Twin Paradox: Special Relativity Theory Explained

[Chris Hillman]

Acceleration

Hi again, MeJennifer,

Well, running the risk of getting stuck in the middle of this discussion, strictly speaking acceleration is not handled by SR for the simple reason that acceleration is mitigated by curved space-time.

In a Lorentzian manifold (curved or flat), the acceleration vector is a property of a curve (or a congruence of curves), and is identified with the covariant derivative of the tangent to the curve, taken along the curve. This has nothing to do with the curvature of the Lorentzian manifold itself, or with general relativity. In fact, the decomposition of a vector field into acceleration, expansion, and vorticity is sometimes called the kinematical decomposition. In a locally flat spacetime, it makes sense to regard this as part of special relativity, although to some extent this is a matter of convention, I suppose. The important point is that no-one should get the idea that this kinematical decomposition "requires general relativity" or is "a technique which only makes sense in a curved manifold"!

 

[MeJennifer]

In a Lorentzian manifold (curved or flat), the acceleration vector is a property of a curve (or a congruence of curves), and is identified with the covariant derivative of the tangent to the curve, taken along the curve. This has nothing to do with the curvature of the Lorentzian manifold itself, or with general relativity. In fact, the decomposition of a vector field into acceleration, expansion, and vorticity is sometimes called the kinematical decomposition. In a locally flat spacetime, it makes sense to regard this as part of special relativity, although to some extent this is a matter of convention, I suppose. The important point is that no-one should get the idea that this kinematical decomposition "requires general relativity" or is "a technique which only makes sense in a curved manifold"!

So you don't think that EM and other forces modify the stress-energy tensor and through this modify the curvature of space-time?

 

[Chris Hillman]

The Lorentz group acts on each fiber of the frame bundle

It's accurate in the sense that the ordinary algebraic equations of SR like $$\tau = t \sqrt{1 - v^2/c^2}$$ can only be used in inertial frames

This is potentially seriously misleading, although I see that you immediately added a caveat:

as I understand it you can put SR into tensor form so it'll apply in any frame, or you can figure out a different set of algebraic equations for an accelerating frame.

Better yet, consider frame fields on any Lorentzian manifold. Each frame field is a section in the frame bundle, a mild elaboration of the notion of the tangent bundle. In the tangent bundle, the fibers are the tangent spaces to each event, which are "bundled" together smoothly to make a smooth manifold. Similarly, in the frame bundle, the fiber over an event is a vector space which allows us to define, at that event, an orthonormal basis of vectors in the tangent space at that event (in a Lorentzian manifold, this will consist of one timelike unit vector and three spacelike unit vectors), and these fibers are "bundled" together to make a smooth manifold. If the "base manifold" is a four dimensional Lorentzian manifold, the tangent bundle is an eight dimensional manifold, and the frame bundle is a ten dimensional manifold (because it only requires six components to specify the orthonormal frame over each event, so the fibers are six dimensional--- three for the timelike unit vector, two for the first spacelike unit vector, one for the second, leaving no remaining degrees of freedom for the third).

The Lorentz group acts on each fiber of the frame bundle, because we can smoothly rotate/boost the frame at each event. In less fancy language, in the context of physics, frame fields provide the generalization of the kinematics of str to any Lorentzian manifold. The point is, we can certainly apply the Lorentz transformations at the level of tangent spaces, or better, in the fiber of the frame bundle.

Frame fields (elaboration of vector field) can be regarded as a generalization to arbitrary manifolds of the "frames" of str, but even in flat spacetime they are significantly more complicated than the frames used in elementary str (which correspond to "constant frame fields", hence the perennial terminological confusion).

In a given Lorentzian manifold, curved or not, a special propery which a frame field may or many not enjoy is the property of being an inertial frame, in the sense that the timelike vector field is a timelike geodesic vector field. Likewise, an independent property which a frame field may or may not enjoy is the propery of being an irrotational frame, in the sense that the vorticity tensor of the timelike vector field vanishes. Still a third property: some frames are nonspinning frames in the sense that the Fermi derivatives of the spacelike vector fields, taken along the timelike vector field, all vanish.

The "nicest" frames are the nonspinning inertial frames; these are close as we can get, in a curved manifold, to the "Lorentz frames" of elementary str. I stress that even in flat spacetime, there are nonspinning inertial frames which are not Lorentz frames! Irrotational frames enjoy another nice property: they are associated with a family of spatial hyperslices. So the very very nicest frames are inertial nonspinning irrotational.

For example: in the Schwarzschild vacuum, the world lines of the Lemaitre observers (freely and radially falling in "from rest at infinity") can be extended to define (in the right exterior and future interior regions only, or left exterior and future interior regions only!) a nonspinning inertial irrotational frame; the hyperslices are then locally isometric to three-dimensional euclidean space. The world lines of the static observers can be extended to define (in the left or right exterior region only!) an nongeodesic nonspinning irrotational frame.

Just as the tangent bundle has a "dual" notion, the cotangent bundle, the frame bundle has a dual notion, the coframe bundle. In a four dimensional Lorentzian manifold, an orthonormal coframe consists of four covector fields (or one-forms) which are orthonormal at each event, and the Lorentz group also acts on each fiber of the coframe bundle. That is, we can apply Lorentz transformations at each event in the cotangent bundle to rotate/boost a one-form or covector field event-wise, so likewise we can apply Lorentz transformations to rotate/boost a coframe at each event.

I am oversimplifying all of this stuff a bit, in order to try to convey some flavor of this essential construction. For some of the details, see for example Nakayama, Geometry, Topology, and Physics.