28 pm (UTC)(no subject)
Date: 2015-06-24 05:20 pm (UTC)(no subject)
Date: 2015-06-24 05:25 pm (UTC)Кстати, в рамках КМ обычно не любят обсуждать неинерциальные системы отсчета. Интересно, почему. Ведь в классической механике с этим нет особой проблемы.
(no subject)
Date: 2015-06-24 05:31 pm (UTC)(1) Does anybody say that the on scales below Planck length, metric is defined? I don't know anything in this field, but g_{\mu\nu} is an operator, and (any) classical metric is emergent.
(2) Why do you want to reformulate field theory using only measurables? E.G. wave function (or density matrix) is not directly measurable. As long as you start measuring it, it collapses (or whatever equivalent you choose), and you don't know it (well, not sure about weak measurements though). Measurables might not be easy objects to work with.
Isn't it correct to say that at any scale below Planck scale, Pseudo-Riemann metric is a good approximation, and no experiments will find out what is more general theory. What about experiments related to Black Holes?
(no subject)
Date: 2015-06-24 05:35 pm (UTC)(no subject)
Date: 2015-06-24 05:44 pm (UTC)(no subject)
Date: 2015-06-24 05:51 pm (UTC)(2) Reformulating the theory in terms of things which are measurable may help identify a problem in the foundations. Currently, the problem of black hole entropy is unsolved, perhaps a solution lies in relaxing assumptions about the geometry of space-time (namely, that is locally Minkowskian) ? Thinking about what is measurable may help to either rule out this possibility, or find a proper generalization of the notion of a pseudoriemannian space-time.
(no subject)
Date: 2015-06-24 05:55 pm (UTC)(no subject)
Date: 2015-06-24 06:24 pm (UTC)(no subject)
Date: 2015-06-24 07:28 pm (UTC)(no subject)
Date: 2015-06-24 08:13 pm (UTC)(no subject)
Date: 2015-06-24 08:54 pm (UTC)Конечно, метрика есть классическое понятие, получающееся из квантового описания как инклюзивная (средняя) картина. Она подразумевает множественные наблюдения и множество наблюдателей, не мешающих наблюдаемому явлению. Любая теория (QМ или СМ) должна выдавать ее на гора, как инклюзивную картину, но уравнения для "ненаблюдаемых" величин могут быть и не обязательно "ковариантными".
(no subject)
Date: 2015-06-24 11:59 pm (UTC)So the black hole entropy problem might be not relevant because our general model is not QM + GR at lower scales than Planck scale.
(no subject)
Date: 2015-06-25 10:15 am (UTC)IMHO, GR has itself severe problems, black hole solution is one of them. When everything is attracting and collapsing - it is not very physical. In Classical Electrodynamics there is also a problem of self-action of a point-like charge leading to non physical solutions. It means the theory formulation is not correct physically. Mass renormalization in CED leaves a wrong "radiation reaction" term (jerk) - with self-accelerating solutions. What is the radiation reaction term in GR and what solutions does it produce? Nobody knows. It is too early to accept GR as a self-consistent theory, unfortunately.
(no subject)
Date: 2015-06-25 12:29 pm (UTC)исчезновение волны
Date: 2015-06-25 01:15 pm (UTC)Взять лагранжиан, зависящий от сигнатуры как параметра порядка (аналогично теории Ландау фазовых переходов второго рода). И вариацией получить сигнатуру Минковского.
Или (раз Вы уж переключились т.т.) рассмотреть метаматериал, в котором при разных внешних условиях (температуре и давлении) будет разная сигнатура. Тогда можно будет наблюдать, как при изменении т. или д. будет исчезать волна с появлением разных неустойчивостей. Веселаго (и до него) полвека назад не побоялся же рассмотреть отрицательный показатель преломления.
(no subject)
Date: 2015-06-25 03:22 pm (UTC)Re: исчезновение волны
Date: 2015-06-25 04:38 pm (UTC)(no subject)
Date: 2015-06-25 06:25 pm (UTC)(no subject)
Date: 2015-06-25 07:07 pm (UTC)(no subject)
Date: 2015-06-28 03:45 am (UTC)Ненулевые g_0i - означают вращение- есть соотв. параграф.
Стационарное гравитационное поле - грави-магнитное поле - тоже есть соответствующий параграф с таким названием.
(no subject)
Date: 2015-06-28 05:52 am (UTC)(no subject)
Date: 2015-06-30 02:12 pm (UTC)(no subject)
Date: 2015-06-30 02:13 pm (UTC)(no subject)
Date: 2015-07-01 02:08 am (UTC)(no subject)
Date: 2015-07-11 08:33 am (UTC)(no subject)
Date: 2015-08-17 11:31 pm (UTC)В 3 томе Ландафшица действительно этого нет. Думаю,всё же имелся в виду 2 том "Теория поля", параграф 45 "Теорема Лармора".
Есть хороший сборник, (http://gen.lib.rus.ec/book/index.php?md5=43bf8166e3c0e1ca9577cf8ba69f65a4) посвящённый вопросу о вращающейся системе отсчёта в теории относительности. Так вот там есть 2 главы, в которых затрагивается и квантовая механика, в частности, глава 17 "Quantum mechanics in a rotating frame".