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In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on ...
All curves through point p have a tangent vector, not only world lines. The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar. Therefore, all tangent vectors for a point p span a linear space, termed the tangent space at point p. For example, taking a 2-dimensional space, like the ...
The subscript star denotes the pushforward (to be introduced later), and it is in this special case simply the identity map (as is the inclusion map). The latter equality holds because a tangent space to a submanifold at a point is in a canonical way a subspace of the tangent space of the manifold itself at the point in question.
On the right, the hyperbolic world line of the rocket. Fig 6-2 Minkowski diagram in a non-inertial reference frame. On the left, the world line of the falling object. On the right, the vertical world line of the rocket. The photon world lines are determined using the metric with =. [31]
A four-velocity is thus the normalized future-directed timelike tangent vector to a world line, and is a contravariant vector. Though it is a vector, addition of two four-velocities does not yield a four-velocity: the space of four-velocities is not itself a vector space. [nb 2]
The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. [2]
Any Hilbert space is a Hilbert manifold with a single global chart given by the identity function on . Moreover, since is a vector space, the tangent space to at any point is canonically isomorphic to itself, and so has a natural inner product, the "same" as the one on .
This defines a local Lorentz frame in the tangent space at each event (in the region covered by our Rindler chart, namely the Rindler wedge). The integral curves of the timelike unit vector field e → 0 {\displaystyle {\vec {e}}_{0}} give a timelike congruence , consisting of the world lines of a family of observers called the Rindler observers .