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Claude Cohen:
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Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe, Quantum Mechanics 2, Wiley J., 2006, ISBN 978-0471569527.
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L. D. Landau, J. M. Lifszyc, Mechanika, PWN, Warszawa 2011.
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Metric connection - artykuł dobrze omawia pojęcie koneksji. Koneksja zależy od metryki, tzn. podczas przesuwania wektora obowiązują reguły: (1) Gdy wektor przesuwany jest wzdłuż linii geodezyjnej, to kąt nachylenia wektora do geodezyjnej pozostaje stały podczas ruchu, przy czym równanie linii geodezyjnej zależy od przyjętej metryki - geodezyjne w płaskiej przestrzeni są prostymi euklidesowymi, ale z zakrzywionej już nie. Inaczej mówiąc - iloczyn skalarny przesuwanego wektora i wektorów stycznych do geodezyjnej nie zmienia się.(2) Pyt. Czy iloczyn skalarny wektorów zależy od metryki??
Odp. NIE! Iloczyn skalarny jest niezmiennikiem - nie zależy od układu współrzędnych, w którym się go liczy (3) Możemy przesuwać wektor wzdłuż dowolnej krzywej - wtedy krzywą dzielimy na małe fragmenty, tak że jej odcinki są styczne z fragmentami linii geodezyjnych. (4) In mathematics, a metric connection is a connection in a vector bundle E equipped with a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. Other common equivalent formulations of a metric connection include:
A connection for which the covariant derivatives of the metric on E vanish.
A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection is the Levi-Civita connection. Here the bundle E is the tangent bundle of a manifold. In addition to being a metric connection, the Levi-Civita connection is required to be torsion free.
GTR - causal structure - struktura przyczynowa
The causal structure of an arbitrary (possibly curved) Lorentzian manifold describes the causal relationships between points in the manifold, i.e. the structure describes which events in spacetime can influence which other events.
Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships.
(a) vectors
timelike vector = wektor czasopodobny
spacelike vector = przestrzennopodobny
null (lightlike) vector = zerowy (światłopodobny)
causal vector = null or timelike = przyczynowy
(b) curves
chronological (or timelike) if the tangent vector is timelike at all points in the curve.
null if the tangent vector is null at all points in the curve.
spacelike if the tangent vector is spacelike at all points in the curve.
causal (or non-spacelike) if the tangent vector is timelike or null at all points in the curve.
(c) planes
spacelike plane = płaszczyzna przestrzennopodobna
(d) time-orientable Lorentzian manifold
A Lorentzian manifold is time-orientable if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.
(e) Vector fields
timelike future-directed vector field = pole wektorowe czasopodobne skierowane ku przyszłości
Tw. 1 NIE ISTNIEJE RÓWNOCZESNOŚĆ CZASOWA W GTR na żadnej z hiperpowierzchni przestrzennopodobnej.
Warunki początkowe - dobiera się na hiperpowierzchni przestrzennopodobnej. In general, we need it to be a Cauchy surface, which exists in globally hyperbolic spacetimes
The edited question seems clearer, and maybe the following will address your question. The short physics answer to your question is that GR doesn't have a global notion of simultaneity or a notion of a global frame of reference. Therefore a spacelike hypersurface is not a surface of simultaneity. What is true is that such a surface locally defines a surface of simultaneity. This works because locally, GR becomes SR, curved spacetime becomes flat spacetime (Minkowski space), a smooth spacelike curve becomes a spacelike hyperplane, and in flat spacetime any spacelike hyperplane defines a notion of simultaneity. It may help to think about what we need to do operationally in order to establish simultaneity of events in SR. For example, inertial observers Alice and Bob send each other flashes of light, and if they find that the time between sending their flash and receiving the other person's flash is the same as the time measured similarly by the other person, they know that they sent the flashes simultaneously. This clearly won't work in GR. For example, Alice could be inside the event horizon of a black hole and Bob outside it. Sometimes we do have a family of observers who are somehow preferred, and then this establishes a notion of simultaneity. For example, in a cosmological spacetime we can have an observer who is at rest with respect to the local matter, and this observer has a preferred status. The existence of these preferred observers defines a preferred time coordinate, which is the proper time of such an observer, measured from the Big Bang. Typically the reason we would care about a spacelike hypersurface in GR is that it could be an appropriate place on which to define initial conditions. Given these initial conditions, we typically expect that we can use the laws of physics to evolve conditions forward in time. When we use a surface for this purpose, it doesn't matter if it represents any reasonable notion of simultaneity. It does matter that it's spacelike, although that's not sufficient. (In general, we need it to be a Cauchy surface, which exists in globally hyperbolic spacetimes.)
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