timelike vector

Given a timelike vector 'v', there is a worldline of constant velocity associated with it.

This can be achieved admitting a unit timelike vector field in addition to the metric tensor of spacetime.

Physical quantities such as mass and electric charge which can be identified with the magnitude of a timelike vector have the geometric dimension of length.

First, a unit timelike vector field can be interpreted as defining the world lines of some family of (possibly noninertial) ideal observers.

Each frame field can be thought of as representing an observer in the spacetime moving along the integral curves of the timelike vector field.

In Minkowski space, if x and y are both timelike vectors lying in the future light cone, the triangle inequality is reversed:

The weak energy condition stipulates that for every timelike vector field , the matter density observed by the corresponding observers is always non-negative:

Physically this designation of the two classes of future- and past-directed timelike vectors corresponds to a choice of an arrow of time at the point.

According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector.

Parabolic Lorentz transformations are often called null rotations, since they preserve null vectors, just as rotations preserve timelike vectors and boosts preserve spacelike vectors.

A plausible structure for the quasipotential can be found by observing that the one-body Klein-Gordon equation takes the form when one introduces a scalar interaction and timelike vector interaction via and .

In physics the Einstein æther theory, also called æ-theory, is a generally covariant modification of general relativity which describes a spacetime endowed with both a metric and a unit timelike vector field named the æther.

The Lie transformations preserve the contact elements, and act transitively on Z. For a given choice of point cycles (the points orthogonal to a chosen timelike vector v), every contact element contains a unique point.

Third, in the case of general relativity, given an arbitrary timelike vector field , again interpreted as describing the motion of a family of ideal observers, the Raychaudhuri scalar is the scalar field obtained by taking the trace of the tidal tensor corresponding to those observers at each event: