<Li> The present often means the single spacetime event being considered . </Li> <P> Since a world line w (τ) ∈ R 4 (\ displaystyle w (\ tau) \ in R ^ (4)) determines a velocity 4 - vector v = d w d τ (\ displaystyle v = (\ frac (dw) (d \ tau))) that is time - like, the Minkowski form η (v, x) (\ displaystyle \ eta (v, x)) determines a linear function R 4 → R (\ displaystyle R ^ (4) \ rightarrow R) by x ↦ η (v, x). (\ displaystyle x \ mapsto \ eta (v, x).) Let N be the null space of this linear functional . Then N is called the simultaneous hyperplane with respect to v . The relativity of simultaneity is a statement that N depends on v. Indeed, N is the orthogonal complement of v with respect to η . When two world lines u and w are related by d u d τ = d w d τ, (\ displaystyle (\ frac (du) (d \ tau)) = (\ frac (dw) (d \ tau)),) then they share the same simultaneous hyperplane . This hyperplane exists mathematically, but physical relations in relativity involve the movement of information by light . For instance, the traditional electro - static force described by Coulomb's law may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve retarded potentials . </P> <P> The use of world lines in general relativity is basically the same as in special relativity, with the difference that spacetime can be curved . A metric exists and its dynamics are determined by the Einstein field equations and are dependent on the mass - energy distribution in spacetime . Again the metric defines lightlike (null), spacelike and timelike curves . Also, in general relativity, world lines are timelike curves in spacetime, where timelike curves fall within the lightcone . However, a lightcone is not necessarily inclined at 45 degrees to the time axis . However, this is an artifact of the chosen coordinate system, and reflects the coordinate freedom (diffeomorphism invariance) of general relativity . Any timelike curve admits a comoving observer whose "time axis" corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis . See also for example Eddington - Finkelstein coordinates . </P> <P> World lines of free - falling particles or objects (such as planets around the Sun or an astronaut in space) are called geodesics . </P>

Why is the world line for light 45 degrees
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