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Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by timelike and null (lightlike) geodesics, respectively. [64]
A vector is timelike if c 2 t 2 > r 2, spacelike if c 2 t 2 < r 2, and null or lightlike if c 2 t 2 = r 2. This can be expressed in terms of the sign of η ( v , v ) , also called scalar product , as well, which depends on the signature.
Consider the case of trying to find a geodesic between two timelike-separated events. Let the action be = where = is the line element. There is a negative sign inside the square root because the curve must be timelike.
causal (or non-spacelike) if the tangent vector is timelike or null at all points in the curve. The requirements of regularity and nondegeneracy of Σ {\displaystyle \Sigma } ensure that closed causal curves (such as those consisting of a single point) are not automatically admitted by all spacetimes.
The extra dimension is timelike. In this article we adopt the convention that the metric in a timelike direction is negative. Image of (1 + 1)-dimensional anti-de Sitter space embedded in flat (1 + 2)-dimensional space. The t 1 - and t 2-axes lie in the plane of rotational symmetry, and the x 1-axis is normal to that plane.
A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations: [3] = (,,,) = + + + = + = where A α is the magnitude component and E α is the basis vector component; note that both are necessary to make a vector, and that when A α is seen alone, it refers strictly to the components of the vector.
In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions x μ (τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates.
In theoretical physics, null infinity is a region at the boundary of asymptotically flat spacetimes.In general relativity, straight paths in spacetime, called geodesics, may be space-like, time-like, or light-like (also called null).