User:Teobruf/Genre (relativity)

In special relativity and general relativity, the type of a quadrivector is determined by the sign of its norm. According to this sign, we speak of time gender, space gender, or light gender. By extension, we speak of a trajectory of time gender, space gender, or light gender when the vector tangent to it is always of the considered gender.

The type of a hypersurface is that of the quadrivector which is orthogonal to it.

Definition
The type of a quadrivector {\displaystyle V} is determined by its standard, i.e. by the quantity :


 * $$N = g_{ab} V^a V^b$$,

where {\displaystyle g} corresponds to the metric of the considered space (Minkowski's metric in restricted relativity, and the metric associated with the differential variety describing space-time in general relativity).

Light gender
When the quantity {\displaystyle N} is null, the vector is said to be of the light type. This is what happens when the vector corresponds to the vector tangent to the trajectory of a photon.

Time and space gender
The definition of space gender and time gender depends on the convention used to describe the signature of the metric. For authors for whom the metric signature is (-+++), then the time gender vectors are such that {\displaystyle N} is negative and the space gender vectors are such that {\displaystyle N} is positive, whereas if we place ourselves in the convention where the metric signature is (+---), the time gender vectors are such that {\displaystyle N} is positive and the space gender vectors are such that {\displaystyle N} is negative.

Irrespective of the convention chosen, a time-like vector corresponds to a trajectory that can be followed by a physical object. A space-type vector corresponds neither to a possible trajectory of an object, nor to that of a particle without mass.

Examples
In Minkowski's metric, one can define the base vectors associated with a Cartesian coordinate system {\displaystyle (t,x,y,z)}. These vectors, which can be denoted as {\displaystyle \partial /\partial t} or t, and so on, and components $$t^a = (1, 0, 0, 0)$$ and so on, have the following standards (in the signature convention (+---)) :


 * $$\eta_{ab} t^a t^b = c^2$$ (c is the speed of light)
 * $$\eta_{ab} x^a x^b = - 1$$,
 * $$\eta_{ab} y^a y^b = - 1$$,
 * $$\eta_{ab} z^a z^b = - 1$$.

The vectors x, y, z are thus space vectors, while t is time vectors. t corresponds in fact to the quadrivity of an immobile observer in the considered coordinate system. Conversely, x, y, or z does not correspond to the quadrivity of a material object, but to a direction orthogonal to it. They describe a direction in space. A vector of the type

{\displaystyle c{\mathbf {x} }+{\mathbf {t}}

is it of zero standard:


 * $$\eta_{ab} (c x^a + t^a) (c x^b + t^b) = 0$$.